Defs

📁 Source: Mathlib/Algebra/Module/Defs.lean

Statistics

Metric	Count
Definitions toDistribMulAction, toMulActionWithZero, toModule, instDistribSMul	4
Theorems combo_eq_smul_sub_add, combo_self, add_smul, eq_zero_of_zero_eq_one, ext, ext', ext_iff, nontrivial, subsingleton, zero_smul, add_smul, neg_one_smul, neg_smul, neg_smul_neg, smul_add_one_sub_smul, sub_smul, two_smul	17
Total	21

Convex

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
combo_eq_smul_sub_add 📖	mathematical	Distrib.toAdd NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne	AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup AddCommGroup.toAddCommMonoid SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction SubNegMonoid.toSub	—	sub_add_add_cancel add_comm smul_sub combo_self
combo_self 📖	mathematical	Distrib.toAdd NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne	AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction	—	add_smul one_smul

Module

Definitions

Name	Category	Theorems
toDistribMulAction 📖	CompOp	7862 math, 1 bridge math: AnalyticWithinAt.const_smul, AdicCompletion.of_surjective_iff, MultilinearMap.map_piecewise_smul, withSeminorms_iff_mem_nhds_isVonNBounded, SchwartzMap.norm_toLp_le_seminorm, range_cfcₙ_nnreal, Matrix.l2_opNorm_toEuclideanCLM, Pi.comul_eq_adjoint, DFinsupp.coprodMap_apply, LinearMap.det_toContinuousLinearMap, LinearEquiv.multilinearMapCongrLeft_symm_apply, LieSubmodule.coe_smul, OreLocalization.left_distrib, units_inv_smul, LieAlgebra.IsKilling.rootSystem_toLinearMap_apply, absorbent_iff_eventually_nhdsNE_zero, LeftPreLieAlgebra.toSMulCommClass, MeasureTheory.Measure.integral_comp_inv_smul, CategoryTheory.ShortComplex.opcyclesMap_smul, ContMDiffAt.coordChangeL, iteratedDeriv_scomp_two, LinearMap.IsSymmetric.clm_adjoint_eq, Finset.expect_ite_mem, ContinuousLinearMap.rayleigh_smul, LinearMap.lTensor_ker_subtype_tensorKerEquiv_symm, NumberField.mixedEmbedding.integral_comp_polarSpaceCoord_symm, ContinuousMap.toLp_inj, InnerProductSpace.Core.inner_smul_right, AffineSpace.asymptoticNhds_smul, Submodule.instIsScalarTowerQuotientIdealSubtypeMemTorsionBySetCoe, KaehlerDifferential.kerCotangentToTensor_toCotangent, ContinuousMultilinearMap.map_piecewise_smul, LinearIsometryEquiv.strictConvex_preimage, TemperedDistribution.lineDerivOpCLM_eq, Finite.instLocalizationLocalizedModule, vsub_midpoint, CliffordAlgebra.contractRight_algebraMap_mul, LinearMap.restrictScalars_toMatrix, Orientation.oangle_sign_smul_add_right, DiffContOnCl.smul, MeasureTheory.Lp.toTemperedDistributionCLM_apply, nhds_hasBasis_absConvex_open, fderiv_iteratedFDeriv, Orientation.kahler_map_complex, ConvexCone.convex, MeasureTheory.setIntegral_tilted, RootPairing.rootForm_self_smul_coroot, CochainComplex.HomComplex.δ_units_smul, FormalMultilinearSeries.comp_rightInv_aux1, MDifferentiable.coordChangeL, Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smul, ContinuousLinearEquiv.conjContinuousAlgEquiv_refl, Seminorm.isBounded_const, TensorAlgebra.ofDirectSum_of_tprod, cfcₙ_def, PresheafOfModules.Sheafify.app_eq_of_isLocallyInjective, FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_changeOrigin, AffineMap.lineMap_apply_module, LinearMap.SeparatingRight.toMatrix₂', KaehlerDifferential.mvPolynomialBasis_apply, ContinuousMultilinearMap.norm_mkPiRing, Matrix.toLinAlgEquiv_one, CFC.posPart_smul_of_nonpos, Finsupp.linearCombination_apply_of_mem_supported, ContinuousLinearMap.derivWithin_of_bilinear, IntermediateField.AdjoinSimple.trace_gen_eq_sum_roots, Submodule.mem_colon_span_singleton, EuclideanGeometry.dist_smul_vadd_sq, HasStrictFDerivAt.isLittleOTVS, SemimoduleCat.Hom.hom₂_apply, Bundle.ContMDiffRiemannianMetric.isVonNBounded, SchwartzMap.lineDerivOpCLM_eq, PointedCone.toConvexCone_closure_pointed, DivisibleHull.archimedeanClassMk_mk_eq, real_inner_div_norm_mul_norm_eq_neg_one_iff, LinearMap.charpoly_def, LieAlgebra.IsKilling.iInf_ker_weight_eq_bot, LinearEquiv.congrRight₂_refl, Field.Emb.Cardinal.equivLim_coherence, InfiniteGalois.normal_iff_isGalois, iSupIndep.dfinsupp_lsum_injective, SymmetricAlgebra.lift_ι_apply, continuous_cfcₙHomSuperset_left, hasDerivWithinAt_Iio_iff_Iic, cfcₙL_integral, LinearMap.BilinForm.isOrtho_smul_right, HasStrictFDerivAt.fun_mul', Derivation.apply_eval_eq, End.exp_mul_of_derivation, PiTensorProduct.lift_reindex, AffineMap.hasStrictDerivAt_lineMap, ContinuousLinearMap.isPositive_iff_eq_sum_rankOne, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, dot_self_cross, Bundle.ContMDiffRiemannianMetric.contMDiff, CliffordAlgebra.contractRight_algebraMap, continuousOn_stereoToFun, InnerProductSpace.isPositive_rankOne_self, AffineMap.lineMap_vsub_left, MonoidAlgebra.liftMagma_apply_apply, MeasureTheory.condExpL1_smul, LieDerivation.instNoetherian, Seminorm.absorbent_closedBall_zero, Matrix.toLin_kronecker, ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable, IntermediateField.exists_algHom_adjoin_of_splits, LinearMap.ortho_smul_right, Matrix.toLin'_symm, ModuleCat.FilteredColimits.colimit_smul_mk_eq, Subalgebra.centralizer_tensorProduct_eq_center_tensorProduct_right, ordinaryHypergeometric_sum_eq, AlgEquiv.moduleEndSelfOp_apply_apply, Matrix.SeparatingLeft.toBilin', ProbabilityTheory.isGaussian_iff_gaussian_charFun, MeasureTheory.Lp.instIsScalarTower, LinearMap.detAux_def'', ContinuousLinearMap.mulLeftRight_isBoundedBilinear, QuadraticMap.smul_toBilin, Ideal.Filtration.submodule_eq_span_le_iff_stable_ge, TensorProduct.LieModule.liftLie_apply, CategoryTheory.ShortComplex.homologyMap_smul, norm_cfcₙHom, Algebra.TensorProduct.instSMulCommClassTensorProduct, Basis.mem_submodule_iff, Polynomial.smeval_monomial_mul, PiTensorProduct.lift.tprod, IntermediateField.aeval_gen_minpoly, RootPairing.pairing_smul_root_eq_of_not_linearIndependent, LieRinehartRing.lie_smul_eq_mul, CliffordAlgebra.contractRight_eq, OrthonormalBasis.orthogonalProjection_eq_sum, cfcₙHom_mono, hasDerivWithinAt_ofNat, RootPairing.four_nsmul_coPolarization_compl_polarization_apply_root, ContinuousLinearMap.analyticWithinAt_bilinear, Complex.restrictScalars_one_smulRight', LinearMap.toMatrixAlgEquiv'_mul, SymmetricAlgebra.rank_eq, Matrix.kroneckerTMulStarAlgEquiv_symm_apply, Matrix.frobeniusIsBoundedSMul, HasCompactSupport.convolution_integrand_bound_left, LinearMap.BilinForm.toMatrix_toBilin, ContinuousLinearMap.compSL_apply, ContinuousLinearMap.continuous₂, Basis.end_repr_apply, MeasureTheory.Martingale.smul, LinearMap.IsSymmetric.inner_map_polarization, CategoryTheory.Functor.mapExtLinearMap_toAddMonoidHom, hasDerivWithinAt_add_const_iff, LinearMap.coe_toSeminorm, FormalMultilinearSeries.comp_partialSum, SameRay.nonneg_smul_left, mellin_eq_fourier, Real.sSup_smul_of_nonpos, OreLocalization.cardinalMk, SchwartzMap.instLineDerivLeftSMulReal, NonUnitalStarSubalgebra.instSMulMemClass, LinearMap.adjoint_adjoint, mem_annihilator, Matrix.SeparatingRight.toLinearMap₂, AnalyticOnNhd.smul, IntermediateField.LinearDisjoint.iff_inf_eq_bot, RootPairing.linearIndepOn_root_baseOf', IntermediateField.normal_inf, PolyEquivTensor.toFunBilinear_apply_eq_sum, Rep.MonoidalClosed.linearHomEquiv_symm_hom, binomialSeries_eq_ordinaryHypergeometricSeries, Polynomial.lcoeff_comp_mapAlgHom_eq, Affine.Simplex.faceOppositeCentroid_vsub_point_eq_smul_sum_vsub, StarAddMonoid.toStarModuleNNRat, StarAlgEquiv.ofLeftInverse'_apply, InnerProductSpace.toLinearIsometry_toDual, SchwartzMap.integral_sesq_fourier_fourier, MultilinearMap.fromDFinsuppEquiv_apply, Bornology.isVonNBounded_iff_tendsto_smallSets_nhds, Submodule.isScalarTower, PowerSeries.IsRestricted.smul, support_quotient, LinearMap.exists_monic_and_aeval_eq_zero, Subspace.dualAnnihilator_dualAnnihilator_eq_map, WithSeminorms.image_isVonNBounded_iff_seminorm_bounded, AnalyticOn.exists_hasFTaylorSeriesUpToOn, Filter.IsBoundedUnder.smul_tendsto_zero, MeasureTheory.charFun_eq_fourierIntegral', Real.hasFDerivAt_fourierChar_neg_bilinear_left, RootPairing.Polarization_apply, PolynomialModule.comp_single, hasStrictDerivAt_intCast, segment_subset_closedBall_right, real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul, MeasureTheory.convolution_eq_right', hasFDerivAt_iff_hasGradientAt, NonUnitalIsometricContinuousFunctionalCalculus.isometric, HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn, MeasureTheory.eLpNormEssSup_const_smul, BilinForm.toMatrix_symm, TensorProduct.AlgebraTensorModule.leftComm_eq, gauge_def', Asymptotics.isBigO_const_smul_right, cfcₙHom_apply_mem_elemental, MeasureTheory.hausdorffMeasure_segment, MeasureTheory.integrable_withDensity_iff_integrable_coe_smul, NonUnitalStarAlgHom.coe_codRestrict, CategoryTheory.Linear.instMonoHSMulHomOfInvertible, SimpleGraph.lapMatrix_toLinearMap₂', curveIntegral_smul, Quaternion.snd_imJ_dualNumberEquiv_symm, IntervalIntegrable.smul_continuousOn, DomMulAct.instSMulCommClassSubtypeAEEqFunMemAddSubgroupLp_1, Orientation.inner_rotation_pi_div_two_right_smul, MultilinearMap.map_update_smul, Orientation.oangle_add_left_smul_rotation_pi_div_two, CFC.monotoneOn_one_sub_one_add_inv, RootPairing.coroot_eq_polarizationEquiv_apply_root, derivWithin_smul, Real.fderiv_fourierIntegral, hasStrictFDerivAt_list_prod_finRange', MeasureTheory.integral_smul_nnreal_measure, ContinuousMultilinearMap.map_smul_univ, Ideal.IsTwoSided.pow_succ, MvPolynomial.rTensor_apply_tmul_apply, InnerProductSpace.rankOne_one_left_eq_innerSL, Submodule.le_annihilator_iff, ContDiff.fderiv_succ, LinearIndependent.pair_add_smul_right_iff, IntermediateField.adjoin.finrank, AlternatingMap.constLinearEquivOfIsEmpty_symm_apply, Seminorm.le_finset_sup_apply, LinearMap.Nondegenerate.congr, FormalMultilinearSeries.smul_apply, NonUnitalCStarAlgebra.toStarModule, Complex.circleTransformDeriv_eq, LinearMap.IsSymmetric.restrictScalars, Seminorm.norm_sub_map_le_sub, bddBelow_smul_iff_of_neg, IsBaseChange.comp_iff, addMonoidHomLequivNat_symm_apply, HasFiniteFPowerSeriesAt.neg, TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm, Submodule.mem_ideal_smul_span_iff_exists_sum, nnnorm_cfcₙ_nnreal_le_iff, LinearMap.isNilRegular_iff_natTrailingDegree_charpoly_eq_nilRank, Matrix.separatingRight_toLinearMap₂'_iff, LinearMap.baseChange_smul, FormalMultilinearSeries.ofScalars_series_eq_zero_of_scalar_zero, MvPolynomial.rTensorAlgEquiv_apply, ContinuousLinearMap.smulRight_one_eq_toSpanSingleton, Algebra.Generators.H1Cotangent.δAux_mul, AffineMap.lineMap_linear, Complex.kahler, Pi.image_update_openSegment, MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left, AlgCat.hom_inv_associator, ContinuousLinearMap.map_add_add, LieAlgebra.IsKilling.restrict_killingForm_eq_sum, SeminormFamily.basisSets_smul_right, MeasureTheory.Measure.integral_comp_mul_right, intervalIntegral.inv_smul_integral_comp_div_add, Antivary.smul_add_smul_le_smul_add_smul, AddSubgroup.zero_smul, MeromorphicOn.divisor_fun_smul, TrivSqZeroExt.lift_inlAlgHom_inrHom, Finite.multilinearMap, FormalMultilinearSeries.changeOriginSeries_sum_eq_partialSum_of_finite, Affine.Simplex.smul_centroid_vsub_point_eq_smul_faceOppositeCentroid_vsub_point, Unitization.inrNonUnitalAlgHom_toFun, AdicCompletion.AdicCauchySequence.mk_eq_mk, Real.hasDerivAt_sigmoid, MeasureTheory.IntegrableOn.continuousOn_smul, taylorWithinEval_hasDerivAt_Ioo, AlternatingMap.map_vecCons_smul, CStarModule.innerₛₗ_apply, NonUnitalSubalgebraClass.coe_subtype, MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul, fderivWithin_fun_smul, ContinuousMultilinearMap.norm_iteratedFDerivComponent_le, End.smul_def, Finset.weightedVSubOfPoint_apply, ContinuousMultilinearMap.analyticAt_uncurry_compContinuousLinearMap, ContinuousAlternatingMap.hasStrictFDerivAt_toContinuousMultilinearMap_comp_iff, MvPolynomial.scalarRTensor_apply_monomial_tmul, Real.fourierIntegralInv_eq', AffineMap.hasDerivAtFilter, ContinuousLinearEquiv.toCompactConvergenceCLM_symm_apply, LinearIndependent.pair_add_smul_left_iff, End.HasEigenvalue.mem_spectrum, QuadraticMap.canLift, LieModule.toEnd_pow_apply_map, Polynomial.toMatrix_sylvesterMap', IntermediateField.coe_algebraMap_over_bot, IntermediateField.instIsAlgClosureAlgebraicClosureSubtypeMemOfIsAlgebraic, fderivWithin_fderivWithin_eq_of_mem_nhdsWithin, Matrix.toLinearMapRight'_mul, RingTheory.Sequence.isWeaklyRegular_cons_iff', ContDiffMapSupportedIn.norm_iteratedFDeriv_apply_le_seminorm, Finset.sum_smul, ContinuousLinearMap.toLinearMap₁₂_injective, ContinuousLinearMapWOT.le_nhds_iff_forall_inner_apply_le_nhds, finsuppTensorFinsupp_apply, Submodule.smul_eq_map₂, AntivaryOn.sum_smul_lt_sum_comp_perm_smul_iff, curveIntegrable_segment, Submodule.toConvexCone_inf, Polynomial.disjoint_ker_aeval_of_isCoprime, LinearMap.IsRefl.domRestrict, ContDiffOn.smul_const, Ideal.iInf_pow_smul_eq_bot_of_isLocalRing, MvPolynomial.universalFactorizationMap_comp_map, Basis.equivFun_symm_apply, LinearMap.isPositive_adjoint_comp_self, ProbabilityTheory.covarianceBilinDual_apply', AffineSubspace.affineSpan_pair_parallel_iff_exists_unit_smul', PolynomialModule.smul_def, Seminorm.bot_eq_zero, CoalgHom.coe_addMonoidHom_injective, Complex.integral_boundary_rect_eq_zero_of_differentiableOn, ContinuousLinearMap.continuousSMul, LinearMap.isAdjointPair_iff_comp_eq_compl₂, hasFTaylorSeriesUpToOn_top_iff_right, Seminorm.smul_comp, CochainComplex.HomComplex.Cocycle.coe_smul, ConvexCone.mem_positive, LinearIsometry.map_smul, Affine.Simplex.orthogonalProjection_vadd_smul_vsub_orthogonalProjection, MeasureTheory.Lp.const_smul_mem_Lp, RootPairing.reflectionPerm_eq_iff_smul_root, Submodule.map_dualCoannihilator_le, IsBaseChange.finsuppPow, strictConvex_Iic, Matrix.separatingRight_toLinearMap₂_iff, AddSubgroupClass.coe_zmod_smul, AdicCompletion.map_val_apply, Ideal.mul_le_right, End.HasUnifEigenvector.pow_apply, AffineSubspace.setOf_sSameSide_eq_image2, ContinuousMultilinearMap.analyticAt_uncurry_of_linear, intervalIntegral.integral_hasDerivWithinAt_right, Matrix.spectrum_toEuclideanLin, ContDiff.iteratedFDeriv_right, Bornology.isVonNBounded_add_of_nonempty, norm_add_eq_iff_real, Monotone.quasiconcaveOn, Matrix.iSup_eigenspace_toLin'_diagonal_eq_top, Complex.integral_comp_pi_polarCoord_symm, FormalMultilinearSeries.radius_pi_eq_iInf, ProperCone.convex, contDiffAt_norm_smul_iff, KaehlerDifferential.polynomialEquiv_symm, Derivation.coe_to_linearMap_comp, addMonoidHomLequivInt_apply, SchwartzMap.pairing_apply_apply, Submodule.instStrictConvexSpace, InnerProductSpace.isIdempotentElem_rankOne_self, norm_iteratedFDerivWithin_smul_le, SymmetricAlgebra.isSymmetricAlgebra_ι, Submodule.tensorEquivSpan_apply_tmul, ContDiffMapSupportedIn.iteratedFDerivLM_apply, Matrix.cstar_nnnorm_def, minpoly.algEquiv_apply, IsLprojection.Lnorm, AddCommGroup.intIsScalarTower, MeasureTheory.weightedSMul_apply, NonUnitalSubalgebra.instIsScalarTower', Ideal.smul_top_eq_map, LineDifferentiableWithinAt.smul, Basis.toDual_linearCombination_left, intervalIntegral.integral_hasFDerivAt_of_tendsto_ae, MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport, Commute.cfcₙ_nnreal, AlgHom.coe_addMonoidHom_injective, IntermediateField.isScalarTower_mid', LinearEquiv.congrRight_symm, LinearMap.mapMatrixLinear_apply, LinearMap.BilinForm.toMatrixAux_eq, CliffordAlgebra.ι_mul_ι_mul_invOf_ι, HomogeneousLocalization.val_nsmul, StrongDual.polarSubmodule_eq_setOf, IsScalarTower.to_smulCommClass', ContDiffMapSupportedIn.structureMapCLM_eq_of_scalars, hasDerivWithinAt_Ioi_iff_Ici, ContinuousLinearMapWOT.tendsto_iff_forall_inner_apply_tendsto, Orientation.tan_oangle_add_left_smul_rotation_pi_div_two, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul', IsExposed.isExtreme, ContinuousLinearMap.isUniformEmbedding_restrictScalars, LieModule.trace_toEnd_eq_zero_of_mem_lcs, Algebra.Generators.cotangentSpaceBasis_apply, RingTheory.Sequence.isWeaklyRegular_cons_iff, ConvexCone.canLift, hasDerivWithinAt_taylorWithinEval, Matrix.toLinearMap₂'_comp, map_ratCast_smul, FormalMultilinearSeries.changeOriginSeriesTerm_apply, PiTensorProduct.lift_reindex_symm, DirectLimit.Module.of_f, Asymptotics.isLittleOTVS_one, kroneckerTMulAlgEquiv_symm_single_tmul, Seminorm.finset_sup_apply, ContinuousLinearMap.hasDerivAtFilter, isTorsionBySet_quotient_set_smul, span_fourier_closure_eq_top, Real.fourierIntegral_eq, LinearMap.toMatrix_apply', MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one, IntermediateField.adjoinRootEquivAdjoin_symm_apply_gen, ContinuousLinearMap.comp_hasFiniteFPowerSeriesOnBall, hasFDerivAt_integral_of_dominated_loc_of_lip_interval, intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_right, IntermediateField.isCyclotomicExtension_singleton_iff_eq_adjoin, TrivSqZeroExt.isNilpotent_iff_isNilpotent_fst, LinearMap.llcomp_apply, CategoryTheory.ShortComplex.Homotopy.smul_h₁, LinearMap.toMatrix₂_mul, HasFPowerSeriesOnBall.add, LinearMap.BilinForm.smul_left_of_tower, Orientation.kahler_comp_rightAngleRotation', Algebra.Extension.H1Cotangent.map_id, circleIntegral_def_Icc, convexJoin_singletons, MeasureTheory.integral_unique, LieAlgebra.LoopAlgebra.twoCochainOfBilinear_apply_apply, setIntegral_withDensity_eq_setIntegral_smul₀, Complex.hasSum_taylorSeries_of_entire, ContinuousMap.toLp_denseRange, ContinuousLinearMap.toContinuousAddMonoidHom_add, Basis.dualBasis_coord_toDualEquiv_apply, Matrix.repr_toLin, Bornology.isVonNBounded_add_self, MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae, intervalIntegral.intervalIntegral_eq_integral_uIoc, AdicCompletion.of_ofAlgEquiv_symm, Seminorm.ball_smul_closedBall, ContinuousLinearMap.integral_apply, CochainComplex.HomComplex.Cochain.leftShift_smul, WeierstrassCurve.Projective.nonsingularLift_some, Orientation.two_zsmul_oangle_smul_left_of_ne_zero, segment_inter_subset_endpoint_of_linearIndependent_sub, LocalizedModule.restrictScalars_map_eq, ContinuousLinearMap.precompUniformConvergenceCLM_apply, circleIntegrable_iff, ProbabilityTheory.covarianceBilinDual_apply, PolynomialModule.map_smul, SchwartzMap.integral_clm_comp_lineDerivOp_right_eq_neg_left, LieAlgebra.mapsTo_toEnd_genWeightSpace_add_of_mem_rootSpace, KaehlerDifferential.mvPolynomialBasis_repr_D_X, convex_iff_sum_mem, AlternatingMap.constLinearEquivOfIsEmpty_apply, NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded, Real.fderiv_fourierChar_neg_bilinear_right_apply, LinearMap.tensorKer_tmul, WithSeminorms.mem_nhds_iff, LinearMap.toMatrixAlgEquiv_comp, quasispectrumRestricts_iff_spectrumRestricts_inr', LinearMap.IsSymmetric.diagonalization_apply_self_apply, TensorProduct.sum_tmul_basis_right_injective, LinearMap.extendScalarsOfIsLocalization_apply, Submodule.mem_colon', Real.fourier_continuousLinearMap_apply, DirectSum.of_smul, FiniteGaloisIntermediateField.instIsGaloisSubtypeMemIntermediateField, Unitization.real_cfcₙ_eq_cfc_inr, real_inner_smul_self_left, LieAlgebra.IsKilling.isCompl_ker_weight_span_coroot, neg_smul, fderivWithin_fun_pow', intervalIntegral.fderiv_integral_of_tendsto_ae, ContDiffAt.fderiv_right_succ, ZMod.invDFT_apply, SMulPosReflectLE_iff_smulPosReflectLT, Representation.asAlgebraHom_single, NormedSpace.isBounded_iff_subset_smul_ball, AffineIsometryEquiv.ofTop_apply, LieAlgebra.finrank_engel, skewAdjointLieSubalgebraEquiv_apply, hasFPowerSeriesOnBall_pi_iff, SmoothPartitionOfUnity.contMDiffAt_finsum, imaginaryPart_I_smul, AnalyticOnNhd.iteratedFDeriv_of_isOpen, Continuous.convolution_integrand_fst, ContinuousAlternatingMap.vecCons_smul, MDifferentiableOn.smul_const_section, ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries, Ray.neg_units_smul, MvPolynomial.leibniz_iff_X, DilationEquiv.smulTorsor_symm_apply, ContDiffMapSupportedIn.structureMapCLM_apply, iteratedFDerivWithin_succ_eq_comp_right, OrthonormalBasis.toBasis_tensorProduct, HasFPowerSeriesWithinAt.const_smul, norm_smul_inv_norm', LipschitzWith.ae_lineDeriv_sum_eq, SchwartzMap.lineDerivOp_fourier_eq, Subspace.dualRestrict_comp_dualLift, LinearMap.BilinForm.congr_apply, diam_smul₀, LinearEquiv.conj_trans, isClosedEmbedding_cfcₙAux, TensorAlgebra.toTrivSqZeroExt_ι, abs_smul, Polynomial.homogenize_smul, setIntegral_withDensity_eq_setIntegral_toReal_smul, LieModule.traceForm_eq_sum_finrank_nsmul', LinearMap.mul_apply_apply, Flat.localizedModule, LinearMap.BilinForm.toMatrix_compRight, innerSL_apply_apply, NonUnitalAlgHom.range_comp_le_range, HasFPowerSeriesWithinOnBall.restrictScalars, NormedSpace.isVonNBounded_iff, segment_translate_preimage, convexHull_eq_iInter, SeminormFamily.filter_eq_iInf, LinearMap.rTensor_smul_action, toMul_uzpow, aemeasurable_smul_const, SchwartzMap.integral_bilin_fourierInv_eq, LinearMap.adjoint_innerₛₗ_apply, TestFunction.toBoundedContinuousFunctionCLM_eq_of_scalars, Algebra.lmul_algebraMap, HasFTaylorSeriesUpTo.hasFDerivAt, TensorProduct.AlgebraTensorModule.rid_tmul, ContinuousAlternatingMap.hasStrictFDerivAt_compContinuousLinearMap, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul, ContMDiffWithinAt.mfderivWithin_const, MeasureTheory.Measure.toSphere_apply_aux, Unitary.conjStarAlgEquiv_unitaryLinearIsometryEquiv, ContinuousLinearMap.innerSL_apply_comp_of_isSymmetric, Submodule.dualQuotEquivDualAnnihilator_apply, PolynomialModule.comp_smul, Unitization.splitMul_apply, tendsto_integral_peak_smul_of_integrable_of_tendsto, inner_eq_norm_mul_iff, TensorAlgebra.ι_comp_lift, ContinuousLinearMap.toBilinForm_apply, WittVector.IsocrystalHom.frob_equivariant, Seminorm.restrictScalars_closedBall, smulPosMono_iff_smulPosStrictMono, ProbabilityTheory.covarianceBilin_apply_eq_cov, LinearEquiv.congrQuadraticMap_symm, ProbabilityTheory.covarianceBilin_self, LinearEquiv.conj_apply_apply, tendsto_smul_comp_nat_floor_of_tendsto_mul, LinearMap.adjoint_eq_toCLM_adjoint, Field.Emb.Cardinal.instInverseSystemWithTopToTypeOrdRankAlgHomSubtypeMemIntermediateFieldCoeOrderEmbeddingFiltrationAlgebraicClosureEmbFunctor, LieAlgebra.IsKilling.instIsReducedSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, Seminorm.bddAbove_range_iff, bilinearIteratedFDerivTwo_eq_iteratedFDeriv, AntivaryOn.sub_smul_sub_nonpos, Basis.basis_singleton_iff, convex_iff_pairwise_pos, NormedSpace.image_isVonNBounded_iff, VectorField.mpullbackWithin_smul, ContinuousMap.range_toLp, eqOn_dslope_sub_smul, Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div, convex_Iic, AddMonoidAlgebra.mapDomainNonUnitalAlgHom_apply, Differentiable.fun_smul, TensorAlgebra.adjoin_range_ι, Subbimodule.smul_mem', LeftInvariantDerivation.left_invariant', hasDerivAt_bernoulliFun, Representation.ofCoinvariantsTprodLeftRegular_mk_tmul_single, FormalMultilinearSeries.le_changeOriginSeries_radius, derivWithin_fderivWithin, Matrix.toLinearMap₂_toMatrix₂, LinearMap.toMatrix_one, WithLp.isBoundedSMulSeminormedAddCommGroupToProd, InfiniteGalois.isOpen_and_normal_iff_finite_and_isGalois, Submodule.smul_mem_span_smul, Field.Emb.Cardinal.filtration_succ, isTorsionBySet_quotient_ideal_smul, IsSelfAdjoint.smul_mem_skewAdjoint, MultilinearMap.currySum_add, DualNumber.range_lift, Unitization.algHom_ext'_iff, QuotSMulTop.equivTensorQuot_naturality, MeasureTheory.condExpIndL1Fin_smul, LieAlgebra.IsKilling.rootSystem_coroot_apply, Affine.Simplex.centroid_eq_smul_sum_vsub_vadd, PosSMulReflectLE_iff_posSMulReflectLT, CliffordAlgebraDualNumber.equiv_symm_eps, Monovary.sum_smul_comp_perm_le_sum_smul, TensorAlgebra.toDirectSum_tensorPower_tprod, ProbabilityTheory.uncenteredCovarianceBilin_apply, InnerProductGeometry.angle_smul_right_of_pos, TemperedDistribution.lineDerivOp_fourier_eq, End.spectrum_intrinsicStar, WeakFEPair.symm_Λ₀_eq, Flat.ker_lTensor_eq, KaehlerDifferential.cotangentComplexBaseChange_tmul, LinearMap.isSelfAdjoint_toContinuousLinearMap_iff, SpecialLinearGroup.mem_center_iff_spec, cross_cross, Seminorm.smul_inf, Matrix.separatingRight_toBilin_iff, Submodule.Quotient.instIsBoundedSMul, intervalIntegral.integral_comp_sub_div, FormalMultilinearSeries.derivSeries_coeff_one, Matrix.spectrum_toLpLin, HasDerivAt.comp_ofReal, tangentConeAt_nonempty_of_properSpace, mdifferentiableOn_symm_coordChangeL, AdicCompletion.instIsScalarTower_1, convexJoin_singleton_left, ProbabilityTheory.hasDerivAt_integral_pow_mul_exp, ContinuousLinearMap.opNorm_lsmul_apply_le, intervalIntegral.integral_comp_mul_left, AnalyticAt.fun_const_smul, Submodule.ideal_span_singleton_smul, Continuous.integral_hasStrictDerivAt, LinearMap.ofIsCompl_smul, OrthonormalBasis.orthogonalProjection_eq_sum_rankOne, IsSymmetricAlgebra.equiv_apply, Submodule.top_eq_ofList_cons_smul_iff, BoxIntegral.unitPartition.mem_smul_span_iff, SemimoduleCat.MonoidalCategory.rightUnitor_hom_apply, intervalIntegral.integral_comp_mul_sub, mdifferentiableWithinAt_smul_const_section, InnerProductSpace.Core.inner_smul_left, SchwartzMap.bilinLeftCLM_apply, AlternatingMap.smulRight_apply, InnerProductSpace.toDual_symm_apply, ContinuousLinearMap.fpowerSeriesBilinear_apply_zero, Dual.eval_comp_comp_evalEquiv_eq, IsConformalMap.is_complex_or_conj_linear, Bornology.isVonNBounded_pi_iff, LieAlgebra.isRegular_iff_natTrailingDegree_charpoly_eq_rank, OreLocalization.universalHom_apply, ContinuousLinearMap.nhds_zero_eq_of_basis, HasGradientAt.hasDerivAt', OreLocalization.nsmul_eq_nsmul, Fintype.expect_dite_eq, Basis.isScalarTower_of_nonempty, LieModule.exists_genWeightSpace_zero_le_ker_of_isNoetherian, LinearMap.toAddMonoidHom_mulLeft, CategoryTheory.Linear.comp_apply, LinearMap.BilinForm.toMatrix'_symm, LinearEquiv.map_mem_invtSubmodule_conj_iff, End.isFinitelySemisimple_sub_algebraMap_iff, Prod.image_mk_openSegment_left, InfiniteGalois.normalAutEquivQuotient_apply, LinearMap.mem_isPairSelfAdjointSubmodule, AffineMap.hasDerivWithinAt, Submodule.dualPairing_apply, Submodule.starProjection_singleton, TrivSqZeroExt.inv_inl, SemilinearMapClass.toMulActionSemiHomClass, NonUnitalAlgHom.coe_lmul_eq_mul, Matrix.toLin_mul_apply, LinearMap.tensorEqLocusEquiv_apply, toMatrix_distrib_mul_action_toLinearMap, LinearMap.domRestrict'_apply, Bundle.RiemannianMetric.isVonNBounded, vadd_right_mem_affineSpan_pair, ContinuousMultilinearMap.norm_compContinuousLinearMapLRight_le, Orientation.tan_oangle_add_right_smul_rotation_pi_div_two, dualMap_dualMap_eq_iff, NormedSpace.toIsBoundedSMul, FiniteGaloisIntermediateField.instIsGaloisSubtypeMemIntermediateFieldMax, Fintype.mem_span_image_iff_exists_fun, ModuleCat.Iso.homCongr_eq_arrowCongr, IsAlgClosure.of_exists_root, HasFPowerSeriesAt.neg, Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, AlgEquiv.linearEquivConj_mulLeft, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply_comp_toReal, curveIntegralFun_smul, Algebra.TensorProduct.tensorTensorTensorComm_symm, addInvariantVectorField_smul, PiToModule.fromEnd_apply, IntermediateField.isAlgebraic_tower_bot, VertexOperator.of_coeff_apply_coeff, neg_strictConvexOn_iff, Basis.smul_eq_zero, smul_lt_smul_of_neg_left, egauge_ball_le_of_one_lt_norm, ProbabilityTheory.integral_gaussianReal_eq_integral_smul, MeasureTheory.charFun_toDual_symm_eq_charFunDual, MvPolynomial.algHom_ext'_iff, freeLocus_localization, Polynomial.natDegree_smul, MonovaryOn.sum_smul_comp_perm_le_sum_smul, hasDerivWithinAt_iff_isLittleO, TensorPower.gMul_eq_coe_linearMap, hasStrictFDerivAt_iff_isLittleOTVS, CStarMatrix.mapₗ_reindexₐ, AnalyticOn.iteratedFDerivWithin, LieSubalgebra.toEnd_eq, derivWithin_fun_const_smul_field, ContinuousAlternatingMap.compContinuousLinearMapₗ_apply, ContinuousLinearEquiv.toCompactConvergenceCLM_apply, LinearMap.BilinForm.IsRefl.groupSMul, LinearMap.BilinForm.toMatrix'_apply, Submodule.mem_smul_span_singleton, WittVector.factorPowSucc_comp_fontaineThetaModPPow, Finsupp.sum_smul_index_linearMap', MeasureTheory.IntegrableOn.smul_continuousOn, ContinuousAlternatingMap.piLIE_apply_apply, AffineSubspace.wOppSide_smul_vsub_vadd_left, ContinuousLinearMap.applySMulCommClass', PreInnerProductSpace.Core.smul_left, inv_intCast_smul_eq, extremePoints_prod, Differentiable.fderiv_norm_rpow, combo_mem_ball_of_ne, lintegral_fderiv_lineMap_eq_edist, UnitAddTorus.mFourierSubalgebra_coe, TemperedDistribution.smulLeftCLM_compL_smulLeftCLM, LinearMap.BilinForm.SeparatingRight.toMatrix, ContinuousLinearMap.integrable_of_bilin_of_bdd_left, AdicCompletion.incl_apply, hasDerivWithinAt_zero, AdicCompletion.map_ext'_iff, CategoryTheory.Abelian.Ext.smul_hom, IntermediateField.botEquiv_symm, AlgCat.hom_inv_rightUnitor, coe_convexAddSubmonoid, Unitization.starAlgHom_ext_iff, LinearMap.nondegenerate_toLinearMap₂'_of_det_ne_zero', AbsolutelyContinuousOnInterval.const_smul, ContDiffAt.hasStrictDerivAt, Seminorm.neg_ball, TrivSqZeroExt.inv_one, LinearEquiv.congrRight₂_apply, FormalMultilinearSeries.hasSum_of_finite, signedDist_right_congr, cross_self, LieAlgebra.of_smul, Rep.to_Module_monoidAlgebra_map_aux, CliffordAlgebra.contractRight_mul_ι, SpecialLinearGroup.mem_center_iff, Ideal.iInf_pow_smul_eq_bot_of_le_jacobson, hasDerivAtFilter_ofNat, Asymptotics.IsBigO.smul, AdicCompletion.val_smul_eq_evalₐ_smul, instStarModuleNNRealOfReal, Matrix.Nondegenerate.toLinearMap₂, rTensor_injective_iff_lcomp_surjective, TensorProduct.gradedMul_assoc, FormalMultilinearSeries.changeOriginSeries_summable_aux₃, curveIntegrable_smul_iff, Complex.hasSum_taylorSeries_on_ball, Submodule.mem_span_image_iff_exists_fun, Subalgebra.centralizer_coe_map_includeLeft_eq_center_tensorProduct, QuadraticMap.map_smul_of_tower, MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff, HasFPowerSeriesOnBall.restrictScalars, smul_nonneg_iff_neg_imp_nonpos, ContinuousLinearMap.isometry_mul_flip, Quaternion.snd_re_dualNumberEquiv_symm, ZSpan.smul, Basis.flag_le_ker_dual, HasStrictFDerivAt.mul', AEval.of_symm_smul, Mathlib.Tactic.Module.NF.eval_neg, Ray.units_smul_of_neg, Matrix.intrinsicStar_toLin', LieModule.isNilpotent_toEnd_genWeightSpace_zero, HasFPowerSeriesWithinAt.hasStrictFDerivWithinAt, AlgebraicGeometry.StructureSheaf.const_add, LocalizedModule.mk_add_mk, conformal_const_smul, Subfield.relfinrank_eq_finrank_of_le, ContinuousLinearEquiv.conjContinuousAlgEquiv_surjective, ContinuousLinearMap.adjoint_id, Ideal.Filtration.smul_le, InnerProductSpace.smul_left, ClosedSubmodule.inf_orthogonal, LinearMap.coe_toContinuousLinearMap', ContDiffWithinAt.fun_smul, balanced_ball_zero, HasFDerivWithinAt.linear_multilinear_comp, KaehlerDifferential.polynomialEquiv_D, SemimoduleCat.Iso.homCongr_eq_arrowCongr, iteratedFDerivWithin_succ_eq_comp_left, supportDim_add_length_eq_supportDim_of_isRegular, isGalois_iff_isGalois_top, minpoly_algHom_toLinearMap, NonUnitalStarSubalgebra.unitizationStarAlgEquiv_apply_coe, ContDiffMapSupportedIn.iteratedFDerivWithOrderLM_apply_of_gt, ConvexCone.coe_zero, CharacterModule.smul_apply, MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.imPart, Free.instFaithfulSMulOfNontrivial, contDiffOn_iff_continuousOn_differentiableOn, CommRing.Pic.mul_eq_tensor, hasFDerivWithinAt_comp_smul_iff_smul, ContinuousLinearMap.isStarNormal_iff_norm_eq_adjoint, convexJoin_segments, LinearEquiv.dualMap_trans, ClosedSubmodule.instSMulMemClass, Basis.coe_dualBasis, Submodule.comap_le_comap_smul, sum_cauchyPowerSeries_eq_integral, SchwartzMap.laplacianCLM_eq, fderiv_fun_smul, CStarModule.innerSL_apply, IsLocalizedModule.iso_symm_apply, QuadraticMap.toBilinHom_apply, MeromorphicOn.fun_smul, ExteriorAlgebra.liftAlternating_ι_mul, TensorProduct.AlgebraTensorModule.rightComm_symm_tmul, LinearMap.tensorKer_coe, LinearIsometryEquiv.conjStarAlgEquiv_apply, MeasureTheory.smul_convolution, Real.vector_fourierIntegral_eq_integral_exp_smul, FiniteGaloisIntermediateField.instFiniteDimensionalSubtypeMemIntermediateFieldMin_1, VectorField.mlieBracketWithin_const_smul_right, PadicInt.mahlerTerm_apply, Algebra.Extension.CotangentSpace.map_id, LinearMap.toMatrixOrthonormal_reindex, AlternatingMap.map_update_smul, LinearMap.toMatrixAlgEquiv_toLinAlgEquiv, WithLp.smul_snd, MeasureTheory.ConvolutionExistsAt.integrable, Fintype.linearIndependent_iff', Complex.taylorSeries_eq_of_entire, HVertexOperator.coeff_isPWOsupport, exists_pos_right_iff_sameRay, QuadraticMap.associated_eq_self_apply, DirectLimit.Module.lift_of, QuadraticMap.toMatrix'_comp, LinearMap.trace_mul_cycle, SemilinearMapClass.distribMulActionSemiHomClass, Real.fourier_real_eq_integral_exp_smul, FiniteGaloisIntermediateField.mem_fixingSubgroup_iff, Monotone.quasilinearOn, Asymptotics.IsBigOWith.const_smul_self, Asymptotics.isLittleOTVS_prodMk_left, ExteriorAlgebra.toTrivSqZeroExt_comp_map, AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff, inner_smul_left_eq_star_smul, ProbabilityTheory.covarianceOperator_apply, BoundedContinuousFunction.toContinuousMapLinearMap_apply, nndist_smul₀, InnerProductGeometry.angle_smul_left_of_neg, basicOpen_subset_freeLocus_iff, MeasureTheory.integral_tilted, TensorProduct.quotTensorEquivQuotSMul_comp_mkQ_rTensor, NonUnitalSubalgebra.inclusion_mk, Submodule.span_smul_eq_of_isUnit, WithSeminorms.continuousSMul, IsSl2Triple.lie_lie_smul_f, LinearMap.ortho_smul_left, hasLineDerivAt_iff_isLittleO_nhds_zero, SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix, deriv.scomp_of_eq, ContinuousMultilinearMap.mkPiRing_apply, LinearMap.lTensor_smul, smul_unitClosedBall, im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four, AddMonoidHom.coe_smul', MeasureTheory.average_mem_openSegment_compl_self, TestFunction.continuous_iff_continuous_comp, HasFiniteFPowerSeriesOnBall.sum, LinearMap.ker_localizedMap_eq_localized'_ker, MeasureTheory.L1.integral_smul, exteriorPower.alternatingMapLinearEquiv_ιMulti, ContinuousLinearMap.opNorm_mul_flip_apply, ContinuousLinearMap.bilinearRestrictScalars_eq_restrictScalarsL_comp_restrictScalars, inv_natCast_smul_eq, DifferentiableOn.smul_const, IsLocalizedModule.iso_apply_mk, IntermediateField.exists_algHom_of_adjoin_splits, Quaternion.imJ_fst_dualNumberEquiv, InnerProductSpace.inner_left_rankOne_apply, cfcₙHom_nnreal_eq_restrict, star_dotProduct_toMatrix₂_mulVec, TestFunction.toBoundedContinuousFunctionCLM_apply, Conformal.const_smul, MeasureTheory.Lp.instFourierInvSMul, IsAlgClosure.of_exist_roots, CochainComplex.HomComplex.Cocycle.coe_units_smul, IsBaseChange.linearMapLeftRightHom_comp_apply, TensorProduct.AlgebraTensorModule.lTensor_comp_cancelBaseChange, SchwartzMap.fderivCLM_apply, hasStrictDerivAt_abs_pos, IntermediateField.normal_iff_forall_map_eq, Subspace.dualAnnihilator_dualAnnihilator_eq, ContinuousLinearMap.smul_compLp, MultilinearMap.currySumEquiv_apply, difference_quotients_converge_uniformly, hasDerivWithinAt_abs_pos, LocalizedModule.divBy_apply, TensorProduct.AlgebraTensorModule.lTensor_id, HasSum.hasSum_at_zero, HasFPowerSeriesAt.add, HasFDerivAt.norm_sq, DivisibleHull.mk_add_mk_left, lineDifferentiableAt_smul_iff, skewAdjointLieSubalgebraEquiv_symm_apply, TrivSqZeroExt.fstHom_comp_map, ContinuousAlternatingMap.curryLeft_smul, derivWithin.scomp_of_eq, fderiv_const_smul_field, Matrix.SpecialLinearGroup.center_equiv_rootsOfUnity'_symm_apply_coe_coe, zero_prodMap_dualTensorHom, Matrix.conjTranspose_natCast_smul, LinearMap.sum_repr_mul_repr_mulₛₗ, ContinuousLinearMap.hasBasis_nhds_zero, Ideal.mul_sup_eq_of_coprime_right, tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support, cfcL_apply, End.apply_smulCommClass', HasFDerivAtFilter.isBigOTVS_sub, TensorProduct.AlgebraTensorModule.rTensor_one, antideriv_bernoulliFun, CochainComplex.HomComplex.Cochain.units_smul_comp, AddCircle.integral_haarAddCircle, InnerProductSpace.HarmonicAt.const_smul, Matrix.Represents.smul, eventually_nhds_norm_smul_sub_lt, CentroidHom.smul_def, IsTorsionFree.isSMulRegular, MeasureTheory.integrableOn_image_iff_integrableOn_abs_det_fderiv_smul, HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn, TensorProduct.zero_smul, Algebra.lmul_injective, QuadraticMap.associated_rightInverse, associatedPrimes.mem_associatePrimes_localizedModule_atPrime_of_mem_associated_primes, Real.fourier_deriv, RightPreLieAlgebra.toIsScalarTower, iteratedDeriv_scomp_three, fderiv_fun_pow', hasDerivWithinAt_pi, Matrix.toBilin_symm, StarAddMonoid.toStarModuleRat, LinearMap.spectrum_toMatrix', LieModule.trace_toEnd_genWeightSpace, Real.integrable_prod_sub, Orientation.oangle_smul_left_of_neg, NormedSpace.isVonNBounded_ball, locallyConvexSpace_iff, MDifferentiable.smul, Real.smul_map_diagonal_volume_pi, strictConvex_Icc, antivary_iff_smul_rearrangement, norm_smul_inv_norm, LinearMap.smulRight_id, ContinuousLinearMap.eq_adjoint_iff, IsLocalizedModule.mk'_eq_zero', ContDiffAt.fderiv_succ, NonUnitalSubalgebra.inclusion_right, toMatrix_rotation, tangentConeAt_univ, PiTensorProduct.toDualContinuousMultilinearMap_apply_apply, convexHull_union_neg_eq_absConvexHull, LinearMap.separatingRight_congr_iff, ContinuousLinearMap.toSpanSingleton_apply, FreeNonUnitalNonAssocAlgebra.of_comp_lift, smul_vsub_rev_vadd_mem_affineSpan_pair, KaehlerDifferential.isScalarTower', Submodule.mem_torsion_iff, eval_apply_injective, derivWithin.scomp, LinearEquiv.congrQuadraticMap_refl, AlgHom.mulLeftRightMatrix.inv_comp, Submodule.dualAnnihilator_map_dualMap_le, DifferentiableOn.circleIntegral_sub_inv_smul, Derivation.linearEquiv_coe_to_linearMap_comp, Submodule.sum_smul_mem, Prod.segment_subset, OrderedFinpartition.norm_compAlongOrderedFinpartitionL_apply_le, smul_nonpos_iff, HasFPowerSeriesAt.comp, le_comap_range_lTensor, Submodule.dualCopairing_eq, EuclideanGeometry.reflection_vadd_smul_vsub_orthogonalProjection, ContinuousLinearMap.IsPositive.smul_of_nonneg, tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto, ContDiffAt.exists_forall_mem_closedBall_exists_eq_forall_mem_Ioo_hasDerivAt₀, InnerProductSpace.rankOne_apply, Convex.combo_eq_smul_sub_add, LieModule.shiftedGenWeightSpace.toEnd_eq, FormalMultilinearSeries.sum_of_finite, RingTheory.Sequence.isRegular_cons_iff, Ideal.finsuppTotal_apply_eq_of_fintype, MeasureTheory.convolution_tendsto_right, ContinuousLinearEquiv.map_smul, extremePoints_pi, IntermediateField.restrictScalars_normal, WittVector.factorPowSucc_fontaineThetaModPPow_eq, Submodule.reflection_singleton_apply, IntermediateField.isScalarTower, smul_unitClosedBall_of_nonneg, HasFTaylorSeriesUpToOn.zero_eq', KaehlerDifferential.quotKerTotalEquiv_symm_comp_D, Fintype.expect_ite_eq, TensorProduct.liftAux_tmul, MultilinearMap.map_insertNth_smul, QuotSMulTop.equivTensorQuot_naturality_mk, LieDerivation.inner_apply_apply, IsLocalRing.map_mkQ_eq, dot_cross_self, smulMemClass, Affine.Triangle.orthocenter_eq_smul_vsub_vadd_circumcenter, LieModule.Cohomology.twoCochain_alt, WeierstrassCurve.Projective.Point.zero_point, MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero, endTensorEndAlgHom_apply, inv_intCast_smul_comm, Matrix.toLpLin_mul_same, ContinuousLinearMap.mkOfIsCompactOperator_mem_compactOperator, LinearMap.mem_skewAdjointSubmodule, signedDist_left_congr, cfcₙ_eq_cfcₙL_mkD, Algebra.FormallySmooth.adjoin_of_algebraicIndependent, MeasureTheory.convolution_integrand_bound_right_of_le_of_subset, ContinuousMap.toLp_norm_le, Ideal.bot_pow, RootPairing.Hom.weight_coweight_transpose, ContinuousMultilinearMap.norm_smulRightL_le, Algebra.TensorProduct.comm_comp_map_apply, AnalyticOnNhd.iteratedFDeriv, Multiset.sum_smul_sum, balancedCore_nonempty_iff, IsAlgClosed.instIsAlgClosure, WeakFEPair.h_feq, LinearMap.ker_restrictScalars, DualBases.coe_dualBasis, Antitone.convex_le, ContinuousLinearMap.precompL_apply, Seminorm.closedBall_smul_ball, egauge_univ, Algebra.normalizedTrace_algebraMap, LieModule.toLinearMap_maxTrivLinearMapEquivLieModuleHom, IsSelfAdjoint.adjoint_eq, Antitone.quasiconvexOn, Submodule.mem_dualAnnihilator, Submodule.smul_mem', isPurelyInseparable_iff_fd_isPurelyInseparable, Submodule.mem_ideal_smul_span_iff_exists_sum', MeasureTheory.Measure.addHaar_preimage_smul, setIntegral_withDensity_eq_setIntegral_toReal_smul', QuadraticForm.equivalent_signType_weighted_sum_squared, LinearMap.toMatrix_baseChange, TrivSqZeroExt.fst_exp, Ideal.FG.pow, DirectSum.mk_smul, ContDiffMapSupportedIn.iteratedFDerivWithOrderLM_eq_of_scalars, curveIntegralFun_segment, SchwartzMap.integral_fourierInv_smul_eq, DifferentiableAt.hasDerivAt, IntermediateField.LinearDisjoint.of_isField', MonovaryOn.sum_comp_perm_smul_le_sum_smul, RootPairing.coroot_eq_smul_coroot_iff, Representation.asAlgebraHom_def, LieAlgebra.IsKilling.le_chainBotCoeff_of_rootSpace_ne_top, Besicovitch.SatelliteConfig.exists_normalized_aux2, Submodule.dualAnnihilator_eq_top_iff, signedDist_triangle, TensorProduct.adjoint_map, exists_nonneg_left_iff_sameRay, Affine.Simplex.point_vsub_faceOppositeCentroid_eq_smul_vsub, LinearEquiv.conj_comp, tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_integrableOn, LinearMap.BilinForm.isOrtho_smul_left, VectorBundleCore.contMDiffOn_coordChange, FormalMultilinearSeries.radius_le_smul, EuclideanGeometry.dist_eq_iff_eq_smul_rotation_pi_div_two_vadd_midpoint, PowerBasis.equivAdjoinSimple_symm_aeval, CliffordAlgebra.contractRight_comm, Ideal.pow_sup_eq_top, Ring.jacobson_smul_top_le, IsArtinian.exists_pow_succ_smul_dvd, TrivSqZeroExt.map_inr, Unitization.starLift_symm_apply_apply, AntivaryOn.smul_add_smul_le_smul_add_smul, NonUnitalSubalgebra.prod_inf_prod, intervalIntegral.integral_hasDerivAt_of_tendsto_ae_right, InfiniteGalois.toAlgEquivAux_eq_proj_of_mem, End.instLieModule, CStarMatrix.toCLM_injective, biUnion_associatedPrimes_eq_compl_regular, Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange, Complex.integral_circleTransform, ContinuousLinearMap.isPositive_self_comp_adjoint, iteratedDeriv_scomp_eq_sum_orderedFinpartition, convexOn_univ_dist, LinearMap.trace_eq_matrix_trace_of_finset, Unitization.starLift_apply_apply, HasFPowerSeriesOnBall.hasSum_iteratedFDeriv, LinearMap.adjoint_toContinuousLinearMap, Matrix.SeparatingLeft.toLinearMap₂', hasDerivAt_iff_isLittleO, IntermediateField.lift_adjoin_simple, LinearMap.rTensor_comp_flip_mk, Submodule.coe_dualCoannihilator_span, ExteriorAlgebra.liftAlternating_comp, TopModuleCat.continuousSMul, LocallyConvexSpace.convex_basis_zero, algebraicClosure.normalClosure_eq_self, NormedSpace.isBounded_iff_subset_smul_closedBall, KaehlerDifferential.isLocalizedModule_of_isLocalizedModule, LinearEquiv.conj_refl, CharacterModule.curry_apply_apply, hasFPowerSeriesAt_const, IntermediateField.AdjoinDouble.normal_algebraicClosure, Subalgebra.inclusion.isScalarTower_left, HasFPowerSeriesOnBall.const_smul, ContinuousLinearMap.toSesqForm_apply_coe, Finsupp.linearCombination_smul, signedDist_linear_apply_apply, Algebra.TensorProduct.assoc_tmul, Basis.toDualEquiv_apply, CliffordAlgebra.changeFormAux_changeFormAux, neg_one_smul, Fintype.bilinearCombination_apply_single, Submodule.pow_smul_top_le, HasFPowerSeriesWithinOnBall.fderivWithin, ContinuousLinearMap.cpolynomialOn_uncurry_of_multilinear, LinearMap.BilinForm.nondegenerate_toMatrix_iff, FormalMultilinearSeries.radius_compNeg, Fintype.expect_ite_zero, LinearMap.toMatrixₛₗ₂'_symm, LieAlgebra.IsKilling.apply_coroot_eq_cast', Ideal.range_mul', IsBoundedBilinearMap.smul_left, setIntegral_withDensity_eq_setIntegral_toReal_smul₀, quasispectrum.mem_iff_of_isUnit, SchwartzMap.integral_bilin_fourierIntegral_eq, QuadraticMap.discr_smul, Real.fourierIntegral_continuousMultilinearMap_apply, ProbabilityTheory.HasGaussianLaw.smul, LinearMap.rank_diagonal, LinearMap.ker_localizedMap_eq_localized₀_ker, QuotSMulTop.map_comp_mkQ, RootPairing.injOn_dualMap_subtype_span_root_coroot, LinearIsometryEquiv.adjoint_eq_symm, smul_mem_closure_star_mul, extDeriv_fun_smul, Ideal.finsuppTotal_apply, IsLocalizedModule.exist_integer_multiples, ContinuousAlternatingMap.toAlternatingMap_curryLeft, TensorProduct.flip_mk_surjective, Ideal.pow_sup_pow_eq_top, LinearMap.separatingRight_toMatrix₂'_iff, Matrix.toEuclideanLin_apply, LieSubmodule.Quotient.toEnd_comp_mk', Finset.expect_ite_zero, Submodule.biSup_eq_range_dfinsupp_lsum, Quaternion.imI_fst_dualNumberEquiv, LieAlgebra.IsKilling.rootSpace_neg_nsmul_add_chainTop_of_lt, ContinuousLinearMap.toLinearMap₁₂_apply, LinearMap.IsPositive.isPositive_smul_iff, Algebra.toMatrix_lmul_eq, analyticOrderAt_smul_eq_top_of_left, VectorFourier.fourierPowSMulRight_eq_comp, ContinuousLinearMap.finiteDimensional, HasFPowerSeriesOnBall.tendsto_partialSum_prod, ContinuousLinearMap.hasFDerivWithinAt_of_bilinear, TensorProduct.tensorQuotEquivQuotSMul_comp_mkQ_lTensor, WithSeminorms.tendsto_nhds', LinearMap.toMatrixAlgEquiv'_id, isAdjointPair_toLinearMap₂, OreLocalization.mul_zero, ContinuousLinearMap.opNorm_mul, MvPowerSeries.smul_eq_C_mul, LieModule.Cohomology.d₂₃_apply, Algebra.Generators.liftBaseChange_injective_of_isLocalizationAway, Seminorm.pullback_apply, collinear_iff_of_mem, sameRay_iff_inv_norm_smul_eq_of_ne, TensorProduct.equivFinsuppOfBasisLeft_apply_tmul_apply, ContinuousAlternatingMap.constOfIsEmptyLIE_apply, CoalgCat.MonoidalCategoryAux.counit_tensorObj_tensorObj_right, norm_sum_lt_of_strictConvexSpace, FGModuleCat.instFiniteHomModuleCatObjIsFG, LieModule.instIsTriangularizableSubtypeEndMemLieSubalgebraRangeToEnd, exteriorPower.alternatingMapLinearEquiv_symm_map, UniformConvexOn.strictConvexOn, TemperedDistribution.smulLeftCLM_neg, Orthonormal.basisTensorProduct, InnerProductSpace.Core.inner_smul_ofReal_right, Orientation.oangle_sign_sub_smul_left, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge', MeasureTheory.SimpleFunc.integral_eq_sum_filter, egauge_zero_zero, algebraicClosure.isAlgebraic, Matrix.conjTranspose_inv_intCast_smul, QuadraticForm.tmul_tensorMap_apply, PointwiseConvergenceCLM.postcomp_apply, segment_subset_convexHull, MeasureTheory.Measure.quasiMeasurePreserving_smul, IsGalois.normalClosure, Unitization.nnnorm_def, LieModule.mem_weightSpace, instSMulCommClassMatrixFinOfNatNatProd, PowerSeries.le_order_smul, ContinuousLinearMap.hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear, Matrix.liftLinear_singleLinearMap, LieModule.Cohomology.d₁₂_apply_apply, Algebra.Extension.Cotangent.smul_eq_zero_of_mem, convexHull_toCone_eq_sInf, NonUnitalSubalgebra.prod_top, Algebra.Generators.cotangentSpaceBasis_repr_one_tmul, ringKrullDim_quotSMulTop_succ_eq_ringKrullDim_of_mem_jacobson, Finset.zero_mem_smul_finset_iff, LinearMap.smulRight_apply, HasFDerivAt.fun_smul, AEval'.X_smul_of, Function.HasTemperateGrowth.toTemperedDistribution_apply, ContinuousLinearMap.flipMultilinear_apply_apply, ContinuousMultilinearMap.uncurrySum_apply, Orthonormal.inner_left_fintype, lie_smul, HasLineDerivAt.tendsto_slope_zero_right, LieModule.mem_posFittingCompOf, TensorProduct.AlgebraTensorModule.map_smul_left, NonUnitalStarAlgHom.range_comp_le_range, Complex.hasDerivAt_GammaIntegral, Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope, LinearMap.ofIsComplProd_apply, LinearMap.mul_apply', compactOperator_topologicalClosure, LocalizedModule.divBy_mul_by, IncidenceAlgebra.instIsScalarTower, Rep.coindFunctorIso_hom_app_hom_hom_apply_hom_hom_apply, LinearMap.toMatrix_symm, Matrix.toBilin_apply, IntermediateField.isAlgebraic_iff, LinearMap.toMatrix₂'_compl₁₂, MonoidAlgebra.coe_liftNCAlgHom, PiTensorProduct.mapL_smul, End.maxGenEigenspace_eq_maxGenEigenspace_zero, SchwartzMap.integral_sesq_fourier_eq, DualNumber.lift_apply_eps, cfcL_integrable, Real.fderiv_fourier, CategoryTheory.ShortComplex.Homotopy.smul_h₂, LieAlgebra.SpecialLinear.singleSubSingle_add_singleSubSingle, ContinuousLinearMap.adjointAux_norm, MeasureTheory.L1.setToL1_smul_left, LinearMap.prodEquiv_apply, Representation.smul_ofModule_asModule, CStarMatrix.norm_def', KaehlerDifferential.instIsScalarTowerTensorProduct, algebra_compatible_smul, MeasureTheory.Integrable.smul_bdd, Fintype.expect_dite_eq', Ideal.map_mul, Basis.toLin_toMatrix, NonUnitalStarAlgHom.injective_codRestrict, MeasureTheory.convolution_smul, TensorProduct.AlgebraTensorModule.restrictScalars_lTensor, neg_smul_neg, hasDerivAtFilter_natCast, Submodule.map_smul'', LinearMap.flip_injective_iff₁, Finsupp.linearCombination_one_tmul, WeierstrassCurve.Projective.addMap_of_Z_eq_zero_left, CategoryTheory.Linear.units_smul_comp, meromorphicOrderAt_ne_top_iff, Ideal.mul_bot, intervalIntegral.hasFDerivAt_integral_of_dominated_loc_of_lip, MeasureTheory.SignedMeasure.singularPart_smul, Pi.counit_eq_adjoint, ConvexCone.coe_map, WithLp.unitization_algebraMap, ProbabilityTheory.covarianceBilin_apply_basisFun_self, HahnSeries.SummableFamily.hsum_smul_module, ContinuousAlternatingMap.instContinuousSMul, LinearMap.ker_smul', DualNumber.inv_eps, LinearMap.toMatrixOrthonormal_apply_apply, LocalizedModule.map_surjective, IsLocalizedModule.map_linearMap_of_isLocalization, HasFDerivAt.restrictScalars, balancedHull_subset_convexHull_union_neg, ContinuousLinearMap.hasStrictFDerivAt_of_bilinear, Matrix.toLin'_mul_apply, tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux, MultilinearMap.fromDirectSumEquiv_symm_apply, SameRay.exists_eq_smul_add, CategoryTheory.Functor.Linear.map_smul, LieModule.isNilpotent_range_toEnd_iff, stereoInvFunAux_apply, Finset.sum_smul_vsub_eq_weightedVSubOfPoint_sub, StrongFEPair.functional_equation, norm_curveIntegral_segment_le, IntermediateField.coe_inclusion, Matrix.toLinearMapₛₗ₂'_single, LieAlgebra.bracket_ofTwoCocycle, VectorFourier.fourierIntegral_continuousLinearMap_apply, AnalyticAt.exists_eventuallyEq_sum_add_pow_mul, LocalizedModule.mk_eq, ContinuousLinearEquiv.conjContinuousAlgEquiv_apply, signedDist_le_dist, IsLinearMap.isLinearMap_smul', AffineSubspace.coe_subtypeₐᵢ, MonoidAlgebra.instIsPushout, Matrix.nondegenerate_toBilin'_iff, Ideal.smul_eq_mul, TensorProduct.AlgebraTensorModule.restrictScalars_curry, HasFDerivAtFilter.isThetaTVS_sub, CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φH, LieAlgebra.rank_le_natTrailingDegree_charpoly_ad, LinearMap.trace_eq_contract_of_basis', TensorProduct.AlgebraTensorModule.rightComm_eq, QuadraticForm.tmul_comp_tensorComm, LinearMap.BilinForm.congr_comp, NonUnitalSubalgebra.inclusion_self, LinearMap.toMatrixAlgEquiv_apply, Finset.mem_convexHull', cfcₙ_apply_mkD, iteratedDeriv_smul_const, circleIntegral.integral_sub_inv_smul_sub_smul, Action.smul_hom, RingTheory.Sequence.isWeaklyRegular_iff_Fin, HasStrictFDerivAt.list_prod', FreeNonUnitalNonAssocAlgebra.lift_unique, SchwartzMap.fourier_lineDerivOp_eq, DifferentiableAt.smul, LieAdmissibleAlgebra.toSMulCommClass, LinearMap.isBigOTVS_rev_comp, cfcₙHom_id, Algebra.Extension.instIsScalarTowerH1CotangentOfCotangent, TemperedDistribution.instLineDerivLeftSMulReal, Submodule.span_singleton_eq_top_iff, CStarMatrix.inner_toCLM_conjTranspose_left, RootPairing.reflection_apply_root', LinearMap.toMatrix'_toLin', hasStrictDerivAt_exp_smul_const, Submodule.FG.restrictScalars, PolynomialModule.monomial_smul_single, Derivation.leibniz_div, MultilinearMap.curryRight_apply, EuclideanGeometry.inversion_vsub_center, isBoundedBilinearMap_comp, instIsAbelianGaloisSubtypeMemIntermediateFieldBot, mapEvalEquiv_apply, LinearMap.range_toAddSubmonoid, set_smul_sphere_zero, MeasureTheory.Integrable.essSup_smul, HasFPowerSeriesWithinOnBall.uniform_geometric_approx, SchwartzMap.smulLeftCLM_compCLMOfContinuousLinearEquiv, MeasureTheory.pdf.integral_pdf_smul, TensorProduct.toMatrix_assoc, CochainComplex.HomComplex.Cochain.rightUnshift_units_smul, Subspace.finrank_dualCoannihilator_eq, ContMDiff.smul, Bialgebra.TensorProduct.comul_eq_algHom_toLinearMap, hasStrictDerivAt_add_const_iff, hasLineDerivWithinAt_smul_iff, Matrix.toLinearMap₂'_apply', Asymptotics.IsBigO.const_smul_left, Asymptotics.IsTheta.smul, TemperedDistribution.fourierInv_lineDerivOp_eq, MDifferentiable.const_smul, LieSubmodule.lie_top_eq_of_span_sup_eq_top, MeasureTheory.condExpL2_indicator_ae_eq_smul, ModuleCat.smulShortComplex_X₃_isAddCommGroup, conformalAt_const_smul, Submodule.span_smul_le, LeftInvariantDerivation.lift_smul, IsMIntegralCurveOn.hasDerivWithinAt, IsLocalizedModule.mk'_add_mk', Algebra.SubmersivePresentation.basisKaehlerOfIsCompl_apply, Polynomial.X_pow_smul_rTensor_monomial, FormalMultilinearSeries.rightInv_removeZero, PiToModule.fromEnd_injective, ContinuousAlternatingMap.ofSubsingletonLIE_symm_apply, ClosedSubmodule.orthogonal_closure'', instIsPushoutFractionRingPolynomial_1, LinearMap.coe_lTensorHom, hasStrictDerivAt_exp_smul_const', KaehlerDifferential.mapBaseChange_tmul, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable, DirectLimit.quotMk_of, openSegment_translate_image, OrderIso.smulRightDual_apply, AffineSubspace.smul_vsub_vadd_mem, LinearMap.toMatrix_id, MeromorphicAt.fun_smul, Real.fourier_iteratedFDeriv, LinearIsometryEquiv.adjoint_toLinearMap_eq_symm, CFC.negPart_smul_of_nonpos, SchwartzMap.smulLeftCLM_real_smul, strictConvex_uIcc, InnerProductSpace.toDualMap_apply_apply, NNRealRMK.lintegral_rieszMeasure, MultilinearMap.currySum_apply', MeasureTheory.Measure.integral_comp_inv_mul_left, LinearMap.BilinForm.apply_eq_dotProduct_toMatrix_mulVec, Orientation.kahler_eq_zero_iff, KaehlerDifferential.kerTotal_eq, LinearMap.eq_adjoint_iff_basis_left, contDiff_succ_iff_fderiv, NormMulClass.toNormSMulClass_op, RootPairing.pairing_one_four_iff', ModuleCat.ExtendRestrictScalarsAdj.Counit.map_hom_apply, MeasureTheory.measure_smul_average, MDifferentiableOn.smul, OreLocalization.zero_smul, TrivSqZeroExt.fst_invOf, Orientation.oangle_sign_sub_smul_right, hasDerivAtFilter_zero, segment_subset_convexJoin, Matrix.toAlgEquiv_kroneckerStarAlgEquiv, Algebra.Generators.H1Cotangent.δAux_C, Complex.integral_comp_polarCoord_symm, smul_closedBall', signedDist_apply_apply, ContinuousLinearMap.one_smulRight_eq_toSpanSingleton, Algebra.Generators.Hom.toExtensionHom_toAlgHom_apply, spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul, HasStrictFDerivAt.hasStrictDerivAt_norm_smul_neg, QuadraticMap.associated_prod, ConvexCone.coe_comap, SemimoduleCat.ofHom₂_hom_apply_hom, LocalizedModule.mk_cancel, add_smul, IntermediateField.minpoly_gen, PresheafOfModules.Sheafify.map_smul_eq, convex_iff_openSegment_subset, Filter.EventuallyEq.hasDerivWithinAt_iff_of_mem, Orientation.oangle_smul_left_self_of_nonneg, Monotone.convex_ge, UnitAddTorus.mFourierCoeff_toLp, NonUnitalSubalgebra.unitization_injective, Submodule.coe_toConvexCone, LinearMap.toMatrix_transpose, WeakDual.toStrongDual_inj, LinearMap.toMatrix₂_toLinearMapₛₗ₂, ContinuousLinearMap.uniformContinuous_restrictScalars, TensorProduct.finsuppLeft_smul', MvPolynomial.sumAlgEquiv_comp_rename_inr, Ideal.adic_module_basis, LieSubmodule.instSMulMemClass, TensorAlgebra.toDirectSum_ofDirectSum, SubmodulesBasis.smul, ContinuousLinearMap.mulLeftRight_apply, mem_openSegment_translate, PowerSeries.coe_smul, isModuleTopologyOfFiniteDimensional, IsLocalizedModule.map_surjective_iff_localizedModuleMap_surjective, MvPolynomial.aevalTower_toAlgHom, LinearMap.extendTo𝕜'_apply, LieAlgebra.ad_eq_lmul_left_sub_lmul_right, curveIntegral_fun_smul, FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum, Seminorm.ball_eq_emptyset, ArchimedeanClass.mk_le_mk_smul, ContinuousLinearMap.holderₗ_apply_apply, taylor_tendsto, dslope_sub_smul, rescale_to_shell_semi_normed_zpow, HasFDerivAt.curveIntegral_segment_source, LinearMap.trace_smulRight, Algebra.TensorProduct.basisAux_map_smul, balancedCore_mem_nhds_zero, convexJoin_segment_singleton, SameRay.exists_eq_smul, Bundle.ContinuousRiemannianMetric.symm, ContinuousMultilinearMap.curryRight_norm, AdicCompletion.factor_eval_eq_evalₐ, hasDerivWithinAt_Iic_of_tendsto_deriv, CentroidHom.instSMulCommClass_2, mellin_hasDerivAt_of_isBigO_rpow, ContinuousLinearMap.analyticOn_bilinear, deriv_fun_smul, ConvexCone.salient_strictlyPositive, CompactlySupportedContinuousMap.toRealLinearMap_apply_apply, LocalizedModule.mem_ker_mkLinearMap_iff, Matrix.det_kroneckerMapBilinear, PresheafOfModules.Derivation.d_mul, AlgebraicGeometry.instIsScalarTowerObjOppositeOpensCarrierTopValStructureSheafInType, LinearMap.IsRefl.ker_eq_bot_iff_ker_flip_eq_bot, MeasureTheory.convolution_lsmul, QuadraticMap.map_smul, Matrix.toLinAlgEquiv_mul, Localization.algHom_ext_iff, SeminormFamily.finset_sup_comp, Submodule.mem_span_finset, Bornology.isVonNBounded_iff_absorbing_le, AEval.instIsScalarTowerOrigPolynomial, IsLocalizedModule.fromLocalizedModule.bij, CategoryTheory.Pretriangulated.Triangle.smul_hom₂, meromorphicNFOn_smul_iff_right_of_analyticOnNhd, Matrix.SpecialLinearGroup.toLin'_symm_to_linearMap, ringKrullDim_quotSMulTop_succ_eq_ringKrullDim, AEval.C_smul, Algebra.Extension.CotangentSpace.map_tmul, Bornology.isVonNBounded_covers, hasDerivAt_abs_neg, Orientation.kahler_swap, coe_innerSL_apply, Matrix.vecMulBilin_apply, element_smul_restrictScalars, strongConcaveOn_zero, integral_withDensity_eq_integral_smul, Representation.ofMulAction_self_smul_eq_mul, intervalIntegral.integral_comp_add_div, Matrix.instSMulCommClassForall_1, Unitization.lift_symm_apply_apply, convex_singleton, cross_anticomm, LinearMap.range_smulRight_apply_of_surjective, deriv_fderiv, MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux1, RegularNormedAlgebra.isometry_mul', convexJoin_singleton_segment, cauchyPowerSeries_apply, KaehlerDifferential.polynomialEquiv_comp_D, ContDiffWithinAt.hasFDerivWithinAt_nhds, KaehlerDifferential.span_range_map_derivation_of_isLocalization, LocalizedModule.equivTensorProduct_symm_apply_tmul_one, LinearMap.dualMap_surjective_of_injective, Rep.ihom_ev_app_hom, LinearEquiv.toSpanNonzeroSingleton_homothety, IsBaseChange.endHom_toMatrix, Free.multilinearMap, BoxIntegral.HasIntegral.smul, CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM, TemperedDistribution.instFourierSMul, Ideal.range_mul, CommRing.Pic.mk_tensor, integral_smul_const, range_cfcₙHom, LocalizedModule.equivTensorProduct_apply_mk, CStarModule.inner_smul_right_real, Field.Emb.Cardinal.succEquiv_coherence, cross_cross_eq_smul_sub_smul, exists_nontrivial_relation_of_finrank_lt_card, Real.map_matrix_volume_pi_eq_smul_volume_pi, NonUnitalStarSubalgebra.map_map, TrivSqZeroExt.algHom_ext'_iff, Submodule.dualAnnihilator_anti, SameRay.inv_norm_smul_eq, GradedTensorProduct.mul_def, IsTorsionBySet.mk_smul, hasDerivWithinAt_abs, hasFPowerSeriesOnBall_const, QuadraticForm.polarBilin_tmul, openSegment_eq_image', ContDiffMapSupportedIn.seminorm_le_iff, LieModule.isNilpotent_toEnd_of_isNilpotent, LieAlgebra.hasCentralRadical_and_of_isIrreducible_of_isFaithful, MultilinearMap.curryFinFinset_apply, LocalizedModule.map_injective, ContinuousLinearMap.hasDerivAt, hasDerivAt_of_tendstoUniformly, NonUnitalStarSubalgebra.unitization_range, Matrix.toAlgEquiv_kroneckerTMulStarAlgEquiv, smul_nonpos_iff_pos_imp_nonpos, LinearMap.BilinForm.Nondegenerate.congr, Rep.MonoidalClosed.linearHomEquivComm_hom, rieszContentAux_image_nonempty, MeasureTheory.Measure.integral_comp_smul_of_nonneg, Algebra.TensorProduct.tensorTensorTensorComm_symm_tmul, VectorFourier.hasFDerivAt_fourierIntegral, DiffContOnCl.circleIntegral_sub_inv_smul, HahnEmbedding.ArchimedeanStrata.ball_sup_stratum_eq, TensorProduct.equivFinsuppOfBasisLeft_symm, SymmetricAlgebra.instModuleFree, Submodule.comap_smul, TensorProduct.Algebra.smul_def, LinearMap.IsProj.eq_conj_prodMap, LinearMap.flip_surjective_iff₁, LinearMap.tensorEqLocus_coe, TensorAlgebra.toExterior_ι, LaurentPolynomial.smeval_C_mul, Seminorm.ball_norm_mul_subset, IsLocalizedModule.exist_integer_multiples_of_finset, AlternatingMap.alternatizeUncurryFin_apply, List.sum_smul, hasFDerivAt_ringInverse, ExteriorAlgebra.liftAlternatingEquiv_symm_apply, ContinuousMultilinearMap.opNorm_smul_le, TensorProduct.curry_injective, Function.HasTemperateGrowth.fun_smul, Filter.HasBasis.isLittleOTVS_iff, Matrix.toLinearMap₂'_apply, MeasureTheory.convolution_precompR_apply, PolynomialModule.single_smul, WeakDual.coe_toStrongDual, lTensorHomEquivHomLTensor_apply, LinearMap.BilinForm.separatingLeft_toMatrix'_iff, MeasureTheory.Lp.simpleFunc.smul_toSimpleFunc, VectorFourier.fourierIntegral_iteratedFDeriv, LinearMap.extendScalarsOfIsLocalizationEquiv_apply, rank_lt_rank_dual', coeSubmodule_differentIdeal_fractionRing, QuotSMulTop.map_surjective, Ideal.Fiber.lift_residueField_surjective, InformationTheory.hasDerivAt_klFun, isBaseChange_tensorProduct_map, IsLocalizedModule.mk'_eq_mk'_iff, InnerProductSpace.nnnorm_rankOne, FreeLieAlgebra.liftAux_map_smul, FiniteGaloisIntermediateField.instFiniteDimensionalSubtypeMemIntermediateFieldMin, Seminorm.closedBall_zero_eq_preimage_closedBall, fderiv_smul, ArithmeticFunction.one_smul', LinearMap.trace_mul_comm, LinearMap.trace_eq_contract, OrthonormalBasis.orthogonalProjection_apply_eq_sum, intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_left, Filter.BoundedAtFilter.smul, Orientation.kahler_comp_rightAngleRotation, LinearMap.toKerLocalized_isLocalizedModule, Seminorm.ball_smul, ContinuousLinearEquiv.arrowCongr_apply, NonUnitalSubalgebra.unitization_range, LieModule.LinearWeights.map_smul, SemimoduleCat.Algebra.instIsScalarTowerCarrier, instIsOrderedModule, ModuleCat.CoextendScalars.smul_apply, CliffordAlgebra.changeFormAux_apply_apply, hasDerivAt_of_tendstoLocallyUniformlyOn, hasDerivAtFilter_iff_tendsto_slope, starConvex_compl_Ici, LocalizedModule.equivTensorProduct_symm_apply_tmul, ContinuousLinearMap.norm_toSpanSingleton, FormalMultilinearSeries.continuousOn_of_finite, LinearEquiv.arrowCongr_symm_apply, intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_left, TensorProduct.finsuppScalarRight_apply_tmul, Submodule.topologicalClosure_map, Submodule.toConvexCone_top, Representation.single_smul, LinearMap.trace_eq_sum_inner, IntermediateField.finSepDegree_adjoin_simple_eq_finrank_iff, ContinuousLinearMap.toUniformConvergenceCLM_apply, FormalMultilinearSeries.ofScalars_series_of_subsingleton, ContinuousAlternatingMap.curryLeft_apply_apply, NormedSpace.isVonNBounded_closedBall, affinity_unitClosedBall, ContinuousLinearMap.flip_smul, solvableByRad.isSolvable, LieSubmodule.coe_map_toEnd_le, FinitePresentation.linearEquivMap_symm_apply, ContDiffMapSupportedIn.coe_smul, PadicInt.mahlerSeries_apply, HasFDerivAt.isBigOTVS_sub, LieModuleHom.coe_smul, IntermediateField.linearDisjoint_comm, MeasureTheory.integral_singleton', HasCompactSupport.hasDerivAt_convolution_right, InnerProductSpace.isSymmetricProjection_rankOne_self, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, Submodule.annihilator_mul, Subspace.dualPairing_nondegenerate, Matrix.isNilpotent_toLin'_iff, HasFTaylorSeriesUpToOn.eventually_hasFDerivAt, Finset.affineCombination_eq_linear_combination, SameRay.pos_smul_left, Coalgebra.TensorProduct.lid_tmul, Orientation.volumeForm_zero_pos, LinearMap.trace_one, Basis.constrL_apply, AEval'.of_symm_X_smul, TensorAlgebra.equivFreeAlgebra_symm_ι, homTensorHomEquiv_apply, ProbabilityTheory.uncenteredCovarianceBilinDual_of_not_memLp, Bundle.ContinuousRiemannianMetric.continuous, QuadraticMap.associated_apply, LieModule.mem_genWeightSpace, ModuleFilterBasis.smul_right, MeasureTheory.ConvolutionExistsAt.integrable_swap, NormedSpace.expSeries_sum_eq, SchwartzMap.laplacianCLM_eq', hasGradientWithinAt_iff_hasFDerivWithinAt, IsModuleTopology.continuous_bilinear_of_pi_fintype, norm_inner_eq_norm_iff, TensorProduct.mk_apply, IsLocalizedModule.finsetIntegerMultiple_image, Real.circleAverage_eq_circleIntegral, HasFiniteFPowerSeriesOnBall.eq_partialSum, image_openSegment, Ring.instIsScalarTowerSubtypeAlgebraicClosureFractionRingMemIntermediateFieldNormalClosure, ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left, Submodule.mul_mem_smul_iff, FormalMultilinearSeries.ofScalars_add, AlternatingMap.alternatizeUncurryFin_add, Matrix.toLinearEquiv'_symm_apply, IntermediateField.equivOfEq_symm, LinearIsometryEquiv.smul_apply, convexOn_iff_forall_pos, nnnorm_cfcₙ_nnreal_lt, inner_eq_norm_mul_iff_div, convex_Ico, Submodule.topologicalClosure_eq_self, Submodule.smul_subset_smul, SameRay.nonneg_smul_right, LinearMap.hasStrictDerivAt, LinearMap.BilinForm.mul_toMatrix'_mul, VectorFourier.norm_iteratedFDeriv_fourierPowSMulRight, Matrix.toLin_symm, AffineMap.hasDerivAt_lineMap, LinearMap.toMatrixAlgEquiv_id, WeakDual.CharacterSpace.instNonUnitalAlgHomClass, ratCast_smul_eq, MeasureTheory.integral_smul, HomogeneousLocalization.val_zsmul, Seminorm.convexOn, mellinInv_eq_fourierIntegralInv, ContinuousMultilinearMap.curryFinFinset_symm_apply_const, ProbabilityTheory.strong_law_ae, Derivation.comp_aeval_eq, Algebra.Extension.Cotangent.val_smul''', CStarModule.inner_smul_right_complex, HasGradientAtFilter.hasDerivAtFilter', IntermediateField.restrictRestrictAlgEquivMapHom_apply, WeakDual.CharacterSpace.continuousMapEval_bijective, instIsPushoutPolynomial_1, CompactlySupportedContinuousMap.coe_toRealLinearMap, SeparatingDual.dualMap_surjective_iff, Seminorm.balanced_closedBall_zero, InnerProductGeometry.norm_toLp_symm_crossProduct, zmod_smul_mem, dualTensorHom_apply, BilinForm.mul_toMatrix_mul, isNoetherian_linearMap_pi, NonUnitalStarSubalgebra.val_inclusion, SmoothPartitionOfUnity.contDiffAt_finsum, NonUnitalAlgebra.range_id, ContMDiffAt.smul, TrivSqZeroExt.map_comp_inlAlgHom, intervalIntegral.integral_hasDerivWithinAt_left, QuadraticForm.tensorRId_symm_apply, ofNat_smul_eq_nsmul, HasFPowerSeriesOnBall.fderiv_eq, Finsupp.weight_single, Seminorm.coe_restrictScalars, strictConvex_closedBall, Invertible.rTensorEquiv_apply_apply, LinearMap.localized'_range_eq_range_localizedMap, NormedSpace.toNormSMulClass, lTensorHomEquivHomLTensor_toLinearMap, Finset.sum_smul_const_vsub_eq_vsub_affineCombination, Ideal.bot_mul, Quaternion.fst_imK_dualNumberEquiv_symm, midpoint_eq_smul_add, Basis.linearMap_apply, Asymptotics.IsLittleOTVS.zero, isBigO_rpow_top_log_smul, Submodule.dualCoannihilator_iSup_eq, Subalgebra.centralizer_coe_image_includeRight_eq_center_tensorProduct, comap_eval_surjective, NormedSpace.inclusionInDoubleDual_norm_eq, FinitePresentation.exists_lift_of_isLocalizedModule, AffineMap.left_vsub_lineMap, IntermediateField.fieldRange_comp_val, SchwartzMap.smulLeftCLM_sum, Matrix.toLinearMap₂_compl₁₂, midpoint_sub_right, LinearMap.dualMap_id, Submodule.smul_mem_span_smul_of_mem, ContinuousLinearEquiv.conjContinuousAlgEquiv_trans, LinearAlgebra.FreeProduct.ι_apply, Submodule.coe_span_smul, CFC.sqrt_map_prod, AffineMap.hasDerivWithinAt_lineMap, IntermediateField.adjoin.powerBasis_gen, NonUnitalSubsemiring.unitization_apply, ContinuousLinearMap.fderiv_of_bilinear, TestFunction.coe_smul, has_fderiv_at_filter_real_equiv, SkewMonoidAlgebra.nonUnitalAlgHom_ext'_iff, ExteriorAlgebra.liftAlternating_ιMulti, AlgCat.hom_hom_associator, Algebra.Extension.Hom.comp_id, linearIndependent_algHom_toLinearMap', cfcₙHom_integral, Ideal.map_mk_comap_factorPow, MeasureTheory.setIntegral_condExpInd, IntermediateField.equivOfEq_rfl, Seminorm.gaugeSeminorm_ball, ContinuousLinearMap.nnnorm_toSpanSingleton, iteratedFDerivWithin_succ_apply_right, ContinuousAlternatingMap.alternatizeUncurryFin_smul, CliffordAlgebra.changeForm_ι_mul, IsLocalizedModule.iso_symm_comp, PresheafOfModules.map_smul, contDiffAt_map_inverse, fderiv_mul_const', tendsto_integral_exp_smul_cocompact, LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf, QuadraticForm.tensorAssoc_toLinearEquiv, Matrix.linftyOpIsBoundedSMul, MeasureTheory.integral_comp_rpow_Ioi_of_pos, TensorProduct.AlgebraTensorModule.rTensor_tensor, Complex.hasDerivAt_Gammaℝ_one, NonUnitalAlgHom.subsingleton, Matrix.toLinearMapₛₗ₂'_symm, LinearMap.localized'_ker_eq_ker_localizedMap, curveIntegral_segment, Ray.linearEquiv_smul_eq_map, Asymptotics.isLittleOTVS_fun_neg_left, TestFunction.postcompCLM_apply, Submodule.mem_annihilator', hasDerivAt_iff_tendsto_slope_left_right, BoundedContinuousFunction.instStarModule, IntermediateField.normal_iff_normalClosure_le, DualNumber.fst_eq_zero_iff_eps_dvd, HasFTaylorSeriesUpToOn.hasFDerivWithinAt, TrivSqZeroExt.eq_smul_exp_of_ne_zero, Asymptotics.isBigOTVS_iff_smallSets, Derivation.leibniz_inv, Real.fourierIntegral_continuousLinearMap_apply, smul_left_inj, AdicCompletion.isAdicCauchy_iff, LinearMap.polyCharpoly_baseChange, LinearMap.isAlt_iff_eq_neg_flip, Submodule.adjoint_subtypeL, TrivSqZeroExt.invertibleEquivInvertibleFst_symm_apply_invOf, LinearMap.toLinearMap_toContPerfPair, TensorProduct.AlgebraTensorModule.rTensor_comp, Mathlib.Tactic.LinearCombination.smul_lt_const, Polynomial.aevalTower_toAlgHom, MatrixEquivTensor.toFunBilinear_apply, AnalyticOn.fderivWithin, LinearMap.isSymmetric_adjoint_mul_self, InnerProductSpace.Core.cauchy_schwarz_aux, Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul, ContinuousLinearMap.opNorm_mul_apply, gelfandStarTransform_symm_apply, ContinuousLinearMap.norm_smulRightL, wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg, Asymptotics.IsBigOWith.const_smul_left, Submodule.dualRestrict_ker_eq_dualAnnihilator, IsBaseChange.transvection, LieSubalgebra.toEnd_mk, Ring.instIsScalarTowerFractionRingSubtypeAlgebraicClosureMemIntermediateFieldNormalClosure_1, Ideal.mul_left_self_sup, LinearMap.toBilin'Aux_toMatrixAux, MeasureTheory.FiniteMeasure.self_eq_mass_smul_normalize, LieAlgebra.IsKilling.invtSubmoduleToLieIdeal_top, convexOn_univ_norm, IsLocalRing.map_tensorProduct_mk_eq_top, HasFDerivWithinAt.fun_smul, Bialgebra.TensorProduct.comulAlgHom_def, AffineMap.lineMap_vsub_right, WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm, End.intCast_def, Dual.eval_apply, ContDiffAt.fderiv_right, strictAnti_smul_left, MeasureTheory.convolution_def, LinearMap.toMatrix₂_compl₂, Pretrivialization.continuousLinearMapCoordChange_apply, ContinuousMultilinearMap.curryLeft_norm, Seminorm.ball_zero_eq_preimage_ball, finsuppTensorFinsuppLid_symm_single_smul, reflection_mul_reflection_pow_apply, DFinsupp.sum_mapRange_index.linearMap, CategoryTheory.Abelian.Ext.mk₀_smul, InnerProductSpace.trace_rankOne, mulRightLinearMap_eq_mulRight, rTensorHomEquivHomRTensor_apply, ContinuousAlternatingMap.alternatizeUncurryFin_constOfIsEmptyLIE_comp, LinearMap.trace_eq_contract_apply, MeasureTheory.Submartingale.smul_nonneg, ExteriorAlgebra.liftAlternatingEquiv_apply, dualTensorHomEquivOfBasis_symm_cancel_right, CStarModule.inner_smul_left_complex, openSegment_subset_ball_of_ne, HasFPowerSeriesWithinOnBall.prod, units_smul_eq_self_iff, NormedSpace.normalize_smul, smul_add_smul_lt_smul_add_smul, MeasureTheory.integrable_withDensity_iff_integrable_smul', Matrix.toLpLin_symm_pow, IsMIntegralCurveOn.comp_mul, Polynomial.mkDerivation_one_eq_derivative', Prod.image_mk_segment_left, Matrix.isBoundedSMul, MvPolynomial.scalarRTensor_apply_tmul, TensorProduct.prodLeft_tmul, Matrix.conjTranspose_inv_natCast_smul, ContinuousAlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinCLM_comp_apply, ConvexBody.coe_smul, IsLocalization.tensorProduct_isLocalizedModule, ContDiffOn.fderiv_of_isOpen, Polynomial.smeval_smul, infEDist_smul₀, LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton', ContinuousLinearMap.continuous_restrictScalars, UniqueDiffWithinAt.dense_tangentConeAt, hasStrictDerivAt_const_add_iff, TensorAlgebra.lift_symm_apply, InnerProductGeometry.angle_eq_zero_iff, RootPairing.EmbeddedG2.twoShortAddLongRoot_eq, smul_add_smul_le_smul_add_smul, Finset.centerMass_insert, isBoundedBilinearMap_compMultilinear, AnalyticAt.smul, hasLineDerivAt_smul_iff, VectorField.lieBracket_const_smul_left, SameRay.sameRay_nonneg_smul_left, LinearIndependent.pair_smul_iff, neg_convexOn_iff, hasFDerivAt_norm_rpow, ENNReal.smul_top, Submodule.mem_smul_iff_inv_mul_mem, IsBaseChange.det_endHom, ContinuousLinearMap.compL_apply, ContinuousLinearMap.lsmul_flip_inj, LinearMap.polyCharpolyAux_map_eval, AlternatingMap.curryLeft_apply_apply, Matrix.toLinearMapₛₗ₂'_aux_eq, ContinuousAlternatingMap.map_piecewise_smul, mellinInv_eq_fourierInv, IsLocalizedModule.mapExtendScalars_apply_apply, units_smul_eq_neg_iff, HasFPowerSeriesOnBall.uniform_geometric_approx, TrivSqZeroExt.isNilpotent_inr, End.algebraMap_isUnit_inv_apply_eq_iff', NonUnitalSubring.unitization_apply, ContDiffMapSupportedIn.structureMapLM_eq_of_scalars, Algebra.TensorProduct.instSMulCommClassTensorProduct_1, KaehlerDifferential.linearMapEquivDerivation_apply_apply, CommRingCat.coproductCocone_inl, LinearMap.BilinForm.tensorDistribEquiv_toLinearMap, IsSl2Triple.mem_toLieSubalgebra_iff, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, EuclideanGeometry.Sphere.tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center, Submodule.mem_torsionBySet_iff, IntermediateField.relfinrank_dvd_finrank_bot, Asymptotics.isBigOTVS_fun_neg_left, preReflection_preReflection, norm_smul_of_nonneg, ContDiffAt.iteratedFDeriv_right, sub_smul, VectorFourier.fourierIntegral_const_smul, HasFPowerSeriesWithinAt.fderivWithin_eq, TensorAlgebra.equivDirectSum_apply, ContDiffBump.convolution_tendsto_right, Matrix.linfty_opNNNorm_toMatrix, Asymptotics.isBigOTVS_fun_neg_right, CompactlySupportedContinuousMap.instSMulCommClass, Basis.end_apply_apply, KaehlerDifferential.tensorKaehlerEquivBase_tmul, MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real, ContDiffAt.contDiffAt_norm_smul, Real.fourier_real_eq, AnalyticOnNhd.fderiv_of_isOpen, Basis.coe_constrL, Matrix.isRepresentation.toEnd_represents, FormalMultilinearSeries.id_comp', Seminorm.rescale_to_shell_zpow, FormalMultilinearSeries.ofScalars_sub, IsHausdorff.iInf_pow_smul, Asymptotics.IsEquivalent.smul, LieModule.exists_genWeightSpace_le_ker_of_isNoetherian, contDiffOn_infty_iff_fderivWithin, ContinuousMultilinearMap.range_toUniformOnFun, QuadraticMap.toQuadraticMap_associated, FormalMultilinearSeries.hasSum, hasDerivAt_abs_rpow, ContinuousLinearMap.IsPositive.conj_adjoint, ContinuousLinearMap.flip_apply, IsLocalizedModule.fromLocalizedModule.inj, MeasureTheory.Measure.integral_comp_inv_smul_of_nonneg, Matrix.toEuclideanCLM_toLp, PiTensorProduct.liftAux.smul, Invertible.bijective_curry, Algebra.traceForm_toMatrix, Seminorm.smul_closedBall_zero, WeakDual.CharacterSpace.homeoEval_naturality, signedDist_eq_zero_of_orthogonal, Seminorm.balanced_ball_zero, LinearMap.trace_eq_zero_of_mapsTo_ne, hasSum_fourier_series_L2, KaehlerDifferential.isLocalizedModule_map, DualNumber.lift_inlAlgHom_eps, DifferentiableWithinAt.hasDerivWithinAt, isSMulRegular_submodule_iff_right_eq_zero_of_smul, IntermediateField.exists_finset_maximalFor_isTranscendenceBasis_separableClosure, CPolynomialOn.iteratedFDeriv, End.apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil, IntermediateField.adjoin_simple_adjoin_simple, Unitization.zero_mem_spectrum_inr, MeasureTheory.L1.SimpleFunc.setToL1S_smul_real, ProbabilityTheory.covarianceBilin_zero, Ioo_smul_sphere_zero, HasFPowerSeriesOnBall.sum, Real.tendsto_integral_gaussian_smul', End.hasUnifEigenvalue_iff_mem_spectrum, contDiffOn_stereoToFun, Real.hasFDerivAt_fourier, VectorFourier.differentiable_fourierIntegral, cross_anticomm', absorbs_iff_eventually_nhdsNE_zero, Matrix.toLin_scalar, Antitone.convex_lt, CategoryTheory.ShortComplex.Homotopy.smul_h₀, LeftInvariantDerivation.evalAt_coe, fderiv_inverse, smul_sphere', SchwartzMap.smulLeftCLM_fun_neg, two_smul, HasFPowerSeriesOnBall.fderiv, LinearMap.adjoint_rTensor, toBilin'Aux_toMatrixAux, MeasureTheory.FiniteMeasure.map_fst_prod, CategoryTheory.ShortComplex.Homotopy.smul_h₃, contDiffOn_nat_iff_continuousOn_differentiableOn, Basis.isScalarTower_finsupp, IntermediateField.restrictRestrictAlgEquivMapHom_surjective, Ideal.FG.mul, CategoryTheory.ShortComplex.cyclesMap_smul, Submodule.LinearDisjoint.linearIndependent_right_of_flat, TensorProduct.finsuppLeft_symm_apply_single, Subalgebra.isScalarTower_right, IntermediateField.AdjoinSimple.norm_gen_eq_one, HilbertBasis.hasSum_orthogonalProjection, ContinuousLinearMap.bilinearRestrictScalars_apply_apply, LieModule.coe_genWeightSpaceOf_zero, Submodule.smul_coe_torsionBy, Fintype.not_linearIndependent_iffₛ, LinearMap.adjoint_inner_right, TensorAlgebra.equivFreeAlgebra_ι_apply, BilinForm.dotProduct_toMatrix_mulVec, InnerProductSpace.adjoint_rankOne, FormalMultilinearSeries.compAlongComposition_nnnorm, LinearMap.BilinForm.tensorDistrib_tmul, map_nnrat_smul, stereoInvFun_apply, IntermediateField.range_val, monovaryOn_iff_forall_smul_nonneg, IntermediateField.instFiniteSubtypeMemBot, TrivSqZeroExt.mul_left_eq_one, MultilinearMap.coe_currySumEquiv_symm, TensorProduct.AlgebraTensorModule.mk_apply, IntermediateField.isCyclotomicExtension_lcm_sup, analyticAt_smul, Seminorm.restrictScalars_ball, Filter.EventuallyEq.hasDerivAt_iff, LinearMap.trace_transpose', InnerProductSpace.rankOne_eq_zero, egauge_le_one, KaehlerDifferential.End_equiv_aux, Real.fourierIntegral_convergent_iff', LinearMap.IntrinsicStar.starLinearEquiv_eq_arrowCongr, LieModuleHom.smul_apply, IsTensorProduct.equiv_symm_apply, Projectivization.cross_mk_of_cross_ne_zero, Submodule.mem_annihilator, hasDerivAt_deriv_iff, RCLike.mul_wInner_left, Matrix.toLpLin_one, CliffordAlgebra.ι_mul_ι_mul_ι, ContinuousAlternatingMap.norm_curryLeft, Algebra.Extension.instIsScalarTowerRing, End.HasEigenvector.apply_eq_smul, MeasureTheory.integral_indicatorConstLp, IntermediateField.linearDisjoint_iff', inner_map_polarization', MultilinearMap.curryFinFinset_symm_apply_piecewise_const, Bundle.ContMDiffRiemannianMetric.symm, ContMDiffOn.coordChangeL, supportDim_quotSMulTop_succ_eq_of_notMem_minimalPrimes_of_mem_maximalIdeal, intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le, InnerProductSpace.inner_right_rankOne_apply, HilbertBasis.hasSum_repr, ContDiff.differentiable_iteratedFDeriv, HasFDerivWithinAt.hasGradientWithinAt, SchwartzMap.pairing_apply, Dual.transpose_apply, QuadraticMap.polar_smul_left_of_tower, ContinuousLinearMapWOT.smul_apply, MeasureTheory.Measure.integral_comp_smul, MeromorphicAt.tendsto_nhds_meromorphicTrailingCoeffAt, Asymptotics.isBigOTVS_iff_isBigO, IntermediateField.splits_iff_mem, LinearMap.isPosSemidef_iff_posSemidef_toMatrix, Asymptotics.IsLittleO.smul, ContinuousAlternatingMap.apply_apply, setIntegral_withDensity_eq_setIntegral_smul₀', LieAlgebra.engel_isBot_of_isMin.lieCharpoly_map_eval, Basis.det_unitsSMul, Bornology.isVonNBounded_sub_of_nonempty, innerSL_inj, AffineIsometryEquiv.coe_ofEq_apply, subset_tangentConeAt_prod_right, TrivSqZeroExt.snd_exp, WithSeminorms.isVonNBounded_iff_seminorm_bddAbove, MeasureTheory.condExpIndSMul_smul, fderivWithin_inv', KaehlerDifferential.isScalarTower_of_tower, TannakaDuality.FiniteGroup.sumSMulInv_apply, TensorProduct.AlgebraTensorModule.rTensor_tmul, hasDerivAt_add_const_iff, MeasureTheory.Measure.hausdorffMeasure_smul₀, Algebra.PreSubmersivePresentation.aevalDifferential_toMatrix'_eq_mapMatrix_jacobiMatrix, real_inner_div_norm_mul_norm_eq_one_iff, TensorProduct.prodLeft_symm_tmul, IsLocalizedModule.map_comp', Monotone.ae_hasDerivAt, StrongDual.toLp_of_not_memLp, star_natCast_smul, ContinuousLinearMap.fderivWithin_of_bilinear, AffineSubspace.isometryEquivMap.toAffineMap_eq, HasFPowerSeriesOnBall.compContinuousLinearMap, Submodule.comap_dualAnnihilator, MeasureTheory.integrable_withDensity_iff_integrable_smul₀', AffineSubspace.signedInfDist_eq_signedDist_of_mem, TensorAlgebra.lift_ι_apply, lineMap_mem_openSegment, KaehlerDifferential.linearMapEquivDerivation_symm_apply, Fintype.sum_single_smul, AlgebraicClosure.instIsAlgClosureOfIsAlgebraic, QuadraticMap.associated_left_inverse, Algebra.normalizedTrace_intermediateField, LinearMap.BilinForm.smul_right_of_tower, RealRMK.rieszMeasure_le_of_eq_one, TensorAlgebra.instModuleFree, ContinuousMultilinearMap.cons_smul, TensorProduct.tmul_eq_smul_one_tmul, DiffContOnCl.deriv_eq_smul_circleIntegral, LinearMap.BilinForm.tensorDistribEquiv_apply, ContinuousLinearMap.applyFaithfulSMul, DifferentiableOn.deriv_eq_smul_circleIntegral, ContDiffMapSupportedIn.uniformSpace_eq_iInf, KaehlerDifferential.span_range_derivation, WeierstrassCurve.Projective.smul_eq, Matrix.piLp_ofLp_toEuclideanLin, Submodule.mem_span_singleton, iteratedFDerivWithin_smul_const_apply, cfcₙAux_id, Algebra.Generators.H1Cotangent.δAux_monomial, LinearMap.BilinForm.toMatrix_apply, LinearMap.IsSymmetric.diagonalization_symm_apply, PiTensorProduct.liftIsometry_tprodL, VectorField.mpullback_smul_apply, DualNumber.isNilpotent_iff_eps_dvd, TrivSqZeroExt.map_comp_inrHom, contDiffWithinAt_iff_of_ne_infty, RootPairing.reflectionPerm_coroot, NormedSpace.inclusionInDoubleDual_norm_le, TensorProduct.map₂_apply_tmul, LieSubmodule.traceForm_eq_zero_of_isTrivial, PowerBasis.constr_pow_algebraMap, lieEquivMatrix'_apply, curveIntegral_segment_const, TensorAlgebra.rank_eq, Algebra.Generators.CotangentSpace.compEquiv_symm_zero, hasDerivWithinAt_one, BilinForm.toMatrix_toBilin, TensorProduct.AlgebraTensorModule.congr_eq, le_egauge_smul_left, sub_smul_slope_vadd, LieRinehartAlgebra.toIsScalarTower, PowerSeries.instIsScalarTower, Algebra.Extension.instIsScalarTowerCotangent, DualNumber.isMaximal_span_singleton_eps, HahnSeries.SummableFamily.smul_hsum, WeakDual.isCompact_closedBall, AffineSubspace.setOf_wOppSide_eq_image2, LieModule.weight_vector_multiplication, BoxIntegral.BoxAdditiveMap.volume_apply, spinGroup.conjAct_smul_range_ι, ConvexCone.isGenerating_bot, ProbabilityTheory.HasGaussianLaw.fun_smul, ModuleCat.Derivation.d_mul, isNormalClosure_normalClosure, VectorFourier.iteratedFDeriv_fourierIntegral, TestFunction.coe_ofSupportedInCLM, AlgHom.toNonUnitalAlgHom_eq_coe, MeasureTheory.Integrable.toL1_smul, SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux, LinearMap.BilinForm.IsRefl.smul, ContDiffBump.normed_convolution_eq_right, AdicCompletion.range_eval, InnerProductSpace.rankOne_eq_rankOne_iff_comm, InnerProductSpace.toDual_apply, realPart_add_I_smul_imaginaryPart, ContinuousLinearMap.toContinuousAddMonoidHom_zero, Ideal.Quotient.smul_top, ContDiffMapSupportedIn.continuous_iff_comp_withOrder, HasLineDerivAt.tendsto_slope_zero, SemilinearMapClass.map_smul_inv, LieDerivation.exp_map_apply, subset_tangentConeAt_prod_left, LocalizedModule.mk_cancel_common_left, Algebra.Generators.cotangentSpaceBasis_repr_tmul, strictConvex_Ioo, ConvexBody.convex, IntermediateField.finrank_eq_one_iff, ContinuousLinearMap.holder_smul_left, ContinuousLinearMap.analyticWithinAt_uncurry_of_multilinear, IntermediateField.lift_top, ContinuousLinearMap.precompR_apply, sameRay_smul_left_iff, RingHom.SurjectiveOnStalks.tensorProductMap, ContinuousAlternatingMap.norm_alternatizeUncurryFinCLM_le, NonUnitalAlgHom.toAlgHom_zero, LinearMap.minpoly_toMatrix', LieSubalgebra.coe_ad_pow, LinearMap.BilinForm.IsPosSemidef.smul, FormalMultilinearSeries.iteratedFDerivSeries_eq_zero, Submodule.FG.restrictScalars_iff, Seminorm.comp_smul, IsGreatest.nnnorm_cfcₙ_nnreal, absorbent_nhds_zero, ContMDiffAt.const_smul_section, MeasureTheory.AEStronglyMeasurable.convolution_integrand_swap_snd', hasDerivAtFilter_one, Real.fourierIntegral_iteratedDeriv, intervalIntegral.smul_integral_comp_mul_sub, Derivation.leibniz', Derivation.leibniz_zpow, iteratedDerivWithin_smul_const, MeasureTheory.hasFDerivAt_convolution_right_with_param, LinearMap.trace_prodMap, ProbabilityTheory.IsGaussian.charFun_eq', Submodule.iSup_dualAnnihilator_le_iInf, fderiv_pow', MeasureTheory.Lp.coeFn_smul, ProbabilityTheory.covarianceBilinDual_eq_covariance, ContMDiffAt.mfderiv_const, ContMDiffSection.coe_smul, Algebra.Generators.Hom.comp_id, LinearMap.liftBaseChangeEquiv_symm_apply, signedDist_zero, Asymptotics.IsLittleO.isLittleOTVS, LieAlgebra.IsKilling.instIsCrystallographicSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, hasDerivAt_ofNat, CommAlgCat.lift_unop_hom, AffineSubspace.signedInfDist_eq_signedDist_orthogonalProjection, iteratedFDerivWithin_eq_equiv_comp, Algebra.WeaklyQuasiFiniteAt.finite_locoalization, CompactlySupportedContinuousMap.toRealLinearMap_apply, LinearMap.toMatrix₂_comp, DifferentiableAt.isBigOTVS_sub, nhds_basis_balanced, LinearMap.restrictScalars_smul, Quotient.mk_smul_mk, HasCompactSupport.hasFDerivAt_convolution_right, isometry_cfcₙHom, ContDiffMapSupportedIn.structureMapCLM_zero_apply, VonNeumannAlgebra.centralizer_centralizer', CoalgCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_left, FormalMultilinearSeries.mkPiRing_coeff_eq, IntermediateField.exists_algHom_adjoin_of_splits_of_aeval, differentialIdeal_le_iff, IsLocalizedModule.exist_integer_multiples_of_finite, SchwartzMap.integral_bilinear_deriv_right_eq_neg_left, RightPreLieAlgebra.toSMulCommClass, ContinuousAlternatingMap.curryLeft_compContinuousAlternatingMap, Unitization.starMap_id, Ideal.sup_pow_eq_top', LinearMap.lsmul_flip_apply, ProbabilityTheory.covarianceBilinDual_zero, IsModuleTopology.instProd, LinearIndependent.smul_left_injective, RealRMK.integral_rieszMeasure, isLocalizedModule_iff, HahnEmbedding.Partial.exists_sub_mem_ball, AlgebraicIndependent.lift_reprField, Matrix.SeparatingRight.toLinearMap₂', Algebra.Extension.Hom.sub_one_tmul, Basis.toDual_ker, Representation.asModuleEquiv_symm_map_smul, Submodule.mem_span_iff_exists_finset_subset, RatFunc.ofFractionRing_add, LinearEquiv.transvection.apply, DualNumber.lift_smul, Basis.dualBasis_apply_self, InnerProductSpace.comp_rankOne, OrthonormalBasis.starProjection_eq_sum_rankOne, IntermediateField.isAlgebraic_adjoin, SameRay.exists_pos_right, MeasureTheory.SimpleFunc.integral_eq_sum, Coalgebra.sum_counit_smul, HahnEmbedding.ArchimedeanStrata.disjoint_ball_stratum, TensorProduct.lift.tmul', iteratedDerivWithin_const_smul_field, WittVector.IsocrystalEquiv.frob_equivariant, Submodule.mem_iSup_iff_exists_dfinsupp, Submodule.map₂_map_left, Quaternion.imK_fst_dualNumberEquiv, QuadraticForm.tensorRId_apply, LinearMap.exists_eq_smul_id_of_forall_notLinearIndependent, InnerProductSpace.toDualMap_apply, MvPolynomial.rTensor_apply_X_tmul, CFC.posPart_smul, TotallyBounded.isVonNBounded, instIsLiouvilleSubtypeMemIntermediateField, smul_min_of_nonpos, Algebra.TensorProduct.lmulEquiv_eq_lidOfCompatibleSMul, LinearMap.pow_eq_aeval_mod_charpoly, LinearMap.prodEquiv_symm_apply, AffineIsometryEquiv.ofTop_symm_apply_coe, NonUnitalSubalgebra.inclusion_injective, NormedSpace.normalize_smul_of_pos, Seminorm.comp_zero, LinearMap.tensorProductEnd_apply, ContinuousLinearMap.adjoint_inner_left, Bialgebra.TensorProduct.assoc_tmul, lift_rank_lt_rank_dual', PointwiseConvergenceCLM.isInducing_inducingFn, Matrix.toLpLin_pow, KaehlerDifferential.tensorKaehlerEquiv_tmul, FormalMultilinearSeries.nnnorm_compContinuousLinearMap_le, ext_iff, AEval'.X_pow_smul_of, LinearMap.trace_eq_contract', LieRinehartAlgebra.Hom.map_smul_apply', Algebra.Generators.H1Cotangent.δAux_toAlgHom, ContinuousLinearEquiv.arrowCongr_symm, Finsupp.linearCombination_single_index, Monovary.smul_add_smul_le_smul_add_smul, CochainComplex.HomComplex.Cochain.shift_smul, ContinuousLinearMap.isBoundedBilinearMap, LinearEquiv.symm_flip, WeakFEPair.Λ₀_eq, LinearMap.restrictScalars_self, ContinuousMultilinearMap.hasBasis_nhds_zero, KaehlerDifferential.D_apply, IsSymmetricAlgebra.equiv_symm_comp, ball_normSeminorm, HasFPowerSeriesWithinOnBall.sum, InnerProductSpace.AlgebraOfCoalgebra.mul_def, LinearMap.BilinForm.separatingRight_toMatrix'_iff, LinearMap.extendTo𝕜_apply, Matrix.isUnit_toLin'_iff, RingTheory.Sequence.isWeaklyRegular_singleton_iff, piEquiv_apply_apply, LinearMap.BilinForm.smul_left, ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_right, compContinuousLinearMap_zero, Finsupp.lift_apply, HasLineDerivWithinAt.smul, ContinuousMultilinearMap.ofSubsingletonₗᵢ_symm_apply, LinearMap.range_dualMap_eq_dualAnnihilator_ker, Submodule.dualCoannihilator_map_linearEquiv_flip, mdifferentiable_smul_const_section, Polynomial.adjSylvester_comp_sylveserMap, Finset.expect_eq_single_of_mem, FormalMultilinearSeries.apply_eq_pow_smul_coeff, FormalMultilinearSeries.le_radius_compContinuousLinearMap, Ideal.ResidueField.liftₐ_comp_toAlgHom, Filter.Tendsto.atTop_smul_nhds_tendsto_asymptoticNhds, Differentiable.fderiv_two, continuousMultilinearCurryFin0_symm_apply, MeasureTheory.setIntegral_condExpIndSMul, totallyBounded_absConvexHull, TensorAlgebra.toClifford_ι, RootPairing.GeckConstruction.trace_toEnd_eq_zero, Asymptotics.isBigOTVS_congr, MeasureTheory.integrable_withDensity_iff_integrable_smul₀, AdicCompletion.AdicCauchySequence.smul_apply, inv_norm_smul_mem_unitClosedBall, Matrix.toLin'_mul, Seminorm.neg_mem_ball_zero, star_ratCast_smul, Submodule.mapsTo_smul_closure, IsArtinian.range_smul_pow_stabilizes, Unitary.conjStarAlgAut_ext_iff', Submodule.iSup_eq_range_dfinsupp_lsum, LinearMap.isScalarTower_of_injective, LinearMap.isNilpotent_toMatrix_iff, LinearMap.BilinForm.lieInvariant_iff, cfcₙHom_predicate, ContinuousLinearEquiv.symm_smul_apply, Asymptotics.isBigOTVS_neg_right, LieModule.Cohomology.twoCochain_skew, MDifferentiableOn.coordChangeL, LieModule.IsTriangularizable.maxGenEigenspace_eq_top, TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply, VectorField.lieBracketWithin_smul_left, LinearMap.BilinForm.dotProduct_toMatrix_mulVec, VectorField.mlieBracketWithin_const_smul_left, convex_halfSpace_ge, AffineMap.right_vsub_lineMap, Quaternion.re_fst_dualNumberEquiv, ContinuousAlternatingMap.ofSubsingletonLIE_apply, End.rTensorAlgHom_apply_apply, LinearMap.det_eq_det_toMatrix_of_finset, ContDiffMapSupportedIn.iteratedFDerivWithOrderLM_apply, FormalMultilinearSeries.compAlongComposition_bound, continuousMultilinearCurryRightEquiv_symm_apply, OrthonormalBasis.equiv_apply, LinearMap.toMatrix_reindexRange, FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le, ExteriorAlgebra.liftAlternating_apply_ιMulti, ContDiff.euclidean_dist, PowerBasis.equivAdjoinSimple_aeval, inner_eq_sum_norm_sq_div_four, AntivaryOn.sum_smul_le_sum_comp_perm_smul, intervalIntegral.integral_comp_add_mul, IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn_real, constFormalMultilinearSeries_apply_zero, IntermediateField.isPurelyInseparable_tower_bot, LinearMap.ker_toAddSubmonoid, ConvexBody.ext_iff, le_separableClosure_iff, DifferentiableAt.fderiv_restrictScalars, starConvex_iff_forall_ne_pos, LieModule.isFaithful_iff, AnalyticAt.fun_smul, hasFTaylorSeriesUpToOn_top_iff', ContinuousLinearMap.lsmul_apply, Matrix.Module.smul_def, AdicCompletion.transitionMap_comp_eval, MeasureTheory.setIntegral_rnDeriv_smul, LinearIsometryEquiv.conjStarAlgEquiv_trans, FormalMultilinearSeries.radius_smul_eq, Bundle.ContMDiffRiemannianMetric.pos, AffineIsometryEquiv.ofEq_symm, Matrix.ofLp_toLpLin, traceForm_dualSubmodule_adjoin, Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded, ContinuousMap.linearIsometryBoundedOfCompact_symm_apply, signedDist_triangle_left, range_piEquiv, ContinuousLinearMap.bilinearComp_zero_right, DivisibleHull.instIsStrictOrderedModuleNNRat, ContinuousLinearMap.toContinuousAddMonoidHom_comp, Matrix.toMatrix₂Aux_toLinearMap₂'Aux, IsContMDiffRiemannianBundle.exists_contMDiff, ContDiffWithinAt.fderivWithin'', Matrix.toBilin'_symm, Basis.dualBasis_equivFun, ModuleCat.smulShortComplex_g, Function.support_smul, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, PiTensorProduct.piTensorHomMap_tprod_tprod, BaireSpace.instBarrelledSpace, Unitization.starLift_range, hasDerivAtFilter_add_const_iff, MultilinearMap.map_update_smul_left, hasDerivAt_jacobiTheta₂_fst, HomologicalComplex.units_smul_f_apply, InnerProductSpace.rank_rankOne, LinearMap.toMatrix_smulBasis_left, MDifferentiableAt.const_smul, Ideal.mul_sup, MvPowerSeries.constantCoeff_smul, CFC.nnrpow_def, VertexOperator.ncoeff_apply, Dual.transpose_comp, ContinuousAlternatingMap.toContinuousMultilinearMapLI_apply, IsSl2Triple.lie_h_e_smul, Algebra.Extension.CotangentSpace.map_toInfinitesimal_bijective, IsAzumaya.mulLeftRight_comp_congr, TrivSqZeroExt.inr_mul_inr, LinearMap.BilinForm.isSymm_iff_flip, convex_openSegment, contMDiffAt_coordChangeL, DiffContOnCl.const_smul, MvPolynomial.algebraTensorAlgEquiv_symm_comp_aeval, Polynomial.sup_aeval_range_eq_top_of_isCoprime, LinearMap.flip_bijective_iff₁, subset_balancedHull, ConvexCone.pointed_positive, VectorFourier.fourierIntegral_convergent_iff, InnerProductSpace.toDual_apply_apply, LinearMap.flip_bijective_iff₂, IsLocalizedModule.subsingleton_iff, ContinuousLinearMap.adjointAux_apply, RCLike.real_smul_eq_coe_smul, MultilinearMap.fromDFinsuppEquiv_single, eq_separableClosure_iff, Algebra.Generators.Hom.toExtensionHom_id, integral_comp_polarCoord_symm, Set.zero_mem_smul_set_iff, LinearMap.range_dualMap_dual_eq_span_singleton, ContinuousLinearMap.inner_map_map_iff_adjoint_comp_self, LinearMap.det_toLin', CompactlySupportedContinuousMap.toRealPositiveLinear_apply, fderivWithin_fun_mul', CategoryTheory.Functor.map_smul, sup_eq_half_smul_add_add_abs_sub', ProperCone.innerDual_singleton, TensorAlgebra.lift_comp_ι, LieSubmodule.toEnd_comp_subtype_mem, Asymptotics.IsLittleO.smul_isBigO, HasFPowerSeriesWithinOnBall.sub, MvPolynomial.tensorEquivSum_one_tmul_C, HasLineDerivAt.tendsto_slope_zero_left, Submodule.map₂_map_right, Matrix.kroneckerMapBilinear_apply_apply, ContinuousLinearMap.cpolyomialOn_uncurry_of_multilinear, LinearMap.minpoly_dvd_charpoly, ContinuousMultilinearMap.hasFiniteFPowerSeriesOnBall, Ideal.mul_mono, LinearMap.SeparatingRight.congr, Ideal.pi_tensorProductMk_quotient_surjective, Matrix.rank_eq_finrank_range_toLin, derivWithin_const_smul_field, MvPolynomial.rTensor_apply_tmul, Antitone.convex_ge, OrderedFinpartition.compAlongOrderedFinpartitionₗ_apply_apply, hasFDerivAt_list_prod', hasFDerivAt_integral_of_dominated_loc_of_lip', CliffordAlgebra.invOf_ι_mul_ι_mul_ι, Matrix.toLinearMapRight'_mul_apply, ConvexCone.coe_strictlyPositive, SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf, Asymptotics.isLittleO_const_smul_left, Submodule.annihilator_smul, InnerProductSpace.rankOne_def, Submodule.jacobson_smul_lt_top, MDifferentiableWithinAt.coordChangeL, BilinForm.toMatrix_comp, smul_vsub_vadd_mem_affineSpan_pair, smul_nonneg_of_nonpos_of_nonpos, LieAlgebra.ad_nilpotent_of_nilpotent, OreLocalization.numeratorRingHom_apply, CliffordAlgebra.EvenHom.contract, ContinuousLinearMap.coe_flipₗᵢ', AlgHom.mulLeftRight_apply, WeakFEPair.h_feq', star_nnrat_smul, Basis.linearCombination_dualBasis, PiTensorProduct.norm_eval_le_injectiveSeminorm, stereoToFun_apply, ContinuousLinearMap.adjointAux_inner_left, uniqueDiffWithinAt_iff, SchwartzMap.convolution_apply, IntermediateField.normal_iff_forall_fieldRange_le, Submodule.IsPrincipal.mem_iff_eq_smul_generator, contDiffOn_succ_iff_fderiv_of_isOpen, PointwiseConvergenceCLM.equivWeakDual_apply, PointedCone.pointed_toConvexCone, fderivWithin_fderivWithin_eq_of_mem_nhds, LinearMap.lTensor_smul_action, transpose_dualTensorHom, Submodule.IsLattice.smul, ContinuousLinearMap.norm_smulRight_apply, ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv, Polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime, Matrix.toLin_mul, TensorProduct.AlgebraTensorModule.assoc_tmul, Set.zero_mem_smul_iff, contDiff_iff_continuous_differentiable, Finsupp.instFaithfulSMulOfNonempty, hasDerivAt_const, dslope_sub_smul_of_ne, TensorAlgebra.ι_range_disjoint_one, LieModule.rank_eq_natTrailingDegree, Localization.mk_sum, Finset.centerMass_smul_left, HasFPowerSeriesWithinOnBall.uniform_geometric_approx', Matrix.toLin_pow, Submodule.coe_tensorSpanEquivSpan_apply_tmul, Algebra.TensorProduct.lmul'_comp_map, LieSubmodule.mem_baseChange_iff, End.aeval_apply_of_hasEigenvector, AdicCompletion.instIsHausdorff, CategoryTheory.Abelian.Ext.smul_comp, QuadraticMap.associated_tmul, LinearMap.smulRight_eq_comp, IsAzumaya.bij, fderiv_mul', IsGalois.normalAutEquivQuotient_apply, UniformSpace.Completion.norm_toComplL, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_comp_inl, IsRegular.smul_right_injective, fderivWithin_const_smul_of_field, ProbabilityTheory.covarianceBilin_comm, Derivation.liftKaehlerDifferential_comp, Matrix.toLin_self, ContinuousAlternatingMap.constOfIsEmptyLIE_symm_apply, ContinuousMap.toLp_def, kroneckerTMulLinearEquiv_one, ContinuousLinearMap.IsStarNormal.ker_adjoint_eq_ker, natCast_le_analyticOrderAt, Complex.hasDerivAt_Gammaℂ_one, Submodule.dualCoannihilator_bot, VectorField.lieBracket_fmul_left, NonUnitalAlgHom.injective_codRestrict, Basis.SmithNormalForm.snf, iteratedDerivWithin_fun_const_smul_field, LieDerivation.exp_apply, sup_eq_half_smul_add_add_abs_sub, RootPairing.pairing_neg_one_neg_four_iff, SchwartzMap.integral_bilin_fourier_eq, KaehlerDifferential.ker_map, intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right, QuadraticForm.equivalent_one_zero_neg_one_weighted_sum_squared, ClosedSubmodule.orthogonal_eq_inter, Algebra.TensorProduct.lift_comp_includeRight, ContinuousMultilinearMap.prodL_apply, Seminorm.add_comp, Orientation.kahler_neg_orientation, TensorProduct.AlgebraTensorModule.rightComm_tmul, segment_eq_image, AdicCompletion.coe_eval, Matrix.conjTranspose_inv_ofNat_smul, affinity_unitBall, convex_eball, instPosSMulStrictMono, LinearMap.separatingLeft_toMatrix₂'_iff, hasFTaylorSeriesUpToOn_succ_iff_right, ContinuousLinearMap.coe_flipMultilinearEquiv, Affine.Triangle.circumsphere_eq_of_dist_of_oangle, Seminorm.bddAbove_iff, NonUnitalAlgHom.coe_Lmul, QuadraticForm.equivalent_weightedSumSquares, ContinuousLinearMap.toSpanSingleton_comp, ContinuousMultilinearMap.norm_map_init_le, LinearEquiv.map_smulₛₗ, IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite, ModuleFilterBasis.smul_left', LeftInvariantDerivation.left_invariant'', CliffordAlgebra.contractLeft_contractLeft, TensorProduct.AlgebraTensorModule.assoc_symm_tmul, TensorProduct.algebraMap_gradedMul, Subalgebra.centralizer_coe_map_includeRight_eq_center_tensorProduct, SchwartzMap.fourierInv_apply_eq, SchwartzMap.smulRightCLM_apply_apply, hasEigenvector_toLin'_diagonal, LinearIsometryEquiv.conjStarAlgEquiv_refl, IntermediateField.rank_adjoin_eq_one_iff, InnerProductSpace.laplacian_smul, MeasureTheory.setToFun_smul_left, IsBaseChange.lift_comp, LinearMap.separatingRight_iff_flip_ker_eq_bot, MvPolynomial.mkDerivationₗ_monomial, MeasureTheory.condExp_stronglyMeasurable_simpleFunc_bilin, ModuleCat.MonoidalCategory.rightUnitor_hom_apply, starConvex_iff_openSegment_subset, mem_segment_iff_sameRay, LinearMap.toMatrixAlgEquiv'_apply, LieAlgebra.ad_pow_lie, Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin, ProbabilityTheory.setIntegral_tilted_mul_eq_mgf', ContinuousAlternatingMap.map_update_smul, eventually_singleton_add_smul_subset, SubMulAction.algebraMap_mem, Dual.instIsReflecive, Matrix.fin_two_smul_prod, LinearMap.IsSymmetric.toSelfAdjoint_apply, TensorProduct.finsuppScalarRight_smul, Ray.units_smul_of_pos, Antivary.sum_smul_lt_sum_comp_perm_smul_iff, mem_extremePoints_iff_forall_segment, LieModule.Cohomology.d₁₂_apply_coe_apply_apply, circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const, Orientation.map_eq_det_inv_smul, fderiv_norm_smul_neg, SchwartzMap.toTemperedDistributionCLM_apply_apply, ContinuousLinearMap.coe_restrict_scalarsL', fderivWithin_fderivWithin_eq_of_eventuallyEq, Unitization.inrNonUnitalStarAlgHom_apply, Real.fourierIntegral_real_eq_integral_exp_smul, Matrix.toLinearMapₛₗ₂'_toMatrix', ConvexCone.mem_map, CochainComplex.HomComplex.Cochain.leftUnshift_smul, LinearMap.toMatrix'_algebraMap, InnerProductSpace.laplacianWithin_smul, isNoetherian_linearMap, DomMulAct.instSMulCommClassSubtypeAEEqFunMemAddSubgroupLp_2, StrongDual.polarSubmodule_eq_polar, TensorProduct.tensorQuotEquivQuotSMul_tmul_mk, AlgCat.hom_hom_rightUnitor, algebraMap_smul, finsuppTensorFinsupp_single, PiTensorProduct.liftEquiv_symm_apply, PiToModule.fromMatrix_apply_single_one, LocalizedModule.coe_map_eq, NonUnitalStarSubalgebra.toNonUnitalSubalgebra_subtype, Representation.linHom_apply, NonUnitalSubalgebra.toSubring_subtype, Algebra.Extension.CotangentSpace.map_sub_map, TestFunction.coe_ofSupportedInLM, HomogeneousLocalization.val_sub, QuadraticMap.associated_toQuadraticMap, Orientation.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero, LinearMap.dualMap_injective_of_surjective, extDerivWithin_fun_smul, Unitization.starMap_inl, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, LinearMap.dualMap_injective_iff, inner_smul_real_right, FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum, Asymptotics.isLittleOTVS_fun_neg_right, FreeLieAlgebra.liftAux_map_add, IsLocalizedModule.mem_ker_iff, ConvexCone.strictlyPositive_le_positive, ContDiffWithinAt.smul, NonUnitalStarSubalgebra.range_val, AlternatingMap.curryLeft_apply_toMultilinearMap, LieModule.isNilpotent_toEnd_of_isNilpotent₂, Basis.eval_range, Seminorm.closedBall_finset_sup, Antivary.sum_smul_le_sum_smul_comp_perm, MeasureTheory.SignedMeasure.re_toComplexMeasure, TemperedDistribution.smulLeftCLM_sub, reflection_mul_reflection_pow, ZMod.algebraOfModule.proof, eq_pos_convex_span_of_mem_convexHull, BilinearForm.toMatrixAux_eq, CliffordAlgebra.contractLeft_ι_mul, QuadraticMap.two_nsmul_associated, TensorProduct.lid_tmul, PolynomialModule.isScalarTower', IntermediateField.finrank_eq_fixingSubgroup_index, IsBoundedLinearMap.smul, Commute.cfcₙHom, LinearMap.dualPairing_nondegenerate, Submodule.localized₀_smul, DualNumber.exists_mul_left_or_mul_right, RootPairing.GeckConstruction.span_range_h'_eq_top, iteratedDerivWithin_scomp_eq_sum_orderedFinpartition, Submodule.norm_subtypeL, SModEq.smul', intervalIntegral.integral_hasStrictFDerivAt_of_tendsto_ae, fourierCoeff.const_smul, Matrix.SeparatingLeft.toBilin, ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear, FreeNonUnitalNonAssocAlgebra.lift_symm_apply, BddBelow.smul_of_nonpos, Matrix.charpoly_toLin, monovary_iff_forall_smul_nonneg, Submodule.smul_le_smul, closure_openSegment, setIntegral_withDensity_eq_setIntegral_toReal_smul₀', LinearMap.toMatrixOrthonormal_symm_apply, Derivation.compAlgebraMapL_apply_apply, biUnion_associatedPrimes_eq_zero_divisors, ContinuousMultilinearMap.cpolyomialOn_uncurry_of_linear, IsLocalizedModule.map_linearCombination, hasSum_cauchyPowerSeries_integral, ContinuousLinearMap.aestronglyMeasurable_comp₂, Flat.eqLocus_lTensor_eq, Subspace.dualRestrict_leftInverse, QuadraticForm.tmul_tensorAssoc_apply, MeasureTheory.Lp.instFourierSMul, SchwartzMap.smulLeftCLM_const, MeasureTheory.integral_fintype, smul_top_inf_eq_smul_of_isSMulRegular_on_quot, LinearEquiv.piRing_symm_apply, MDifferentiableAt.smul_section, trace_eq_trace_adjoin, Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge, CliffordAlgebra.even.lift_ι, Set.OrdConnected.convex, LinearMap.toMatrix_mulVec_repr, OreLocalization.zsmul_eq_zsmul, LinearMap.toMatrix_innerₛₗ_apply, Mathlib.Tactic.LinearCombination.smul_le_const, strictConvexSpace_iff, intervalIntegral.smul_integral_comp_sub_mul, AffineSubspace.toContinuousAffineMap_subtypeₐᵢ, QuadraticMap.associated_left_inverse', HasFPowerSeriesWithinOnBall.hasSum_derivSeries_of_hasFDerivWithinAt, ContinuousLinearMap.IsStarNormal.adjoint_apply_eq_zero_iff, MeromorphicAt.iff_eventuallyEq_zpow_smul_analyticAt, LieModule.Cohomology.mem_twoCochain_iff, PointwiseConvergenceCLM.equivWeakDual_symm_apply, AdicCompletion.ofTensorProductEquivOfFiniteNoetherian_symm_of, hasDerivAt_of_tendstoUniformlyOn, Asymptotics.isLittleO_const_smul_right, smul_right_injective, Unitization.starMap_inr, TensorProduct.lift.tmul, AffineMap.hasDerivAt, Ideal.sup_mul_right_self, Matrix.toLin_apply_eq_zero_iff, hasDerivWithinAt_congr_set, Orientation.inner_rotation_pi_div_two_left_smul, sameRay_smul_left_iff_of_ne, TrivSqZeroExt.invertibleEquivInvertibleFst_apply_invOf, Nonneg.instNonUnitalIsometricContinuousFunctionalCalculus, Algebra.Extension.Hom.toRingHom_algebraMap, WeakDual.CharacterSpace.coe_toNonUnitalAlgHom, smul_sphere, TensorProduct.lTensorHomToHomLTensor_apply, LieSubmodule.mapsTo_pow_toEnd_sub_algebraMap, FormalMultilinearSeries.radius_compContinuousLinearMap_linearIsometryEquiv_eq, strictConvex_Iio, MeasureTheory.Measure.toSignedMeasure_smul, star_inv_intCast_smul, HVertexOperator.compHahnSeries_smul, Matrix.nondegenerate_toBilin_iff, ConvexCone.toPointedCone_top, fourierIntegral_eq_half_sub_half_period_translate, LinearMap.range_toContinuousLinearMap, linearDepOn_iff, Ideal.exact_mulQuot_quotOfMul, MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux2, Polynomial.mkDerivationEquiv_symm_apply, AdicCompletion.mk_smul_mk, TensorProduct.AlgebraTensorModule.cancelBaseChange_tmul, Ideal.Cotangent.smul_eq_zero_of_mem, meromorphicOrderAt_eq_int_iff, ContinuousLinearMap.norm_precompR_le, hasFDerivWithinAt_pow', ContDiffMapSupportedIn.structureMapLM_apply_withOrder, Matrix.toLin'_reindex, Ideal.mul_le, TrivSqZeroExt.isNilpotent_inl_iff, hasStrictDerivAt_finCons', Submodule.mem_colon_iff_le, LocalizedModule.subsingleton_iff_ker_eq_top, LinearMap.transvection.apply, InfiniteGalois.proj_adjoin_singleton_val, hasDerivWithinAt_taylorWithinEval_at_Icc, CliffordAlgebra.contractRight_mul_algebraMap, IntermediateField.lift_inf, FinitePresentation.exists_free_localizedModule_powers, IsSemiprimaryRing.isNilpotent, ContinuousLinearMap.isBoundedLinearMap_comp_right, Matrix.PosSemidef.toLinearMap₂'_zero_iff, LinearMap.BilinForm.toMatrix'_compLeft, Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_aux₂, Matrix.toLinearMapₛₗ₂_apply, ContinuousLinearMap.hasFPowerSeriesOnBall_bilinear, HasFPowerSeriesWithinAt.pi, IsLocalizedModule.fromLocalizedModule_mk, Basis.isUnitSMul_apply, NormedSpace.isVonNBounded_iff', LinearMap.mulRight_tmul, VonNeumannAlgebra.mem_carrier, WeakDual.isSeqCompact_closedBall, NonUnitalStarSubalgebra.inclusion_mk, hasStrictDerivAt_finCons, PiTensorProduct.liftEquiv_apply, LinearMap.smul_lTensor, LinearMap.lsmul_injective, has_antideriv_at_fourier_neg, Ideal.quotientMulEquivQuotientProd_snd, hasFTaylorSeriesUpTo_top_iff', MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul, Set.neg_smul, IsReflexive.bijective_dual_eval', fderivWithin_iteratedFDerivWithin, Unitization.algebraMap_eq_inl, SpecialLinearGroup.centerEquivRootsOfUnity_symm_apply, HahnModule.instIsScalarTowerHahnSeries_1, rescale_to_shell, HahnSeries.SummableFamily.powerSeriesFamily_hsum_zero, WithSeminorms.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup, LinearIsometry.toSpanSingleton_apply, Submodule.mem_span_set, AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_apply, GradedMonoid.smulCommClass_right, ExteriorAlgebra.liftAlternating_one, Subspace.flip_quotDualCoannihilatorToDual_bijective, LinearMap.isSelfAdjoint_iff', PiTensorProduct.liftAux_tprodCoeff, Derivation.commutator_coe_linear_map, ContinuousLinearMap.deriv_of_bilinear, balanced_neg, Submodule.apply_mem_map₂, Orientation.areaForm'_apply, segment_inter_eq_endpoint_of_linearIndependent_of_ne, openSegment_eq_image, Basis.end_apply, innerSLFlip_apply_apply, Continuous.cfcₙ_nnreal, ContinuousLinearMap.nnnorm_holder_apply_apply_le, TrivSqZeroExt.mul_inv_cancel, NonUnitalSubalgebraClass.subtype_injective, Submodule.set_smul_eq_map, Polynomial.coe_smul, AffineMap.isCentralScalar, IntermediateField.exists_algHom_of_splits, Unitization.val_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe, MultilinearMap.curryMid_apply_apply, Asymptotics.isTheta_const_smul_left, intervalIntegral.integral_smul_measure, LieHom.toNonUnitalAlgHom_toFun, TensorProduct.isPushout, upperBounds_smul_of_neg, signedDist_smul_of_pos, Distribution.mapCLM_apply, rankAtStalk_eq_finrank_tensorProduct, FiniteGaloisIntermediateField.instIsSeparableSubtypeMemIntermediateFieldMin_1, FormalMultilinearSeries.norm_compContinuousLinearMap_le, End.isSemisimple_sub_algebraMap_iff, PowerBasis.equivAdjoinSimple_gen, Polynomial.aeval_apply_smul_mem_of_le_comap, Matrix.Nondegenerate.toBilin, WeakDual.CharacterSpace.toAlgHom_apply, TensorProduct.finsuppScalarRight_apply_tmul_apply, LinearMap.BilinForm.nondegenerate_toBilin'_iff_det_ne_zero, Derivation.coe_comp, ModuleCat.instIsScalarTowerLocalizationCarrierLocalizedModule, infEdist_smul₀, Field.Emb.Cardinal.equivSucc_coherence, ContinuousLinearMap.bilinear_hasTemperateGrowth, LinearMap.toMatrix'_apply, IsBoundedBilinearMap.isBoundedLinearMap_deriv, IsContinuousRiemannianBundle.exists_continuous, Bornology.isVonNBounded_iff_smul_tendsto_zero, ContinuousAlternatingMap.fderivCompContinuousLinearMap_eq_alternatizeUncurryFin, Monotone.convex_lt, QuadraticForm.tmul_tensorComm_apply, Basis.repr_symm_single, Matrix.toLpLin_toLp, RingEquiv.moduleEndSelf_apply, apply_eq_dotProduct_toMatrix₂_mulVec, ContinuousLinearMap.postcomp_apply, Ideal.fst_comp_quotientMulEquivQuotientProd, reflection_reflection_iterate, TrivSqZeroExt.inv_mul_cancel, Submodule.smul_toAddSubmonoid, LinearMap.coe_toContinuousLinearMap, ContinuousLinearMap.prodMapL_apply, Ideal.span_mul_span', ContinuousLinearMap.uncurryLeft_apply, HilbertBasis.dense_span, IsLocalizedModule.map_injective, LinearMap.BilinForm.separatingRight_toMatrix_iff, RootPairing.prod_rootForm_smul_coroot_mem_range_domRestrict, AffineMap.lineMap_apply_module', LinearMap.flip_apply, MultilinearMap.freeDFinsuppEquiv_def, map_equiv_traceDual, TensorProduct.finsuppLeft_apply, IntermediateField.restrictScalars_adjoin_eq_sup, Submodule.orthogonal_orthogonal_eq_closure, AnalyticOn.const_smul, LinearMap.BilinForm.SeparatingLeft.toMatrix', isSMulRegular_iff_torsionBy_eq_bot, AffineSubspace.convex, TensorProduct.mapBilinear_apply, Submodule.smul_starProjection_singleton, finsuppTensorFinsuppRid_symm_single_smul, FiniteGaloisIntermediateField.isGalois, ContinuousLinearMap.map_smulₛₗ, SMulCommClass.complexToReal, contMDiff_smul, MeasureTheory.dist_convolution_le, smul_add_one_sub_smul, LinearEquiv.conj_apply, PointedCone.convex, Basis.toDual_range, PiTensorProduct.dualDistribInvOfBasis_apply, intervalIntegral.integral_hasStrictFDerivAt, with_gaugeSeminormFamily, Representation.asAlgebraHom_single_one, CStarAlgebra.linear_combination_nonneg, MeasureTheory.FiniteMeasure.instContinuousSMulNNReal, fderivWithin_mul_const', Submodule.mem_span_finset', Unitization.val_inv_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe, ContinuousAlternatingMap.curryLeft_compContinuousLinearMap, NonUnitalStarSubalgebra.unitization_injective, smul_left_injective, right_mem_openSegment_iff, LieAlgebra.IsKilling.apply_coroot_eq_cast, inv_natCast_smul_comm, NonUnitalSubalgebra.smul_mem', ZMod.invDFT_def', ConvexCone.blunt_strictlyPositive, signedDist_lineMap_lineMap, toCentroidHom_apply_toFun, contMDiffOn_continuousLinearMapCoordChange, fderivWithin_smul, Submodule.map₂_eq_span_image2, NonUnitalStarSubalgebra.unitization_apply, ContinuousMultilinearMap.curryFinFinset_apply_const, InnerProductSpace.enorm_rankOne, aestronglyMeasurable_smul_const_iff, Rep.ihom_obj_ρ_apply, absConvexHull_add_subset, MvPowerSeries.le_weightedOrder_smul, InnerProductSpace.continuousLinearMapOfBilin_zero, Matrix.ker_toLin_eq_bot, differentiableAt_complex_iff_differentiableAt_real, ContinuousMultilinearMap.compContinuousLinearMapLRight_apply, IsRegular.isSMulRegular, Convex.exists_forall_hasDerivWithinAt, cfcHom_real_eq_restrict, intervalIntegral.inv_smul_integral_comp_sub_div, ConvexCone.IsGenerating.top_le_span, TensorProduct.AlgebraTensorModule.dualDistrib_apply, hasDerivWithinAt_taylor_coeff_within, IntermediateField.Lifts.lt_iff, ContinuousLinearMap.toLinearMap_innerSL_apply, WeierstrassCurve.Projective.addMap_eq, UniformConvexOn.convexOn, HasFPowerSeriesAt.hasStrictDerivAt, AnalyticOn.smul, ContinuousLinearMap.continuousMultilinearMapOption_apply_eq_self, Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, LieSubalgebra.coe_incl', IntermediateField.extendScalars_le_iff, Algebra.TensorProduct.toLinearEquiv_tensorTensorTensorComm, instSeminormClassAlgebraNormSubtypeAlgebraicClosureMemIntermediateFieldNormalClosure, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, Submodule.dualAnnihilator_sup_eq, ContinuousAt.cfcₙ_nnreal, LieModule.Cohomology.d₁₂_apply_apply_ofTrivial, WithSeminorms.tendsto_nhds, hasFDerivAt_list_prod_finRange', GradedTensorProduct.auxEquiv_mul, ContDiff.smulRight, Fintype.linearIndependent_iff'ₛ, QuadraticForm.tmul_tensorRId_apply, LinearMap.IsSymmetric.adjoint_eq, not_linearIndependent_iffₒₛ, ContinuousMultilinearMap.flipLinear_apply_apply, SymmetricAlgebra.algebraMap_eq_zero_iff, HasGradientAt.hasFDerivAt, Submodule.lTensorOne'_tmul, RingTheory.Sequence.IsWeaklyRegular.regular_mod_prev, Representation.instIsScalarTowerMonoidAlgebraAsModule, ContinuousLinearMap.comp_hasFPowerSeriesWithinOnBall, FinitePresentation.linearEquivMap_apply, ContinuousMap.coe_smul', CochainComplex.HomComplex.δ_smul, LinearMap.hasDerivWithinAt, WithSeminorms.isOpen_iff_mem_balls, CStarMatrix.toCLM_apply_single, hasSum_two_pi_I_cauchyPowerSeries_integral, Real.fourier_fderiv, convexHull_pair, Set.Subsingleton.strictConvex, rescale_to_shell_semi_normed, MvPolynomial.rTensorAlgHom_toLinearMap, FormalMultilinearSeries.comp_id, range_cfcₙ_eq_range_cfcₙHom, fderivWithin_comp_smul_eq_fderivWithin_smul, ContinuousMap.nnnorm_smul_const, smul_eq_zero_iff_left, ContinuousMap.adjoin_id_eq_span_one_union, CategoryTheory.Linear.comp_units_smul, signedDist_vadd_left_swap, FormalMultilinearSeries.cpolynomialAt_changeOrigin_of_finite, MeasureTheory.condExp_smul, AlgebraicGeometry.StructureSheaf.smul_const, TrivSqZeroExt.invOf_eq_inv, LinearMap.llcomp_apply', Submodule.dualCoannihilator_sup_eq, Seminorm.const_isBounded, Splits.algebraicClosure, hasFTaylorSeriesUpTo_succ_nat_iff_right, PowerBasis.equivAdjoinSimple_symm_gen, LieModule.exists_forall_pow_toEnd_eq_zero, Seminorm.preimage_metric_closedBall, tangentConeAt_of_mem_nhds, neg_concaveOn_iff, FreeLieAlgebra.liftAux_map_mul, ContinuousMultilinearMap.isUniformEmbedding_toUniformOnFun, NonUnitalSubalgebra.instSMulCommClass, Algebra.Extension.CotangentSpace.map_cotangentComplex, LocalizedModule.mk_cancel_common_right, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, HasFDerivWithinAt.norm_sq, IntermediateField.coe_val, Complex.restrictScalars_one_smulRight, LieAlgebra.IsKilling.instIsRootSystemSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, Ideal.pow_le_self, QuotSMulTop.equivQuotTensor_naturality, LinearMap.smulRightₗ_apply, LinearMap.IsNonneg.smul, IsModuleTopology.isoₛₗ, Submodule.rTensorOne'_tmul, Ideal.relNorm_algebraMap', JacobsonNoether.exist_pow_eq_zero_of_le, ClosedSubmodule.iInf_orthogonal, Asymptotics.isLittleOTVS_iff_smallSets, Orientation.oangle_eq_iff_eq_pos_smul_rotation_or_eq_zero, hasDerivAt_fourier_neg, nhds_hasBasis_absConvex, smul_add_smul_lt_smul_add_smul', MeasureTheory.smul_extend, LinearMap.toMatrix'_symm, End.IsSemisimple.aeval, HahnEmbedding.Partial.eval_smul, symm_dualMap_evalEquiv, Localization.finite_of_primesOver_eq_singleton, LinearMap.flip_mul, HasFPowerSeriesOnBall.neg, VectorFourier.fderiv_fourierIntegral, iteratedDerivWithin_scomp_three, ProperCone.dual_singleton, Algebra.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex, ContinuousLinearMap.uncurryBilinear_apply, LinearMap.IsSymmetric.isSymmetric_smul_iff, NonUnitalStarSubalgebraClass.coe_subtype, AdicCompletion.of_bijective_iff, LinearMap.mapMatrix_smul, HasFDerivWithinAt.smul_const, ContinuousLinearEquiv.continuousAlternatingMapCongrRight_symm, AntivaryOn.card_smul_sum_le_sum_smul_sum, Seminorm.sSup_empty, KaehlerDifferential.derivationQuotKerTotal_apply, InnerProductGeometry.norm_ofLp_crossProduct, AdicCompletion.transitionMap_comp_reduceModIdeal, MeasureTheory.integral_posConvolution, Asymptotics.isLittleOTVS_iff_tendsto_div, hasStrictDerivAt_abs, TensorProduct.AlgebraTensorModule.uncurry_apply, MeasureTheory.LocallyIntegrableOn.smul_continuousOn, SchwartzMap.fourierInv_lineDerivOp_eq, VectorFourier.integral_fourierIntegral_swap, IsOpen.smul_sphere, EuclideanGeometry.Sphere.secondInter_smul, AlternatingMap.domCoprod'_apply, Bundle.RiemannianMetric.symm, monovary_iff_smul_rearrangement, AEval.X_pow_smul_of, ContinuousLinearMap.cpolynomialAt_uncurry_of_multilinear, NormedSpace.isVonNBounded_of_isBounded, AdicCompletion.instIsScalarTowerQuotientIdealHSMulTopSubmodule, strongConcaveOn_iff_convex, iteratedFDeriv_eq_equiv_comp, LinearMap.polyCharpolyAux_map_eq_charpoly, Real.smul_iInf_of_nonpos, LinearMap.toContinuousLinearMap_eq_iff_eq_toLinearMap, MeasureTheory.condExpIndL1_smul, Monovary.sum_comp_perm_smul_lt_sum_smul_iff, Submodule.baseChange_eq_span, linearMap_toMatrix_mul_basis_toMatrix, tensorKaehlerQuotKerSqEquiv_symm_tmul_D, Matrix.conjTranspose_ratCast_smul, Unitization.norm_eq_sup, MonovaryOn.sum_smul_sum_le_card_smul_sum, Submodule.dualCopairing_apply, LinearEquiv.toSpanNonzeroSingleton_apply, Matrix.kroneckerTMulAlgEquiv_symm_apply, ContinuousMap.toLp_norm_eq_toLp_norm_coe, dualPairing_apply, Submodule.mem_span_insert', Matrix.ofLp_toEuclideanCLM, Ideal.coe_restrictScalars, Polynomial.mkDerivationEquiv_apply, hasFTaylorSeriesUpToOn_succ_nat_iff_right, Fourier.fourierIntegral_const_smul, Derivation.liftKaehlerDifferential_comp_D, hasDerivAtFilter_iff_isLittleO, AlgebraicClosure.instIsAlgClosure, LinearMap.polyCharpolyAux_eval_eq_toMatrix_charpoly_coeff, CStarMatrix.toCLM_apply_eq_sum, TensorProduct.AlgebraTensorModule.coe_lTensor, Set.Subsingleton.convex, Submodule.span_algebraMap_image_of_tower, HasStrictFDerivAt.restrictScalars, LinearMap.exact_smul_id_smul_top_mkQ, Coalgebra.rTensor_counit_comp_comul, MeasureTheory.Measure.ae_smul_measure_iff, LinearMap.toMatrix_transpose_apply, balancedHull_add_subset, Projectivization.mk_eq_mk_iff, IntermediateField.finrank_eq_finrank_subalgebra, Matrix.kroneckerTMulStarAlgEquiv_apply, LieSubmodule.toEnd_restrict_eq_toEnd, HasFPowerSeriesWithinAt.sub, ClosedSubmodule.sup_orthogonal, strictConvex_Ici, hasDerivAt_exp_smul_const, AddMonoidAlgebra.instIsPushout, IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable', Meromorphic.fun_smul, instFinitePresentationLocalizationLocalizedModule, toMatrix_dualTensorHom, Asymptotics.isBigOTVS_neg_left, MeasureTheory.Measure.NullMeasurableSet.const_smul, TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_eq, LinearEquiv.arrowCongr_apply, MeasureTheory.eLpNormEssSup_smul_measure, ContDiffMapSupportedIn.locallyConvexSpace, hasFDerivWithinAt_comp_smul_smul_iff, fourierCoeff_toLp, IsLocalizedModule.mk'_cancel', IntermediateField.restrictRestrictAlgEquivMapHom_injective, exteriorPower.alternatingMapLinearEquiv_apply_ιMulti, ContinuousLinearMap.toPMap_adjoint_eq_adjoint_toPMap_of_dense, Algebra.Generators.H1Cotangent.δAux_ofComp, Ideal.pow_right_mono, ContinuousMultilinearMap.smul_prod_smul, CategoryTheory.Functor.mapExactFunctor_smul, smul_unitBall_of_pos, smul_lt_smul_iff_of_neg_left, ProbabilityTheory.hasDerivAt_complexMGF, TensorProduct.AlgebraTensorModule.homTensorHomMap_apply, AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff, map_real_smul, RootPairing.reflectionPerm_root, DifferentiableWithinAt.isBigOTVS_sub, ContMDiffWithinAt.smul, LinearMap.isPositive_toContinuousLinearMap_iff, Matrix.range_toLin', HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn', LinearIsometryEquiv.toSpanUnitSingleton_apply, Asymptotics.IsBigOWith.smul, ContinuousMapZero.nonUnitalStarAlgHom_precomp_apply, IsScalarTower.rat, LinearMap.prodMapLinear_apply, MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd', LieModule.IsFaithful.injective_toEnd, map_inv_natCast_smul, iteratedDeriv_comp_const_smul, ProbabilityTheory.covarianceBilin_eq_covarianceBilinDual, norm_inner_eq_norm_tfae, Real.circleAverage_def, Submodule.map₂_span_singleton_eq_map_flip, CategoryTheory.ShortComplex.smul_τ₃, LinearMap.pi_apply_eq_sum_univ, DilationEquiv.smulTorsor_apply, FormalMultilinearSeries.changeOrigin_eval_of_finite, sum_smul_eq_zero_of_isTrivialRelation, Ideal.iSup_mul, intervalIntegral.integral_smul, ZMod.dft_apply, PiTensorProduct.piTensorHomMap_tprod_eq_map, Quaternion.fst_imJ_dualNumberEquiv_symm, exists_nonneg_right_iff_sameRay, LinearMap.dualMap_apply', iteratedFDerivWithin_two_apply', LinearMap.mul_toMatrix', signedDist_vadd_right, LinearMap.toMatrix₂_basisFun, Algebra.Presentation.differentials.comm₂₃', cfcₙ_tsub, deriv.scomp, AnalyticAt.analyticOrderAt_eq_natCast, TensorProduct.LieModule.coe_liftLie_eq_lift_coe, inr_comp_cfcₙHom_eq_cfcₙAux, Submodule.finite_quotient_smul, NonUnitalSubalgebra.unitizationAlgEquiv_apply_coe, FGModuleCat.ihom_obj, PiTensorProduct.lift_comp_reindex, hasDerivAt_taylorWithinEval_succ, IsMIntegralCurveAt.hasMFDerivAt, Bornology.IsBounded.smul₀, LinearMap.dualMap_bijective_iff, MemHolder.smul, HVertexOperator.of_coeff_coeff, PiTensorProduct.map₂_tprod_tprod, LinearMap.star_eq_adjoint, Basis.mem_submodule_iff', skewAdjoint.negISMul_apply_coe, LinearMap.instLieModule, Submodule.smithNormalFormBotBasis_def, Basis.sum_dual_apply_smul_coord, insert_endpoints_openSegment, PMF.integral_eq_tsum, Ideal.Filtration.Stable.exists_pow_smul_eq, not_linearIndependent_iffₛ, HomogeneousLocalization.val_add, contDiffWithinAt_succ_iff_hasFDerivWithinAt', DiffContOnCl.circleAverage_smul_div, ContinuousLinearMap.range_smulRight_apply, IsLocalFrameOn.eventually_eq_sum_coeff_smul, LinearMap.toMatrix_smulBasis_right, Matrix.toLpLin_symm_id, contDiffOn_succ_iff_hasFDerivWithinAt, OreLocalization.right_distrib, Basis.toDual_eq_repr, Matrix.range_toLin_eq_top, AdicCompletion.smul_mk, AlternatingMap.curryLeft_zero, ContinuousLinearMap.continuousWithinAt_uncurry_of_multilinear, AddMonoidAlgebra.instIsPushout', LieModule.toLinearMap_maxTrivLinearMapEquivLieModuleHom_symm, Affine.Simplex.smul_faceOppositeCentroid_vsub_point_eq_sum_vsub, Basis.baseChange_end, intervalIntegral.integral_comp_div, IntermediateField.finrank_adjoin_eq_one_iff, TensorProduct.lid'_apply_tmul, Matrix.toLin_finTwoProd_apply, FDRep.char_linHom, KaehlerDifferential.instSMulCommClassTensorProduct_1, Localization.mkAddMonoidHom_apply, Algebra.Extension.Cotangent.val_smul', Finsupp.weight_single_index, FormalMultilinearSeries.radius_unshift, TensorProduct.map_smul_right, HVertexOperator.coeff_inj_iff, HasFPowerSeriesAt.fderiv_eq, CategoryTheory.endomorphism_simple_eq_smul_id, Orientation.oangle_sign_smul_left, IntermediateField.normal_iff_forall_map_le', Finset.sum_Ico_by_parts, Matrix.isPositive_toEuclideanLin_iff, ContinuousWithinAt.cfcₙ_nnreal, ContinuousLinearMap.coe_restrictScalarsIsometry, MeasureTheory.Integrable.bdd_smul, Finsupp.linearCombination_apply, rank_eq_one_iff, ContinuousLinearMap.memLp_of_bilin, TensorAlgebra.ofDirectSum_toDirectSum, HasFPowerSeriesOnBall.tendstoUniformlyOn, IntermediateField.normal_iff_forall_fieldRange_eq, MatrixModCat.mem_toModuleCatObj, FreeAlgebra.toTensor_ι, ContinuousLinearMap.toLinearMap_eq_iff_eq_toContinuousLinearMap, MeasureTheory.setAverage_eq, SchwartzMap.smulLeftCLM_smulLeftCLM_apply, ConvexBody.isBounded, Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent, Convex.combo_self, CStarMatrix.reindexₐ_apply, WeakBilin.instContinuousSMul, continuousMultilinearCurryFin0_apply, smul_le_smul_iff_of_neg_left, EuclideanGeometry.Sphere.inv_tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center, HasFDerivWithinAt.mapsTo_tangent_cone, DivisibleHull.coeAddMonoidHom_apply, LocalizedModule.smul'_mk, IsLocalizedModule.map_mk', PiTensorProduct.dualSeminorms_bounded, DivisibleHull.instIsOrderedCancelAddMonoid, NonUnitalSubalgebra.inclusion_inclusion, FormalMultilinearSeries.radius_neg, Flat.isSMulRegular_of_isRegular, HasFDerivWithinAt.mul_const', InnerProductSpace.gramSchmidtOrthonormalBasis_apply_of_orthogonal, extDeriv_smul, LinearMap.mul_toMatrix₂, homeomorphUnitSphereProd_symm_apply_coe, PointwiseConvergenceCLM.coeLMₛₗ_apply, ContinuousLinearMap.postcompCompactConvergenceCLM_apply, ProbabilityTheory.covarianceBilin_self_nonneg, algebraMap_end_eq_smul_id, Unitization.val_inv_unitsFstOne_mulEquiv_quasiregular_apply, Algebra.Extension.Cotangent.map_comp, taylorWithinEval_succ, convex_Ioo, Submodule.mulRightMap_eq_mulMap_comp, Algebra.TensorProduct.basis_repr_symm_apply', LieAlgebra.IsKilling.corootForm_rootSystem_eq_killing, ContinuousLinearMap.opNorm_extend_le, Submodule.orthogonal_closure, IsSelfAdjoint.conj_adjoint, Ideal.mul_mem_mul, ContinuousMultilinearMap.curryMidEquiv_apply, TensorProduct.AlgebraTensorModule.rid_eq_rid, StrictConvexSpace.strictConvex_closedBall, Real.fourier_eq, IsBaseChange.prodMap, kroneckerTMulLinearEquiv_mul, Algebra.trace_comp_trace_of_basis, QuadraticMap.discr_comp, Basis.linearMap_repr_apply, AffineMap.lineMap_vadd_apply, Mathlib.Tactic.LinearCombination.smul_lt_const_weak, exteriorPower.alternatingMapLinearEquiv_comp_ιMulti, differentialIdeal_le_fractionalIdeal_iff, LinearMap.lsmul_eq_DistribMulAction_toLinearMap, eval_apply_eq_zero_iff, IntermediateField.finrank_bot, kroneckerTMulLinearEquiv_symm_kroneckerTMul, SMulPosReflectLT.toSMulPosReflectLE, MultilinearMap.curryFinFinset_apply_const, sameRay_neg_smul_left_iff_of_ne, AdicCompletion.map_mk, Set.OrdConnected.convex_of_chain, ContinuousLinearMap.opNorm_mulLeftRight_apply_le, ContinuousLinearMap.analyticOnNhd_bilinear, MultilinearMap.freeFinsuppEquiv_def, localizedModuleIsLocalizedModule, Prod.openSegment_subset, TensorProduct.AlgebraTensorModule.map_eq, WeakBilin.instIsScalarTower, Orientation.oangle_sign_smul_add_smul_smul_add_smul, IsModuleTopology.continuous_bilinear_of_finite_left, DoubleCentralizer.algebraMap_snd, Finset.expect_dite_eq', Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner_naturality, SeminormFamily.withSeminorms_iff_uniformSpace_eq_iInf, LieModule.trace_comp_toEnd_genWeightSpace_eq, MeasureTheory.Measure.integral_comp_mul_left, LinearMap.nondegenerate_congr_iff, AnalyticAt.analyticOrderNatAt_eq_iff, TensorProduct.gradedMul_one, ProbabilityTheory.covarianceBilinDual_self_eq_variance, QuadraticForm.tensorComm_toLinearEquiv, CliffordAlgebra.contractLeft_comm, CompactlySupportedContinuousMap.integralLinearMap_apply, MeasureTheory.integrable_smul_const, hasFiniteFPowerSeriesOnBall_const, InfiniteGalois.restrict_fixedField, End.smulLeft_apply, HahnModule.SMulCommClass, Ideal.mul_sup_eq_of_coprime_left, ZMod.invDFT_apply', iteratedFDerivWithin_succ_apply_left, hasStrictFDerivAt_norm_sq, Submodule.starProjection_unit_singleton, Matrix.dualNumberEquiv_symm_apply, Finset.weightedVSub_const_smul, LinearMap.toMatrix'_mul, Algebra.TensorProduct.productMap_eq_comp_map, SchwartzMap.fourier_evalCLM_eq, IsBaseChange.end, LinearMap.adjoint_toSpanSingleton, mul_basis_toMatrix, ContinuousWithinAt.integral_sub_linear_isLittleO_ae, convex_convexHull, ContinuousMultilinearMap.cpolynomialAt_uncurry_of_linear, Basis.toDual_toDual, DualNumber.exp_eps, Submodule.mul_smul_mul_eq_smul_mul_smul, IntermediateField.sepDegree_bot, normalClosure.instIsScalarTowerSubtypeMemIntermediateFieldNormalClosure, KaehlerDifferential.tensorKaehlerEquiv_left_inv, contDiffOn_infty_iff_fderiv_of_isOpen, hasDerivWithinAt_natCast, IsTopologicalSemiring.toOppositeIsModuleTopology, Submodule.rTensorOne_tmul, Orientation.oangle_sign_smul_sub_left, IsLocalizedModule.rTensor, LinearEquiv.conj_symm_conj, fderiv_pow_ring', LinearMap.toSpanSingleton_apply, ContinuousLinearMap.map_sub₂, MeasureTheory.memL1_smul_of_L1_withDensity, ContinuousLinearMap.bilinearComp_apply, analyticAt_iff_analytic_fun_smul, killingForm_apply_apply, imaginaryPart_eq_neg_I_smul_skewAdjointPart, LinearMap.trace_id, LinearMap.toMatrix_algebraMap, midpoint_vsub_left, MeasureTheory.Measure.toSphere_apply', QuadraticModuleCat.toIsometry_hom_rightUnitor, TemperedDistribution.instLineDerivSMulReal, LinearMap.spectrum_toMatrix, AdicCompletion.ofAlgEquiv_apply, iteratedDeriv_comp_neg, PowerBasis.constr_pow_gen, WithVal.instIsScalarTower_1, LinearMap.BilinForm.congr_refl, NonUnitalSubalgebra.map_toNonUnitalSubsemiring, LieAlgebra.smulCommClass, range_cfcₙ_nnreal_eq_image_cfcₙ_real, SchwartzMap.evalCLM_apply_apply, Matrix.Module.instIsScalarTowerForall, LieSubalgebra.mem_engel_iff, Finset.sum_smul_const_vsub_eq_neg_weightedVSub, Polynomial.smeval_C_mul, Finite.exists_smul_of_comp_eq_of_isLocalizedModule, Algebra.norm_eq_norm_adjoin, LieAlgebra.lie_smul, LinearMap.SeparatingRight.toMatrix₂, InnerProductGeometry.angle_smul_right_of_neg, Affine.Triangle.inv_tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_circumcenter, Matrix.isUnit_toLin_iff, DiffContOnCl.smul_const, MeasureTheory.condExpIndL1_smul', Submodule.singleton_smul, Subspace.quotAnnihilatorEquiv_apply, LinearMap.conjugate_apply, Unitization.mem_spectrum_inr_of_not_isUnit, IntermediateField.isAlgebraic_adjoin_iff_isAlgebraic, Filter.Tendsto.cfcₙ_nnreal, tendsto_integral_exp_smul_cocompact_of_inner_product, Submodule.toConvexCone_le_toConvexCone, tsum_const_smul'', vadd_segment, Projectivization.mk_eq_mk_iff_crossProduct_eq_zero, Matrix.Module.scalar_smul, Matrix.diagonal_toLin', LinearMap.re_inner_adjoint_mul_self_nonneg, PointwiseConvergenceCLM.coeLM_apply, mem_parallelepiped_iff, isTorsionBySet_iff_subseteq_ker_lsmul, Submodule.instIsScalarTowerQuotientSpanSingletonSetSubtypeMemTorsionBy, contDiff_smul_const, MeasureTheory.integral_biUnion_eq_sum_powerset, HasFTaylorSeriesUpToOn.fderivWithin, Prod.image_mk_openSegment_right, CStarMatrix.toCLM_apply, KaehlerDifferential.kerTotal_map, IntermediateField.restrictScalars_bot_eq_self, End.mem_genEigenspace_top, LieModule.lie_smul, InnerProductSpace.nullSubmodule_le_ker_toDualMap_left, DoubleCentralizer.algebraMap_toProd, LinearMap.isPositive_self_comp_adjoint, Antitone.cauchySeq_series_mul_of_tendsto_zero_of_bounded, Algebra.TensorProduct.mul_one, RingTheory.Sequence.isWeaklyRegular_append_iff', Matrix.charpoly_toLin', Quaternion.fst_imI_dualNumberEquiv_symm, LinearMap.BilinForm.toMatrix_symm, differentiableAt_norm_smul, exists_nontrivial_relation_sum_zero_of_not_affine_ind, FormalMultilinearSeries.id_comp, circleIntegral.integral_smul, ContinuousLinearMap.norm_map_tail_le, LieModule.iterate_toEnd_mem_lowerCentralSeries₂, LinearMap.BilinMap.toQuadraticMap_smul, invOf_two_smul_add_invOf_two_smul, Real.circleAverage_eq_integral_add, Matrix.toLinearMap₂'_toMatrix', isSMulRegular_algebraMap_iff, HasFPowerSeriesAt.restrictScalars, contMDiffOn_symm_coordChangeL, hasDerivAt_integral_of_dominated_loc_of_deriv_le, intervalIntegral.inv_smul_integral_comp_div_sub, rank_dual_eq_card_dual_of_aleph0_le_rank', ContinuousLinearMap.instStarModuleId, ContDiff.smul_const, Matrix.toLinearMapₛₗ₂'_apply, edist_smul₀, Seminorm.convex_ball, CategoryTheory.ShiftedHom.comp_smul, PolynomialModule.eval_lsingle, MultilinearMap.map_nonempty, Ideal.relNorm_algebraMap, Algebra.Extension.H1Cotangent.equivOfFormallySmooth_apply, DoubleCentralizer.instIsCentralScalar, IsLocalizedModule.map_apply, sectionOfRetractionKerToTensorAux_prop, End.genEigenspace_one, ContinuousLinearMapWOT.instContinuousSMul, Algebra.TensorProduct.mk_one_injective_of_isScalarTower, CochainComplex.HomComplex.Cochain.smul_comp, Submodule.smul_le_right, Submodule.span_eq_closure, Matrix.toLin'_pow, PiTensorProduct.injectiveSeminorm_apply, ContinuousLinearMap.isBoundedLinearMap_comp_left, PartitionOfUnity.sum_finsupport_smul_eq_finsum, LieAdmissibleAlgebra.toIsScalarTower, IsSymmetricAlgebra.equiv_symm_apply, LinearIsometryEquiv.toLinearEquiv_smul, LieModule.Cohomology.mem_twoCocycle_iff, CliffordAlgebra.ofBaseChangeAux_ι, Ideal.ker_tensorProductMk_quotient, WittVector.inv_pair₂, norm_fderiv_iteratedFDeriv, IsCompact.extremePoints_nonempty, MeasureTheory.SignedMeasure.haveLebesgueDecomposition_smul_real, Matrix.instSMulCommClassMulOppositeForallOfIsScalarTower_1, Unitization.lift_range, CommAlgCat.tensorHom_hom, DistribMulActionHom.instLinearMapClassId, LinearMap.BilinForm.SeparatingLeft.toMatrix, meromorphicOrderAt_smul, Asymptotics.isLittleOTVS_map, instIsLocalizedModuleQuotientSubmoduleLocalizedModuleLocalizationLocalizedToLocalizedQuotient, Derivation.liftKaehlerDifferential_unique_iff, IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable, ProbabilityTheory.covarianceBilinDual_self_nonneg, ContinuousAlternatingMap.fderivCompContinuousLinearMapCLM_apply, Path.range_segment, instIsStrictOrderedModuleOfIsCancelAdd, segment_subset_Icc, Derivation.liftKaehlerDifferential_D, TensorProduct.AlgebraTensorModule.mapBilinear_apply, Ideal.span_mul_span, LinearMap.BilinMap.toQuadraticMapLinearMap_apply, TemperedDistribution.fourier_lineDerivOp_eq, Affine.Simplex.point_vsub_faceOppositeCentroid_eq_smul_sum_vsub, Matrix.toLinAlgEquiv'_apply, groupHomology.inhomogeneousChains.d_single, ProbabilityTheory.setIntegral_tilted_mul_eq_cgf', Matrix.l2_opNNNorm_def, ContinuousLinearEquiv.strictConvex_image, ContinuousMultilinearMap.instIsBoundedSMul, IntermediateField.mem_lift, IntermediateField.isAlgebraic_adjoin_pair, cfcₙHom_isClosedEmbedding, LocalizedModule.mk_neg, hasDerivAtFilter_sub_const_iff, Basis.dualBasis_apply, spectrum.hasDerivAt_resolvent, Basis.sum_toMatrix_smul_self, ContinuousLinearMap.adjointAux_adjointAux, apply_eq_star_dotProduct_toMatrix₂_mulVec, tensorKaehlerQuotKerSqEquiv_tmul_D, MvPolynomial.mkDerivation_monomial, LinearMap.aeval_eq_aeval_mod_charpoly, LinearMap.toMatrix_mul, ContDiffMapSupportedIn.seminorm_le_iff_withOrder, smul_nonpos_iff_neg_imp_nonneg, Finsupp.linearCombination_single, realPart_I_smul, Submodule.exists_smul_notMem_of_rank_lt, LinearMap.toMatrix₂_compl₁₂, LinearMap.BilinForm.IsSymm.smul, linearIndependent_smul, Seminorm.exists_apply_eq_finset_sup, LinearMap.hasEigenvalue_zero_tfae, Matrix.minpoly_toLin', WeierstrassCurve.Projective.Point.fromAffine_some, PolynomialModule.smul_apply, QuadraticForm.tensorDistrib_tmul, signedDist_lineMap_left, gauge_def, Subalgebra.centralizer_tensorProduct_eq_center_tensorProduct_left, Complex.hasDerivAt_logTaylor, Multiset.sum_smul, WithSeminorms.isVonNBounded_iff_finset_seminorm_bounded, MeasureTheory.FiniteMeasure.smul_testAgainstNN_apply, ProbabilityTheory.covarianceBilin_real_self, LinearMap.eq_adjoint_iff, Mathlib.Tactic.Module.NF.zero_sub_eq_eval, deriv_smul, nnnorm_cfcₙ_nnreal_lt_iff, Prod.image_mk_segment_right, GradedTensorProduct.mulHom_apply, VectorField.mpullbackWithin_smul_apply, PositiveLinearMap.gnsStarAlgHom_apply, IsSl2Triple.HasPrimitiveVectorWith.pow_toEnd_f_eq_zero_of_eq_nat, LieModule.toEnd_eq_iff, BilinForm.toMatrix_compRight, Real.fourier_iteratedDeriv, MeasureTheory.Integrable.smul_of_top_left, LinearMap.BilinForm.nondegenerate_toMatrix'_iff, MeasureTheory.integral_comp_rpow_Ioi, ContinuousAlternatingMap.toAlternatingMap_alternatizeUncurryFin, Polynomial.wronskianBilin_apply, convex_closedEBall, hasDerivAt_iff_tendsto_slope_zero, LinearMap.toMatrix₂_toLinearMap₂, IntermediateField.finrank_sup_le, LinearMap.detAux_def', TensorAlgebra.algebraMap_eq_one_iff, MultilinearMap.fromDirectSumEquiv_lof, LinearMap.toMatrix_transpose_apply', AlgCat.hom_whiskerLeft, LinearMap.norm_extendTo𝕜'_apply_sq, Asymptotics.isLittleOTVS_iff_isLittleO, Affine.Simplex.faceOppositeCentroid_vsub_point_eq_smul_vsub, AffineSubspace.wSameSide_smul_vsub_vadd_left, TrivSqZeroExt.inv_inv, SchwartzMap.integral_pow_mul_iteratedFDeriv_le, HasFTaylorSeriesUpToOn.hasFDerivAt, WeakFEPair.hf_zero, LinearMap.toMatrix'_intrinsicStar, MultilinearMap.domCoprod'_apply, convex_ball, real_inner_smul_right, LinearMap.BilinForm.congr_congr, ContinuousLinearMap.flip_add, WeierstrassCurve.Projective.negMap_of_Z_ne_zero, QuadraticMap.polar_smul_left, exists_pos_left_iff_sameRay, IntermediateField.adjoinRootEquivAdjoin_apply_root, LinearMap.finiteDimensional, hasFiniteFPowerSeriesAt_const, Algebra.Extension.Cotangent.map_id, MeasureTheory.Integrable.toL1_smul', ModuleCat.ExtendScalars.smul_tmul, QuadraticForm.tensorComm_apply, AffineSubspace.isometryEquivMap.coe_apply, infDist_smul₀, Submodule.smul_mem_iff_of_isUnit, ContinuousAlternatingMap.hasBasis_nhds_zero, Orientation.oangle_sub_right_smul_rotation_pi_div_two, MeasureTheory.LocallyIntegrable.integrable_smul_left_of_hasCompactSupport, Bornology.isBounded_iff_isVonNBounded, Antivary.sum_smul_lt_sum_smul_comp_perm_iff, Convex.min_le_combo, cross_apply, monovaryOn_iff_smul_rearrangement, cfcₙHom_le_iff, LinearEquiv.lieConj_apply, VectorFourier.fourierSMulRight_apply, LieModule.isNilpotent_toEnd_of_mem_rootSpace, IsBaseChange.linearMapLeftRightHom_toMatrix, Derivation.leibniz, ContinuousLinearMap.isSelfAdjoint_iff', HasFDerivWithinAt.restrictScalars, LinearIsometry.map_smulₛₗ, UnitAddTorus.instContinuousSMulComplexSubtypeAEEqFunVolumeMemAddSubgroupLp, OrthonormalBasis.equiv_apply_euclideanSpace, RootPairing.PolarizationIn_apply, AlternatingMap.eq_smul_basis_det, LinearMap.BilinForm.toMatrixAux_apply, PiTensorProduct.mapMultilinear_apply, StrongConcaveOn.strictConcaveOn, CochainComplex.HomComplex.Cochain.rightShift_smul, TemperedDistribution.lineDerivOp_toTemperedDistributionCLM_eq, ContinuousLinearMap.toContinuousAddMonoidHom_injective, IntermediateField.finSepDegree_adjoin_simple_eq_natSepDegree, MeasureTheory.condExpL2_indicator_eq_toSpanSingleton_comp, ContinuousMultilinearMap.uncurryRight_apply, Seminorm.closedBall_smul, IntermediateField.rank_bot, Matrix.SpecialLinearGroup.toLin_equiv.symm_toLinearMap_eq, ContinuousLinearMap.isVonNBounded_iff, LinearEquiv.dualMap_symm, InnerProductSpace.Core.inner_smul_ofReal_left, Invertible.toModuleEnd_bijective, Real.isScalarTower, WeierstrassCurve.Projective.nonsingularLift_iff, Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, differentiableWithinAt_iff_restrictScalars, IsModuleTopology.iso, AdicCompletion.of_surjective, NormedSpace.normalize_smul_of_neg, TrivSqZeroExt.exp_def, derivWithin_comp_mul_left, Algebra.Generators.Cotangent.exact, Seminorm.coe_bot, hasDerivAt_circleMap, LinearMap.toMatrixAlgEquiv'_toLinAlgEquiv', LinearMap.toMatrix_smulRight, IntermediateField.isPurelyInseparable_adjoin_iff_pow_mem, vectorSpan_segment, kroneckerTMulLinearEquiv_tmul, stabilizer_units_eq_bot_of_ne_zero, LinearEquiv.alternatingMapCongrRight_apply_apply, ProperCone.smul_mem, derivWithin_fun_smul, ContinuousLinearMap.bilinearComp_zero, MeasureTheory.integral_fun_norm_addHaar, LinearMap.toMatrix₂Aux_eq, HasFDerivAt.fun_pow', LocalizedModule.subsingleton_iff, MeasureTheory.setIntegral_ge_of_const_le, le_restrictScalars_separableClosure, signedDist_anticomm, AddMonoidAlgebra.coe_liftNCAlgHom, imaginaryPart_apply_coe, segment_same, End.natCast_def, MultilinearMap.curryLeft_apply, iteratedDeriv_eq_equiv_comp, SchwartzMap.smulLeftCLM_add, pinGroup.conjAct_smul_range_ι, LinearMap.lsmul_apply, HasMFDerivWithinAt.mul', intervalIntegral.integral_sub_linear_isLittleO_of_tendsto_ae, TensorProduct.AlgebraTensorModule.restrictScalars_rTensor, LieDerivation.toLinearMapLieHom_injective, segment_subset_closedBall_left, UnitAddTorus.hasSum_mFourier_series_L2, FreeLieAlgebra.liftAux_spec, norm_fderivWithin_iteratedFDerivWithin, Submodule.smul_closure_subset, HasFTaylorSeriesUpToOn.shift_of_succ, CentroidHom.instSMulCommClass_1, vecMulVecBilin_apply_apply, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply, QuadraticForm.tmul_comp_tensorMap, isAdjointPair_toLinearMap₂', ConvexCone.mem_add, SemimoduleCat.Algebra.instSMulCommClassCarrier, Matrix.toLin'_apply', QuotSMulTop.map_exact, Representation.asModuleEquiv_symm_map_rho, MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.rePart, LinearMap.isPairSelfAdjoint_equiv, LinearMap.toMatrix₂_symm, contDiffOn_fderiv_coord_change, ContinuousLinearEquiv.continuousMultilinearMapCongrRight_symm, End.commute_exp_left_of_commute, IntermediateField.instIsScalarTowerSubtypeMem_1, Configuration.ofField.crossProduct_eq_zero_of_dotProduct_eq_zero, TensorPower.toTensorAlgebra_gMul, sum_mem_ball_of_strictConvexSpace, IsModuleTopology.continuous_of_distribMulActionHomₑ, Matrix.ker_toLin'_eq_bot_iff, ContinuousAlternatingMap.toContinuousMultilinearMapCLM_comp_fderivCompContinuousLinearMap, le_egauge_smul_right, ContinuousMultilinearMap.integral_apply, NonUnitalSubalgebra.unitization_apply, LinearMap.vecCons₂_apply, IsAzumaya.coe_tensorEquivEnd, SeminormFamily.basisSets_smul, DualNumber.lift_apply_apply, LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg, LocalizedModule.smul_def, LinearMap.eval_charpoly, LinearMap.SeparatingLeft.toMatrix₂, Submodule.span_eq_iUnion_nat, LinearMap.toMatrix'_toLinearMap₂', SameRay.sameRay_nonneg_smul_right, TensorProduct.lift_comp_map, TensorPower.toTensorAlgebra_gOne, Derivation.llcomp_apply, HasFiniteFPowerSeriesOnBall.neg, ModuleTopology.eq_coinduced_of_surjectiveₛₗ, CentroidHom.instIsScalarTower_1, Basis.multilinearMap_apply_apply, LinearMap.BilinForm.toLin'Flip_apply, TensorProduct.tensorQuotEquivQuotSMul_symm_mk, LinearMap.reduceModIdeal_apply, RingTheory.Sequence.isWeaklyRegular_append_iff, MeasureTheory.average_union_mem_segment, cfc_eq_cfcL_mkD, lineDifferentiableWithinAt_smul_iff, Fintype.expect_ite_mem, instIsTorsionFreeLocalizationAlgebraMapSubmonoidPrimeCompl_1, BilinForm.toMatrix_mul_basis_toMatrix, DifferentiableWithinAt.smul_const, SMulCommClass.rat', MultilinearMap.dfinsuppFamilyₗ_apply, WeakDual.isClosed_closedBall, VectorFourier.pow_mul_norm_iteratedFDeriv_fourierIntegral_le, cfcₙ_real_eq_nnreal, Submodule.le_span_singleton_iff, hasDerivAt_fourier, Ideal.IsTwoSided.instHPowNat, Unitization.lift_symm_apply, HahnSeries.SummableFamily.coeff_smul, QuadraticMap.half_moduleEnd_apply_eq_half_smul, smul_lie, Seminorm.neg_closedBall, LinearMap.BilinForm.mul_toMatrix', IntermediateField.finrank_bot_mul_relfinrank, LieModule.toEnd_lie, cfcₙ_map_prod, QuadraticForm.tensorAssoc_symm_apply, EuclideanGeometry.Sphere.direction_orthRadius_le_iff, AffineMap.lineMap_apply, Finsupp.linearCombination_unique, smul_eq_zero_iff_right, AntivaryOn.sum_smul_comp_perm_eq_sum_smul_iff, LinearMap.map_mul_iff, MultilinearMap.map_update_smul', Real.fourier_bilin_convolution_eq, BoxIntegral.BoxAdditiveMap.toSMul_apply, SymmetricAlgebra.algHom_ext_iff, IsLocalizedModule.mk'_cancel_right, segment_eq_image', CStarMatrix.mul_entry_mul_eq_inner_toCLM, Affine.Simplex.faceOppositeCentroid_eq_sum_vsub_vadd, ContinuousOn.cfcₙ_nnreal_of_mem_nhdsSet, Matrix.toLinearMap₂_symm, homeomorphSphereProd_symm_apply_coe, MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left, Submodule.convex, End.isRoot_of_hasEigenvalue, ExteriorAlgebra.toTrivSqZeroExt_ι, ContinuousLinearMap.le_opNorm₂, ContinuousLinearMap.lpPairing_eq_integral, TrivSqZeroExt.snd_invOf, Matrix.coe_toEuclideanCLM_eq_toEuclideanLin, slope_fun_def, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, Seminorm.continuous_iff, HasFDerivAt.smul_const, MeasureTheory.integrable_tilted_iff, ContinuousLinearMap.norm_uncurryMid, Submodule.torsionBy.mk_ideal_smul, SymmetricAlgebra.equivMvPolynomial_symm_X, Submodule.mem_span_iff_of_fintype, InnerProductSpace.rankOne_def', ContinuousLinearMap.opNorm_mul_le, MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one, ContinuousLinearMap.uncurryLeft_norm, LeftPreLieAlgebra.ext_iff, LinearMap.toMatrixAlgEquiv_symm, ContDiffWithinAt.fderivWithin', star_inv_natCast_smul, LinearMap.toMatrixRight'_id, mellin_eq_fourierIntegral, leibniz_cross, mem_vectorSpan_pair_rev, PiTensorProduct.smul_tprodCoeff_aux, hasStrictDerivAt_abs_neg, ContinuousLinearMap.holder_smul_right, WithSeminorms.continuous_seminorm, HasFPowerSeriesOnBall.tendstoUniformlyOn', HasFPowerSeriesWithinOnBall.const_smul, HVertexOperator.coeff_inj, PolynomialModule.smul_single_apply, AddMonoid.End.natCast_def, TensorProduct.rid_tmul, ProbabilityTheory.norm_uncenteredCovarianceBilinDual_le, Submodule.coe_torsionBySet, IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite, LinearMap.minpoly_toMatrix, LinearMap.det_dualMap, PiToModule.fromEnd_apply_single_one, Ideal.pow_le_pow_right, HasFPowerSeriesOnBall.sub, SchwartzMap.integral_sesq_fourierIntegral_eq, CharacterModule.homEquiv_symm_apply_apply_apply, IntermediateField.finiteDimensional_adjoin, contDiffAt_succ_iff_hasFDerivAt, Real.fourierIntegral_eq', WittVector.inv_pair₁, Real.circleAverage_fun_smul, RootPairing.coroot_root_two, ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_left, CategoryTheory.ShortComplex.cyclesMap'_smul, QuadraticModuleCat.toIsometry_inv_rightUnitor, PiTensorProduct.piTensorHomMapFun₂_smul, LinearMap.adjoint_lTensor, LieModule.Cohomology.add_apply_apply, Matrix.nondegenerate_toLinearMap₂'_iff_nondegenerate_toLinearMap₂, EuclideanGeometry.orthogonalProjection_vadd_smul_vsub_orthogonalProjection, Complex.hasSum_taylorSeries_on_emetric_ball, LinearEquiv.smulOfNeZero_apply, MeasureTheory.integrable_withDensity_iff_integrable_smul, IntermediateField.lift_injective, multilinearCurryLeftEquiv_apply, MeasureTheory.Integrable.convolution_integrand, LieModule.exists_forall_lie_eq_smul, FormalMultilinearSeries.radius_compContinuousLinearMap_eq, norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul, Projectivization.exists_smul_eq_mk_rep, reflection_mul_reflection_pow_apply_self, CliffordAlgebra.evenToNeg_ι, LinearMap.toMatrixOrthonormal_apply, DivisibleHull.qsmul_def, hasFPowerSeriesOn_cauchy_integral, HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod, MvPolynomial.coeff_rTensorAlgHom_tmul, antitone_smul_left, Representation.asAlgebraHom_of, cfcₙHom_eq_cfcₙ_extend, LieRinehartRing.leibniz_smul_right, EuclideanGeometry.inversion_def, LinearMap.IsPositive.smul_of_nonneg, NormedSpace.vonNBornology_eq, ProbabilityTheory.covarianceBilin_apply_pi, KaehlerDifferential.isBaseChange, Function.Periodic.const_smul₀, SchwartzMap.integral_bilinear_laplacian_right_eq_left, LinearMap.lTensor_comp_mk, Basis.toDual_injective, AlgHom.instNonUnitalAlgHomClassOfAlgHomClass, KaehlerDifferential.instSMulCommClassTensorProduct, iteratedDeriv_fun_const_smul_field, Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge, HasFDerivAt.lim, Seminorm.comp_apply, hasDerivWithinAt_const, openSegment_eq_image_lineMap, FormalMultilinearSeries.compAlongComposition_norm, ContinuousLinearMap.isometry_mul, Complex.circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable, VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral', intervalIntegral.integral_comp_sub_mul, NonUnitalCommCStarAlgebra.toStarModule, multiple_mem_span_of_mem_localization_span, OrthonormalBasis.sum_repr_symm, homeomorphUnitSphereProd_apply_fst_coe, IntermediateField.isSeparable_adjoin_simple_iff_isSeparable, LieAlgebra.IsKilling.disjoint_ker_weight_corootSpace, CFC.posPart_smul_of_nonneg, MultilinearMap.cons_smul, StrongDual.toLpₗ_apply, Orientation.kahler_apply_self, DifferentiableWithinAt.restrictScalars_fderivWithin, Invertible.rTensorInv_leftInverse, Ideal.span_singleton_mul_span_singleton, RootPairing.Hom.weight_coweight_transpose_apply, MultilinearMap.curryFinFinset_symm_apply_const, KaehlerDifferential.mvPolynomialBasis_repr_apply, LinearMap.prodMapAlgHom_apply_apply, signedDist_triangle_right, MonoidAlgebra.nonUnitalAlgHom_ext'_iff, End.eigenspace_def, isCompact_closedAbsConvexHull_of_totallyBounded, IntermediateField.AdjoinSimple.trace_gen_eq_zero, Orthonormal.inner_left_sum, NonUnitalSubalgebra.instSMulMemClass, Orientation.inner_smul_rotation_pi_div_two_right, MeasureTheory.FiniteMeasure.normalize_eq_inv_mass_smul_of_nonzero, ContinuousLinearMap.continuous_uncurry_of_multilinear, UniformConcaveOn.concaveOn, SymmetricAlgebra.lift_comp_ι, constFormalMultilinearSeries_apply_of_nonzero, RootPairing.CoPolarization_apply, egauge_ball_one_le_of_one_lt_norm, HasFTaylorSeriesUpToOn.add, MeasureTheory.dist_convolution_le', Matrix.linftyOpNormSMulClass, finsuppTensorFinsuppRid_single_tmul_single, AddMonoidAlgebra.nonUnitalAlgHom_ext'_iff, Real.circleAverage_smul, LieModule.shiftedGenWeightSpace.coe_lie_shiftedGenWeightSpace_apply, AlternatingMap.curryLeftLinearMap_apply, star_qsmul, LinearMap.flip_separatingRight, SymmetricAlgebra.lift_ι, Localization.mk_list_sum, Subalgebra.inclusion.isScalarTower_right, ContDiffMapSupportedIn.isUniformEmbedding_pi_structureMapCLM, WithLp.normSMulClassSeminormedAddCommGroupToProd, ContinuousLinearMap.isSelfAdjoint_toLinearMap_iff, PiTensorProduct.piTensorHomMap₂_tprod_tprod_tprod, VectorField.lieBracket_const_smul_right, preReflection_apply, Algebra.toMatrix_lmul', LieAlgebra.ad_apply, PiLp.smul_apply, LieAlgebra.toEnd_pow_apply_mem, ProbabilityTheory.hasDerivAt_neg_exp_mul_exp, abs_signedDist_eq_dist_iff_vsub_mem_span, InnerProductSpace.innerSL_norm, CommAlgCat.whiskerLeft_hom, MeasureTheory.uniformIntegrable_average, LinearIsometryEquiv.strictConvex_image, ContinuousMultilinearMap.isEmbedding_toUniformOnFun, CochainComplex.HomComplex.Cochain.shift_units_smul, AffineMap.homothety_linear, Algebra.Extension.contangentEquiv_tmul, Unitization.norm_def, convex_uIcc, IsLocalization.instCompatibleSMulLocalizationOfIsScalarTower_1, ModuleTopology.eq_coinduced_of_surjective, LieAlgebra.IsKilling.lie_eq_smul_of_mem_rootSpace, Ideal.quotOfMul_surjective, Subalgebra.coe_smul, map_natCast_smul, Submodule.map₂_span_span, SchwartzMap.eLpNorm_le_seminorm, WeierstrassCurve.Projective.addMap_of_Y_eq, MDifferentiableAt.smul, Affine.Simplex.centroid_eq_smul_vsub_vadd_point, IsLocalExtrOn.linear_dependent_of_hasStrictFDerivAt, IsLocalizedModule.map_id, Orientation.kahler_rotation_left', Representation.linHom.invariantsEquivRepHom_symm_apply_coe, hasFDerivAtFilter_iff_isLittleOTVS, MeasureTheory.setIntegral_rnDeriv_smul', IntermediateField.finSepDegree_adjoin_simple_le_finrank, FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp, TensorProduct.finsuppScalarLeft_apply_tmul_apply, Matrix.ker_diagonal_toLin', ContinuousAlternatingMap.curryLeft_same, TensorProduct.CompatibleSMul.int, ContinuousMultilinearMap.curryRight_apply, ContDiff.fourierPowSMulRight, CategoryTheory.Abelian.Ext.comp_smul, ArithmeticFunction.mul_smul', ContMDiffWithinAt.mfderivWithin, LinearMap.polyCharpolyAux_baseChange, CategoryTheory.Linear.comp_smul, star_nnqsmul, End.mem_iInf_maxGenEigenspace_iff, HasFPowerSeriesOnBall.uniform_geometric_approx', image_segment, intervalIntegral.smul_integral_comp_mul_left, LinearMap.coe_rTensorHom, Ideal.snd_comp_quotientMulEquivQuotientProd, Function.support_const_smul_of_ne_zero, ContinuousLinearMap.extendTo𝕜_apply, Submodule.starConvex, ContDiffMapSupportedIn.toBoundedContinuousFunctionLM_eq_of_scalars, Algebra.TensorProduct.assoc_toLinearEquiv, Basis.symmetricAlgebra_repr_apply, LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective, Coalgebra.lift_lsmul_comp_counit_comp_comul, ContinuousMultilinearMap.curryFinFinset_symm_apply_piecewise_const, Algebra.TensorProduct.assoc_symm_tmul, PowerSeries.coeff_smul, AEval.of_symm_X_smul, IntermediateField.card_algHom_adjoin_integral, Derivation.leibniz_of_mul_eq_one, FGModuleCat.Iso.conj_eq_conj, Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq, uniformConcaveOn_zero, AffineSubspace.mem_affineSpan_insert_iff, ConvexCone.pointed_zero, AffineSubspace.subtypeₐᵢ_linear, LieModule.traceForm_eq_sum_finrank_nsmul, Matrix.toBilin'_apply, Seminorm.closedBall_eq_emptyset, Affine.Simplex.mongePoint_eq_smul_vsub_vadd_circumcenter, IntermediateField.relrank_dvd_rank_bot, LinearMap.det_restrictScalars, hasStrictFDerivAt_pow', convex_Icc, ContinuousLinearMap.applySMulCommClass, strictConvex_Ioc, LieAlgebra.conj_ad_apply, LinearMap.flip_injective_iff₂, QuadraticForm.tensorRId_toLinearEquiv, RestrictScalars.ringEquiv_map_smul, InnerProductGeometry.angle_smul_smul, RootPairing.prod_rootFormIn_smul_coroot_mem_range_PolarizationIn, HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope, RestrictScalars.lsmul_apply_apply, IsIntegralClosure.range_le_span_dualBasis, ContinuousMapZero.toContinuousMapHom_toNNReal, LinearMap.toMatrix_prodMap, Matrix.toLin_apply, Matrix.toLin'_one, AdicCompletion.mk_surjective, ModelWithCorners.convex_range, ContDiffMapSupportedIn.postcompCLM_apply, TemperedDistribution.smulLeftCLM_add, DoubleCentralizer.coeHom_apply, Algebra.TensorProduct.instCompatibleSMulRat, ContinuousAlternatingMap.curryLeft_zero, signedDist_neg, ProbabilityTheory.covarianceBilin_apply_basisFun, RootPairing.reflection_dualMap_eq_coreflection, Matrix.Module.smul_apply, DualNumber.coe_lift_symm_apply, ContDiffBump.integral_normed_smul, ContinuousMultilinearMap.analyticOn_uncurry_of_linear, ContinuousAlternatingMap.restrictScalarsLI_apply, Complex.restrictScalars_toSpanSingleton, Orientation.oangle_sign_smul_right, Orientation.normSq_kahler, AlgEquiv.linearEquivConj_mulLeftRight, LinearMap.IsRefl.flip_isRefl_iff, Basis.constr_apply_fintype, hasGradientAt_iff_hasFDerivAt, fwdDiff_smul, Asymptotics.IsTheta.isThetaTVS, LinearEquiv.symm_conj_apply, LieSubmodule.Quotient.isCentralScalar, AddMonoidAlgebra.liftMagma_apply_apply, TensorProduct.LieModule.lift_apply, Ideal.Filtration.pow_smul_le_pow_smul, IsLocalizedModule.fromLocalizedModule'_add, ContinuousAlternatingMap.cons_smul, LinearMap.isOrtho_flip, NumberField.mixedEmbedding.fundamentalCone.hasDerivAt_expMap_single, HasFPowerSeriesOnBall.tendsto_partialSum, inhomogeneousCochains.d_hom_apply, Algebra.Generators.toKaehler_tmul_D, ZMod.dft_dft, SmoothBumpFunction.contMDiff_smul, LieDerivation.IsKilling.rangeAdOrthogonal_carrier, Quaternion.imJ_snd_dualNumberEquiv, cfcₙHom_map_quasispectrum, Filter.Tendsto.zero_smul_isBoundedUnder_le, CoalgCat.MonoidalCategoryAux.counit_tensorObj, MultilinearMap.piFamilyₗ_apply, MultilinearMap.fromDirectSumEquiv_apply, fderiv_fun_mul', TrivSqZeroExt.liftEquivOfComm_apply, Finset.centerMass_segment', TensorProduct.isBaseChange, ContinuousAlternatingMap.toContinuousMultilinearMap_curryLeft, NormedSpace.expSeries_sum_eq_rat, isTorsionBy_iff_mem_ker_lsmul, Submodule.top_le_span_range_iff_forall_exists_fun, HasFPowerSeriesWithinAt.prod, OreLocalization.zero_mul, LinearPMap.map_smul, VectorFourier.contDiff_fourierIntegral, injOn_dualMap_subtype_span_range_range, End.finite_spectrum, ContMDiffMap.coe_smul, Set.neg_smul_set, Derivation.instIsCentralScalar, Submodule.map₂_le, WeierstrassCurve.Projective.negMap_eq, Ideal.mul_assoc, IsLocalizedModule.exists_integer_multiple, iteratedFDeriv_succ_apply_right, fourierSubalgebra_coe, NonUnitalStarSubalgebra.prod_inf_prod, FormalMultilinearSeries.leftInv_coeff_one, LinearMap.IsSymmetric.smul, Basis.smul_eq_map, instFlatAtPrime, MultilinearMap.smulRight_apply, PosSMulReflectLT.toPosSMulReflectLE, sameRay_iff_inv_norm_smul_eq, Matrix.toLin_conjTranspose, mdifferentiableOn_continuousLinearMapCoordChange, CommAlgCat.whiskerRight_hom, instIsPushoutPolynomial, AdicCompletion.map_ext''_iff, hasDerivAtFilter_intCast, MonoidAlgebra.instIsPushout', cfcHom_nnreal_eq_restrict, BoxIntegral.integral_smul, CategoryTheory.ShortComplex.RightHomologyMapData.smul_φH, RatFunc.ofFractionRing_sub, Submodule.exists_smith_normal_form_of_rank_eq, hasFPowerSeriesAt_pi_iff, MeasureTheory.eLpNorm'_const_smul, LinearEquiv.conj_conj_symm, IsBaseChange.iff_lift_unique, LieAlgebra.IsKilling.span_weight_isNonZero_eq_top, IntermediateField.LinearDisjoint.of_inf_eq_bot, Algebra.Extension.Cotangent.map_toInfinitesimal_bijective, QuadraticMap.canLift', ProbabilityTheory.integral_tilted_mul_eq_mgf, MeasureTheory.integral_simpleFunc_larger_space, Antitone.convex_gt, hasDerivWithinAt_univ, Asymptotics.isLittleOTVS_neg_right, finrank_eq_one_iff', ContDiffBump.apply, TrivSqZeroExt.isUnit_inv_iff, Seminorm.ball_zero_absorbs_ball_zero, Basis.smulTower'_apply, ModuleCat.smulShortComplex_f, Submodule.pointed_toConvexCone, TensorProduct.add_smul, Orientation.oangle_sign_add_smul_left, Submodule.span_range_update_sub_smul, HasFDerivAt.lim_real, PointwiseConvergenceCLM.precomp_apply, TensorProduct.AlgebraTensorModule.lTensor_comp, signedDist_vsub_self_rev, LieAlgebra.IsKilling.chainTopCoeff_add_chainBotCoeff, ContinuousLinearMap.toContinuousAddMonoidHom_inj, ContinuousLinearMap.analyticOnNhd_uncurry_of_multilinear, homTensorHomEquiv_toLinearMap, smul_upperBounds_subset_lowerBounds_smul, InnerProductSpace.isStarProjection_rankOne_self, coord_norm', ProperCone.relative_hyperplane_separation, OrderedFinpartition.compAlongOrderedFinpartitionL_apply, ContinuousAlternatingMap.piLIE_symm_apply_apply, affineSegment_eq_segment, endVecAlgEquivMatrixEnd_apply_apply, Submodule.coe_torsionBy, Quaternion.snd_imK_dualNumberEquiv_symm, contDiff_clm_apply_iff, strictConvex_Ico, HasFPowerSeriesAt.hasFDerivAt, Submodule.dualAnnihilator_gc, SpecialLinearGroup.centerEquivRootsOfUnity_apply, Affine.Simplex.centroid_vsub_faceOppositeCentroid_eq_smul_vsub, TensorAlgebra.toDirectSum_comp_ofDirectSum, ConvexCone.coe_positive, KaehlerDifferential.submodule_span_range_eq_ideal, analyticAt_iff_analytic_smul, spec_cfcₙAux, LinearEquiv.smul_apply, instInvertibleReprₛ, AlgebraicIndependent.liftAlgHom_comp_reprField, UniformConcaveOn.strictConcaveOn, Subalgebra.inclusion.faithfulSMul, sectionOfRetractionKerToTensor_algebraMap, HasDerivAt.ofReal_comp, HasFPowerSeriesWithinOnBall.compContinuousLinearMap, ContinuousOn.cfcₙ_nnreal', Bialgebra.TensorProduct.coalgebra_rid_eq_algebra_rid_apply, Algebra.TensorProduct.map_range, DivisibleHull.qsmul_of_nonneg, IntermediateField.finrank_top', LinearMap.BilinForm.nondegenerate_congr_iff, NNRealRMK.le_rieszMeasure_of_isCompact_tsupport_subset, RatFunc.ofFractionRing_neg, CochainComplex.HomComplex.Cochain.units_smul_v, LinearMap.aeval_self_charpoly, exteriorPower.alternatingMapLinearEquiv_comp, Matrix.trace_toLin'_eq, MvPowerSeries.le_order_smul, signedDist_linear_apply, HomologicalComplex.smul_f_apply, Orientation.rotation_eq_matrix_toLin, IsLocalizedModule.smul_inj, FormalMultilinearSeries.zero_radius, smul_unitBall, IsLocalization.linearMap_compatibleSMul, intervalIntegral.integral_const_of_cdf, Algebra.Extension.H1Cotangent.val_smul, LinearMap.BilinForm.flip_apply, Subalgebra.isScalarTower_mid, PolynomialModule.instIsScalarTower, AlternatingMap.curryLeft_smul, WithSeminorms.congr_equiv, IntermediateField.inclusion_self, ContinuousLinearMap.hasFPowerSeriesAt_bilinear, intervalIntegral.integral_const', TrivSqZeroExt.inv_neg, eval_ker, IntermediateField.lift_bot, VertexOperator.ext_iff, Complex.differentiable_on_off_countable_deriv_eq_smul_circleIntegral, hasStrictDerivAt_const, Set.smul_mem_center, LinearMap.isNilpotent_trace_of_isNilpotent, DivisibleHull.coeOrderAddMonoidHom_apply, PosSMulMono.toSMulPosMono, SymmetricAlgebra.equivMvPolynomial_ι_apply, LieModule.isRegular_iff_natTrailingDegree_charpoly_eq_rank, algebraMap_end_apply, AlgebraicGeometry.structureSheafInType.add_apply, Submodule.sup_dualAnnihilator_le_inf, hasStrictFDerivAt_list_prod_attach', FinitePresentation.linearEquivMapExtendScalars_apply, IsLocalizedModule.iso_symm_apply_aux, IsBaseChange.endHom_apply, LinearMap.isSkewAdjoint_iff_neg_self_adjoint, IntermediateField.normalClosure_of_normal, StrongDual.norm_toLpₗ_le, Orientation.oangle_sub_left_smul_rotation_pi_div_two, HasFPowerSeriesWithinAt.restrictScalars, tangentConeAt_real_subset_isRCLikeNormedField, LinearMap.polyCharpoly_eq_of_basis, Basis.parallelepiped_eq_map, LieAlgebra.rootSpace_comap_eq_genWeightSpace, Invertible.rTensorInv_injective, TensorProduct.AlgebraTensorModule.lTensor_mul, ContDiffMapSupportedIn.norm_apply_le_seminorm, Algebra.TensorProduct.map_comp_includeRight, integral_withDensity_eq_integral_toReal_smul, LinearEquiv.toLinearMap_smul, contDiffOn_clm_apply, ZLattice.covolume.tendsto_card_div_pow', MvPolynomial.ker_eval₂Hom_universalFactorizationMap, WithCStarModule.instContinuousSMul, MeasureTheory.Lp.instIsCentralScalar, ContinuousLinearMap.restrictScalars_smul, DivisibleHull.nsmul_mk, Matrix.scalar_comm_iff, SchwartzMap.hasDerivAt, End.mem_eigenspace_iff, Algebra.TensorProduct.one_mul, Seminorm.inf_apply, Ideal.constr_basisSpanSingleton, TrivSqZeroExt.sndHom_comp_map, PiTensorProduct.mapLMultilinear_apply_apply, Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_or_eq_zero, IsSl2Triple.HasPrimitiveVectorWith.lie_f_pow_toEnd_f, Units.neg_smul, LinearEquiv.isUnit_det, signedDist_self, LinearPMap.supSpanSingleton_apply_smul_self, Localization.instIsTorsionFreeAtPrimeAlgebraMapSubmonoidPrimeComplOfIsDomain, RootPairing.range_polarizationIn, CategoryTheory.Functor.map_units_smul, isBoundedBilinearMap_apply, InfiniteGalois.isOpen_iff_finite, FormalMultilinearSeries.analyticOnNhd, Ideal.pow_sup_eq_top', ContinuousLinearEquiv.map_smulₛₗ, Submodule.mem_span_range_iff_exists_fun, Unitization.nnreal_cfcₙ_eq_cfc_inr, CliffordAlgebra.contractLeft_ι, exteriorPower.presentation_relation, LinearMap.polyCharpolyAux_map_aeval, hasDerivAtFilter_iff_tendsto, convex_hyperplane, tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos, ContinuousLinearMap.IsInvertible.contDiffAt_map_inverse, CStarModule.inner_smul_left_real, HVertexOperator.ext_iff, Finset.weightedVSubOfPoint_apply_const, PosSMulMono.toPosSMulStrictMono, InnerProductSpace.HarmonicOnNhd.const_smul, hasStrictFDerivAt_ringInverse, KaehlerDifferential.map_compDer, Subspace.dualAnnihilator_iInf_eq, MeasureTheory.setIntegral_indicatorConstLp, signedDist_vsub_self, LinearMap.liftBaseChange_tmul, ContinuousLinearMap.mul_apply', starConvex_singleton, AffineSubspace.wOppSide_smul_vsub_vadd_right, Algebra.traceMatrix_of_basis, TrivSqZeroExt.map_comp_map, MeasureTheory.SignedMeasure.rnDeriv_smul, TensorProduct.quotTensorEquivQuotSMul_symm_mk, CStarMatrix.reindexₐ_symm, ContDiff.smul, AdicCompletion.of_ofLinearEquiv_symm, AdicCompletion.kerProj_of, LinearMap.instStarModuleId, IsPicardLindelof.exists_forall_mem_closedBall_eq_hasDerivWithinAt_continuousOn, MeasureTheory.hausdorffMeasure_smul_right_image, NNReal.convex_iff, LieDerivation.commutator_coe_linear_map, signedDist_smul, MonoidAlgebra.liftMagma_symm_apply, IntermediateField.algHomAdjoinIntegralEquiv_symm_apply_gen, LieModule.Cohomology.smul_apply_apply, MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_sub, HasStrictFDerivAt.hasStrictDerivAt_norm_smul_pos, AddMonoidAlgebra.liftMagma_symm_apply, LinearIsometryEquiv.symm_conjStarAlgEquiv_apply_apply, InnerProductSpace.rankOne_comp_rankOne, smulAddHom_apply, MeasureTheory.convolution_lsmul_swap, Orientation.oangle_add_right_smul_rotation_pi_div_two, Matrix.spectrum_toLin, TensorAlgebra.ι_leftInverse, IntermediateField.relfinrank_bot_left, HasFPowerSeriesWithinOnBall.fderivWithin_of_mem, Algebra.IsPushout.tensorProduct_tensorProduct, Basis.eval_injective, PolyEquivTensor.toFunBilinear_apply_apply, Unitization.starLift_apply, LinearMap.toMatrix'_comp, Matrix.toLinearEquiv'_apply, LinearMap.isUnit_toMatrix'_iff, Orientation.inner_smul_rotation_pi_div_two_left, CommAlgCat.associator_inv_hom, ModuleCat.ihom_coev_app, Ideal.span_singleton_mul_left_mono, Matrix.toBilin_toMatrix, hasDerivAtFilter_finCons', MeasureTheory.Supermartingale.smul_nonpos, MeasureTheory.SignedMeasure.haveLebesgueDecomposition_smul, AEval.mem_mapSubmodule_symm_apply, PolyEquivTensor.toFunLinear_tmul_apply, ContinuousMap.toLp_injective, midpoint_vsub_right, ContinuousLinearMap.comp_smulₛₗ, setIntegral_withDensity_eq_setIntegral_smul, IsLocalizedModule.mk'_smul_mk', Matrix.nondegenerate_toLinearMap₂_iff, CoalgCat.MonoidalCategoryAux.counit_tensorObj_tensorObj_left, StrongDual.toLpₗ_of_not_memLp, isConformalMap_complex_linear_conj, MeromorphicAt.meromorphicTrailingCoeffAt_smul, fderiv_smul_const, LinearMap.BilinForm.comp_congr, hasDerivWithinAt_finCons', AffineMap.coe_lineMap, Matrix.toBilin'_comp, mellin_comp_rpow, Matrix.isHermitian_iff_isSymmetric, TrivSqZeroExt.instL1IsBoundedSMul, LieModule.Cohomology.instLinearMapClassSubtypeLinearMapIdMemSubmoduleTwoCochain, Subspace.dualRestrict_surjective, cfcₙHomSuperset_id, LieModule.toEnd_pow_lie, MeasureTheory.Submartingale.sum_smul_sub', IsBaseChange.endHom_one, Asymptotics.Filter.Tendsto.isBigOTVS_one, LinearMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal', AdicCompletion.evalOneₐ_of, antivaryOn_iff_smul_rearrangement, convex_RCLike_iff_convex_real, PointedCone.mem_span_set, End.HasUnifEigenvector.apply_eq_smul, Algebra.TensorProduct.tensorTensorTensorComm_toLinearEquiv, intervalIntegral.smul_integral_comp_mul_add, IntermediateField.le_extendScalars_iff, TannakaDuality.FiniteGroup.sumSMulInv_single_id, AEval.X_smul_of, Submodule.map₂_span_singleton_eq_map, InnerProductSpace.rankOne_comp, CochainComplex.HomComplex.Cochain.rightShift_units_smul, LinearMap.lift_lsmul_mul_eq_lsmul_lift_lsmul, StrongDual.mem_polarSubmodule, End.apply_smulCommClass, Asymptotics.isLittleOTVS_fun_comm, FormalMultilinearSeries.apply_eq_prod_smul_coeff, Unitization.starMap_comp, Submodule.map₂_flip, ClassGroup.equivPic_symm_apply, Finset.sum_range_by_parts, Asymptotics.IsTheta.const_smul_left, CliffordAlgebra.invOf_ι, DifferentiableWithinAt.fun_smul, Orientation.kahler_rightAngleRotation_left, Asymptotics.isLittleOTVS_comm, intervalIntegral.hasFDerivAt_integral_of_dominated_of_fderiv_le, Rep.leftRegularHom_hom_single, Algebra.TensorProduct.ext_iff, ContinuousLinearMap.hasBasis_nhds_zero_of_basis, hasFPowerSeriesWithinAt_pi_iff, Ideal.IsTwoSided.mul_one, IsLocalizedModule.exists_of_eq, MeasureTheory.eLpNorm_const_smul, LinearMap.separatingLeft_congr_iff, Basis.end_repr_symm_apply, TensorAlgebra.GradedAlgebra.ι_apply, LieSubmodule.coe_toEnd, hasDerivWithinAt_comp_mul_left_smul_iff, Ideal.restrictScalars_mul, IsLocalRing.instIsScalarTowerResidueFieldCotangentSpace, hasDerivAt_iff_isLittleO_nhds_zero, evalEquiv_apply, inner_smul_right, Filter.Tendsto.atTop_smul_const_tendsto_asymptoticNhds, QuadraticMap.toFun_smul, SchwartzMap.tsupport_smulLeftCLM_subset, Orientation.kahler_comp_linearIsometryEquiv, DistribMulActionHom.coe_toLinearMap, Real.fourierIntegral_fderiv, Ideal.sup_mul_eq_of_coprime_right, CliffordAlgebra.contractRight_contractRight, InnerProductSpace.toMatrix_rankOne, LinearMap.equivariantProjection_apply, cfcₙHom_real_eq_restrict, MvPolynomial.rTensor_symm_apply_single, FinitePresentation.exists_lift_equiv_of_isLocalizedModule, LinearMap.polar_AbsConvex, ContDiffMapSupportedIn.structureMapCLM_apply_withOrder, LieModule.toEnd_apply_apply, vadd_openSegment, MeasureTheory.ComplexMeasure.im_apply, LinearMap.IsPerfPair.bijective_right, PiTensorProduct.liftIsometry_comp_mapL, iteratedFDeriv_smul_const_apply, FractionalIdeal.den_mul_self_eq_num, MeasureTheory.Measure.addHaar_smul_of_nonneg, add_smul, Ideal.pow_sup_pow_eq_top', CochainComplex.HomComplex.Cochain.rightUnshift_smul, LinearMap.mul_toMatrix₂_mul, Submodule.map_smul', RootPairing.pairingIn_one_four_iff, MeasureTheory.Lp.instIsBoundedSMul, RootPairing.Equiv.instSMulCommClassAut, InnerProductSpace.isSymmetric_rankOne_self, Orientation.oangle_smul_right_of_neg, ContinuousLinearMap.toSpanSingleton_comp_toSpanSingleton, Seminorm.ball_smul_ball, TensorAlgebra.hom_ext_iff, ContinuousMap.coe_toLp, WeierstrassCurve.Projective.addMap_of_Z_eq_zero_right, Real.fourierIntegral_convergent_iff, skewAdjointPart_apply_coe, instIsTorsionFreeLocalizationAlgebraMapSubmonoidPrimeCompl, continuousMultilinearCurryLeftEquiv_apply, LinearMap.BilinForm.tensorDistribEquiv_tmul, instFiniteAtPrimeLocalizationAlgebraMapSubmonoidPrimeCompl, LinearMap.trace_tensorProduct', linearIndependent_add_smul_iff, MonoidAlgebra.GroupSMul.linearMap_apply, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, LinearMap.eq_adjoint_iff_basis, MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff, precomp_compactConvergenceCLM_apply, ConvexBody.nonempty, MeasureTheory.integral_rnDeriv_smul, LieModule.commute_toEnd_of_mem_center_right, LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff, KaehlerDifferential.D_tensorProductTo, Asymptotics.IsBigOTVS.rfl, fourierCoeff_eq_intervalIntegral, IntermediateField.inclusion_injective, Ideal.mul_le_left, Differential.instDifferentialAlgebraSubtypeMemIntermediateField, Seminorm.bddBelow_range_add, NonUnitalSubalgebra.instSMulCommClass', CStarAlgebra.conjugate_le_norm_smul', ContinuousAlternatingMap.restrictScalarsCLM_apply, Asymptotics.IsLittleOTVS.bot, IsSemisimpleModule.exists_end_algEquiv, SchwartzMap.tsupport_fderivCLM_subset, ContinuousAlgEquiv.eq_continuousLinearEquivConjContinuousAlgEquiv, ContinuousLinearMap.isBigOTVS_comp, ContDiffAt.smul_const, IntermediateField.LinearDisjoint.bot_right, TensorAlgebra.range_lift, IsLocalizedModule.map_bijective_iff_localizedModuleMap_bijective, supportDim_le_supportDim_quotSMulTop_succ, AnalyticAt.exists_eq_sum_add_pow_mul, QuotSMulTop.equivQuotTensor_naturality_mk, inner_sum_smul_sum_smul_of_sum_eq_zero, Ideal.ResidueField.algHom_ext_iff, ContinuousLinearMap.restrictScalarsIsometry_toLinearMap, SymmetricAlgebra.algebraMap_eq_one_iff, WithLp.smul_fst, iteratedDerivWithin_comp_const_smul, ProbabilityTheory.hasDerivAt_iteratedDeriv_mgf, CompactlySupportedContinuousMap.toNNRealLinear_apply, Matrix.toLpLinAlgEquiv_apply_apply_ofLp, ContinuousMultilinearMap.alternatization_apply_toContinuousMultilinearMap, Complex.taylorSeries_eq_on_eball, MeasureTheory.Measure.map_addHaar_smul, LinearMap.toMatrix'_id, InnerProductSpace.gramSchmidt_def'', Real.hasDerivAt_fourierChar, Algebra.TensorProduct.algEquivIncludeRange_toAlgHom, ContinuousLinearMap.intervalIntegral_apply, LinearMap.polyCharpolyAux_coeff_eval, AnalyticOnNhd.fderiv, LinearMap.range_smul', WStarAlgebra.exists_predual, Function.noZeroSMulDivisors, span_fourierLp_closure_eq_top, LieModule.toEnd_baseChange, selfAdjointPartL_apply_coe, PMF.integral_eq_sum, MeasureTheory.average_add_measure, LocalizedModule.mul_by_divBy, hasDerivAt_intCast, Orientation.kahler_rotation_left, Seminorm.absorbent_ball_zero, IsRegular.smul_eq_zero_iff_right, PiTensorProduct.liftIsometry_symm_apply, Derivation.tensorProductTo_mul, nnnorm_cfcₙHom, Matrix.toLinearMapₛₗ₂_symm, SchwartzMap.smulLeftCLM_neg, intervalIntegral.inv_smul_integral_comp_div, ContinuousLinearMap.opNNNorm_mul, ContinuousMultilinearMap.iteratedFDerivComponent_apply, SemimoduleCat.Iso.conj_eq_conj, RootPairing.pairingIn_neg_one_neg_four_iff, IntermediateField.normal_iff_forall_map_le, LinearMap.BilinMap.tensorDistrib_tmul, Polynomial.smeval_at_zero, Projectivization.cross_mk, RestrictScalars.addEquiv_symm_map_smul_smul, ContinuousLinearMap.toContinuousAddMonoidHom_neg, AdicCompletion.mk_smul_top_ofAlgEquiv_symm, LinearMap.BilinForm.mul_toMatrix_mul, LinearMap.trace_transpose, ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin, ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear, PowerSeries.constantCoeff_smul, LieModule.smul_lie, IsBaseChange.map_id_lsmul_eq_lsmul_algebraMap, Submodule.span_range_update_add_smul, ValuationSubring.ofPrime_scalar_tower, IsLocalization.smul_mem_iff, Polynomial.mkDerivation_X, Continuous.cfcₙ_nnreal_of_mem_nhdsSet, HasFPowerSeriesAt.const_smul, ContDiffMapSupportedIn.norm_iteratedFDeriv_apply_le_seminorm_withOrder, iteratedFDeriv_succ_eq_comp_right, ContinuousMultilinearMap.changeOriginSeries_support, WeakFEPair.functional_equation₀, separableClosure.isAlgebraic, Finsupp.linearCombination_comp_addSingleEquiv, IsAlgClosure.ofAlgebraic, Real.tendsto_integral_cexp_sq_smul, QuadraticForm.comp_tensorRId_eq, Asymptotics.isBigOTVS_fun_comm, TrivSqZeroExt.algebraMap_eq_inl, ContDiffMapSupportedIn.seminorm_apply, IsBaseChange.endHom_comp_apply, LinearMap.BilinForm.toMatrix_mul_basis_toMatrix, AffineSubspace.sSameSide_smul_vsub_vadd_right, LinearAlgebra.FreeProduct.rel_id, Matrix.toLinAlgEquiv_apply, WithSeminorms.tendsto_nhds_atTop, Submodule.mem_smul_top_iff, aemeasurable_withDensity_iff, ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, SubmodulesRingBasis.leftMul, Fintype.not_linearIndependent_iff, LinearMap.nondegenerate_toMatrix₂'_iff, Monovary.sum_smul_comp_perm_lt_sum_smul_iff, MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff, PointedCone.canLift, CochainComplex.HomComplex.Cochain.smul_v, ODE.hasDerivWithinAt_picard_Icc, support_quotSMulTop, signedDist_smul_of_neg, LinearMap.BilinForm.toMatrix'_comp, Submodule.map_map₂, MeasureTheory.lpNorm_const_smul, VectorField.lieBracket_smul_left, DualNumber.range_inlAlgHom_sup_adjoin_eps, Submodule.smul_one_eq_span, ContinuousLinearMap.rotation_apply, Real.iteratedFDeriv_fourier, Seminorm.smul_closedBall_subset, Submodule.span_singleton_smul_eq, circleAverage_sub_sub_inv_smul_of_differentiable_on, Algebra.Generators.CotangentSpace.exact, Algebra.Generators.H1Cotangent.δAux_X, DoubleCentralizer.algebraMap_fst, LinearMap.ker_dualMap_eq_dualCoannihilator_range, hasDerivWithinAt_diff_singleton, fderiv_continuousLinearEquiv_comp, CharacterModule.dual_rTensor_conj_homEquiv, SchwartzMap.convolution_continuous_left, ContinuousLinearMap.adjoint_toSpanSingleton, ContinuousLinearMap.toBilinForm_injective, quasispectrumRestricts_iff_spectrumRestricts_inr, Submodule.mem_annihilator_span, IsArtinianRing.isNilpotent_jacobson_bot, mulLinearMap_apply_apply, DifferentiableAt.fun_smul, circleIntegral.integral_smul_const, Submodule.mulLeftMap_eq_mulMap_comp, inner_smul_real_left, CliffordAlgebra.changeForm_contractLeft, ContDiffOn.fderivWithin, Submodule.map_pointwise_smul, exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card, ContinuousLinearMap.continuousOn_uncurry_of_multilinear, IsLocalization.map_eq_toLinearMap_mapₐ, IntermediateField.rank_eq_one_iff, BoxIntegral.norm_volume_sub_integral_face_upper_sub_lower_smul_le, WittVector.StandardOneDimIsocrystal.frobenius_apply, HasFPowerSeriesWithinAt.isBigO_sub_partialSum_pow, triple_product_eq_det, LinearEquiv.dualMap_apply, Matrix.toLinearMap₂_basisFun, ContDiffMapSupportedIn.seminorm_eq_bot_of_gt, TensorAlgebra.algebraMap_inj, QuadraticMap.coe_associatedHom, LieAlgebra.Extension.d₁₂_oneCochainOfTwoSplitting, Real.hasDerivAt_Gamma_one_half, IsClosed.balancedCore, LinearMap.mem_selfAdjointSubmodule, ProbabilityTheory.isPosSemidef_covarianceBilinDual, toSpanSingleton_derivWithin, LinearMap.finrank_range_dualMap_eq_finrank_range, fderiv_norm_smul, LocalizedModule.map_exact, LinearMap.mulLeft_tmul, isMIntegralCurveAt_comp_mul_ne_zero, ContinuousLinearMap.isEmbedding_restrictScalars, lowerBounds_smul_of_neg, MeasureTheory.SimpleFunc.integral_eq_sum_of_subset, HasFDerivAt.hasFDerivAt_norm_smul, KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination, IntermediateField.adjoin_rank_le_of_isAlgebraic_right, bilinearIteratedFDerivWithinTwo_eq_iteratedFDeriv, instFiniteFractionRingLocalizationAlgebraMapSubmonoidNonZeroDivisors, HasFiniteFPowerSeriesOnBall.eq_partialSum', bijective_dual_eval, VectorField.mlieBracket_const_smul_left, EuclideanGeometry.dist_sq_smul_orthogonal_vadd_smul_orthogonal_vadd, IntermediateField.LinearDisjoint.adjoin_rank_eq_rank_left_of_isAlgebraic_left, CategoryTheory.ShortComplex.homologyMap'_smul, StarModule.complexToReal, End.eigenspace_div, IsScalarTower.to_smulCommClass, LieHom.toNonUnitalAlgHom_apply, Complex.wedgeIntegral_add_wedgeIntegral_eq, Algebra.Extension.Hom.subToKer_apply_coe, IsMIntegralCurve.comp_mul, IsBoundedBilinearMap.toContinuousLinearMap_apply, Matrix.toLinearMapRight'_apply, annihilator_eq_bot, Bornology.isVonNBounded_sub, ModuleFilterBasis.smul', LinearMap.BilinForm.toMatrix_mul, MatrixModCat.isScalarTower_toModuleCat, ModuleCat.restrictScalars.smul_def, HasFPowerSeriesOnBall.pi, spectralNorm.eq_of_normalClosure, ContinuousLinearEquiv.continuousAlternatingMapCongrLeft_apply, Submodule.toSubMulAction_strictMono, HasFPowerSeriesWithinOnBall.fderivWithin_eq, jacobson_density, nhds_basis_closed_balanced, FaithfullyFlat.tensorProduct_mk_injective, Polynomial.span_minpoly_eq_annihilator, ContinuousMultilinearMap.piₗᵢ_symm_apply, ray_pos_smul, Unitization.fstHom_apply, Matrix.instIsScalarTowerForall, convex_parallelepiped, ContDiff.iteratedFDeriv_right', LinearMap.nonUnitalAlgHom_comp_convMul_distrib, End.genEigenrange_nat, AnalyticAt.analyticOrderAt_ne_top, HasFDerivWithinAt.pow', SchwartzMap.pairing_continuous_left, LieAlgebra.SpecialLinear.val_single, CochainComplex.HomComplex.Cochain.comp_units_smul, Matrix.ofLp_toEuclideanLin_apply, ContinuousLinearMap.smulRight_comp_smulRight, balancedCoreAux_subset, spectralNorm.eq_of_normalClosure', signedDist_vadd_right_swap, Submodule.mem_span_pair, Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable, Matrix.toLinearMap₂_apply, Filter.Tendsto.isVonNBounded_range, AdicCompletion.of_inj, Real.hasFDerivAt_fourierIntegral, intervalIntegral.integral_hasDerivAt_of_tendsto_ae_left, SchwartzMap.integralCLM_apply, NonUnitalStarSubalgebra.inclusion_right, LieModule.isNilpotent_iff_forall, Polynomial.mkDerivation_one_eq_derivative, InnerProductSpace.rankOne_one_right_eq_toSpanSingleton, PolynormableSpace.withSeminorms', Ideal.quotientMulEquivQuotientProd_fst, ConvexBody.isClosed, hasDerivAt_one, SameRay.exists_nonneg_left, CliffordAlgebra.forall_mul_self_eq_iff, LinearMap.IsSymmetric.apply_eigenvectorBasis, IsOpen.strictConvex_iff, RootPairing.coroot_root_eq_pairing, FormalMultilinearSeries.radius_shift, Matrix.Module.single_smul, Representation.smul_tprod_one_asModule, LieModule.isNilpotent_toEnd_sub_algebraMap, rTensorHomEquivHomRTensor_toLinearMap, NonUnitalStarSubalgebraClass.subtype_injective, IsBaseChange.ofEquiv, AdicCompletion.ofTensorProduct_tmul, mulLeftLinearMap_eq_mulLeft, isClosedEmbedding_cfcₙHom_of_cfcHom, LinearEquiv.toSpanNonzeroSingleton_symm_apply_smul, HasFTaylorSeriesUpToOn.hasStrictFDerivAt, AnalyticAt.const_smul, smul_nonneg_iff, KaehlerDifferential.kerTotal_mkQ_single_smul, MultilinearMap.uncurryRight_apply, LinearMap.nondegenerate_toLinearMap₂'_iff_det_ne_zero, innerSLFlip_apply, MvPolynomial.rTensor_apply_monomial_tmul, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_injective, FormalMultilinearSeries.div_le_radius_compContinuousLinearMap, NonUnitalStarSubalgebra.coe_smul, HahnSeries.SummableFamily.binomialFamily_apply, PiTensorProduct.piTensorHomMapFun₂_add, ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis, Matrix.mulVecBilin_apply, nat_smul_eq_nsmul, ZMod.map_smul, Algebra.Generators.repr_CotangentSpaceMap, LinearEquiv.congrQuadraticMap_symm_apply, Basis.dualBasis_repr, ContinuousLinearMap.toContinuousAddMonoidHom_restrictScalars, hasLineDerivAt_iff_tendsto_slope_zero, CategoryTheory.Abelian.Ext.bilinearCompOfLinear_apply_apply, AlternatingMap.uncurryFin_smul, Ideal.mem_iInf_smul_pow_eq_bot_iff, lieEquivMatrix'_symm_apply, InnerProductSpace.toContinuousLinearMap_toDualMap, HasFDerivAt.list_prod', separableClosure.isSepClosure, MeasureTheory.SignedMeasure.singularPart_smul_nnreal, ProbabilityTheory.covarianceBilin_real, LieSubalgebra.instIsNoetherianSubtypeMem, InfiniteGalois.toAlgEquivAux_eq_liftNormal, Matrix.proj_diagonal, ProbabilityTheory.strong_law_Lp, separatingDual_iff_injective, fderivWithin_const_smul_field, Pi.openSegment_subset, TensorProduct.dualDistribInvOfBasis_apply, Ideal.Filtration.pow_smul_le, MeasureTheory.integral_const, isNilpotent_tensor_residueField_iff, LinearEquiv.isSMulRegular_congr', LinearMap.trace_eq_sum_trace_restrict, Basis.smulTower_apply, LinearMap.map_smul₂, IsSelfAdjoint.commute_cfcₙHom, IsLocalizedModule.isScalarTower_module, LieAlgebra.SpecialLinear.singleSubSingle_sub_singleSubSingle, AffineMap.homothety_apply, BialgHomClass.map_comp_comulAlgHom, Polynomial.sup_ker_aeval_le_ker_aeval_mul, CStarAlgebra.star_left_conjugate_le_norm_smul, LinearMap.intrinsicStarModule, NNRealRMK.le_rieszMeasure_of_tsupport_subset, IntermediateField.normalClosure_def'', DifferentiableOn.smul, contDiff_nat_iff_continuous_differentiable, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, hasFDerivAt_pow', dualTensorHom_prodMap_zero, ContinuousAlternatingMap.hasBasis_nhds_zero_of_basis, sameRay_smul_right_iff_of_ne, absorbent_ball, MeasureTheory.Lp.simpleFunc.coe_smul, Unitization.starMap_apply, Seminorm.zero_or_exists_apply_eq_finset_sup, Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, SchwartzMap.lineDerivOp_fourierInv_eq, LinearIsometryEquiv.symm_conjStarAlgEquiv, Submodule.topologicalClosure_eq_top_iff, hasDerivAt_abs, mdifferentiableOn_coordChangeL, LinearMap.IsContPerfPair.bijective_right, HasFiniteFPowerSeriesOnBall.add, ContinuousLinearMap.opNorm_mulLeftRight_apply_apply_le, IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots, iteratedDeriv_comp_const_sub, LinearMap.isUnit_toMatrix_iff, LinearMap.det_smul, Subfield.relrank_eq_rank_of_le, ContinuousLinearEquiv.continuousAlternatingMapCongrRight_apply, Submodule.coe_smul, ContinuousLinearMap.opNorm_lsmul_le, LinearMapClass.map_smul, LocalizedModule.map_id, NonUnitalAlgHom.comp_mul', Seminorm.ball_comp, intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae, TensorProduct.AlgebraTensorModule.rTensor_mul, Orientation.inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two, MultilinearMap.curryMidLinearEquiv_apply, ProbabilityTheory.covarianceBilin_of_not_memLp, LinearMap.BilinForm.toMatrix_compLeft, WithLp.instProdNormSMulClass, LinearMap.trace_eq_matrix_trace, RootPairing.polarizationEquiv_symm_apply_coroot, BilinForm.toMatrix_apply, WeakDual.CharacterSpace.norm_le_norm_one, DualNumber.commute_eps_left, IntermediateField.isSeparable_tower_bot, KaehlerDifferential.isLocalizedModule, Matrix.linfty_opNorm_toMatrix, IntermediateField.minpoly_eq, hasDerivAt_natCast, cfcL_integral, IsMprojection.Mnorm, smul_vsub_rev_mem_vectorSpan_pair, ContinuousMap.coeFn_toLp, Seminorm.rescale_to_shell, AffineSubspace.direction_affineSpan_pair_le_iff_exists_smul, Matrix.toBilin'_apply', gaugeRescale_def, Basis.linearCombination_coord, ConvexBody.isCompact, innerSL_apply_norm, Real.hasDerivAt_Gamma_one, segment_translate_image, InnerProductSpace.toDual_apply_eq_toDualMap_apply, Real.fourier_bilin_convolution_eq_integral, Derivation.map_aeval, Matrix.toLinearMap₂'_single, closedBall_normSeminorm, Ideal.smul_restrictScalars, ExteriorAlgebra.liftAlternating_comp_ιMulti, Complex.coe_smul, Matrix.smul_eq_mulVec, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, ContinuousLinearMap.coe_restrictScalarsL, LinearMap.separatingRight_toLinearMap₂'_of_det_ne_zero', Algebra.TensorProduct.mul_assoc, ModuleFilterBasis.smul_left, LinearMap.adjoint_inner_left, MemHolder.nnHolderNorm_smul, IntermediateField.isSplittingField_iff, IsTorsionBySet.isScalarTower, MeasureTheory.Integrable.integral_smul, Basis.toDual_apply, Matrix.star_dotProduct_gram_mulVec, OpenPartialHomeomorph.univUnitBall_apply, LocalizedModule.instIsScalarTower, LinearMap.BilinForm.mul_toMatrix, ContinuousLinearMap.norm_compL_le, ContinuousMultilinearMap.hasStrictFDerivAt_compContinuousLinearMap, SeminormFamily.withSeminorms_iff_nhds_eq_iInf, Basis.tensorAlgebra_repr_apply, ContMDiffVectorBundle.contMDiffOn_coordChangeL, EulerSine.antideriv_cos_comp_const_mul, intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae, isTorsionBySet_quotient_iff, Algebra.Generators.toKaehler_cotangentSpaceBasis, TrivSqZeroExt.isUnit_inr_iff, ContDiffMapSupportedIn.toBoundedContinuousFunctionCLM_eq_of_scalars, sameRay_neg_smul_right_iff_of_ne, meromorphicOrderAt_smul_of_ne_zero, Submodule.mem_smul_span, Matrix.normSMulClass, Bornology.isVonNBounded_add, Subspace.dualLift_rightInverse, DualNumber.algHom_ext_iff, IntermediateField.topEquiv_symm_apply_coe, HasFPowerSeriesAt.sub, MvPolynomial.support_smul_eq, QuadraticMap.associated_linMulLin, MeasureTheory.JordanDecomposition.toSignedMeasure_smul, MeasureTheory.Lp.instSMulCommClass, BoxIntegral.integralSum_smul, FiniteGaloisIntermediateField.adjoin_val, LieAlgebra.rank_le_finrank_engel, Finset.centerMass_segment, dualTensorHomEquivOfBasis_toLinearMap, Finsupp.sum_option_index_smul, OrthogonalFamily.range_linearIsometry, QuadraticForm.associated_isSymm, IsBaseChange.dual, apply_evalEquiv_symm_apply, TensorProduct.finsuppScalarLeft_apply_tmul, Real.tendsto_integral_exp_smul_cocompact, Submodule.coe_dualAnnihilator_span, mellin_convergent_iff_norm, RestrictScalars.isScalarTower, MeasureTheory.setIntegral_tilted', VectorBundleCore.IsContMDiff.contMDiffOn_coordChange, TensorAlgebra.tprod_apply, Orientation.kahler_mul, RootPairing.EmbeddedG2.threeShortAddTwoLongRoot_eq, LinearMap.trace_conj', ProbabilityTheory.covarianceBilinDual_comm, Mathlib.Tactic.LinearCombination.smul_const_lt, Matrix.SpecialLinearGroup.toLin'_to_linearMap, ContinuousOn.integral_sub_linear_isLittleO_ae, SchwartzMap.integral_smul_lineDerivOp_right_eq_neg_left, MvPolynomial.tensorEquivSum_X_tmul_X, MeasureTheory.L1.SimpleFunc.integral_smul, iteratedDerivWithin_smul, NonUnitalStarSubalgebra.instSMulCommClass', ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv, ContinuousLinearMap.opNNNorm_mul_flip_apply, LinearMap.mul_toMatrix₂'_mul, smul_le_smul_of_nonpos_left, IsNilpotent.exp_eq_sum, Rep.coinvariantsTensor_hom_ext_iff, LinearMap.toMatrix₂'_mul, Nonneg.instNonUnitalContinuousFunctionalCalculus, HasFPowerSeriesOnBall.hasFDerivAt, MeasureTheory.charFun_eq_charFunDual_toDualMap, IntermediateField.rank_top', IntermediateField.isAlgebraic_adjoin_simple, SchwartzMap.instLocallyConvexSpace, egauge_eq_zero_iff, AdicCompletion.transitionMap_map_pow, Orientation.oangle_smul_smul_self_of_nonneg, Algebra.Extension.Cotangent.val_smul, MeromorphicNFAt.smul_analytic, ContinuousLinearEquiv.conjContinuousAlgEquiv_ext_iff, zero_mem_absConvexHull, IsTensorProduct.lift_eq, LinearMap.BilinForm.flip_flip, ContinuousMultilinearMap.curryMid_apply, IsSl2Triple.HasPrimitiveVectorWith.lie_e_pow_succ_toEnd_f, inv_smul_lt_iff_of_neg, IntermediateField.AdjoinSimple.normal_algebraicClosure, Algebra.toMatrix_lsmul, convex_Ici, toMatrix_innerSL_apply, Finset.weightedVSub_eq_linear_combination, CoalgCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_right, Basis.map_orientation_eq_det_inv_smul, ClosedSubmodule.sInf_orthogonal, Algebra.SubmersivePresentation.basisKaehler_apply, LinearMap.flip_nondegenerate, intervalIntegral.fderivWithin_integral_of_tendsto_ae, MvPolynomial.coeff_rTensorAlgHom_monomial_tmul, concaveOn_iff_pairwise_pos, ContinuousMap.toLp_comp_toContinuousMap, Function.HasTemperateGrowth.smul, KaehlerDifferential.polynomial_D_apply, PiTensorProduct.toDualContinuousMultilinearMap_le_projectiveSeminorm, Ideal.subtype_isoBaseOfIsPrincipal_eq_mul, StarAlgHom.coe_toNonUnitalStarAlgHom, IntermediateField.normal_iff_forall_map_eq', HasStrictFDerivAt.isBigOTVS_sub, Matrix.separatingLeft_toLinearMap₂'_iff, TensorProduct.AlgebraTensorModule.map_smul_right, rescale_to_shell_zpow, ConvexCone.mem_zero, PolynomialModule.eval_apply, LieAlgebra.isScalarTower, triple_product_permutation, LinearMap.separatingLeft_toLinearMap₂'_of_det_ne_zero', balanced_closedBall_zero, integrableOn_peak_smul_of_integrableOn_of_tendsto, TrivSqZeroExt.isUnit_or_isNilpotent_of_isMaximal_isNilpotent, IntermediateField.intermediateFieldMap_apply_coe, IntermediateField.topEquiv_apply, LinearMap.intrinsicStar_smulRight, Finset.neg_smul_finset, IsSl2Triple.HasPrimitiveVectorWith.lie_h_pow_toEnd_f, DivisibleHull.mk_add_mk, ModelWithCorners.convex_range', MDifferentiableAt.coordChangeL, MeasureTheory.Lp.smul_comm, Submodule.smul_mem_iff, End.baseChangeHom_apply_apply, isBoundedBilinearMap_smul, LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpace, Matrix.isRepresentation.toEnd_surjective, BilinForm.toMatrix_mul, IntermediateField.exists_algHom_adjoin_of_splits', EulerSine.antideriv_sin_comp_const_mul, LinearMap.map_smul, Subspace.dualCopairing_nondegenerate, Finset.sum_smul_sum_eq_sum_perm, strictConvex_singleton, Algebra.Generators.CotangentSpace.compEquiv_symm_inr, exists_lt_rieszContentAux_add_pos, ContinuousLinearMap.norm_restrictScalars, Affine.Simplex.sum_inv_height_sq_smul_vsub_eq_zero, LinearMap.toMatrix_comp, coe_normSeminorm, MvPolynomial.tensorEquivSum_C_tmul_one, Submodule.restrictScalars.isScalarTower, ContinuousLinearEquiv.arrowCongrSL_apply, reflection_mul_reflection_zpow_apply_self, Algebra.normalizedTrace_def, signedDist_vadd_left, associatedPrimes.mem_associatedPrimes_atPrime_of_mem_associatedPrimes, ConvexCone.coe_add, HasFPowerSeriesWithinOnBall.tendstoUniformlyOn, parallelepiped_eq_sum_segment, CochainComplex.HomComplex.Cochain.leftUnshift_units_smul, convex_Iio, Matrix.toLinAlgEquiv_self, RootPairing.coreflection_apply, convex_halfSpace_gt, ContinuousLinearMap.hasFDerivAt_uncurry_of_multilinear, Basis.coe_toDual_self, ContDiffBump.convolution_eq_right, Matrix.kroneckerMapBilinear_mul_mul, ContinuousMultilinearMap.map_update_smul, TemperedDistribution.instFourierInvSMul, sameRay_neg_smul_right_iff, MeasureTheory.AEEqFun.Integrable.smul, Submodule.mul_smul, ContinuousLinearMap.toContinuousAddMonoidHom_id, ExistsContDiffBumpBase.w_def, FiniteField.minpoly_frobeniusAlgHom, MeasureTheory.Measure.setIntegral_comp_smul, Matrix.separatingLeft_toBilin'_iff, ContDiffOn.smul, mem_vectorSpan_pair, WeakFEPair.hf_modif_FE, Submodule.dualAnnihilator_top, Matrix.trace_toLin_eq, AEval'_def, FormalMultilinearSeries.analyticAt_changeOrigin, FiniteDimensional.exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card, mem_convexHull_iff_exists_fintype, ContinuousLinearMap.opNorm_mul_apply_le, deriv_smul_const, Coalgebra.TensorProduct.assoc_tmul, intervalIntegral.integral_hasFDerivWithinAt, IntermediateField.AdjoinSimple.isIntegral_gen, HasFPowerSeriesAt.tendsto_partialSum, ContinuousLinearMap.smul_def, LinearMap.toMatrix₂_mul_basis_toMatrix, ContinuousLinearMap.orthogonal_range, Antitone.quasilinearOn, HasFiniteFPowerSeriesAt.add, Matrix.conjTranspose_intCast_smul, HasFPowerSeriesAt.hasDerivAt, TensorProduct.AlgebraTensorModule.lift_apply, ContinuousLinearMap.toContinuousAddMonoidHom_sub, AffineMap.coe_homothetyAffine, hasDerivWithinAt_congr_set', End.IsSemisimple_smul, Finsupp.mem_span_range_iff_exists_finsupp, LieAlgebra.IsKilling.rootSystem_pairing_apply, Basis.toMatrix_eq_toMatrix_constr, Finite.toModuleEnd_moduleEnd_surjective, LieAlgebra.IsKilling.chainTopCoeff_zero_right, TensorProduct.comul_tmul, NormedSpace.norm_smul_le, ProbabilityTheory.covarianceBilin_apply, ContinuousLinearMap.isThetaTVS_comp, DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, DirectLimit.Module.hom_ext_iff, fourierIntegral_half_period_translate, ediam_smul₀, LinearMap.polyCharpoly_map_eq_charpoly, CategoryTheory.ShiftedHom.map_smul, HasFDerivAt.linear_multilinear_comp, Ideal.pi_mkQ_rTensor, AddCommMonoid.nat_isScalarTower, MeasureTheory.withDensitySMulLI_apply, ContinuousMapZero.toContinuousMapHom_apply, IntermediateField.exists_lt_finrank_of_infinite_dimensional, Asymptotics.isLittleOTVS_congr, HasStrictFDerivAt.isThetaTVS_sub, hasDerivWithinAt_const_add_iff, convexHull_toCone_isLeast, ConvexCone.IsGenerating.span_eq_top, MeasureTheory.Measure.addHaar_nnreal_smul, NonUnitalSubring.unitization_range, Complex.taylorSeries_eq_on_emetric_ball, finsuppTensorFinsuppLid_apply_apply, EuclideanGeometry.hasFDerivAt_inversion, Orientation.kahler_map, MeasureTheory.convolutionExistsAt_iff_integrable_swap, toModuleEnd_apply, AlgebraicGeometry.StructureSheaf.Localizations.comapFun_mk, MonoidAlgebra.mapDomainNonUnitalAlgHom_apply, dist_div_norm_sq_smul, TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm_tmul, AdicCompletion.of_injective, mem_segment_iff_wbtw, CliffordAlgebra.contractRight_one, torusIntegral_smul, PolyEquivTensor.toFunBilinear_apply_eq_smul, LinearEquiv.smulOfNeZero_symm_apply, NonUnitalStarSubalgebra.toSubring_subtype, VectorField.mpullback_smul, hasDerivAt_ofReal_cpow_const, LinearMap.isOrthoᵢ_def, Monovary.sum_comp_perm_smul_eq_sum_smul_iff, Monovary.sum_comp_perm_smul_le_sum_smul, CliffordAlgebraDualNumber.equiv_ι, WeierstrassCurve.Projective.Point.zero_def, Derivation.leibniz_pow, convex_iff_div, hasDerivAt_norm_rpow, PiTensorProduct.lift_tprod, Algebra.IsAlgebraic.isNormalClosure_normalClosure, TensorProduct.lcurry_apply, IsLocalizedModule.smul_injective, TrivSqZeroExt.snd_map, NonUnitalAlgebra.map_top, TensorProduct.curry_apply, IntermediateField.isPurelyInseparable_bot, Ideal.mem_torsionOf_iff, smul_algebra_smul_comm, ContinuousLinearEquiv.continuousMultilinearMapCongrLeft_apply, ModuleCat.ExtendRestrictScalarsAdj.HomEquiv.evalAt_apply, IsHausdorff.eq_iff_smodEq, Submodule.mem_span_triple, dualTensorHomEquivOfBasis_symm_cancel_left, Matrix.instSMulCommClassForall, DivisibleHull.archimedeanClassOrderIso_apply, End.hasEigenvalue_iff_isRoot, Submodule.mem_torsionBy_iff, SmoothPartitionOfUnity.sum_finsupport_smul_eq_finsum, ContinuousMultilinearMap.compContinuousLinearMapL_apply, ProbabilityTheory.uncenteredCovarianceBilin_of_not_memLp, TensorProduct.smul_tmul_smul, signedDist_apply_linear_apply, Invertible.instFaithfulSMul, Seminorm.smul_closedBall_preimage, Invertible.rTensorEquiv_symm_apply_apply, HasFDerivWithinAt.unique_on, TensorProduct.comul_def, MeromorphicOn.extract_zeros_poles, CategoryTheory.Free.lift_map_single, ProbabilityTheory.IsGaussianProcess.smul, cfcₙAux_mem_range_inr, Asymptotics.IsLittleO.tendsto_inv_smul_nhds_zero, LinearMap.eqLocus_toAddSubmonoid, ContinuousLinearMap.norm_bilinearRestrictScalars, HasFPowerSeriesWithinOnBall.iteratedFDerivWithin, ContinuousLinearMap.nhds_zero_eq, Flat.iff_lift_lsmul_comp_subtype_injective, PiTensorProduct.liftIsometry_apply_apply, ContinuousMultilinearMap.prodL_symm_apply, ContinuousMap.adjoin_id_eq_span_one_add, DistribSMul.toAddMonoidHom_eq_zsmulAddGroupHom, Ideal.range_finsuppTotal, MeasureTheory.Measure.addHaar_singleton_add_smul_div_singleton_add_smul, AlgebraicIndependent.aevalEquivField_apply_coe, ContinuousMultilinearMap.analyticWithinAt_uncurry_of_linear, CliffordAlgebra.foldr'Aux_apply_apply, ProbabilityTheory.covarianceBilin_map, LieAlgebra.IsKilling.orthogonal_span_coroot_eq_ker, zero_mem_tangentConeAt, SchwartzMap.instFourierSMul, CategoryTheory.Abelian.Ext.smul_eq_comp_mk₀, NonUnitalAlgHom.toAlgHom_apply, LinearMap.exists_mem_center_apply_eq_smul_of_forall_notLinearIndependent, MeasureTheory.measureReal_abs_inner_gt_le_integral_charFun, HasFPowerSeriesOnBall.prod, Unitization.nnnorm_eq_sup, Orientation.two_zsmul_oangle_smul_right_self, Finsupp.sum_smul_index_semilinearMap', MvPowerSeries.instIsScalarTower, SchwartzMap.fourierTransformCLM_apply, HasFDerivAt.curveIntegral_segment_source', AffineSubspace.signedInfDist_singleton, LinearEquiv.isSMulRegular_congr, HasFPowerSeriesWithinOnBall.neg, selfAdjointPart_apply_coe, IntermediateField.LinearDisjoint.exists_field_of_isDomain, Affine.Simplex.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection, Finset.expect_univ, WeakSpace.instContinuousSMul, LinearEquiv.conj_id, AlgEquiv.moduleEndSelfOp_symm_apply, TensorProduct.directSumRight'_restrict, MultilinearMap.uncurrySum_add, skewAdjointPartL_apply_coe, minpoly_algEquiv_toLinearMap, HomogeneousLocalization.val_neg, Finset.weightedVSub_apply, ContinuousLinearMap.bilinearComp_zero_left, ContinuousLinearMap.nnnorm_smulRight_apply, Surreal.dyadicMap_apply, RootPairing.pairing_neg_one_neg_four_iff', hasFPowerSeriesWithinOnBall_pi_iff, RingTheory.Sequence.IsRegular.quot_ofList_smul_nontrivial, hasDerivAt_zero, RingHom.Finite.tensorProductMap, Matrix.toBilin'Aux_eq, SchwartzMap.fderivCLM_fourier_eq, TensorProduct.lift.equiv_symm_apply, ContinuousMultilinearMap.instContinuousSMul, IsLocalizedModule.mk'_smul, CStarModule.inner_mul_inner_swap_le, addMonoidHomLequivNat_apply, Bundle.ContinuousRiemannianMetric.pos, mellin_const_smul, LieAlgebra.IsKilling.cartanEquivDual_apply_apply, Subalgebra.centralizer_coe_range_includeLeft_eq_center_tensorProduct, ContDiffAt.exists_forall_mem_closedBall_exists_eq_forall_mem_Ioo_hasDerivAt, ContMDiffOn.const_smul_section, LinearIndependent.units_smul, FormalMultilinearSeries.ofScalars_smul, SchwartzMap.derivCLM_apply, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', ContDiffAt.fderiv, Matrix.toLin_finTwoProd_toContinuousLinearMap, IsBaseChange.linearMapLeftRight, Seminorm.gauge_ball, IntermediateField.isSeparable_adjoin_pair_of_isSeparable, IsLocalization.mapExtendScalars_eq_toLinearMap_mapₐ, End.IsSemisimple_smul_iff, Localization.add_mk, IsCoercive.continuousLinearEquivOfBilin_apply, Fintype.sum_smul_sum, Ideal.top_mul, AlternatingMap.curryLeft_compAlternatingMap, MeasureTheory.integral_fun_fst, Orientation.volumeForm_zero_neg, SmoothPartitionOfUnity.contMDiff_finsum_smul, hasDerivWithinAt_inter', MeasureTheory.pdf.integrable_pdf_smul_iff, Seminorm.coe_comp, IntermediateField.finiteDimensional_left, LinearMap.map_smulₛₗ, DirectSum.GAlgebra.smul_def, AffineSubspace.subtypeₐᵢ_linearIsometry, LinearMap.toMatrixAlgEquiv'_symm, ContDiffBump.dist_normed_convolution_le, instIsPushout_1, MeasureTheory.FiniteMeasure.map_smul, Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero, VectorField.lieBracketWithin_smul_right, Algebra.Extension.Hom.sub_aux, DualNumber.commute_eps_right, Algebra.IsAlgebraic.rank_fractionRing, hasDerivAt_exp_smul_const_of_mem_ball', extremePoints_convexHull_subset, PolynomialModule.eval_smul, Finset.weightedVSubOfPoint_const_smul, LinearEquiv.isScalarTower, instIsPushoutFractionRingPolynomial, ContinuousMap.linearIsometryBoundedOfCompact_apply_apply, deriv_comp_mul_left, hasDerivAtFilter_finCons, ContinuousLinearMap.adjoint_comp, PiTensorProduct.norm_eval_le_projectiveSeminorm, iteratedFDeriv_zero_eq_comp, End.mem_genEigenspace_one, LieModule.toEnd_eq_zero_iff, intervalIntegral.integral_hasDerivAt_right, hasFTaylorSeriesUpToOn_succ_iff_left, instFiniteLocalizationAlgebraMapSubmonoidPrimeCompl, mem_segment_iff_div, WithSeminorms.image_isVonNBounded_iff_finset_seminorm_bounded, HasCompactSupport.hasDerivAt_convolution_left, KaehlerDifferential.derivationTensorProduct_algebraMap, AlgebraicClosure.instIsScalarTower, le_perfectClosure_iff, perfectClosure.isAlgebraic, LinearMap.IsProj.trace, MvPolynomial.aevalTower_comp_toAlgHom, Submodule.smul_comap_le_comap_smul, DualNumber.instIsLocalRing, VectorFourier.integral_bilin_fourierIntegral_eq_flip, LinearMap.charpoly_sub_smul, smul_inv_le_iff_of_neg, LinearEquiv.congrRight₂_trans, Subspace.map_le_dualAnnihilator_dualAnnihilator, End.apply_faithfulSMul, SchwartzMap.integral_clm_comp_deriv_right_eq_neg_left, HasFPowerSeriesWithinOnBall.add, exposedPoints_subset_extremePoints, isTorsionBy_quotient_iff, CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φK, IntermediateField.isSeparable_sup, contDiffOn_succ_iff_fderivWithin, Finset.centerMass_pair, Algebra.TensorProduct.lTensor_ker, contDiff_smul, exists_maximal_linearIndepOn, FiniteGaloisIntermediateField.finiteDimensional, TensorProduct.AlgebraTensorModule.rTensor_id, AffineMap.hasStrictDerivAt, Matrix.kroneckerTMulBilinear_apply, InnerProductSpace.toLinearMap_rankOne, openSegment_translate_preimage, ContinuousLinearMap.comp_hasFPowerSeriesOnBall, meromorphicNFAt_smul_iff_right_of_analyticAt, reflection_mul_reflection_zpow_apply, KaehlerDifferential.tensorProductTo_surjective, LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff_mem, ContinuousLinearMap.adjointAux_inner_right, HasDerivWithinAt.Ioi_iff_Ioo, MeromorphicOn.smul, Seminorm.finset_sup_smul, Complex.taylorSeries_eq_on_ball, Bundle.RiemannianMetric.pos, ContDiffWithinAt.iteratedFDerivWithin_right, TemperedDistribution.instLineDerivSMulComplex, LinearMap.restrict_smul_one, InfiniteGalois.finGaloisGroupFunctor_map_proj_eq_proj, CStarAlgebra.conjugate_le_norm_smul, Algebra.norm_apply, Submodule.image2_subset_map₂, RingHom.smulOneHom_apply, Algebra.norm_eq_zero_iff', Basis.linearMap_apply_apply, IntermediateField.coe_smul, Nat.cast_smul_eq_nsmul, finsuppTensorFinsupp_symm_single, smul_lowerBounds_subset_upperBounds_smul, addMonoidHomLequivInt_symm_apply, Seminorm.isBounded_sup, norm_midpoint_lt_iff, LinearMap.prodMap_smul, IsSemisimpleModule.exists_end_algEquiv_pi_matrix_end, LinearMap.trace_tensorProduct, continuousMultilinearCurryRightEquiv_symm_apply', comp_dualTensorHom, MeasureTheory.integral_comp_mul_right_Ioi, MeasureTheory.condExpInd_smul', NormedSpace.expSeries_sum_eq_div, IsSepClosure.isAlgClosure_of_perfectField_top, Summable.tsum_smul_const, Homotopy.smul_hom, Invertible.instLocalizationLocalizedModule, ContinuousAlternatingMap.fderivCompContinuousLinearMap_apply, HahnSeries.SummableFamily.powerSeriesFamily_of_orderTop_pos, measurable_smul_const, AffineSubspace.wSameSide_smul_vsub_vadd_right, ContDiffWithinAt.fderivWithin, WeakFEPair.functional_equation, MeasureTheory.Submartingale.sum_smul_sub, hasEigenvalue_toLin'_diagonal_iff, SchwartzMap.smul_apply, ContDiffMapSupportedIn.continuousSMul, ContinuousLinearMap.prodL_apply, LinearEquiv.map_smul, LieAlgebra.isNilpotent_ad_of_mem_rootSpace, ContDiffMapSupportedIn.structureMapLM_zero_injective, toSpanSingleton_deriv, IsLocalizedModule.mk_eq_mk', IsLocalizedModule.map_injective_iff_localizedModuleMap_injective, Coalgebra.TensorProduct.assoc_symm_tmul, convex_Ioc, LinearEquiv.smul_id_of_finrank_eq_one_apply, MvPolynomial.pderiv_inr_universalFactorizationMap_X, LinearMap.toMatrix'_toLinearMapₛₗ₂', Submodule.smul_torsionBy, TrivSqZeroExt.fst_map, AdicCompletion.transitionMap_ideal_mk, LinearMap.BilinForm.smul_right, LinearMap.isStarProjection_toContinuousLinearMap_iff, MeromorphicOn.divisor_smul, Algebra.TensorProduct.tensorTensorTensorComm_tmul, IsSelfAdjoint.linearly_dependent_of_isLocalExtrOn, IsLinearMap.isLinearMap_smul, LocalizedModule.mk'_smul_mk, HasFDerivAtFilter.isLittleOTVS, Rep.MonoidalClosed.linearHomEquiv_hom, Representation.ofModule_asAlgebraHom_apply_apply, Ideal.mul_le_inf, IsCyclic.mulAutMulEquiv_symm_apply_symm_apply, Submodule.smul_le, Matrix.toLpLin_mul, MeasureTheory.integrable_withDensity_iff_integrable_coe_smul₀, iteratedDerivWithin_eq_equiv_comp, TensorProduct.map_convMul_map, TensorAlgebra.equivDirectSum_symm_apply, MvPowerSeries.coeff_smul, InfiniteGalois.restrictNormalHom_continuous, Matrix.toLin'_submatrix, TensorProduct.gradedMul_algebraMap, slope_sub_smul, iteratedFDerivWithin_two_apply, Matrix.conjTranspose_ofNat_smul, Affine.Simplex.smul_centroid_vsub_point_eq_sum_vsub, extDerivWithin_smul, AddAction.automorphize_smul_left, End.isIntegral, Derivation.compAEval_eq, Pretrivialization.continuousOn_continuousLinearMapCoordChange, ContDiffMapSupportedIn.seminorm_postcompLM_le, AlgebraicGeometry.tilde.isUnit_algebraMap_end_basicOpen, ContinuousMultilinearMap.cpolynomialOn_uncurry_compContinuousLinearMap, Matrix.separatingLeft_toLinearMap₂_iff, CategoryTheory.MonoidalLinear.smul_whiskerRight, Derivation.tensorProductTo_tmul, End.mem_maxGenEigenspace, MeasureTheory.integral_countable, DoubleCentralizer.coe_fst, HasStrictFDerivAt.fun_smul, AdicCompletion.transitionMap_map_mul, ContinuousMultilinearMap.analyticWithinAt_uncurry_compContinuousLinearMap, AffineSubspace.affineSpan_pair_parallel_iff_exists_unit_smul, Polynomial.aeval_endomorphism, MvPolynomial.instIsPushout_1, End.genEigenspace_div, DivisibleHull.zsmul_mk, smul_inv_lt_iff_of_neg, ContDiffPointwiseHolderAt.fderiv, CompactlySupportedContinuousMap.toReal_smul, IsGalois.of_separable_splitting_field_aux, WeierstrassCurve.Projective.negMap_of_Z_eq_zero, IntermediateField.coe_equivMap_apply, Derivation.map_smul, IsPicardLindelof.exists_eq_forall_mem_Icc_hasDerivWithinAt, norm_cfcₙ_one_sub_one_add_inv_lt_one, Matrix.toLin_toMatrix, PiTensorProduct.lift_comp_map, Seminorm.closedBall_bot, LocallyConvexSpace.convex_open_basis_zero, Submodule.adjoint_orthogonalProjection, CategoryTheory.MonoidalLinear.whiskerLeft_smul, TensorProduct.homTensorHomMap_apply, Filter.HasBasis.isBigOTVS_iff, Real.sInf_smul_of_nonpos, LinearMap.tensorEqLocus_tmul, Matrix.isRepresentation.eq_toEnd_of_represents, wbtw_smul_vadd_smul_vadd_of_nonneg_of_le, HasFDerivAt.pow', continuousOn_cfcₙ_nnreal, LieModule.traceForm_apply_apply, Function.Odd.smul_even, HasStrictFDerivAt.pow', cross_dot_cross, LinearMap.toMatrix₂_apply, ModuleCat.Hom.hom₂_apply, ContDiffWithinAt.smul_const, surjOn_extremePoints_image, Complex.IsConservativeOn.hasDerivAt_wedgeIntegral, hasFDerivWithinAt_iff_hasGradientWithinAt, Real.fourierIntegral_continuousLinearMap_apply', CurveIntegrable.smul, AddMonoid.End.intCast_def, sectionOfRetractionKerToTensorAux_algebraMap, AffineIsometryEquiv.ofEq_rfl, IntermediateField.finInsepDegree_bot, LieAlgebra.IsKilling.isSemisimple_ad_of_mem_isCartanSubalgebra, BoxIntegral.Integrable.smul, RootPairing.rootFormIn_self_smul_coroot, LinearMap.IsSymmetric.trace_eq_sum_eigenvalues, Finset.centroid_pair, TopModuleCat.hom_smul, isMIntegralCurveOn_comp_mul_ne_zero, ContinuousLinearMap.hasFDerivAt_of_bilinear, cfcₙHom_continuous, starConvex_iff_forall_pos, Submodule.tensorToSpan_apply_tmul, LinearMap.adjoint_comp, FormalMultilinearSeries.min_radius_le_radius_add, krullTopology_mem_nhds_one_iff, hasStrictDerivAt_one, FormalMultilinearSeries.ofScalars_comp_neg_id, Affine.Simplex.point_vsub_centroid_eq_smul_vsub, RootPairing.corootForm_self_smul_root, PiToModule.fromMatrix_apply, SymmetricAlgebra.algHom_surjective, signedDist_right_lineMap, Algebra.baseChange_lmul, KaehlerDifferential.mvPolynomialBasis_repr_D, QuadraticForm.tensorAssoc_apply, ContinuousLinearMap.closed_range_of_antilipschitz, ProbabilityTheory.uncenteredCovarianceBilin_zero, tendsto_integral_comp_smul_smul_of_integrable, Matrix.frobeniusNormSMulClass, LinearMap.toContPerfPair_apply, LieModule.trace_toEnd_genWeightSpaceChain_eq_zero, IsCyclotomicExtension.nonempty_algEquiv_adjoin_of_exists_isPrimitiveRoot, strictConvex_iff_div, VonNeumannAlgebra.coe_toStarSubalgebra, Matrix.toEuclideanLin_apply_piLp_toLp, BddAbove.smul_of_nonpos, lie_eq_smul, SeminormFamily.basisSets_singleton_mem, Ideal.ofList_cons_smul, AEval.of_aeval_smul, Finset.centerMass_eq_of_sum_1, Asymptotics.isLittleOTVS_iff_tendsto_inv_smul, IntermediateField.algebraAdjoinAdjoin.instIsScalarTowerSubtypeMemSubalgebraAdjoinAdjoin, LinearMap.dualMap_apply, LieModule.Cohomology.twoCochain_val_apply, LieAlgebra.ext_iff, WeierstrassCurve.Projective.addMap_of_Z_ne_zero, ContinuousAlternatingMap.curryLeft_add, Ideal.sup_mul_eq_of_coprime_left, hasFDerivAt_integral_of_dominated_of_fderiv_le, IsScalarTower.complexToReal, Rep.FiniteCyclicGroup.homResolutionIso_inv_f_hom_apply_hom_hom_apply, HasFPowerSeriesAt.pi, ProbabilityTheory.setIntegral_tilted_mul_eq_mgf, TensorAlgebra.ι_inj, LinearMap.dualMap_surjective_iff, multilinearCurryLeftEquiv_symm_apply, Unitization.quasispectrum_eq_spectrum_inr', smul_ball, Subalgebra.instIsScalarTowerSubtypeMem, HasFPowerSeriesWithinOnBall.tendstoUniformlyOn', map_intCast_smul, AddAut.mulLeft_apply_apply, IntervalIntegrable.continuousOn_smul, Seminorm.smul_ball_preimage, ContDiffMapSupportedIn.toBoundedContinuousFunctionLM_apply, hasStrictDerivAt_pi, ContDiff.hasStrictDerivAt, LinearIsometry.strictConvexSpace_range_iff, SameRay.exists_pos, VectorFourier.fourierPowSMulRight_apply, cfcₙHomSuperset_continuous, RatFunc.neg_def, HasFDerivWithinAt.smul, skewAdjointPart_eq_I_smul_imaginaryPart, MultilinearMap.piLinearMap_apply_apply_apply, finsuppTensorFinsuppRid_apply_apply, Basis.repr_sum_self, TensorProduct.toLinearMap_symm_lid, inner_eq_norm_mul_iff_real, TensorAlgebra.ι_eq_zero_iff, NonUnitalSubalgebra.coe_smul, CoalgCat.MonoidalCategoryAux.comul_tensorObj, ContinuousAlternatingMap.prodLIE_apply, Finsupp.linearCombination_option, Ideal.mul_iSup, PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal, LieAlgebra.IsKilling.chainBotCoeff_of_eq_zsmul_add, MeasureTheory.Submartingale.smul_nonpos, supportDim_le_supportDim_quotSMulTop_succ_of_mem_jacobson, QuadraticMap.associated_sq, LinearMap.dualMap_def, MultilinearMap.coe_currySumEquiv, MultilinearMap.uncurrySum_apply, iSupIndep_iff_dfinsupp_lsum_injective, image_extremePoints, TrivSqZeroExt.liftEquivOfComm_symm_apply_coe, Summable.smul_const, RootPairing.toPerfPair_conj_reflection, QuadraticMap.associated_comp, MeasureTheory.Lp.simpleFunc.toLp_smul, segment_eq_image_lineMap, KaehlerDifferential.ker_map_of_surjective, TensorProduct.tmul_of_gradedMul_of_tmul, sub_smul_slope, dist_smul₀, CPolynomialOn.fderiv, zero_smul, concaveOn_iff_forall_pos, IsLocalizedModule.map_exact, PadicInt.mahlerEquiv_apply, Submodule.span_singleton_eq_range, SchwartzMap.convolution_flip, QuadraticMap.associated_isOrtho, ContinuousLinearMap.opNorm_flip, LieRinehartAlgebra.Hom.map_smul_apply, Basis.parallelepiped_map, Algebra.H1Cotangent.isLocalizedModule, Algebra.TensorProduct.lift_comp_includeRight', cfcₙ_apply_pi, CliffordAlgebra.contractLeft_mul_algebraMap, continuousOn_integral_bilinear_of_locally_integrable_of_compact_support, ContinuousLinearMap.precompCompactConvergenceCLM_apply, map_dualTensorHom, Filter.Tendsto.integral_sub_linear_isLittleO_ae, FormalMultilinearSeries.radius_compContinuousLinearMap_le, hasDerivWithinAt_derivWithin_iff, Submodule.range_dualMap_mkQ_eq, hasDerivAt_sub_const_iff, midpoint_mem_segment, HasStrictFDerivAt.mul_const', ContinuousLinearMap.IsPositive.adjoint_conj, MultilinearMap.curryMidLinearEquiv_symm_apply, Polynomial.toMatrix_sylvesterMap, LinearMap.smulRightₗ_apply_apply, krullTopology_mem_nhds_one_iff_of_normal, Complex.isometryOfOrthonormal_apply, IsConformalMap.smul, NonUnitalStarAlgHom.subtype_comp_codRestrict, intervalIntegral.smul_integral_comp_mul_right, LinearMap.IsRefl.eq_iff, AlternatingMap.uncurryFin_add, TensorProduct.AlgebraTensorModule.lTensor_one, DivisibleHull.neg_mk, Matrix.dualNumberEquiv_apply, intervalIntegral.integral_comp_div_add, ConvexCone.isGenerating_top, CategoryTheory.Pretriangulated.Triangle.smul_hom₃, Matrix.separatingRight_toBilin'_iff, Complex.hasDerivAt_Gamma_one, LieModule.maxGenEigenSpace_toEnd_eq_top, LinearMap.restrictScalars_extendScalarsOfIsLocalization, LinearMap.vecEmpty₂_apply, LieModule.coe_maxTrivLinearMapEquivLieModuleHom_symm, intervalIntegral.integral_smul_const, Matrix.toLin_finTwoProd, FormalMultilinearSeries.derivSeries_eq_zero, InnerProductSpace.laplacianWithin_smul_nhds, KaehlerDifferential.tensorKaehlerEquivOfFormallyEtale_symm_D_algebraMap, Unitization.inrRangeEquiv_apply_coe, QuotientGroup.automorphize_smul_left, Matrix.SpecialLinearGroup.toLin_equiv.toLinearMap_eq, Unitization.instStarModule, Algebra.Generators.H1Cotangent.δ_eq_δAux, ContDiffWithinAt.smulRight, ProbabilityTheory.hasDerivAt_iteratedDeriv_complexMGF, Basis.linearMap_repr_symm_apply, Matrix.toLinAlgEquiv'_toMatrixAlgEquiv', NonUnitalSubsemiring.unitization_range, LinearEquiv.lieConj_symm, Orientation.two_zsmul_oangle_smul_right_of_ne_zero, ContinuousLinearMap.star_eq_adjoint, CliffordAlgebra.even.lift.aux_ι, SchwartzMap.integral_fourier_smul_eq, Real.dimH_segment, KaehlerDifferential.mvPolynomialBasis_repr_comp_D, LieAlgebra.rank_eq_natTrailingDegree, Affine.Simplex.faceOppositeCentroid_eq_smul_vsub_vadd_point, Complex.circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, IsBaseChange.equiv_tmul, ContinuousMultilinearMap.curryLeft_apply, End.ker_aeval_ring_hom'_unit_polynomial, Finset.centerMass_smul, Algebra.lmul_isUnit_iff, LinearEquiv.charpoly_conj, LinearMap.tensorProduct_apply, Monotone.quasiconvexOn, Submodule.mem_toConvexCone, AffineSpace.asymptoticNhds_eq_smul, smul_max_of_nonpos, AffineSubspace.direction_smul, Rep.coinvariantsTensorMk_apply, ContMDiffWithinAt.coordChangeL, ContDiffAt.fun_smul, HasFPowerSeriesWithinAt.neg, RootPairing.pairing_smul_coroot_eq, HasFDerivAt.hasFDerivAt_norm_smul_neg, IntermediateField.isCyclotomicExtension_adjoin_of_exists_isPrimitiveRoot, QuotSMulTop.map_comp, finsuppTensorFinsuppLid_single_tmul_single, Asymptotics.isThetaTVS_comm, Projectivization.cross_mk_of_ne, fderivWithin_list_prod', Algebra.tensorH1CotangentOfIsLocalization_toLinearMap, Matrix.kroneckerTMulAlgEquiv_apply, LinearMap.not_separatingLeft_zero, NonUnitalStarAlgebra.range_id, End.genEigenspace_nat, Unitization.complex_cfcₙ_eq_cfc_inr, ker_algebraMap_end, separableClosure.isSeparable, LieAdmissibleAlgebra.ext_iff, FormalMultilinearSeries.radius_le_radius_derivSeries, ContinuousAlternatingMap.curryLeftLI_apply, smul_pos_iff_of_neg_left, endVecAlgEquivMatrixEnd_symm_apply_apply, LieAlgebra.IsKilling.chainBotCoeff_le_chainLength, CharacterModule.uncurry_apply, ContMDiffWithinAt.const_smul_section, LinearMap.restrictScalarsField_apply_apply, ContinuousLinearEquiv.smul_apply, AdicCompletion.of_bijective, Bialgebra.TensorProduct.assoc_toCoalgEquiv, iteratedFDerivWithin_zero_eq_comp, HasFiniteFPowerSeriesAt.sub, Ideal.span_singleton_mul_left_inj, HasFDerivWithinAt.fun_mul', ContinuousLinearMap.norm_compSL_le, IsLocalization.tensorProduct_compatibleSMul, QuadraticMap.Ring.associated_pi, ContinuousAlternatingMap.instIsCentralScalar, fderivWithin_smul_const, Unitization.inrNonUnitalAlgHom_apply, VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral, NonUnitalAlgebra.comap_top, LinearMap.nondegenerate_toMatrix₂_iff, Matrix.trace_kroneckerMapBilinear, ContDiffMapSupportedIn.withSeminorms, ConvexCone.mem_strictlyPositive, AlgEquiv.linearEquivConj_mulRight, AnalyticOn.fun_const_smul, MeasureTheory.withDensityᵥ_smul, wbtw_or_wbtw_smul_vadd_of_nonneg, ContDiffAt.exists_eventually_eq_hasDerivAt, MvPolynomial.instIsPushout, DivisibleHull.qsmul_of_nonpos, Bornology.isVonNBounded_vadd, ModuleCat.Iso.conj_eq_conj, MeasureTheory.integrable_comp_smul_iff, derivWithin_fun_const_smul', IsModuleTopology.continuous_of_distribMulActionHom, convex_halfSpace_le, LieAlgebra.mem_corootSpace', CategoryTheory.ShortComplex.RightHomologyMapData.smul_φQ, cfcₙ_nnreal_eq_real, Matrix.instSMulCommClassMulOppositeForallOfIsScalarTower, ProbabilityTheory.setIntegral_tilted_mul_eq_cgf, HasFDerivWithinAt.isBigOTVS_sub, Matrix.toLinAlgEquiv'_one, innerSL_apply, DirectSum.support_smul, convexHull_isClosed, abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul, Matrix.toLin_one, SchwartzMap.integral_bilinear_lineDerivOp_right_eq_neg_left, CliffordAlgebra.contractLeft_algebraMap_mul, iteratedDeriv_const_smul_field, IntermediateField.adjoin.finiteDimensional, IntermediateField.botEquiv_def, SemilinearEquivClass.map_smulₛₗ, IsMIntegralCurveAt.comp_mul_ne_zero, CategoryTheory.NatTrans.app_smul, Submodule.FG.jacobson_smul_lt, inf_eq_half_smul_add_sub_abs_sub, InnerProductSpace.isIdempotentElem_rankOne_self_iff, le_algebraicClosure_iff, fderivWithin_continuousLinearEquiv_comp, ContinuousMultilinearMap.toUniformOnFun_toFun, Quaternion.re_snd_dualNumberEquiv, cfc_eq_cfcL, LinearMap.trace_eq_contract_of_basis, LinearMap.range_dualMap_le_dualAnnihilator_ker, Matrix.toLin_transpose, IntermediateField.adjoin_adjoin_comm, IsLocalization.instCompatibleSMulLocalizationOfIsScalarTower, Submodule.span_singleton_mul, div_le_egauge_ball, TrivSqZeroExt.algebraMap_eq_inlHom, hasDerivAtFilter_pi, stereographic_apply, Complex.toMatrix_conjAe, StrongDual.toLp_apply, Ideal.mem_mul_span_singleton, Orientation.rotation_apply, ContinuousLinearMap.map_smul₂, ContinuousLinearMap.coe_flipₗᵢ, Submodule.finite_dualAnnihilator_iff, ContinuousAlternatingMap.fderivCompContinuousLinearMap_of_isEmpty, SameRay.norm_smul_eq, Representation.leftRegular_norm_apply, segment_subset_uIcc, BilinForm.mul_toMatrix, TensorProduct.liftAux.smul, inner_map_polarization, contDiffOn_succ_iff_hasFDerivWithinAt_of_uniqueDiffOn, Ideal.IsTwoSided.pow_add, not_linearIndepOn_finset_iff, ContMDiffAt.mfderiv, CompactlySupportedContinuousMap.eq_toRealPositiveLinear_toReal, End.IsSemisimple.minpoly_squarefree, IntermediateField.val_mk, NonUnitalAlgHom.range_comp, OrthonormalBasis.sum_repr, hasDerivAtFilter_const_add_iff, TemperedDistribution.derivCLM_toTemperedDistributionCLM_eq, Real.iteratedFDeriv_fourierIntegral, SMulPosMono.toSMulPosStrictMono, StieltjesFunction.measure_smul, MeasureTheory.Lp.coe_LpSubmodule, Submodule.exists_of_finrank_lt, Bornology.isVonNBounded_insert, Algebra.Generators.CotangentSpace.map_toComp_injective, hasDerivWithinAt_abs_neg, TestFunction.instLocallyConvexSpaceReal, TensorProduct.mk_surjective, AdicCompletion.evalₐ_of, End.lTensorAlgHom_apply_apply, StieltjesFunction.ae_hasDerivAt, InnerProductSpaceable.innerProp, IntermediateField.rank_sup_le, Matrix.toLpLin_symm_comp, IntermediateField.LinearDisjoint.self_right, LinearIsometry.strictConvexSpace_range, Algebra.Extension.H1Cotangent.equivOfFormallySmooth_toLinearMap, Basis.toDual_linearCombination_right, MeasureTheory.withDensityᵥ_rnDeriv_smul, Matrix.toLinearMapₛₗ₂_apply_basis, FormalMultilinearSeries.ofScalars_series_eq_zero, CochainComplex.HomComplex.Cochain.leftShift_units_smul, Antivary.sum_smul_comp_perm_eq_sum_smul_iff, LieAlgebra.ad_lie, fderivWithin_pow_ring', ContinuousLinearMap.analyticOn_uncurry_of_multilinear, isPrecomplete_iff, Affine.Simplex.faceOppositeCentroid_vsub_centroid_eq_smul_vsub, LocalizedModule.mk_smul_mk, TensorProduct.finsuppLeft_apply_tmul, LinearEquiv.coord_apply_smul, FormalMultilinearSeries.changeOrigin_eval, Unitization.lift_range_le, VertexOperator.ncoeff_of_coeff, LinearMap.toSeminorm_ball_zero, AffineMap.homothety_def, ContinuousAlternatingMap.norm_alternatizeUncurryFin_le, ExteriorAlgebra.liftAlternating_algebraMap, inner_smul_left_eq_smul, Finset.expect_dite_eq, StrongConvexOn.strictConvexOn, CategoryTheory.Free.lift_map, Orientation.kahler_apply_apply, midpoint_sub_left, Submodule.mem_span_set', ContinuousLinearMap.reApplyInnerSelf_smul, NonUnitalCStarAlgebra.toIsScalarTower, MeasureTheory.integral_indicator_const, isQuasiregular_iff_isUnit', LinearMap.toMatrix_toSpanSingleton, RootPairing.toPerfPair_flip_conj_coreflection, IsSelfAdjoint.adjoint_conj, DirectLimit.Module.lift_apply, not_linearIndependent_iff, closedAbsConvexHull_eq_closure_absConvexHull, LinearMap.toMatrix₂_symm', MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff, NonUnitalSubalgebra.toNonUnitalSubsemiring_subtype, DivisibleHull.archimedeanClassOrderIso_symm_apply, RootPairing.pairing_smul_root_eq, FiniteGaloisIntermediateField.instFiniteDimensionalSubtypeMemIntermediateField, VectorFourier.fourierIntegral_continuousMultilinearMap_apply, HasFPowerSeriesAt.prod, TensorProduct.lidOfCompatibleSMul_tmul, midpoint_vsub, LinearMap.toSeminorm_apply, AlgEquiv.restrictNormalHom_apply, LieModule.isNilpotent_iff_forall', InfiniteGalois.mk_toAlgEquivAux, LinearIndependent.units_smul_iff, linearDepOn_iffₛ, hasDerivAt_ofReal_cpow_const', hasStrictDerivAt_natCast, Complex.hasDerivAt_Gamma_one_half, IsLocalizedModule.fromLocalizedModule'_mk, VectorPrebundle.contMDiffOn_contMDiffCoordChange, LinearMap.mulRight_toAddMonoidHom, MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff, instIsAbelianGaloisSubtypeMemIntermediateField, HasFPowerSeriesAt.isBigO_sub_partialSum_pow, ZMod.invDFT_def, Orientation.oangle_smul_add_right_eq_zero_or_eq_pi_iff, lineDeriv_smul, ContinuousMap.smul_apply', MeasureTheory.integrable_rnDeriv_smul_iff, LinearMap.map_smul', IntermediateField.adjoin_simple_comm, algebraicClosure.isAlgClosure, Real.differentiable_fourierChar_neg_bilinear_right, OrthonormalBasis.sum_rankOne_eq_id, End.mem_genEigenspace_nat, ProbabilityTheory.strong_law_ae_simpleFunc_comp, MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one_aux, MeasureTheory.Lp.ker_toTemperedDistributionCLM_eq_bot, mdifferentiableAt_coordChangeL, hasDerivAt_finCons', segment.lift, MeasureTheory.Lp.toTemperedDistribution_toLp_eq, SymmetricAlgebra.algebraMap_leftInverse, absorbs_iff_eventually_nhds_zero, LinearMap.Nondegenerate.toMatrix₂', Finsupp.support_smul_eq, AffineSubspace.signedInfDist_def, WeakDual.CharacterSpace.continuousMapEval_apply_apply, Localization.neg_mk, TensorProduct.equivFinsuppOfBasisRight_symm, GradedMonoid.isScalarTower_right, HasFPowerSeriesWithinAt.add, Antivary.card_smul_sum_le_sum_smul_sum, IntermediateField.inclusion_inclusion, inf_eq_half_smul_add_sub_abs_sub', ContinuousLinearMap.adjoint_inner_right, exists_extension_norm_eq, Real.tendsto_integral_gaussian_smul, map_nnratCast_smul, HVertexOperator.compHahnSeries_coeff, LinearMap.liftBaseChange_comp, HahnSeries.map_smul, IsSMulRegular.subsingleton_linearMap_iff, Subspace.dualAnnihilator_le_dualAnnihilator_iff, hasFDerivAt_integral_of_dominated_of_fderiv_le'', HasFDerivWithinAt.fun_pow', StarAlgEquiv.ofInjective'_apply, LieSubmodule.coe_toEnd_pow, Basis.orientation_unitsSMul, Submodule.lTensorOne_tmul, IntervalIntegrable.integral_smul, Matrix.Nondegenerate.toBilin', HasFiniteFPowerSeriesOnBall.hasStrictFDerivAt, FDRep.Iso.conj_ρ, IntermediateField.extendScalars_restrictScalars, LinearMap.IsAlt.eq_of_add_add_eq_zero, VectorFourier.fourierIntegral_comp_add_right, LinearMap.isIntegral, Ideal.mul_mono_left, ContDiffMapSupportedIn.postcompLM_apply, LinearMap.BilinForm.congr_trans, LinearMap.toMatrix_singleton, Localization.mk_multiset_sum, LinearMap.BilinForm.toMatrix'_mul, Submodule.mem_colon_singleton, UnitAddTorus.span_mFourier_closure_eq_top, signedDist_lineMap_right, analyticOrderAt_smul, ContinuousLinearMap.norm_smulRightL_le, Real.smul_iSup_of_nonpos, Convex.combo_le_max, MeasureTheory.withDensityᵥ_smul', TensorAlgebra.algebraMap_leftInverse, NonUnitalAlgebra.map_bot, LinearMap.trace_restrict_eq_of_forall_mem, differentiableAt_iff_restrictScalars, TemperedDistribution.smulLeftCLM_smulLeftCLM_apply, hasDerivAtFilter_const, smul_mem_asymptoticCone_iff, LinearIsometryEquiv.symm_smul_apply, SchwartzMap.tsupport_derivCLM_subset, DirectSum.smul_apply, Filter.Tendsto.cesaro_smul, Algebra.Extension.Hom.sub_tmul, deriv_fun_const_smul_field, LieAlgebra.isNilpotent_range_ad_iff, SubMulAction.val_neg, tendsto_integral_exp_inner_smul_cocompact, exteriorPower.alternatingMapLinearEquiv_symm_apply, QuadraticMap.polar_smul_right_of_tower, CategoryTheory.Linear.instEpiHSMulHomOfInvertible, Finset.centroid_pair_fin, ContinuousLinearMap.coe_symm_flipMultilinearEquiv, inner_smul_right_eq_smul, Rep.coinvariantsTensorIndHom_mk_tmul_indVMk, LieModule.mem_genWeightSpaceOf, DistribMulActionHom.ext_ring_iff, ProperCone.pointed_toConvexCone, Rep.ihom_coev_app_hom, QuadraticMap.apply_linearCombination, Submodule.top_smul, LinearMap.toMatrix₂'_compl₂, MulAction.automorphize_smul_left, DualNumber.exp_smul_eps, SchwartzMap.integral_smul_laplacian_right_eq_left, AdicCompletion.factor_evalₐ_eq_eval, ContinuousMultilinearMap.smulRight_apply, Basis.multilinearMap_apply, Algebra.TensorProduct.comm_comp_map, FiniteGaloisIntermediateField.instIsGaloisSubtypeMemIntermediateFieldMin, Set.OrdConnected.strictConvex, TrivSqZeroExt.map_inl, FormalMultilinearSeries.radius_prod_eq_min, Matrix.toLinAlgEquiv'_symm, imaginaryPart_smul, Quaternion.snd_imI_dualNumberEquiv_symm, OrthonormalBasis.coe_equiv_euclideanSpace, VectorFourier.fourierIntegral_probChar, ContinuousLinearEquiv.continuousAlternatingMapCongr_apply, ModuleCat.restrictScalars.smul_def', Matrix.instIsScalarTowerMulOppositeForallOfSMulCommClass, VectorFourier.norm_fourierSMulRight_le, openSegment_same, Submodule.le_dualAnnihilator_iff_le_dualCoannihilator, Seminorm.preimage_metric_ball, NonUnitalCommCStarAlgebra.toSMulCommClass, isAlgClosure_iff, LocallyConvexSpace.convex_basis, HasFDerivAt.mul_const', LieDerivation.instLieModule, MvPolynomial.derivation_C_mul, essSup_smul_measure, LieAlgebra.IsKilling.traceForm_coroot, mem_segment_add_sub, IsCyclic.mulAutMulEquiv_symm_apply_apply, toAddMonoidEnd_apply_apply, InfiniteGalois.proj_of_le, HasFiniteFPowerSeriesOnBall.fderiv', Orientation.two_zsmul_oangle_smul_left_self, Submodule.span_smul_eq, Orientation.kahler_rotation_right, End.mulSemiringActionToAlgEquiv_conjAct_surjective, Fourier.fourierIntegral_def, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le, NonUnitalStarSubalgebra.instIsScalarTower, LinearMap.rid_comp_lTensor, ContinuousMultilinearMap.currySum_apply, Submodule.mem_colon, NormSMulClass.toIsBoundedSMul, Matrix.Module.diagonal_const_smul, CompactlySupportedContinuousMap.integralPositiveLinearMap_toFun, MeasureTheory.ComplexMeasure.re_apply, FormalMultilinearSeries.ofScalars_comp_neg, MvPolynomial.tensorEquivSum_one_tmul_X, SeminormedSpace.Core.norm_smul, PiTensorProduct.lift.unique', Matrix.nondegenerate_toLinearMap₂'_iff, signedDist_apply, Unitization.norm_splitMul_snd_sq, LinearMap.separatingRight_toMatrix₂_iff, SameRay.sameRay_pos_smul_right, LieModule.traceForm_eq_sum_finrank_nsmul_mul, HasStrictFDerivAt.fun_pow', CFC.nnrpow_eq_nnrpow_prod, ContinuousLinearMap.smulRight_id, continuousMultilinearCurryRightEquiv_apply', ContinuousLinearMap.flipₗᵢ'_symm, le_egauge_ball_one, Ideal.instNoZeroDivisors, Coalgebra.lTensor_counit_comp_comul, rank_eq_zero_iff, KaehlerDifferential.isBaseChange_of_formallyEtale, IsLocalizedModule.map_units, NonUnitalSubalgebra.instIsScalarTowerSubtypeMem, TemperedDistribution.lineDerivOp_fourierInv_eq, AlternatingMap.curryLeft_compLinearMap, LieAlgebra.nilpotent_ad_of_nilpotent_algebra, QuotientAddGroup.automorphize_smul_left, fourierCoeffOn_eq_integral, DivisibleHull.instIsStrictOrderedModuleRat, CliffordAlgebra.changeForm.associated_neg_proof, MeasureTheory.SimpleFunc.integral_const, CompactConvergenceCLM.continuousSMul, HasFiniteFPowerSeriesOnBall.hasFDerivAt, CliffordAlgebra.ofEven_ι, LinearMap.BilinForm.IsNonneg.smul, IntermediateField.lift_adjoin, HasFDerivAt.hasGradientAt, hasStrictDerivAt_zero, ProbabilityTheory.hasDerivAt_integral_pow_mul_exp_real, ContinuousAlternatingMap.alternatizeUncurryFin_add, ContinuousLinearMap.isBigOTVS_fun_comp, Orientation.oangle_sign_smul_add_smul_right, signedDist_eq_dist_iff_vsub_mem_span, Coalgebra.TensorProduct.rid_tmul, Pi.image_update_segment, Finset.sum_smul_vsub_const_eq_weightedVSub, le_egauge_of_forall_ne_zero, Finset.sum_smul_vsub_const_eq_affineCombination_vsub, realPart_apply_coe, FormalMultilinearSeries.coeff_iterate_fslope, QuotSMulTop.map_apply_mk, LieModule.instIsFaithfulEnd, HasFPowerSeriesAt.eq_zero, AbsolutelyContinuousOnInterval.fun_smul, Matrix.cramer_smul, Matrix.represents_iff', ContinuousLinearMap.flip_zero, FormalMultilinearSeries.leftInv_removeZero, LieAlgebra.IsKilling.chainBotCoeff_zero_right, deriv_const_smul', Submodule.dualAnnihilator_bot, DualNumber.lift_apply_inl, intervalIntegral.integral_comp_mul_right, Representation.linHom.mem_invariants_iff_comm, hasStrictDerivAt_exp_smul_const_of_mem_ball', tangentConeAt_subset_zero, ContinuousLinearMap.extendTo𝕜'_apply, LocalizedModule.smul_eq_iff_of_mem, Matrix.conjTranspose_rat_smul, NonUnitalStarSubalgebra.inclusion_injective, ContinuousLinearMap.isAdjointPair_inner, LinearMap.coe_smulRight, HasStrictFDerivAt.hasStrictFDerivAt_norm_smul, VectorFourier.norm_fourierPowSMulRight_le, EuclideanGeometry.dist_smul_vadd_eq_dist, PolynomialModule.monomial_smul_apply, Algebra.TensorProduct.map_restrictScalars_comp_includeRight, posSMulMono_iff_posSMulStrictMono, MeasureTheory.integral_singleton, ContinuousLinearEquiv.continuousMultilinearMapCongrRight_apply, AdicCompletion.eval_of, intervalIntegral.integral_comp_div_sub, DivisibleHull.coe_add, isSelfAdjoint_smul_of_mem_skewAdjoint, CommAlgCat.associator_hom_hom, Submodule.setSemiring_smul_def, FormalMultilinearSeries.changeOriginSeries_finite_of_finite, ProbabilityTheory.norm_uncenteredCovarianceBilin_le, ConvexBody.hausdorffDist_coe, LieSubmodule.exists_smul_add_of_span_sup_eq_top, IsLocalizedModule.map_comp, Real.hasFDerivAt_fourierChar_neg_bilinear_right, Dual.eval_naturality, not_linearIndepOn_finset_iffₛ, openSegment_subset_Ioo, HVertexOperator.coeff_comp, DifferentiableAt.smul_const, convexHull_smul, KaehlerDifferential.mvPolynomialBasis_repr_symm_single, Finset.sum_Ioc_by_parts, Real.fourier_continuousMultilinearMap_apply, NonUnitalSubalgebra.coe_inclusion, ModuleCat.ofHom₂_hom_apply_hom, IsSl2Triple.HasPrimitiveVectorWith.lie_h, Unitization.splitMul_injective, WeakFEPair.f_modif_aux1, MDifferentiableAt.smul_const_section, IntermediateField.rank_eq_rank_subalgebra, NonUnitalSubalgebraClass.subtype_apply, Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule_1, FormalMultilinearSeries.ofScalars_series_injective, ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id, collinear_iff_exists_forall_eq_smul_vadd, MeasureTheory.Lp.smul_assoc, ContDiffMapSupportedIn.continuous_iff_comp, Asymptotics.IsTheta.const_smul_right, ContinuousLinearMap.smulRight_comp, cfcₙ_eq_cfcₙL, Algebra.normalizedTrace_trans, DualNumber.lift_op_smul, LinearMap.trace_conj, Submodule.map_range_rTensor_subtype_lid, QuadraticMap.Isometry.tmul_apply, ContMDiffOn.smul_section_of_tsupport, VectorFourier.norm_fourierSMulRight, StarModule.smul_lt_smul_of_pos, autEquivRootsOfUnity_apply_rootOfSplit, ContMDiffWithinAt.smul_section, FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_of_finite, isSMulRegular_on_quot_iff_lsmul_comap_eq, ContinuousLinearEquiv.coord_norm', TrivSqZeroExt.mul_right_eq_one, LinearMap.mul'_tensor, ContinuousMultilinearMap.instIsCentralScalar, OrderIso.smulRightDual_symm_apply, IsBaseChange.comp, smul_mem_asymptoticCone, IsLocalizedModule.iso_localizedModule_eq_refl, LieAlgebra.IsKilling.chainTopCoeff_of_eq_zsmul_add, absConvexHull_eq_convexHull_balancedHull, IsLocalization.map_linearMap_eq_toLinearMap_mapₐ, jacobi_cross, ProbabilityTheory.hasDerivAt_mgf, AlgebraicIndependent.aevalEquivField_algebraMap_apply_coe, HasFPowerSeriesAt.eq_zero_of_eventually, LinearMap.separatingLeft_toMatrix₂_iff, SpecialLinearGroup.centerEquivRootsOfUnity_apply_apply, NormedSpace.norm_smul_normalize, Set.OrdConnected.starConvex, Algebra.IsAlgebraic.rank_fractionRing_mvPolynomial, Derivation.apply_aeval_eq', smul_add_smul_le_smul_add_smul', NonUnitalAlgebra.range_eq_top, map_eval_injective, HVertexOperator.coeff_apply_apply, div_le_egauge_closedBall, SymmetricAlgebra.algebraMap_inj, intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left, hasStrictFDerivAt_inv', MeasureTheory.Integrable.smul_prod, IntermediateField.lift_sup, basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix, AdicCompletion.evalₐ_mk, Basis.toDualFlip_apply, LinearMap.mem_span_iff_bound, cfcₙL_apply, Representation.asModuleEquiv_map_smul, Set.Finite.convexHull_eq_image, homeomorphSphereProd_apply_fst_coe, ContinuousOn.cfcₙ_nnreal, Subalgebra.centralizer_coe_image_includeLeft_eq_center_tensorProduct, WithSeminorms.partial_sups, Submodule.coe_set_smul, SameRay.exists_pos_left, LieAlgebra.IsKilling.chainBotCoeff_add_chainTopCoeff, MeasureTheory.Lp.toTemperedDistribution_apply, SchwartzMap.smulLeftCLM_sub, AlternatingMap.uncurryFin_apply, TemperedDistribution.smulLeftCLM_const, SchwartzMap.smulLeftCLM_apply_apply, strongConvexOn_iff_convex, wbtw_or_wbtw_smul_vadd_of_nonpos, ConvexCone.salient_positive, normalClosure.normal, intervalIntegral.integral_hasStrictDerivAt_right, strongConvexOn_zero, Algebra.IsStandardSmooth.iff_exists_basis_kaehlerDifferential, Affine.Simplex.faceOppositeCentroid_vsub_faceOppositeCentroid, fromModuleCatToModuleCatLinearEquiv_apply, SchwartzMap.fourier_fderivCLM_eq, Unitary.conjStarAlgAut_symm_unitaryLinearIsometryEquiv, DistribSMul.toAddMonoidHom_eq_nsmulAddMonoidHom, Asymptotics.isBigO_const_smul_left, ConvexBody.hausdorffEDist_coe, LinearMap.BilinForm.toMatrix'_toBilin', Fintype.linearCombination_apply_single, LinearEquiv.symm_smul_apply, HVertexOperator.of_coeff_apply, MDifferentiableOn.smul_section, MeasureTheory.SimpleFunc.integral_eq, AlgHom.toRingHom_toAddMonoidHom, integral_withDensity_eq_integral_toReal_smul₀, QuadraticMap.PosDef.smul, SchwartzMap.fourier_convolution_apply, Seminorm.smul_ball_zero, Fintype.linearCombination_apply, mem_openSegment_iff_div, Asymptotics.isBigOTVS_map, neg_cross, Projectivization.generalLinearGroup_smul_def, iteratedFDeriv_two_apply, starConvex_compl_Iic, ContinuousMultilinearMap.analyticOnNhd_uncurry_compContinuousLinearMap, fromModuleCatToModuleCatLinearEquiv_symm_apply_coe, PointedCone.mem_toConvexCone, LocalizedModule.map_mk, CategoryTheory.Presieve.FamilyOfElements.isCompatible_map_smul_aux, PiTensorProduct.lift_comp_reindex_symm, AffineSubspace.isometryEquivMap.apply_symm_apply, PiTensorProduct.map_update_smul, real_inner_smul_left, instIsTorsionFreeLocalizationAlgebraMapSubmonoidPrimeCompl_2, LieModule.coe_lcs_range_toEnd_eq, ContDiffAt.smul, ModuleCat.ofHom₂_compr₂, LinearMap.flip_separatingLeft, CFC.cfcₙ_rpowIntegrand₀₁_eq_cfcₙ_rpowIntegrand₀₁_one, LieModule.toEnd_module_end, IntermediateField.adjoin_adjoin_left, LieAlgebra.mem_zeroRootSubalgebra, TemperedDistribution.smulLeftCLM_sum, PowerBasis.constr_pow_mul, intervalIntegral.integral_hasStrictDerivAt_left, TensorProduct.map_smul_left, SubMulAction.mem_one', AddAut.mulLeft_apply_symm_apply, ModuleCat.Algebra.instSMulCommClassCarrier, LinearMap.det_toMatrix', LieAlgebra.isRegular_iff_finrank_engel_eq_rank, IntermediateField.normalClosure_mono, antivaryOn_iff_forall_smul_nonpos, Matrix.separatingLeft_toBilin_iff, End.apply_isScalarTower, WeakDual.toStrongDual_apply, ConvexCone.isGenerating_bot_iff, LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul, LinearMap.eq_adjoint_iff_basis_right, InnerProductSpace.laplacian_smul_nhds, LinearEquiv.arrowCongr_trans, Real.differentiable_fourierChar_neg_bilinear_left, Unitization.starLift_range_le, FormalMultilinearSeries.enorm_compContinuousLinearMap_le, ConvexBody.hausdorffEdist_coe, RatFunc.sub_def, sameRay_smul_right_iff, LinearMap.Nondegenerate.toMatrix₂, Pi.segment_subset, MultilinearMap.mkPiRing_apply, LinearMap.toMatrixAlgEquiv_apply', IsLocalizedModule.eq_iff_exists, VectorPrebundle.IsContMDiff.exists_contMDiffCoordChange, fderiv_norm_smul_pos, Fintype.expect_ite_eq', FormalMultilinearSeries.le_radius_pi, AffineSpace.asymptoticNhds_eq_smul_vadd, continuousMultilinearCurryLeftEquiv_symm_apply, isSemiprimaryRing_iff, Ideal.pow_mem_pow, MultilinearMap.map_smul_univ, WithLp.unitization_mul, hasDerivAt_exp_smul_const', CategoryTheory.ShortComplex.smul_τ₁, IsModuleTopology.continuous_bilinear_of_finite_right, DoubleCentralizer.instStarModule, TensorProduct.tensorQuotEquivQuotSMul_comp_mk, LinearMap.existsUnique_eq_smul_id_of_finrank_eq_one, solvableByRad.isIntegral, ContinuousMultilinearMap.smulRightL_apply, NNReal.instIsStrictOrderedModule, IntermediateField.restrictNormalHom_ker, FiniteGaloisIntermediateField.instIsSeparableSubtypeMemIntermediateFieldMin, Representation.smul_one_tprod_asModule, CStarAlgebra.star_right_conjugate_le_norm_smul, smul_nonneg_iff_pos_imp_nonneg, ContinuousLinearMap.isUniformEmbedding_toUniformOnFun, NNReal.instContinuousMapZero.UniqueHom, iteratedDerivWithin_scomp_two, exists_pos_left_iff_sameRay_and_ne_zero, CochainComplex.HomComplex.Cochain.comp_smul, sameRay_neg_smul_left_iff, LieSubalgebra.instIsScalarTowerSubtypeMem, RatFunc.add_def, TestFunction.instContinuousSMulReal, MeasureTheory.integral_comp_mul_left_Ioi, IntermediateField.adjoin_rank_le_of_isAlgebraic_left, ContinuousMap.norm_smul_const, Rep.freeLift_hom_single_single, iteratedFDeriv_succ_eq_comp_left, ContinuousMapZero.toNNReal_neg_smul, LinearMap.BilinForm.Nondegenerate.toMatrix', Submodule.smul_mem_smul, antivary_iff_forall_smul_nonpos, ContinuousLinearMap.smul_compLpL, VectorFourier.fourierPowSMulRight_iteratedFDeriv_fourierIntegral, MeasureTheory.SimpleFunc.map_integral, infinite_range_add_smul_iff, IntermediateField.insepDegree_bot, DivisibleHull.mk_zero, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_inl, LinearMap.toMatrix_basis_equiv, IsLocalizedModule.smul_mem_finsetIntegerMultiple_span, Ideal.top_pow, contMDiffOn_coordChangeL, cfcₙ_const_mul, IntermediateField.equivOfEq_apply, ContinuousLinearEquiv.arrowCongrSL_symm_apply, AffineMap.homothety_add, NormedSpace.ext_iff, SkewPolynomial.instSMulCommClass, CategoryTheory.ShortComplex.smul_τ₂, evalEquiv_toLinearMap, DivisibleHull.zero_qsmul, ContinuousLinearMap.opNorm_mulLeftRight_le, MultilinearMap.compLinearMapMultilinear_apply, gauge_smul_left, PointwiseConvergenceCLM.evalCLM_apply, Finset.sum_smul_vsub_eq_affineCombination_vsub, Submodule.map_toAddSubgroup, MeasureTheory.measure_smul_setAverage, ClosedSubmodule.orthogonal_closure, ContDiffPointwiseHolderAt.iteratedFDeriv, DualNumber.lift_comp_inlHom, LinearMap.trace_prodMap', ModuleFilterBasis.smul, AntivaryOn.sum_smul_le_sum_smul_comp_perm, AlgHom.linearMapMk_toAddHom, MeasureTheory.Integrable.comp_smul, LinearMap.toSeminormFamily_apply, MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure, End.mem_spectrum_iff_isRoot_charpoly, IsLocalizedModule.fromLocalizedModule.surj, ContDiffMapSupportedIn.structureMapCLM_zero_injective, Seminorm.ball_finset_sup_eq_iInter, LinearMap.dualMap_comp_dualMap, LinearMap.separatingLeft_iff_ker_eq_bot, Matrix.SeparatingRight.toBilin', LinearMap.lflip_symm, LinearMap.compQuadraticMap_polarBilin, ContinuousMultilinearMap.piₗᵢ_apply, meromorphicAt_smul_iff_of_ne_zero, Basis.opNNNorm_le, Derivation.leibniz_div_const, End.smulLeft_eq, Algebra.injective_lift_lsmul, BilinForm.apply_eq_dotProduct_toMatrix_mulVec, iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod, AdicCompletion.ofLinearEquiv_apply, TensorProduct.lift_compr₂, ContinuousMultilinearMap.ofSubsingletonₗᵢ_apply, Ideal.sup_pow_eq_top, KaehlerDifferential.range_kerCotangentToTensor, Ideal.mul_top, norm_iteratedFDerivWithin_fderivWithin, IntermediateField.fieldRange_val, IntermediateField.liftAlgEquiv_apply, LinearMap.lid_comp_rTensor, DualNumber.algHom_ext'_iff, Ideal.pow_eq_top_iff, MeasureTheory.Integrable.fourier_smul, LinearMap.coe_toContinuousLinearMap_symm, Matrix.range_diagonal, Unitization.lift_apply_apply, AdicCompletion.mk_apply_coe, hasStrictFDerivAt_list_prod', finsum_smul', right_sub_midpoint, balancedHull.balanced, Submodule.torsionBySet.mk_smul, continuousMultilinearCurryFin1_symm_apply, MeasureTheory.Measure.setIntegral_comp_smul_of_pos, AnalyticAt.exists_eventuallyEq_pow_smul_nonzero_iff, MeasureTheory.condExp_aestronglyMeasurable_bilin_of_bound, LinearMap.IsSymmetric.re_trace_eq_sum_eigenvalues, balancedCore_zero_mem, intervalIntegral.integral_comp_mul_add, HasSum.smul_const, separableClosure_le_separableClosure_iff, LieModule.Cohomology.mem_twoCocycle_iff_of_trivial, Set.smul_mem_centralizer, FormalMultilinearSeries.radius_pi_le, DualNumber.ideal_trichotomy, Submodule.orthogonal_eq_inter, VectorFourier.integral_fourierIntegral_smul_eq_flip, IsSymmetricAlgebra.comp_equiv, AffineSubspace.sOppSide_smul_vsub_vadd_left, LieModule.iterate_toEnd_mem_lowerCentralSeries, PadicInt.mahlerEquiv_symm_apply, LinearMap.extendScalarsOfIsLocalizationEquiv_symm_apply, ContinuousLinearMap.lsmul_flip_apply, smul_vsub_mem_vectorSpan_pair, RootPairing.Base.cartanMatrixIn_mul_diagonal_eq, Orientation.inner_smul_rotation_pi_div_two_smul_left, Submodule.mem_annihilator_span_singleton, reflection_apply, CategoryTheory.Functor.linear_iff, hasDerivAt_iff_tendsto, ContinuousLinearEquiv.toSpanNonzeroSingleton_apply_coe, Submodule.mem_span_set_iff_exists_finsupp_le_finrank, HasStrictFDerivAt.smul_const, SchwartzMap.coe_apply, Finset.sum_smul_const_vsub_eq_sub_weightedVSubOfPoint, LinearMap.dualMap_eq_lcomp, real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul, sub_smul_dslope, IsLocalRing.map_mkQ_eq_top, ContinuousLinearMap.compContinuousAlternatingMapCLM_apply_apply, isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule, LinearMap.trace_comp_comm, KaehlerDifferential.map_D, MeasureTheory.condExp_bot', WeakSpace.instIsScalarTower, Subbimodule.smul_mem, SchwartzMap.smulLeftCLM_ofReal, LinearMap.range_smulRight_apply, Orientation.oangle_smul_right_of_pos, norm_iteratedFDeriv_smul_le, KaehlerDifferential.instIsScalarTowerTensorProduct_1, realPart_smul, LinearMap.trace_eq_sum_trace_restrict', exists_eq_smul_of_parallel, ContDiffAt.differentiableAt_iteratedFDeriv, BilinForm.toMatrix_compLeft, TrivSqZeroExt.inv_zero, LinearMap.IsPositive.adjoint_eq, SMulCommClass.nnrat, abs_real_inner_div_norm_mul_norm_eq_one_iff, HasFDerivAt.hasFDerivAt_norm_smul_pos, Matrix.maxGenEigenspace_toLin_diagonal_eq_eigenspace, MeasureTheory.Integrable.smul_const, Ideal.iInf_pow_smul_eq_bot_of_isTorsionFree, Algebra.coe_lmul_eq_mul, Complex.integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn, ContinuousLinearMap.prodₗ_apply, mem_tangentConeAt_iff_exists_seq_norm_tendsto_atTop, Monotone.convex_le, ContinuousLinearEquiv.coord_norm, Basis.toDual_apply_left, LinearMap.nilRank_le_natTrailingDegree_charpoly, fderiv_norm_rpow, ContinuousMultilinearMap.nnnorm_smulRight, Ring.uniformContinuousConstSMul_op, intervalIntegral.integral_hasDerivAt_left, KaehlerDifferential.DLinearMap_apply, ContinuousLinearMap.norm_map_iff_adjoint_comp_self, zero_mem_tangentConeAt_iff, NonUnitalSubalgebra.map_map, ContDiffOn.smulRight, QuadraticMap.sum_repr_sq_add_sum_repr_mul_polar, LinearIsometryEquiv.map_smul, Algebra.TensorProduct.lmul''_eq_lid_comp_mapOfCompatibleSMul, DualNumber.eps_mul_eps, MeasureTheory.lpMeasToLpTrim_smul, RootPairing.pairing_one_four_iff, LinearMap.toMatrix₂'_comp, MvPolynomial.rTensor_apply, LinearEquiv.congrQuadraticMap_apply, ContinuousMultilinearMap.cpolynomialAt_uncurry_compContinuousLinearMap, Matrix.toLinearMap₂_apply_basis, MDifferentiableOn.smul_section_of_tsupport, MeasureTheory.integral_finset, LinearEquiv.conjAlgEquiv_surjective, ContinuousLinearMap.norm_adjoint_comp_self, IsPicardLindelof.exists_forall_mem_closedBall_eq_hasDerivWithinAt_lipschitzOnWith, Pi.evalStarAlgHom_apply, TensorProduct.lift_comp_comm_eq, Orientation.inner_smul_rotation_pi_div_two_smul_right, ContinuousLinearMap.map_add₂, hasDerivWithinAt_intCast, Rep.MonoidalClosed.linearHomEquivComm_symm_hom, Asymptotics.IsBigO.const_smul_self, ContinuousAt.integral_sub_linear_isLittleO_ae, CircleIntegrable.out, NonUnitalStarSubalgebra.prod_top, End.hasEigenvalue_iff_mem_spectrum, PolynomialModule.monomial_smul_lsingle, birkhoffAverage_apply_sub_birkhoffAverage, IsLinearTopology.tendsto_smul_zero, locallyConvexSpace_iff_zero, TensorPower.toTensorAlgebra_galgebra_toFun, TensorProduct.Algebra.moduleAux_apply, ContinuousLinearMap.continuousConstSMul_apply, HasFPowerSeriesWithinAt.compContinuousLinearMap, LieAlgebra.IsKilling.chainTopCoeff_le_chainLength, RingTheory.Sequence.map_first_exact_on_four_term_right_exact_of_isSMulRegular_last, LinearMap.not_hasEigenvalue_zero_tfae, ContinuousLinearMap.isCentralScalar, ContinuousMultilinearMap.norm_smulRight, LinearMap.toMatrix_pow, TensorAlgebra.algebraMap_eq_zero_iff, LSeries_hasDerivAt, LinearPMap.closure_inverse_graph, RingTheory.Sequence.isWeaklyRegular_iff, FinitePresentation.isLocalizedModule_mapExtendScalars, IntermediateField.rank_bot_mul_relrank, ContinuousLinearEquiv.continuousMultilinearMapCongrLeft_symm, Antivary.sum_smul_le_sum_comp_perm_smul, SameRay.sameRay_pos_smul_left, Antitone.quasiconcaveOn, NNRealRMK.integral_rieszMeasure, ContinuousMapZero.elemental_eq_top, LieDerivation.coe_ad_apply_eq_ad_apply, ContinuousMapZero.adjoin_id_dense, ContinuousLinearEquiv.strictConvex_preimage, CategoryTheory.ShortComplex.rightHomologyMap'_smul, LinearMap.BilinMap.polarBilin_toQuadraticMap, Mathlib.Tactic.LinearCombination.smul_const_le, IsBaseChange.linearMapLeftRightHom_apply, Algebra.trace_apply, ContinuousLinearMap.hasDerivWithinAt, Submodule.Convex.semilinear_range, Antivary.sub_smul_sub_nonpos, TensorAlgebra.toDirectSum_ι, FormalMultilinearSeries.derivSeries_apply_diag, CFC.negPart_smul, MultilinearMap.currySum_smul, CompactlySupportedContinuousMap.monotone_of_nnreal, Submodule.norm_orthogonalProjection, LinearMap.BilinForm.toMatrix_comp, SameRay.exists_nonneg_right, LinearEquiv.map_mem_invtSubmodule_iff, AnalyticAt.fderiv, ExteriorAlgebra.ιMulti_succ_curryLeft, Matrix.SpecialLinearGroup.toLin'_symm_apply, ContinuousMapZero.toNNReal_smul, fderiv_comp_smul, LinearMap.map_smul_inv, convex_vadd, ContinuousLinearMap.opNorm_lsmul, IsLocalizedModule.map_integerMultiple, Submodule.comap_smul', Seminorm.zero_comp, ProbabilityTheory.covarianceBilinDual_of_not_memLp, Algebra.Generators.CotangentSpace.fst_compEquiv_apply, Submodule.mem_set_smul, FormalMultilinearSeries.norm_changeOriginSeriesTerm, TemperedDistribution.smulLeftCLM_smul, ContinuousMultilinearMap.fderivCompContinuousLinearMap_apply, left_mem_openSegment_iff, IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn, LeftPreLieAlgebra.toIsScalarTower, TrivSqZeroExt.isUnit_inl_iff, finsum_smul, LinearMap.trace_comp_cycle, TrivSqZeroExt.instIsScalarTower, VectorField.lieBracketWithin_const_smul_right, HasGradientAt.hasDerivAt, ContinuousLinearEquiv.symm_conjContinuousAlgEquiv, ProbabilityTheory.strong_law_ae_of_measurable, InnerProductGeometry.angle_eq_pi_iff, Algebra.FormallyUnramified.comp_sec, HasFTaylorSeriesUpTo.zero_eq', Basis.toDual_apply_right, AffineSubspace.setOf_sOppSide_eq_image2, AlgCat.hom_tensorHom, TrivSqZeroExt.isUnit_iff_isUnit_fst, LinearMap.trace_comp_cycle', LinearMap.toMatrix'_one, Representation.dual_apply, LieAlgebra.SpecialLinear.val_singleSubSingle, Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, ConvexBody.coe_add, InnerProductSpace.continuousLinearMapOfBilin_apply, Localization.flat, ContinuousAlternatingMap.map_smul_univ, Polynomial.contentIdeal_mul_le_mul_contentIdeal, AdicCompletion.transitionMap_comp_eval_apply, map_rat_smul, Finsupp.linearCombination_comapDomain, NormedSpace.double_dual_bound, AdicCompletion.of_injective_iff, Submodule.comap_smul'', fourierCoeffOn.const_smul, Matrix.cstar_norm_def, ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id, PointwiseConvergenceCLM.withSeminorms, Submodule.baseChange_span, Function.Odd.smul_odd, ModuleCat.smulShortComplex_X₃_isModule, inv_smul_le_iff_of_neg, MeasureTheory.convolution_eq_swap, KaehlerDifferential.tensorKaehlerEquiv_tmul_D, ContinuousLinearMap.map_zero₂, Real.fourierInv_eq, Basis.opNorm_le, surjective_piEquiv_apply_iff, LinearMap.nondegenerate_restrict_of_disjoint_orthogonal, UniformCauchySeqOnFilter.one_smulRight, isConformalMap_complex_linear, mulInvariantVectorField_smul, ContDiff.fderiv_right, hasDerivAt_const_add_iff, Unitization.starLift_symm_apply, Submodule.dualAnnihilator_iSup_eq, ContinuousMultilinearMap.curryFinFinset_symm_apply, Submodule.map₂_map_map, HasFPowerSeriesWithinOnBall.pi, HasFPowerSeriesWithinOnBall.tendsto_partialSum, LieAlgebra.isNilpotent_iff_forall, Dual.eq_of_preReflection_mapsTo', SchwartzMap.smulLeftCLM_apply, ContDiffMapSupportedIn.withSeminorms', Submodule.dualAnnihilator_eq_bot_iff, TensorAlgebra.lift_unique, ContDiffMapSupportedIn.toBoundedContinuousFunctionCLM_apply, IntermediateField.finiteDimensional_map, TensorProduct.quotTensorEquivQuotSMul_comp_mk, Submodule.map_toAddSubmonoid', LieAlgebra.ad_ker_eq_bot_of_hasTrivialRadical, RatFunc.toFractionRingRingEquiv_symm_eq, Projectivization.mk_eq_mk_iff', ProbabilityTheory.uncenteredCovarianceBilinDual_apply, Orientation.kahler_rightAngleRotation_right, InnerProductSpace.Core.re_inner_smul_ofReal_smul_self, MvRatFunc.rank_eq_max_lift, Matrix.iSup_eigenspace_toLin_diagonal_eq_top, Coalgebra.TensorProduct.assoc_toLinearEquiv, LinearEquiv.dualMap_refl, MeasureTheory.condExpIndSMul_ae_eq_smul, HasFTaylorSeriesUpTo.fderiv, concaveOn_iff_convex_hypograph, cross_cross_eq_smul_sub_smul', Seminorm.ball_finset_sup, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, EuclideanGeometry.Sphere.orthRadius_parallel_orthRadius_iff, QuadraticForm.tmul_comp_tensorAssoc, TensorProduct.rTensorHomToHomRTensor_apply, IsSepClosure.isAlgClosure_of_perfectField, neg_strictConcaveOn_iff, LieAlgebra.IsKilling.span_weight_eq_top, LinearMap.tensorKerEquiv_apply, StarAlgEquiv.ofLeftInverse'_symm_apply, IsRegular.smul_right_inj, basis_toMatrix_mul_linearMap_toMatrix, LinearMap.IsAlt.self_eq_zero, Submodule.span_smul_span, AlternatingMap.alternatizeUncurryFin_smul, LinearMap.toMatrixₛₗ₂'_apply, Representation.dualTensorHom_comm, ContinuousLinearMap.precomp_apply, LinearMap.exact_lcomp_of_exact_of_surjective, ContinuousLinearMap.norm_holderL_le, mem_freeLocus, SmoothPartitionOfUnity.contMDiff_smul, IntermediateField.LinearDisjoint.adjoin_rank_eq_rank_left_of_isAlgebraic_right, Ideal.mul_mono_right, PolynormableSpace.withSeminorms, contDiff_infty_iff_fderiv, SchwartzMap.smulLeftCLM_compL_smulLeftCLM, interval_average_eq, Submodule.mem_biSup_iff_exists_dfinsupp, ContinuousAffineMap.toConstProdContinuousLinearMap_fst, Asymptotics.isBigOTVS_prodMk_left, segment_subset_closure_openSegment, gaugeSeminormFamily_ball, Matrix.toLinearMapRight'_one, cfcₙL_integrable, HahnModule.instIsTorsionFree, mdifferentiable_smul_section, isSMulRegular_iff_ker_lsmul_eq_bot, SubMulAction.neg_mem_iff, Matrix.maxGenEigenspace_toLin'_diagonal_eq_eigenspace, HasFDerivWithinAt.mul', LieAlgebra.LoopAlgebra.twoCocycleOfBilinear_coe, PresheafOfModules.germ_ringCat_smul, HasFPowerSeriesWithinOnBall.fderivWithin_of_mem_of_analyticOn, IntermediateField.normalClosure_le_iff_of_normal, rank_le_one_iff, IsLocalizedModule.map_surjective, QuadraticForm.associated_tmul, QuadraticMap.polarSym2_map_smul, intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_lip, SchwartzMap.instFourierInvSMul, Seminorm.comp_smul_apply, differentiableWithinAt_complex_iff_differentiableWithinAt_real, LinearEquiv.alternatingMapCongrRight_symm_apply_apply, instIsPushout, TemperedDistribution.derivCLM_apply_apply, AffineBasis.linear_combination_coord_eq_self, MultilinearMap.curryFinFinset_symm_apply, TorusIntegrable.function_integrable, AffineSubspace.setOf_wSameSide_eq_image2, Submodule.coe_toSubMulAction, AlgHom.fieldRange_of_normal, LinearMap.toMatrixAlgEquiv_reindexRange, ContinuousLinearMap.precomp_uniformConvergenceCLM_apply, IntermediateField.Lifts.eq_iff, MeasureTheory.Supermartingale.smul_nonneg, Finset.sum_smul_vsub_const_eq_weightedVSubOfPoint_sub, abs_signedDist_le_dist, ClassGroup.equivPic_apply, Orientation.norm_kahler, hasDerivWithinAt_Ici_of_tendsto_deriv, supportDim_quotSMulTop_succ_eq_of_notMem_minimalPrimes_of_mem_jacobson, Submodule.dualRestrict_apply, MeasureTheory.condExpIndSMul_smul', LinearMap.hasDerivAt, Submodule.smulMemClass, CategoryTheory.ShortComplex.rightHomologyMap_smul, TensorProduct.one_gradedMul, MDifferentiableWithinAt.smul, IsLocalizedModule.map_iso_commute, Basis.sum_equivFun, End.mem_genEigenspace, TensorProduct.AlgebraTensorModule.smul_eq_lsmul_rTensor, ContinuousLinearMap.postcompUniformConvergenceCLM_apply, Submodule.index_smul_le, egauge_smul_right, AddCommGroup.smul_top_eq_top_of_divisibleBy_int, MeasureTheory.weightedSMul_smul_measure, VectorField.lieBracket_smul_right, midpoint_vsub_midpoint_same_left, Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix, TensorProduct.piScalarRightHom_tmul, AlgebraicGeometry.Scheme.Modules.Hom.app_smul, HasFDerivAtFilter.restrictScalars, TensorAlgebra.ofDirectSum_comp_toDirectSum, LinearMap.hasDerivAtFilter, iteratedFDerivWithin_apply_eq_iteratedDerivWithin_mul_prod, QuadraticForm.tensorComm_symm, convex_halfSpace_lt, intervalIntegral.integral_hasFDerivAt, RatFunc.toFractionRingRingEquiv_apply, LinearMap.flip_surjective_iff₂, HasFiniteFPowerSeriesOnBall.fderiv_eq, Bialgebra.mul_compr₂_comul, smul_closedBall, PointedCone.to_isOrderedModule, ContMDiffOn.smul, MeasureTheory.condExp_stronglyMeasurable_bilin_of_bound, IsLocalFrameOn.coeff_sum_eq, IsBaseChange.linearMapRight, IntermediateField.normalClosure_map_eq, Complex.hasDerivAt_log_sub_logTaylor, MeasureTheory.SimpleFunc.setToSimpleFunc_smul_real, WithSeminorms.isVonNBounded_iff_seminorm_bounded, DualNumber.snd_mul, Submodule.dualAnnihilator_eq_bot_iff', LinearMap.ker_lsmul, ContinuousLinearMap.smulRight_apply, HasFPowerSeriesAt.locally_zero_iff, LocalizedModule.zero_mk, hasDerivAt_iff_tendsto_slope, TrivSqZeroExt.isUnit_or_isNilpotent, intervalIntegral.fderiv_integral, right_vsub_midpoint, MeasureTheory.IntegrableOn.continuousOn_smul_of_subset, RootPairing.reflection_apply, ContinuousLinearMap.analyticAt_bilinear, Matrix.toLpLin_apply, integral_withDensity_eq_integral_smul₀, IntermediateField.relrank_bot_left, ProbabilityTheory.IsGaussian.charFunDual_eq', IsModuleFiltration.toGradedSMul, HasFPowerSeriesAt.hasStrictFDerivAt, Polynomial.aevalTower_comp_toAlgHom, IsPrimitiveRoot.intermediateField_adjoin_isCyclotomicExtension, AnalyticOnNhd.fun_const_smul, Matrix.toLin'_apply, MeasureTheory.integral_prod_smul, ContinuousLinearMap.toPointwiseConvergenceCLM_apply, CliffordAlgebra.contractRight_ι, Set.add_smul_subset, QuotSMulTop.map_id, ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right, Unitization.algebraMap_eq_inlRingHom, isBoundedLinearMap_continuousMultilinearMap_comp_linear, LinearEquiv.arrowCongr_comp, isSMulRegular_on_quot_iff_lsmul_comap_le, Ideal.absNorm_algebraMap, FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm, MeasureTheory.integrableOn_Ioi_comp_rpow_iff, LieAlgebra.isNilpotent_ad_of_isNilpotent, RealRMK.le_rieszMeasure_tsupport_subset, innerSL_apply_coe, Submodule.map_dualCoannihilator_linearEquiv_flip, ContinuousLinearMap.bilinearRestrictScalars_eq_restrictScalars_restrictScalarsL_comp, Basis.constr_apply, fderivWithin_mul', AffineSubspace.sSameSide_smul_vsub_vadd_left, Derivation.leibniz_invOf, hasStrictDerivAt_exp_smul_const_of_mem_ball, Baer.ExtensionOfMaxAdjoin.eqn, smul_neg_iff_of_neg_left, ContMDiff.const_smul_section, LieModule.Weight.toLinear_neg, Ideal.map_pointwise_smul, Real.hasDerivAt_Gamma_nat, LinearMap.rTensor_smul, TensorProduct.isPushout', Orientation.oangle_smul_right_self_of_nonneg, MeasureTheory.FiniteMeasure.map_snd_prod, FreeNonUnitalNonAssocAlgebra.lift_of_apply, LinearMap.BilinForm.SeparatingRight.toMatrix', LinearPMap.supSpanSingleton_apply_mk, IsLocalizedModule.instIsTorsionFreeLocalizationLocalizedModuleOfIsDomain, TensorProduct.toLinearMap_symm_rid, hasDerivAt_pi, ContinuousLinearMap.toUniformConvergenceCLM_symm_apply, Submodule.smul_top_le_comap_smul_top, strictConvex_uIoc, MeasureTheory.ContinuousMap.inner_toLp, convexOn_const, Submodule.set_smul_top_eq_span, LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul', ContDiffMapSupportedIn.iteratedFDerivLM_eq_of_scalars, RestrictScalars.isCentralScalar, IsSymmetricAlgebra.equiv_toAlgHom, MeasureTheory.integrable_fun_norm_addHaar, Ideal.sup_mul, CategoryTheory.Linear.toCatCenter_apply_app, skewAdjoint.I_smul_neg_I, Derivation.linearEquiv_coe_comp, Dual.instLieModule, hasFDerivAt_inv', LinearMap.toMatrix_adjoint, ModuleCat.sMulCommClass_mk, PowerSeries.smul_eq_C_mul, SeminormFamily.basisSets_mem, LinearMap.flip_flip, MvPolynomial.tensorEquivSum_X_tmul_one, Finset.zero_mem_smul_iff, hasDerivWithinAt_finCons, Submodule.dualQuotEquivDualAnnihilator_symm_apply_mk, TensorProduct.AlgebraTensorModule.coe_rTensor, Ideal.quotTorsionOfEquivSpanSingleton_apply_mk, PresheafOfModules.germ_smul, Submodule.smul_le_span, LinearMap.diag_toMatrix_directSum_collectedBasis_eq_zero_of_mapsTo_ne, IntermediateField.le_normalClosure, CFC.abs_smul, Projectivization.map_mk, exists_continuousLinearEquiv_fderiv_symm_eq, QuadraticMap.tensorDistrib_tmul, Unitization.val_unitsFstOne_mulEquiv_quasiregular_apply, Subspace.map_dualCoannihilator, FormalMultilinearSeries.rightInv_coeff, Meromorphic.smul, OpenPartialHomeomorph.unitBallBall_symm_apply, LieAlgebra.Extension.twoCocycleOf_coe_coe, LinearMap.separatingLeft_toLinearMap₂'_iff_det_ne_zero, TensorAlgebra.ringQuot_mkAlgHom_freeAlgebra_ι_eq_ι, MeasureTheory.AEStronglyMeasurable.convolution_integrand', Basis.dvd_coord_smul, Orientation.oangle_sign_smul_add_smul_left, Matrix.Module.smul_def', tendsto_integral_comp_smul_smul_of_integrable', IsLinearMap.map_smul, CharacterModule.homEquiv_apply_apply, cfcₙHom_of_cfcHom_map_quasispectrum, LinearMap.map_smulₛₗ₂, SModEq.smul, IntermediateField.equivOfEq_trans, ExteriorAlgebra.invertibleAlgebraMapEquiv_symm_apply_invOf_toQuot, LinearMap.BilinForm.Nondegenerate.toMatrix, Ideal.IsTwoSided.instHMul, not_linearIndepOn_finset_iffₒₛ, isClosedMap_smul_left, ContinuousLinearMap.isPositive_adjoint_comp_self, SchwartzMap.integral_clm_comp_laplacian_right_eq_left, IntermediateField.normalClosure_def', LinearMap.dualCoannihilator_range_eq_ker_flip, ContDiffOn.differentiableOn_iteratedFDerivWithin, LinearMap.im_inner_adjoint_mul_self_eq_zero, AlgCat.hom_whiskerRight, QuadraticMap.associated_flip, MeasureTheory.average_union_mem_openSegment, PosSMulStrictMono.toSMulPosStrictMono, LinearMap.BilinForm.congr_symm, Algebra.TensorProduct.map_bijective, HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn', ClosedSubmodule.orthogonal_closure', ContinuousLinearMap.map_neg₂, LinearMap.compr₂ₛₗ_apply, Matrix.l2_opNorm_def, ZMod.instIsScalarTower, Real.fourierInv_eq', RestrictScalars.addEquiv_symm_map_algebraMap_smul, Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div', LinearMap.toMatrixAlgEquiv_mul, Ideal.span_pair_mul_span_pair, signedDist_apply_linear, ConvexBody.coe_nsmul, HasFiniteFPowerSeriesOnBall.sub, ContinuousMultilinearMap.norm_compContinuousLinearMapL_le, Bornology.isVonNBounded_singleton, Matrix.toEuclideanLin_toLp, Bundle.RiemannianMetric.continuousAt, ModuleCat.isScalarTower_of_algebra_moduleCat, mem_segment_translate, TensorProduct.lift_mk_compr₂, Algebra.Extension.Cotangent.map_mk, RootPairing.CoPolarizationIn_apply, MellinConvergent.cpow_smul, ContDiffOn.fun_smul, Real.fourierIntegral_deriv, Seminorm.closedBall_comp, Submodule.LinearDisjoint.linearIndependent_left_of_flat, ContinuousLinearMap.holderL_apply_apply, FormalMultilinearSeries.comp_rightInv_aux2, CFC.negPart_smul_of_nonneg, TrivSqZeroExt.mul_inv_rev, Algebra.Extension.CotangentSpace.map_comp, Basis.units_smul_span_eq_top, AdicCompletion.eval_surjective, MeasureTheory.integral_fun_snd, IsPicardLindelof.exists_forall_mem_closedBall_eq_forall_mem_Icc_hasDerivWithinAt, Unitization.quasispectrum_eq_spectrum_inr, fwdDiff_smul_const, ZLattice.covolume.tendsto_card_div_pow'', Asymptotics.isLittleOTVS_insert, Polynomial.sylveserMap_comp_adjSylvester, isClosedEmbedding_smul_left, ModuleCat.Algebra.instIsScalarTowerCarrier, MeasureTheory.condExpL1CLM_smul, Matrix.entryLinearMap_toAddMonoidHom, isConformalMap_const_smul, real_inner_smul_self_right, Submodule.smul_mem_iff'', hasDerivWithinAt_iff_tendsto_slope', MeasureTheory.AEStronglyMeasurable.convolution_integrand, dist_smul_add_one_sub_smul_le, Ring.instIsScalarTowerNormalClosureSubtypeAlgebraicClosureFractionRingMemIntermediateFieldNormalClosure_1, norm_innerSL_le, MeromorphicAt.smul, SeminormFamily.basisSets_iff, PowerBasis.constr_pow_aeval, Submodule.FG.smul, AffineSubspace.subtypeₐᵢ_toAffineMap, NonUnitalAlgHom.coe_codRestrict, VectorFourier.hasFDerivAt_fourierChar_smul, IntermediateField.mem_restrict, Ideal.iInf_pow_smul_eq_bot_of_noZeroSMulDivisors, MeasureTheory.LocallyIntegrableOn.continuousOn_smul, Finset.neg_smul, ConvexCone.IsReproducing.sub_eq_univ, LinearMap.IsAdjointPair.smul, Real.fourierIntegral_real_eq, Matrix.scalar_commute_iff, fderiv_const_smul_of_field, cfcₙHomSuperset_apply, PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal', FormalMultilinearSeries.changeOriginSeriesTerm_changeOriginIndexEquiv_symm, HasLineDerivAt.smul, Real.exists_extension_norm_eq, OrthonormalBasis.sum_repr', CliffordAlgebra.EvenHom.contract_mid, Finset.expect_ite_eq, continuousMultilinearCurryRightEquiv_apply, Seminorm.convex_closedBall, cfcₙ_apply, LinearMap.det_toMatrix, ContinuousLinearMap.integrable_of_bilin_of_bdd_right, Algebra.traceForm_toMatrix_powerBasis, Matrix.minpoly_toLin, basis_toMatrix_mul, Presentation.tautologicalRelations_relation, End.HasEigenvector.pow_apply, ContDiffMapSupportedIn.structureMapLM_zero_apply, nnnorm_cfcₙ_nnreal_le, LieDerivation.maxTrivSubmodule_eq_bot_of_center_eq_bot, Rep.coinvariantsTensorIndInv_mk_tmul_indMk, NumberField.mixedEmbedding.integral_comp_polarCoord_symm, MeasureTheory.average_union, CoalgHom.coe_toAddMonoidHom, OpenPartialHomeomorph.univUnitBall_symm_apply, Matrix.toLin'_toMatrix', Ideal.sup_mul_left_self, SmoothBumpCovering.embeddingPiTangent_coe, ContinuousAlternatingMap.prodLIE_symm_apply, analyticOrderAt_smul_eq_top_of_right, circleAverage_sub_sub_inv_smul_of_differentiable_on_off_countable, convex_zero, ContinuousLinearEquiv.arrowCongrSL_toLinearEquiv_apply, concaveOn_const, AlternatingMap.alternatizeUncurryFinLM_apply, AnalyticOnNhd.const_smul, DifferentiableOn.hasDerivAt, LinearMap.det_toLin, Bornology.sUnion_isVonNBounded_eq_univ, finrank_eq_one_iff_of_nonzero', SchwartzMap.integral_smul_deriv_right_eq_neg_left, QuadraticMap.polar_smul_right, ContinuousLinearMap.apply_apply', HasFPowerSeriesWithinAt.hasFDerivWithinAt, apply_le_nnnorm_cfcₙ_nnreal, IntermediateField.LinearDisjoint.isDomain, ProbabilityTheory.covarianceBilinDual_of_not_memLp', LinearMap.posSemidef_toMatrix_iff, Representation.linHom.invariantsEquivRepHom_apply_hom, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le', Orientation.two_zsmul_oangle_smul_smul_self, ContDiff.fderiv, Submodule.localized'_smul, ContMDiffAt.smul_section, instIsLocalizedModuleLinearMapIdLocalizationLocalizedModuleMapOfFinitePresentation, fderiv_list_prod', SchwartzMap.fourier_convolution, tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn, LinearMap.trace_mul_cycle', Basis.unitsSMul_apply, SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul, UnitAddTorus.span_mFourierLp_closure_eq_top, fderivWithin_comp_smul, Orientation.rotation_symm_apply, nnratCast_smul_eq, strictConvex_Ioi, MeasureTheory.indicatorConstLp_eq_toSpanSingleton_compLp, ContinuousMultilinearMap.analyticOnNhd_uncurry_of_linear, IntermediateField.finrank_dvd_of_le_right, LinearMap.BilinForm.nondegenerate_toBilin'_of_det_ne_zero', Orthonormal.inner_right_fintype, MultilinearMap.snoc_smul, NonUnitalStarSubalgebra.instSMulCommClass, LinearEquiv.smul_refl, HasMFDerivAt.mul', Basis.coord_unitsSMul, ContinuousLinearMap.flipₗᵢ_symm, HasFPowerSeriesAt.compContinuousLinearMap, MvPolynomial.scalarRTensor_apply_tmul_apply, LinearMap.isOrthoᵢ_flip, Asymptotics.IsBigOTVS.refl, Matrix.Represents.algebraMap, LieSubalgebra.instSMulMemClass, convex_closedConvexHull, WeakFEPair.f_modif_aux2, cfcₙ_comp_const_mul, Asymptotics.isTheta_const_smul_right, continuousOn_cfcₙ_nnreal_setProd, IsLocalizedModule.mk'_cancel, Algebra.trace_comp_trace, QuadraticMap.associated_isSymm, ContinuousLinearMap.postcomp_uniformConvergenceCLM_apply, exists_continuousLinearEquiv_fderivWithin_symm_eq, Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op, eHolderNorm_smul, SemimoduleCat.MonoidalCategory.leftUnitor_hom_apply, ContinuousLinearMap.bijective_iff_dense_range_and_antilipschitz, hasDerivAt_of_tendsto_locally_uniformly_on', FGModuleCat.Iso.conj_hom_eq_conj, AdicCompletion.transitionMap_map_one, mellin_cpow_smul, HasGradientWithinAt.hasFDerivWithinAt, Basis.toDual_eq_equivFun, FormalMultilinearSeries.rightInv_coeff_one, LinearMap.SeparatingLeft.toMatrix₂', TensorProduct.toMatrix_comm, Basis.sum_repr, Bialgebra.mul_compr₂_counit, ContinuousMapZero.coe_toContinuousMapHom, Differentiable.smul_const, WithLp.instProdIsBoundedSMul, RootPairing.coreflection_apply_coroot, TensorAlgebra.ι_eq_algebraMap_iff, IntermediateField.finrank_le_of_le_right, dualTensorHomEquivOfBasis_apply, LinearMap.toMatrix₂'_apply, Ideal.mul_right_self_sup, RightPreLieAlgebra.ext_iff, Submodule.toSubMulAction_one, ContinuousLinearMap.isBigOTVS_id, End.genEigenspace_le_smul, LinearMap.isAdjointPair_inner, MeasureTheory.condExpIndL1Fin_smul', LinearMap.IsPositive.adjoint_conj, ContinuousLinearMap.norm_holder_apply_apply_le, LieModule.Cohomology.d₂₃_comp_d₁₂, Unitization.unitsFstOne_val_val_fst, LinearMap.ker_tensorProductMk, ringKrullDim_le_ringKrullDim_quotSMulTop_succ, Unitization.lift_apply, LinearMap.orthogonal_span_singleton_eq_to_lin_ker, supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes, UniformConvergenceCLM.isVonNBounded_iff, DualNumber.instIsPrincipalIdealRing, Algebra.Generators.CotangentSpace.map_ofComp_surjective, LinearMap.ker_dualMap_eq_dualAnnihilator_range, absorbent_ball_zero, SMulCommClass.nnrat', Matrix.kroneckerStarAlgEquiv_apply, LieModule.commute_toEnd_of_mem_center_left, LinearMap.exists_mem_center_apply_eq_smul_of_forall_notLinearIndependent_of_basis, Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂, Submodule.norm_subtypeL_le, LinearEquiv.trans_dualMap_symm_flip, RootPairing.Equiv.instSMulCommClassMulOppositeAut, ContinuousLinearMap.hasStrictDerivAt, ContinuousLinearMap.analyticAt_uncurry_of_multilinear, isGalois_bot, hasDerivWithinAt_iff_tendsto_slope, LinearMap.BilinForm.separatingLeft_toMatrix_iff, MvPolynomial.tensorEquivSum_C_tmul_C, LinearMap.trace_comp_comm', constFormalMultilinearSeries_apply_succ, TensorProduct.sum_tmul_basis_left_injective, signedDist_left_lineMap, OreLocalization.add_smul, FormalMultilinearSeries.continuousOn, IntermediateField.intermediateFieldMap_symm_apply_coe, ZMod.smul_mem, NonUnitalStarAlgHom.subsingleton, ZMod.dft_smul_const, ProbabilityTheory.Kernel.integral_withDensity, ContinuousMultilinearMap.analyticOn_uncurry_compContinuousLinearMap, AEval.mem_mapSubmodule_apply, Orientation.oangle_sign_smul_sub_right, AlgebraicGeometry.structureSheafInType.smul_apply, PiTensorProduct.lift.unique, LieAlgebra.of_symm_smul, ContinuousMultilinearMap.isUniformInducing_toUniformOnFun, LieAlgebra.IsKilling.mem_sl2SubalgebraOfRoot_iff, Submodule.basis_of_pid_aux, Submodule.span_singleton_eq_span_singleton, QuadraticForm.associated_baseChange, MonotoneOn.nnnorm_cfcₙ, Representation.IntertwiningMap.coe_smul, Real.volume_preserving_transvectionStruct, VectorFourier.fourierIntegral_fderiv, ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv, ContinuousLinearMap.fpowerSeries_apply_one, fderiv_continuousLinearEquiv_comp', ContinuousLinearMap.smulRightL_apply_apply, Representation.asAlgebraHom_mem_of_forall_mem, KaehlerDifferential.map_liftBaseChange_smul, WeakFEPair.Λ_residue_zero, ContinuousLinearMapWOT.ContinuousLinearMap.continuous_toWOT, LinearMap.range_smul, ContinuousLinearMap.toSpanSingletonCLE_apply_apply, Matrix.toLpLinAlgEquiv_symm_apply, HasFDerivAt.fun_mul', Matrix.toBilin'_toMatrix', LieSubalgebra.instIsCentralScalarSubtypeMem, Seminorm.continuous_iff_continuous_comp, LieAlgebra.ad_ker_eq_self_module_ker, Algebra.Generators.Hom.id_comp, NonUnitalStarSubalgebraClass.subtype_apply, Algebra.Generators.H1Cotangent.map_comp_cotangentComplex_baseChange, AdicCompletion.eval_apply, gaugeRescale_smul, constFormalMultilinearSeries_zero, intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right, Ideal.mul_eq_bot, OpenPartialHomeomorph.unitBallBall_apply, LinearMap.trace_baseChange, Real.fourierIntegralInv_eq, LinearEquiv.multilinearMapCongrLeft_apply, Submodule.instSMulCommClassSubtypeMemTorsion', cfcₙ_const_mul_id, Asymptotics.IsBigO.isBigOTVS, Int.cast_smul_eq_zsmul, IsScalarTower.nnrat, CliffordAlgebra.contractLeftAux_apply_apply, taylor_within_apply, Asymptotics.isLittleOTVS_neg_left, IntermediateField.FG.exists_finset_maximalFor_isTranscendenceBasis_separableClosure, ContDiffBump.convolution_tendsto_right_of_continuous, Matrix.toLinearMapₛₗ₂_toMatrix₂, LinearIsometryEquiv.toContinuousLinearEquiv_smul, ContinuousAlternatingMap.compContinuousLinearMapCLM_apply, LinearMap.BilinForm.IsAlt.smul, ContinuousLinearMap.opNNNorm_mul_apply, CliffordAlgebra.contractLeft_one, le_comap_range_rTensor, LaurentPolynomial.instIsScalarTowerPolynomial, ContinuousMultilinearMap.curryFinFinset_apply, MultilinearMap.domCoprodDep'_apply, ContDiffAt.smulRight, LinearMap.toMatrix'_mulVec, LinearMap.trace_eq_sum_trace_restrict_of_eq_biSup, TensorProduct.AlgebraTensorModule.rightComm_symm, Unitization.inrRangeEquiv_symm_apply, Affine.Simplex.ninePointCircle_center, TestFunction.injective_toBoundedContinuousFunctionCLM, Submodule.le_traceDual_iff_map_le_one, Subspace.dualAnnihilator_inf_eq, VertexOperator.coeff_eq_ncoeff, derivWithin_smul_const, FormalMultilinearSeries.changeOriginSeriesTerm_bound, IsLocalizedModule.map_LocalizedModules, Matrix.SeparatingLeft.toLinearMap₂, MultilinearMap.uncurrySum_smul, IntermediateField.finiteDimensional_adjoin_pair, instFiniteDimensionalFractionRingOfFinite, MeasureTheory.integral_smul_measure, Fintype.not_linearIndependent_iffₒₛ, KaehlerDifferential.kerTotal_map', convex_Ioi, ModuleCat.RestrictionCoextensionAdj.HomEquiv.fromRestriction_hom_apply_apply, SchwartzMap.denseRange_toLpCLM, IntermediateField.normal_sup, isMIntegralCurve_comp_mul_ne_zero, ConformalAt.const_smul, SchwartzMap.toZeroAtInftyCLM_apply, FormalMultilinearSeries.constFormalMultilinearSeries_radius, Asymptotics.isLittleOTVS_sup, wbtw_smul_vadd_smul_vadd_of_nonpos_of_le, IntermediateField.coe_algebraMap_apply, hasEigenvector_toLin_diagonal, LinearMap.separatingRight_toLinearMap₂'_iff_det_ne_zero, cfcₙHom_nonneg_iff, Bialgebra.TensorProduct.assoc_symm_tmul, ContinuousLinearMap.toSpanSingletonCLE_symm_apply, VectorFourier.norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le, smul_pos_of_neg_of_neg, TensorProduct.quotTensorEquivQuotSMul_symm_comp_mkQ, ContinuousLinearMap.norm_precompL_le, Real.fderiv_fourierChar_neg_bilinear_left_apply, Surreal.dyadicMap_apply_pow, HasFTaylorSeriesUpToOn.restrictScalars, Differentiable.smul, isConformalMap_iff_is_complex_or_conj_linear, ZMod.dft_def, starConvex_iff_div, CStarMatrix.mapₙₐ_apply, ContinuousLinearMap.apply_apply, AbsolutelyContinuousOnInterval.smul, egauge_zero_right, dotProduct_toMatrix₂_mulVec, Presentation.tautological_relation, Asymptotics.isThetaTVS_iff_isTheta, Algebra.Extension.CotangentSpace.map_comp_apply, ContinuousLinearMap.norm_map_removeNth_le, End.exists_isNilpotent_isSemisimple, ContinuousMultilinearMap.hasStrictFDerivAt_uncurry, separableClosure.isGalois, Real.fourierIntegral_continuousMultilinearMap_apply', ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear_apply_apply, HasGradientAtFilter.hasDerivAtFilter, HVertexOperator.comp_apply, TensorProduct.gradedComm_gradedMul, IsSymmSndFDerivWithinAt.eq, ContinuousLinearMap.map_smulₛₗ₂, End.HasUnifEigenvalue.mem_spectrum, PiTensorProduct.mapLMultilinear_opNorm, hasDerivWithinAt_inter, IntermediateField.isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one, Bialgebra.TensorProduct.assoc_toAlgEquiv, hasFPowerSeriesAt_iff', Matrix.diagonal_comp_single, CategoryTheory.ShiftedHom.smul_comp, FDRep.dualTensorIsoLinHom_hom_hom, Real.fourier_eq', CompactlySupportedContinuousMap.instIsScalarTower, ContinuousLinearMap.instModuleFinite, IsPicardLindelof.exists_eq_forall_mem_Icc_hasDerivWithinAt₀, FreeNonUnitalNonAssocAlgebra.hom_ext_iff, hasDerivAt_exp_smul_const_of_mem_ball, IsTorsionBy.mk_smul, LinearIsometryEquiv.conjStarAlgEquiv_apply_apply, TensorPower.toTensorAlgebra_tprod, RingTheory.Sequence.isRegular_cons_iff', Real.fourierIntegral_iteratedFDeriv, hasMellin_const_smul, IntermediateField.restrictScalars_adjoin, Algebra.Extension.CotangentSpace.map_comp_cotangentComplex, LinearMap.toMatrixAlgEquiv_transpose_apply, star_rat_smul, deriv_const_smul_field, NonUnitalCStarAlgebra.toSMulCommClass, mem_convexJoin, Submodule.orthogonalProjection_norm_le, OreLocalization.universalHom_commutes, HasFDerivAt.mul', WeakFEPair.Λ_residue_k, convex_iff_forall_pos, IntermediateField.finrank_adjoin_simple_eq_one_iff, MeasureTheory.StronglyMeasurable.norm_approxBounded_le, WithSeminorms.hasBasis_ball, MultilinearMap.compLinearMapₗ_apply, ContDiff.fun_smul, Monovary.sum_smul_sum_le_card_smul_sum, Algebra.IsAlgebraic.rank_fractionRing_polynomial, exteriorPower.presentation.relations_relation, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, NumberField.mixedEmbedding.integral_comp_polarCoordReal_symm, Orientation.rotationAux_apply, Fourier.fourierIntegral_comp_add_right, Finset.sum_smul_sum, NonUnitalSubalgebra.map_id, smul_finsum, Quaternion.fst_re_dualNumberEquiv_symm, MeasureTheory.MemLp.toLp_const_smul, QuadraticMap.basisRepr_apply, Set.Finite.isVonNBounded, slope_def_module, openSegment_subset_union, smul_right_inj, ContinuousLinearEquiv.arrowCongrSL_toLinearEquiv_symm_apply, IsBaseChange.endHom_comp, End.eigenspace_aeval_polynomial_degree_1, le_egauge_closedBall_one, hasFDerivAt_list_prod_attach', Complex.affine_of_mapsTo_ball_of_norm_dslope_eq_div, Quaternion.imI_snd_dualNumberEquiv, MeasureTheory.AEStronglyMeasurable.convolution_integrand_swap_snd, NonUnitalStarSubalgebra.inclusion_self, MeasureTheory.IntegrableOn.smul_continuousOn_of_subset, MeasureTheory.norm_condExpL2_le_one, HasCompactSupport.convolution_integrand_bound_right, LinearMap.traceAux_def, bddAbove_smul_iff_of_neg, HasFPowerSeriesWithinOnBall.hasFDerivWithinAt, MonovaryOn.smul_add_smul_le_smul_add_smul, integral_comp_pi_polarCoord_symm, MultilinearMap.fromDFinsuppEquiv_symm_apply, CategoryTheory.ShiftedHom.mk₀_smul, SkewPolynomial.smul_monomial, Submodule.mem_span_insert, DirectSum.decompose_smul, Algebra.leftMulMatrix_apply, Asymptotics.isBigOTVS_comm, Submodule.toConvexCone_bot, Seminorm.finset_sup_le_sum, normalClosure.is_finiteDimensional, ContinuousLinearEquiv.conjContinuousAlgEquiv_apply_apply, IsLocalizedModule.isInteger_smul, convexJoin_singleton_right, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, IsSMulRegular.of_ne_zero, AdicCompletion.ofLinearEquiv_symm_of, RCLike.re_extendTo𝕜'ₗ, InnerProductSpace.symm_toEuclideanLin_rankOne, Subspace.isCompl_dualAnnihilator, LinearMap.BilinForm.toMatrix'_compRight, isBoundedBilinearMap_smulRight, ModuleFilterBasis.smul_right', MeasureTheory.charFun_map_smul, LocalizedModule.mkLinearMap_apply, convexHull_eq_self, convex_closedBall, IntermediateField.finiteDimensional_sup, NonUnitalStarSubalgebra.map_id, Localization.sub_mk, Algebra.Extension.Cotangent.ker_mk, IntermediateField.Lifts.le_iff, LinearMap.toAddMonoidHom_mulRight, IsLocalizedModule.mk'_cancel_left, Submodule.map_toAddSubmonoid, Matrix.op_smul_eq_vecMul, ProbabilityTheory.uncenteredCovarianceBilinDual_zero, convexOn_iff_convex_epigraph, ContinuousLinearMap.adjoint_innerSL_apply, SameRay.pos_smul_right, sameRay_iff_norm_smul_eq, LSeries.hasDerivAt_term, Finset.expect_ite_eq', Basis.eval_ker, IsBoundedBilinearMap.smul_right, LinearPMap.IsClosable.graph_closure_eq_closure_graph, Algebra.Extension.Hom.toAlgHom_id, finrank_eq_zero_iff, WeierstrassCurve.Projective.nonsingularLift_zero, DifferentiableWithinAt.smul, ContDiffMapSupportedIn.structureMapLM_apply, ExteriorAlgebra.liftAlternating_ι, TensorProduct.AlgebraTensorModule.lcurry_apply, LieDerivation.instSMulBase, trace_adjoinSimpleGen, derivWithin_const_smul', isBoundedLinearMap_prod_multilinear, LieAlgebra.SpecialLinear.singleSubSingle_sub_singleSubSingle', Matrix.SeparatingRight.toBilin, Ideal.span_singleton_pow, reflection_mul_reflection_mul_reflection_pow_apply_self, LieModule.coe_maxTrivLinearMapEquivLieModuleHom, AlgEquiv.eq_linearEquivConjAlgEquiv, Orthonormal.inner_sum, End.algebraMap_isUnit_inv_apply_eq_iff, IntermediateField.normal_iff_normalClosure_eq, HasCompactSupport.convolution_integrand_bound_right_of_subset, IntermediateField.rank_adjoin_simple_eq_one_iff, Finsupp.weight_apply, HasStrictDerivAt_ofNat, Function.Periodic.const_inv_smul₀, Real.fourier_smul_convolution_eq, reflection_mul_reflection_mul_reflection_zpow_apply_self, PointedCone.smul_mem, CliffordAlgebra.contractLeftAux_contractLeftAux, reflection_mul_reflection_zpow, lipschitzGroup.conjAct_smul_range_ι, balancedCore_eq_iInter, postcomp_compactConvergenceCLM_apply, HasFiniteFPowerSeriesAt.comp, Subspace.comap_dualAnnihilator_dualAnnihilator, Submodule.exists_smith_normal_form_of_le, LinearMap.toMatrix_apply, AlgebraicGeometry.Proj.lift_awayMapₐ_awayMapₐ_surjective, hasDerivWithinAt_iff_tendsto, AlgHom.coe_toAddMonoidHom, balancedHull_convexHull_subseteq_absConvexHull, erange_coe, TensorProduct.toMatrix_map, Matrix.toLinAlgEquiv_toMatrixAlgEquiv, MeasureTheory.integral_countable', finrank_le_one_iff, Matrix.rank_vecMulVec, TemperedDistribution.laplacianCLM_apply, LieAlgebra.IsKilling.rootSystem_root_apply, isBigO_rpow_zero_log_smul, ConvexBody.coe_smul', AdicCompletion.map_of, ContinuousLinearMap.toWOT_apply, IsLocalization.mem_span_iff, IntermediateField.isIntegral_iff, Monovary.sub_smul_sub_nonneg, PresheafOfModules.Sheafify.SMulCandidate.h, MeasureTheory.SignedMeasure.toJordanDecomposition_smul, Wbtw.mem_segment, Rep.coinvariantsTensorFreeToFinsupp_mk_tmul_single, mem_contMDiffFiberwiseLinear_iff, Submodule.le_dualCoannihilator_dualAnnihilator, segment_inter_eq_endpoint_of_linearIndependent_sub, IsLocalizedModule.mk'_eq_iff, NonUnitalStarSubalgebra.instIsScalarTower', ContinuousLinearMap.norm_smulRightL_apply, ContinuousAlternatingMap.alternatizeUncurryFin_apply, Submodule.torsionBy.mk_smul, hasDerivAt_finCons, Asymptotics.IsThetaTVS.rfl, Algebra.Extension.Hom.toAlgHom_apply, HasFDerivAt.smul, CStarAlgebra.exists_sum_four_nonneg, RootPairing.EmbeddedG2.threeShortAddLongRoot_eq, FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin, CategoryTheory.Linear.smul_comp, Mathlib.Tactic.LinearCombination.smul_const_lt_weak, Submodule.toSubMulAction_injective, CategoryTheory.Pretriangulated.Triangle.smul_hom₁, ConvexBody.coe_zero, Matrix.toLinAlgEquiv_symm, SchwartzMap.smulLeftCLM_smul, LinearIndependent.pair_add_smul_add_smul_iff, FinitePresentation.exists_basis_localizedModule_powers, KaehlerDifferential.instIsScalarTowerTensorProduct_2, DoubleCentralizer.coe_snd, MemHolder.smul_iff, hasFPowerSeriesAt_iff, ContinuousLinearMap.fpowerSeriesBilinear_apply_one, MeasureTheory.integral_convolution, LinearMap.toMatrix_id_eq_basis_toMatrix, Seminorm.closedBall_finset_sup_eq_iInter, MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le, CategoryTheory.ShortComplex.leftHomologyMap_smul, Seminorm.comp_add_le, fderiv_norm_sq_apply, DivisibleHull.nnqsmul_mk, separableClosure.normalClosure_eq_self, LinearMap.toMatrixAlgEquiv_transpose_apply', Affine.Simplex.centroid_vsub_point_eq_smul_vsub, Polynomial.mkDerivation_apply, CFC.nnrpow_map_prod, LinearMap.baseChangeHom_apply, mulLinearMap_eq_mul, RightDerivMeasurableAux.le_of_mem_A, LinearMap.toMatrixRight'_comp, Derivation.evalAt_apply, LinearMap.isOrtho_def, Submodule.isPrimary_iff_zero_divisor_quotient_imp_nilpotent_smul, sub_div_sub_smul_slope_add_sub_div_sub_smul_slope, SemimoduleCat.forget₂_map, LieSubalgebra.instIsArtinianSubtypeMem, ContinuousMultilinearMap.norm_curryMid, Submodule.smul_mem, Matrix.kroneckerStarAlgEquiv_symm_apply, zero_mem_tangentCone, Localization.add_mk_self, Algebra.Generators.H1Cotangent.δ_map, AlternatingMap.curryLeft_add, ContinuousAffineMap.instIsCentralScalar, CategoryTheory.ShortComplex.opcyclesMap'_smul, LinearMap.uncurryLeft_apply, aestronglyMeasurable_withDensity_iff, LinearMap.trace_map, IntermediateField.adjoin.powerBasis_dim, Affine.Simplex.points_vsub_eulerPoint, AdicCompletion.of_apply, Submodule.starProjection_tendsto_closure_iSup, LieSubalgebra.smul_mem, InnerProductGeometry.angle_smul_left_of_pos, TensorProduct.AlgebraTensorModule.assoc_eq, SMulCommClass.rat, norm_iteratedFDeriv_fderiv, TrivSqZeroExt.inv_inr, MeasureTheory.Lp.toTemperedDistribution_smul_eq, NormedSpace.dual_def, QuadraticMap.toMatrix'_smul, Unitization.mem_unitsFstOne, FinitePresentation.isLocalizedModule_map, IsGalois.of_fixedField_normal_subgroup, Algebra.TensorProduct.lift_includeLeft_includeRight, HVertexOperator.coeff_of_coeff, IsAddUnit.smul_right, MeasureTheory.Lp_toLp_restrict_smul, MeasureTheory.Measure.integral_comp_div, intervalIntegral.integral_const, const_smul_mfderiv, fderiv_norm_sq, Filter.EventuallyEq.hasDerivAtFilter_iff, LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm, LinearMap.applyₗ_apply_apply, IntermediateField.essFiniteType_iff, LinearMap.IsAlt.neg, Field.Emb.cardinal_separableClosure, HasMFDerivAt.const_smul, Subspace.finrank_add_finrank_dualAnnihilator_eq, Matrix.toBilin_comp, IntermediateField.isPurelyInseparable_sup, NonUnitalCommCStarAlgebra.toIsScalarTower, AlternatingMap.curryLeft_same, MeasureTheory.condExp_bot_ae_eq, map_inv_intCast_smul, LinearMap.polyCharpoly_coeff_eval, FormalMultilinearSeries.hasFPowerSeriesOnBall, KaehlerDifferential.mulActionBaseChange_smul_tmul, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj, ModuleFilterBasis.continuousSMul, ContinuousLinearMap.coe_mulₗᵢ, deriv_fun_const_smul', LieModuleHom.map_lie₂, IntermediateField.splits_of_splits, LinearMap.BilinForm.apply_smul_sub_smul_sub_eq, SimpleGraph.mem_ker_toLin'_lapMatrix_of_connectedComponent, lsum_comp_mapRange_toSpanSingleton, TensorProduct.instIsCentralScalar, balancedCoreAux_empty, NonUnitalSubalgebra.range_val, isTorsionBy_quotient_element_smul, Seminorm.ball_bot, LocalizedModule.oreEqv_eq_r, LinearMap.det_toMatrix_eq_det_toMatrix, Asymptotics.IsBigO.smul_isLittleO, left_sub_midpoint, Algebra.TensorProduct.mul_apply, UniversalEnvelopingAlgebra.lift_ι_apply', ContinuousLinearMap.innerSL_apply_comp, Matrix.SpecialLinearGroup.toLin'_apply, Finite.of_isComplemented_codomain, Balanced.smul_mono, DualNumber.isNilpotent_eps, CategoryTheory.ShortComplex.leftHomologyMap'_smul, ContinuousLinearMap.toSesqForm_apply_norm_le, MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd, hasDerivAt_abs_pos, mem_convexAddSubmonoid, midpoint_vsub_midpoint_same_right, ContinuousLinearMap.coeFn_holder, ContDiffMapSupportedIn.structureMapLM_eq, Submodule.piQuotientLift_mk, Algebra.lsmul_coe, MeasureTheory.L1.SimpleFunc.setToL1S_smul_left, Affine.Simplex.centroid_vsub_eq, NonUnitalStarSubalgebra.inclusion_inclusion, LinearMap.toKerIsLocalized_apply_coe, Orientation.oangle_smul_left_of_pos, openSegment.lift, HilbertBasis.hasSum_repr_symm, ContinuousLinearMap.map_smul, CStarMatrix.norm_def, ContinuousAlternatingMap.smulRight_apply, AlternatingMap.instIsCentralScalar, ContinuousLinearMap.coe_deriv₂, IsMIntegralCurveAt.eventually_hasDerivAt, ContinuousLinearEquiv.symm_conjContinuousAlgEquiv_apply_apply, openSegment_subset_iff_segment_subset, OrderedFinpartition.norm_compAlongOrderedFinpartitionL_le, RestrictScalars.smul_def, ModuleCat.ihom_ev_app, LinearMap.adjoint_id, inner_smul_left, contDiff_succ_iff_hasFDerivAt, MeasureTheory.integrableOn_Ioi_comp_rpow_iff', TrivSqZeroExt.eq_smul_exp_of_invertible, InnerProductSpace.nullSubmodule_le_ker_toDualMap_right, Submodule.restrictScalars_self, MultilinearMap.currySumEquiv_symm_apply, LieModule.Weight.hasEigenvalueAt, LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton, TensorProduct.finsuppLeft'_apply, Unitization.unitsFstOne_val_inv_val_fst, ContinuousAlternatingMap.alternatizeUncurryFinCLM_apply, mem_segment_sub_add, ContinuousLinearMap.adjoint_adjoint, Ideal.exists_pow_inf_eq_pow_smul, HasCompactSupport.hasFDerivAt_convolution_left, TrivSqZeroExt.map_id, Submodule.flip_quotDualCoannihilatorToDual_injective, Algebra.Extension.H1Cotangent.map_comp, LinearMap.lflip_apply, HolderWith.smul_iff, MeasureTheory.Integrable.smul_of_top_right, Bundle.ContinuousRiemannianMetric.isVonNBounded, TensorProduct.lift.equiv_apply, MonovaryOn.sub_smul_sub_nonneg, IsBaseChange.linearMapLeftRightHom_comp, Algebra.Extension.cotangentComplex_mk, IsLocalizedModule.fromLocalizedModule'_smul, TemperedDistribution.smulLeftCLM_apply_apply, Subalgebra.centralizer_range_includeRight_eq_center_tensorProduct, Subspace.dualPairing_eq, Submodule.toSubMulAction_mono, TensorProduct.Algebra.mul'_comp_tensorTensorTensorComm, LinearMap.SeparatingLeft.congr, NonUnitalAlgHom.subtype_comp_codRestrict, Matrix.Nondegenerate.toLinearMap₂', AlgHom.mulLeftRightMatrix.comp_inv, WithSeminorms.hasBasis_zero_ball, AdicCompletion.eval_comp_of, Finset.sum_smul_vsub_eq_weightedVSub_sub, wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos, retractionOfSectionOfKerSqZero_tmul_D, Circle.norm_smul, instIsPushoutFractionRingMvPolynomial_1, locallyConvexSpace_iff_exists_convex_subset_zero, KaehlerDifferential.linearCombination_surjective, uniformConvexOn_zero, IsLocalizedModule.map_lTensor, ContMDiff.coordChangeL, Submodule.restrictScalars_localized'_smul, Matrix.separatingRight_toLinearMap₂'_iff_separatingRight_toLinearMap₂, LinearMap.mulLeft_toAddMonoidHom, VectorField.lieBracketWithin_const_smul_left, mellin_comp_mul_left, DifferentiableOn.fun_smul, Submodule.map_smul, Basis.coord_toDualEquiv_symm_apply, convexOn_iff_pairwise_pos, Convex.combo_affine_apply, convex_segment, vadd_left_mem_affineSpan_pair, Finsupp.linearCombination_onFinset, Basis.baseChange_linearMap, MeasureTheory.SignedMeasure.im_toComplexMeasure, int_smul_eq_zsmul, HasStrictFDerivAt.smul, ContinuousLinearMap.orthogonal_ker, left_vsub_midpoint, ContDiffWithinAt.fderivWithin_right, LinearMap.IsRefl.ker_flip, AnalyticAt.hasStrictDerivAt, LinearMap.toMatrixAlgEquiv'_comp, LinearMap.ker_smul, BoxIntegral.unitPartition.tag_mem_smul_span, signedDist_vadd_vadd, Asymptotics.IsThetaTVS.refl, ContinuousLinearMap.continuousAt_uncurry_of_multilinear, Asymptotics.IsLittleO.const_smul_left, ContDiffMapSupportedIn.norm_toBoundedContinuousFunction, AffineMap.coe_homothety, ContMDiff.smul_section, TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_tmul, Continuous.cfcₙ_nnreal', LinearPMap.IsClosable.existsUnique, contDiffWithinAt_succ_iff_hasFDerivWithinAt, Filter.EventuallyEq.hasDerivWithinAt_iff, Seminorm.neg_mem_closedBall_zero, LinearMap.toMatrix₂Aux_apply, intervalIntegral.inv_smul_integral_comp_add_div, AlgebraicGeometry.Scheme.Modules.map_smul, SeminormFamily.basisSets_smul_left, mapEvalEquiv_symm_apply, LieModule.rank_le_natTrailingDegree_charpoly_ad, InnerProductSpace.norm_rankOne, MeasureTheory.Measure.ae_smul_measure_eq, TensorProduct.quotTensorEquivQuotSMul_mk_tmul, Complex.hasDerivAt_Gamma_nat, RestrictScalars.addEquiv_map_smul, derivationQuotKerSq_mk, hasFDerivAt_integral_of_dominated_loc_of_lip, LieSubalgebra.coe_ad, smul_eq_zero, tendsto_smul_comp_nat_floor_of_tendsto_nsmul, ArchimedeanClass.mk_smul, DistribMulActionHom.instSemilinearMapClass, IntermediateField.isPurelyInseparable_adjoin_simple_iff_pow_mem, TensorProduct.uncurry_apply, Submodule.map_le_smul_top, ContDiffMapSupportedIn.iteratedFDerivWithOrderLM_apply_of_le, Antivary.sum_comp_perm_smul_eq_sum_smul_iff, PiTensorProduct.mapLMultilinear_toFun_apply, IsLocalizedModule.eq_zero_iff, TensorProduct.tensorQuotEquivQuotSMul_symm_comp_mkQ, CStarMatrix.inner_toCLM_conjTranspose_right, instSMulMemClassHomogeneousSubmodule, LinearIsometryEquiv.map_smulₛₗ, HasFiniteFPowerSeriesOnBall.fderiv, IsLocalizedModule.surj', LieModule.toEnd_matrix, real_inner_I_smul_self, AnalyticWithinAt.smul, Seminorm.closedBall_smul_closedBall, ContinuousAffineMap.toConstProdContinuousLinearMap_snd, balancedCore_subset_balancedCoreAux, Orthonormal.inner_right_sum, mellin_comp_mul_right, IsLocalizedModule.surj, PointedCone.dual_singleton, RingFilterBasis.SubmodulesBasis.smul, ContinuousMultilinearMap.curryMidEquiv_symm_apply, FormalMultilinearSeries.comp_summable_nnreal, bernsteinApproximation.apply, LinearMap.sum_repr_mul_repr_mul, fderiv_eq_smul_deriv, CStarMatrix.toCLM_apply_single_apply, LinearMap.IsPositive.conj_adjoint, ContinuousMultilinearMap.uncurryRight_norm, ContinuousLinearMap.flipAlternating_apply_apply, LinearMap.IsPerfPair.dualEval, LinearMap.ker_toContinuousLinearMap, TensorProduct.AlgebraTensorModule.lTensor_tmul, PointedCone.salient_iff_inter_neg_eq_singleton, IntermediateField.adjoin_rootSet_isSplittingField, IsLocalizedModule.iso_mk_one, IntermediateField.nonempty_algHom_adjoin_of_splits, Submodule.span_smul_of_span_eq_top, rieszContentAux_le, IntermediateField.isScalarTower_mid, IntermediateField.finSepDegree_bot, QuadraticModuleCat.hom_hom_associator, Polynomial.C_smul_derivation_apply, LinearMap.charpoly_toMatrix, exists_pos_right_iff_sameRay_and_ne_zero, closedConvexHull_isClosed, LieAlgebra.IsKilling.instIsIrreducibleSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystemOfIsSimple, LinearMap.toLinearMap₂'Aux_toMatrix₂Aux, LieModule.toEnd_pow_comp_lieHom, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge, ContinuousSMul.induced, LieHom.toNonUnitalAlgHom_injective, PointedCone.toConvexCone_injective, LinearMap.isSymm_iff_isHermitian_toMatrix, Monotone.convex_gt, LinearMap.trace_lie, Unitization.cfcₙ_eq_cfc_inr, ContinuousLinearMap.algebraMap_apply, LinearMap.lsmul_eq_distribSMultoLinearMap, MeasureTheory.average_eq, IsAzumaya.AlgHom.mulLeftRight_bij, MDifferentiableWithinAt.smul_section, Matrix.toBilin'_single, IsCyclotomicExtension.nonempty_algEquiv_adjoin_of_isSepClosed, ContMDiffOn.smul_section, perfectClosure.isPurelyInseparable, TensorProduct.liftAux.smulₛₗ, AnalyticWithinAt.exists_hasFTaylorSeriesUpToOn, FinitePresentation.linearEquivMapExtendScalars_symm_apply, iteratedFDeriv_succ_apply_left, LinearMap.separatingLeft_dualProd, IsSymmSndFDerivAt.eq, WithSeminorms.finset_sups, LinearEquiv.conjAlgEquiv_ext_iff, LinearMap.toMatrix_toLin, Submodule.restrictScalars_map, IntermediateField.isSeparable_adjoin_iff_isSeparable, hasDerivWithinAt_sub_const_iff, ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux, TensorProduct.includeRight_lid, intervalIntegral.smul_integral_comp_add_mul, FreeNonUnitalNonAssocAlgebra.lift_comp_of, Complex.restrictScalars_toSpanSingleton', AffineSubspace.sOppSide_smul_vsub_vadd_right, HasFDerivWithinAt.list_prod', Complex.hasSum_taylorSeries_on_eball, ofMul_uzpow, exists_smul_eq_of_finrank_eq_one, Quaternion.imK_snd_dualNumberEquiv, LinearMap.range_localizedMap_eq_localized₀_range, prod_galRestrict_eq_norm, Flat.isSMulRegular_of_nonZeroDivisors, fderiv_inv', MeasureTheory.Measure.integral_comp_inv_mul_right, TensorProduct.finsuppLeft_apply_tmul_apply, ContinuousLinearMapWOT.ext_inner_iff, Besicovitch.SatelliteConfig.exists_normalized_aux3, NormedSpace.exp_eq_expSeries_sum, MultilinearMap.currySum_apply, TwoSidedIdeal.coe_mop_smul, Monovary.sum_smul_comp_perm_eq_sum_smul_iff, instIsPushoutFractionRingMvPolynomial, IsTensorProduct.map_eq, PolynomialModule.eval_single, VectorField.mlieBracket_const_smul_right, norm_inner_div_norm_mul_norm_eq_one_iff, Submodule.disjoint_span_singleton'', ProbabilityTheory.integral_tilted_mul_eq_cgf, FormalMultilinearSeries.comp_assoc, egauge_smul_left, DirectSum.gMulLHom_apply_apply, MeasureTheory.Measure.addHaar_smul, HasFPowerSeriesWithinAt.comp, LinearMap.polyCharpolyAux_map_eq_toMatrix_charpoly, IsLocalizedModule.mk'_sub_mk', MvPolynomial.pderiv_inl_universalFactorizationMap_X, Algebra.Extension.H1Cotangent.map_toInfinitesimal_bijective, autEquivRootsOfUnity_smul, fderivWithin_pow', Algebra.Extension.Cotangent.val_smul'', QuadraticModuleCat.hom_inv_associator, ModuleCat.forget₂_map, hasEigenvalue_toLin_diagonal_iff, balanced_zero, continuousMultilinearCurryFin1_apply, PiTensorProduct.lift_symm, NonUnitalStarAlgHom.range_comp, locallyConvexSpace_iff_exists_convex_subset, RootPairing.reflection_apply_root, LineDifferentiableAt.smul, IsLocalizedModule.injective_iff_isRegular, CliffordAlgebra.contractLeft_algebraMap, Matrix.spectrum_toLin', star_intCast_smul, RootPairing.reflectionPerm_eq_iff_smul_coroot, LinearMap.dualAnnihilator_ker_eq_range_flip, AdicCompletion.ofAlgEquiv_symm_of, HilbertBasis.finite_spans_dense, Derivation.apply_aeval_eq, notMem_support_iff', LieSubalgebra.ad_comp_incl_eq, MeasureTheory.setIntegral_const, norm_combo_lt_of_ne, Matrix.GeneralLinearGroup.fin_two_smul_prod, Ideal.mulQuot_injective, ConvexCone.mem_comap, MvPolynomial.rTensorAlgHom_apply_eq bridge: BoxIntegral.hasIntegralIndicatorConst
toMulActionWithZero 📖	CompOp	68 math math: Polynomial.ascPochhammer_smeval_neg_eq_descPochhammer, Polynomial.smeval_monomial_mul, Polynomial.smeval_commute, Polynomial.smeval_mul_X, ZeroAtInftyContinuousMap.instIsScalarTower, Ring.choose_eq_smul, HahnSeries.SummableFamily.hsum_smul_module, Polynomial.smeval_assoc_X_pow, TensorProduct.equivFinsuppOfBasisRight_apply_tmul, TensorProduct.equivFinsuppOfBasisLeft_apply_tmul, LaurentPolynomial.smeval_C_mul, Polynomial.smeval.linearMap_apply, Polynomial.smeval_neg_nat, Polynomial.smeval_smul, Polynomial.smeval_sub, Polynomial.smeval_natCast, HahnSeries.SummableFamily.smul_hsum, Polynomial.descPochhammer_smeval_eq_descFactorial, Polynomial.smeval_mul, ConvexCone.strictlyPositive_le_positive, Ring.factorial_nsmul_multichoose_eq_ascPochhammer, Polynomial.smeval_X_pow_mul, MeasureTheory.Lp.smul_zero, Ring.descPochhammer_smeval_add, Polynomial.aeval_eq_smeval, inr_comp_cfcₙHom_eq_cfcₙAux, Polynomial.smeval_at_natCast, LaurentPolynomial.leval_apply, Polynomial.smeval_C_mul, Ring.ascPochhammer_succ_succ, MeasureTheory.Lp.add_smul, BinomialRing.factorial_nsmul_multichoose, Polynomial.smeval_X_mul, HahnSeries.SummableFamily.coeff_smul, Ring.smeval_ascPochhammer_int_ofNat, ZeroAtInftyContinuousMap.instSMulCommClass, Polynomial.aeval_coe_eq_smeval, Ring.smeval_ascPochhammer_nat_cast, Polynomial.ascPochhammer_smeval_cast, Polynomial.smeval_at_zero, MeasureTheory.Lp.zero_smul, Polynomial.smeval_pow, MeasureTheory.Lp.smul_neg, MeasureTheory.Lp.smul_comm, Ring.smeval_ascPochhammer_neg_add, MeasureTheory.Lp.smul_add, Polynomial.smeval_comp, Polynomial.smeval_X_pow_assoc, MeasureTheory.Lp.simpleFunc.toLp_smul, Ring.smeval_ascPochhammer_succ_neg, MeasureTheory.Lp.neg_smul, Polynomial.ascPochhammer_smeval_eq_eval, Polynomial.smeval_add, Ring.smeval_ascPochhammer_neg_of_lt, MeasureTheory.Lp.smul_assoc, Ring.descPochhammer_eq_factorial_smul_choose, MeasureTheory.Lp.neg_smul_neg, Ring.descPochhammer_succ_succ_smeval, Polynomial.descPochhammer_smeval_eq_ascPochhammer, Polynomial.smeval_neg, Ring.smeval_ascPochhammer_self_neg, LaurentPolynomial.smeval_add, Polynomial.smeval_commute_left, HahnSeries.SummableFamily.powerSeriesFamily_smul, MeasureTheory.MemLp.toLp_const_smul, MeasureTheory.Lp.toTemperedDistribution_smul_eq, Polynomial.smeval_mul_X_pow, Polynomial.eval₂_smulOneHom_eq_smeval

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
add_smul 📖	mathematical	—	SemigroupAction.toSMul Monoid.toSemigroup MonoidWithZero.toMonoid Semiring.toMonoidWithZero MulAction.toSemigroupAction DistribMulAction.toMulAction AddCommMonoid.toAddMonoid toDistribMulAction Distrib.toAdd NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup	—	—
eq_zero_of_zero_eq_one 📖	mathematical	MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne	AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid	—	one_smul zero_smul
ext 📖	—	SemigroupAction.toSMul Monoid.toSemigroup MonoidWithZero.toMonoid Semiring.toMonoidWithZero MulAction.toSemigroupAction DistribMulAction.toMulAction AddCommMonoid.toAddMonoid toDistribMulAction	—	—	—
ext' 📖	—	SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero toDistribMulAction	—	—	ext
ext_iff 📖	mathematical	—	SemigroupAction.toSMul Monoid.toSemigroup MonoidWithZero.toMonoid Semiring.toMonoidWithZero MulAction.toSemigroupAction DistribMulAction.toMulAction AddCommMonoid.toAddMonoid toDistribMulAction	—	ext
nontrivial 📖	mathematical	—	Nontrivial	—	MulActionWithZero.nontrivial
subsingleton 📖	—	—	—	—	MulActionWithZero.subsingleton
zero_smul 📖	mathematical	—	SemigroupAction.toSMul Monoid.toSemigroup MonoidWithZero.toMonoid Semiring.toMonoidWithZero MulAction.toSemigroupAction DistribMulAction.toMulAction AddCommMonoid.toAddMonoid toDistribMulAction MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass	—	—

Semiring

Definitions

Name	Category	Theorems
toModule 📖	CompOp	6840 math math: AffineSubspace.wOppSide_lineMap_right, Polynomial.fderiv, PowerSeries.coeff_mul_eq_coeff_trunc_mul_trunc₂, LinearMap.subset_bipolar, MvPolynomial.pUnitAlgEquiv_symm_monomial, MvPowerSeries.coeff_zero_eq_constantCoeff_apply, CliffordAlgebra.ι_range_map_map, Ideal.mem_bot, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_left_iff, OreLocalization.left_distrib, Polynomial.support_monomial, LieAlgebra.IsKilling.rootSystem_toLinearMap_apply, IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, CuspFormClass.qExpansion_isBigO, Matrix.UnitaryGroup.toLin'_one, FixedPoints.instFiniteDimensionalSubtypeMemSubfieldSubfield, SemimoduleCat.MonoidalCategory.triangle, geometric_hahn_banach_open, Ideal.quotientEquivAlgOfEq_coe_eq_factorₐ, MvPolynomial.vanishingIdeal_anti_mono, MvPowerSeries.hasSubst_def, Ideal.isPrime_ideal_prod_top, Ideal.toCotangent_to_quotient_square, LinearMap.comp_toSpanSingleton, LinearMap.isSymm_zero, FractionalIdeal.coeIdeal_top, Submodule.instIsScalarTowerQuotientIdealSubtypeMemTorsionBySetCoe, Polynomial.not_ker_le_map_C_of_surjective_of_quasiFiniteAt, KaehlerDifferential.kerCotangentToTensor_toCotangent, Ideal.Quotient.isScalarTower, PrimeSpectrum.closure_range_comap, CoalgHom.ext_of_ring_iff, Finset.weightedVSub_sdiff, RootPairing.InvariantForm.apply_reflection_reflection, MvPolynomial.dvd_monomial_one_iff_exists, IsLocalizedModule.iso_symm_apply', CliffordAlgebra.contractRight_algebraMap_mul, Polynomial.degreeLT.addLinearEquiv_symm_apply_inr, LinearMap.restrictScalars_toMatrix, Lagrange.interpolate_one, Algebra.normalizedTrace_map, RootPairing.rootForm_self_smul_coroot, CliffordAlgebra.toBaseChange_reverse, Polynomial.taylor_eval, Affine.Simplex.affineCombination_mem_closedInterior_iff, IsDedekindDomain.differentIdeal_dvd_map_differentIdeal, CliffordAlgebra.map_id, IsAdjoinRoot.repr_add_sub_repr_add_repr_mem_span, LinearMap.BilinForm.linMulLin_compRight, rank_fun_infinite, cfcₙ_def, Submodule.colon_empty, AffineMap.lineMap_apply_module, LieDerivation.IsKilling.ad_mem_ker_killingForm_ad_range_of_mem_orthogonal, LinearMap.apply_symm_toPerfPair_self, PowerSeries.WithPiTopology.hasSum_iff_hasSum_coeff, ContinuousLinearMap.toSpanSingleton_zero, ContinuousMultilinearMap.norm_mkPiRing, PowerSeries.hasSum_aeval, IntermediateField.AdjoinSimple.trace_gen_eq_sum_roots, Submodule.mem_colon_span_singleton, RootPairing.RootPositiveForm.algebraMap_rootLength, Finset.weightedVSubOfPoint_map, Algebra.SubmersivePresentation.cotangentComplexAux_injective, MeasureTheory.OuterMeasure.map_iInf, LinearMap.baseChange_eq_ltensor, Algebra.normalizedTrace_algebraMap_eq_id, Module.Basis.det_comp_basis, Module.Basis.equivFun_self, TensorProduct.enorm_lid, uniqueDiffOn_Icc, LinearMap.charpoly_def, LieAlgebra.IsKilling.iInf_ker_weight_eq_bot, Lagrange.eval_iterate_derivative_eq_sum, Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_left_iff, TrivSqZeroExt.fstCLM_apply, AdicCompletion.val_sub_apply, MonomialOrder.sPolynomial_leadingTerm_mul', Algebra.exists_unramified_of_isUnramifiedAt, LinearMap.BilinForm.dualSubmoduleParing_spec, Polynomial.eraseLead_monomial, PowerSeries.IsWeierstrassFactorizationAt.algEquivQuotient_symm_apply, continuous_cfcₙHomSuperset_left, cfcₙL_integral, Ideal.comap_fiberIsoOfBijectiveResidueField_apply, Derivation.apply_eval_eq, Algebra.PreSubmersivePresentation.jacobiMatrix_naive, PiLp.basisFun_equivFun, Rep.coe_linearization_obj_ρ, AffineMap.hasStrictDerivAt_lineMap, Algebra.idealMap_eq_ofEq_comp_toLocalized₀, MvPowerSeries.coeff_zero_eq_constantCoeff, LinearMap.BilinForm.zero_right, dot_self_cross, Finset.affineCombination_vsub, CliffordAlgebra.contractRight_algebraMap, ExteriorAlgebra.isLocalHom_algebraMap, AffineMap.lineMap_vsub_left, MonoidAlgebra.liftMagma_apply_apply, Ideal.span_le, Matrix.toLin_kronecker, Lagrange.interpolate_eq_of_values_eq_on, Matrix.toLin'_symm, QuaternionAlgebra.coe_linearEquivTuple_symm, Orthonormal.inner_left_finsupp, MvPolynomial.totalDegree_monomial, Ideal.comap_bot_le_of_injective, Polynomial.derivative_X_sq, Matrix.SeparatingLeft.toBilin', Irreducible.maximalIdeal_eq_setOf_le_v_coe, LinearMap.detAux_def'', RootPairing.posRootForm_posForm_apply_apply, Ideal.selfBasis_def, Ideal.Filtration.submodule_eq_span_le_iff_stable_ge, PowerSeries.coeff_mul_one_sub_of_lt_order, norm_cfcₙHom, Algebra.TensorProduct.instSMulCommClassTensorProduct, LinearMap.BilinForm.dualSubmoduleToDual_apply_apply, Polynomial.smeval_monomial_mul, AlgebraicGeometry.Scheme.zeroLocus_iInf, LinearMap.IsPerfPair.restrict, Ideal.dvd_iff_le, Coalgebra.lTensor_counit_comul, CliffordAlgebra.contractRight_eq, cfcₙHom_mono, RootPairing.four_nsmul_coPolarization_compl_polarization_apply_root, Complex.restrictScalars_one_smulRight', LinearMap.toMatrixAlgEquiv'_mul, MvPolynomial.radical_le_vanishingIdeal_zeroLocus, LinearMap.BilinForm.toMatrix_toBilin, Subfield.rank_comap, Module.Basis.end_repr_apply, RCLike.geometric_hahn_banach_point_open, Ideal.height_le_spanFinrank, Complex.coe_realPart, Polynomial.ofFn_zero, Ideal.eq_jacobson_iff_sInf_maximal, Algebra.Presentation.differentials.comm₁₂_single, IsDedekindDomain.HeightOneSpectrum.equivPrimesOver_apply, Ideal.subset_union, ValuationSubring.valuation_lt_one_iff, MvPowerSeries.coeff_zero, MvPowerSeries.weightedOrder_monomial, Algebra.Extension.Cotangent.span_eq_top_of_span_eq_ker, Module.Basis.map_equivFun, OreLocalization.cardinalMk, MvPolynomial.DirectSum.coeLinearMap_eq_dfinsuppSum, Ideal.height_le_spanRank, Module.mem_annihilator, Matrix.SeparatingRight.toLinearMap₂, PowerSeries.coeff_mul_of_lt_order, LinearMap.IsReflective.regular, RootPairing.linearIndepOn_root_baseOf', LinearMap.BilinForm.symmCompOfNondegenerate_left_apply, FixedPoints.finrank_eq_card, PolyEquivTensor.toFunBilinear_apply_eq_sum, LinearMap.BilinForm.dualSubmoduleToDual_injective, Polynomial.lcoeff_comp_mapAlgHom_eq, ExteriorAlgebra.GradedAlgebra.liftι_eq, Polynomial.coe_taylorAlgHom, InnerProductSpace.toLinearIsometry_toDual, Ideal.pow_multiset_sum_mem_span_pow, PowerSeries.IsRestricted.smul, Module.support_quotient, Finset.affineCombination_affineCombinationSingleWeights, Ideal.Quotient.map_ker_stabilizer_subtype, HomogeneousIdeal.toIdeal_mul, Subspace.dualAnnihilator_dualAnnihilator_eq_map, PadicInt.lift_sub_val_mem_span, Polynomial.monomial_eq_monomial_iff, Valuation.ltIdeal_le_leIdeal, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, RootPairing.Polarization_apply, geometric_hahn_banach_point_point, LinearEquiv.flip_apply, Valuation.supp_quot_supp, Polynomial.mem_image_comap_C_basicOpen, MvPolynomial.weightedHomogeneousComponent_of_isWeightedHomogeneous_same, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, Polynomial.coeff_derivative, hasFDerivAt_iff_hasGradientAt, TensorProduct.AlgebraTensorModule.rid_symm_apply, NonUnitalIsometricContinuousFunctionalCalculus.isometric, IntermediateField.LinearDisjoint.linearIndependent_right', PowerSeries.coeff_of_lt_order_toNat, Orientation.ne_iff_eq_neg, RootPairing.root'In_corootSpanMem_eq_pairingIn, RCLike.reCLM_apply, BilinForm.toMatrix_symm, PowerSeries.constantCoeff_subst, ContinuousAffineMap.lineMap_apply', ExteriorAlgebra.ι_eq_algebraMap_iff, smooth_barycentric_coord, cfcₙHom_apply_mem_elemental, RootPairing.rootForm_restrict_nondegenerate_of_ordered, PowerSeries.WithPiTopology.continuous_coeff, Polynomial.degreeLT.instFreeSubtypeMemSubmodule, TannakaDuality.FiniteGroup.toRightFDRepComp_in_rightRegular, SimpleGraph.lapMatrix_toLinearMap₂', Quaternion.snd_imJ_dualNumberEquiv_symm, ValuationSubring.image_maximalIdeal, ModularFormClass.qExpansionFormalMultilinearSeries_apply_norm, PowerSeries.coeff_mul_C, tendsto_tsum_div_pow_atTop_integral, cardQuot_mul, RCLike.separate_convex_open_set, RootPairing.coroot_eq_polarizationEquiv_apply_root, Polynomial.Sequence.basis_natDegree_strictMono, Ideal.finrank_eq_finrank, Algebra.PreSubmersivePresentation.cotangentComplexAux_apply, Ideal.IsTwoSided.pow_succ, MvPolynomial.rTensor_apply_tmul_apply, Submodule.le_annihilator_iff, MvPolynomial.degrees_monomial, Polynomial.Monic.free_quotient, PrimeSpectrum.asIdeal_bot, Ideal.span_singleton_mul_le_span_singleton_mul, exteriorPower.basis_apply, Ideal.mem_span_iff_exists_isHomogeneous, CliffordAlgebraQuaternion.j_quaternionBasis, ExteriorAlgebra.ι_inj, Ideal.exists_smith_normal_form, dotProductEquiv_symm_apply, Submodule.mem_ideal_smul_span_iff_exists_sum, Polynomial.hasseDeriv_zero', Matrix.separatingRight_toLinearMap₂'_iff, LinearMap.baseChange_smul, MvPolynomial.one_def, ContinuousLinearMap.smulRight_one_eq_toSpanSingleton, ClassGroup.mk0_eq_mk0_iff, AffineMap.lineMap_linear, MvPowerSeries.coeff_mul_monomial, LieAlgebra.IsKilling.restrict_killingForm_eq_sum, PiTensorProduct.isEmptyEquiv_apply_tprod, Lagrange.eval_interpolate_not_at_node', Valuation.mem_maximalIdeal_iff, Differential.implicitDeriv_C, QuadraticMap.sum_polar_sub_repr_sq, RootPairing.RootPositiveForm.zero_lt_posForm_apply_root, CliffordAlgebraComplex.ofComplex_conj, BialgCat.rightUnitor_def, exteriorPower.linearMap_ext_iff, TrivSqZeroExt.lift_inlAlgHom_inrHom, HopfAlgCat.rightUnitor_def, Module.nonempty_linearEquiv_of_finrank_eq_one, Matrix.det_smul_inv_mulVec_eq_cramer, IsLocalization.coeSubmodule_mul, PrimeSpectrum.localization_specComap_range, Unitization.inrNonUnitalAlgHom_toFun, Ideal.toCotangent_eq, Ideal.subset_span, AdicCompletion.AdicCauchySequence.mk_eq_mk, AddMonoidAlgebra.of'_mem_span, PiTensorProduct.dualDistribEquivOfBasis_symm_apply, Filter.Tendsto.lineMap, ConvexOn.monotoneOn_slope_lt, Polynomial.derivWithin_aeval, Matrix.UnitaryGroup.toGL_mul, Module.Baer.ExtensionOfMaxAdjoin.extensionToFun_wd, NormedSpace.Dual.isClosed_image_polar_of_mem_nhds, Valuation.mem_leSubmodule_iff, Finset.weightedVSubOfPoint_apply, PowerSeries.coeff_rescale, Ideal.comap_inf, MvPolynomial.scalarRTensor_apply_monomial_tmul, MvPolynomial.homogeneousComponent_eq_zero', QuadraticMap.canLift, Polynomial.derivative_intCast_mul, Polynomial.toMatrix_sylvesterMap', Module.basisUnique_repr_eq_zero_iff, Finset.affineCombination_apply_eq_lineMap_sum, IntermediateField.instIsAlgClosureAlgebraicClosureSubtypeMemOfIsAlgebraic, Valuation.mem_supp_iff, IdealFilter.ringFilterBasis_sets, Matrix.toLinearMapRight'_mul, eval_minpolyDiv_self, Algebra.normalizedTrace_surjective, Finset.centroid_eq_affineCombination_fintype, Differential.algHom_deriv, Submodule.smul_eq_map₂, Subspace.instModuleDualFiniteDimensional, curveIntegrable_segment, Algebra.normalizedTrace_eq_of_finiteDimensional, Ideal.mem_span_singleton', Ideal.mem_ofPolynomial, Ideal.iInf_pow_smul_eq_bot_of_isLocalRing, MvPolynomial.universalFactorizationMap_comp_map, Representation.ofMulActionSelfAsModuleEquiv_symm_apply, Module.Basis.equivFun_symm_apply, SkewPolynomial.monomial_add, ProbabilityTheory.covarianceBilinDual_apply', Ideal.height_le_iff_exists_minimalPrimes, LinearMap.BilinForm.IsAlt.neg_eq, Rep.diagonalSuccIsoFree_inv_hom_single, Ideal.under_bot, IsAdjoinRoot.mem_ker_map, Submodule.isInternal_prime_power_torsion_of_pid, AlgebraicGeometry.StructureSheaf.const_mul_rev, Ideal.relNorm_smul, LinearMap.IsPerfPair.bijective_left, MeasureTheory.charFunDual_apply, MonomialOrder.degree_monomial, IsLocalization.AtPrime.coe_orderIsoOfPrime_apply_coe, ClassGroup.mk_eq_mk_of_coe_ideal, differentIdeal_eq_differentIdeal_mul_differentIdeal, IsFractionRing.coeSubmodule_strictMono, uniqueDiffWithinAt_Iic, Submodule.map_dualCoannihilator_le, smoothSheafCommRing.nonunits_stalk, Matrix.separatingRight_toLinearMap₂_iff, Matrix.cramer_transpose_row_self, LinearEquiv.coe_toLinearMap_flip, CommRing.Pic.mk_dual, AdicCompletion.map_val_apply, PowerSeries.coeff_expand_mul, Ideal.mul_le_right, expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul, mem_reesAlgebra_iff_support, Ideal.isHomogeneous_iff_subset_iInter, right_lt_lineMap_iff_one_lt, NumberField.Units.fun_eq_repr, Polynomial.derivative_C_mul, ProbabilityTheory.hasGaussianLaw_iff_charFunDual_map_eq, eval₂_minpolyDiv_self, Matrix.iSup_eigenspace_toLin'_diagonal_eq_top, PrimeSpectrum.zeroLocus_nilradical, RingEquiv.idealComapOrderIso_apply, LinearMap.BilinMap.baseChange_isSymm, KaehlerDifferential.polynomialEquiv_symm, ProbabilityTheory.IsGaussian.integrable_dual, Ideal.exists_mem_span_singleton_map_residueField_eq, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, algebraicIndependent_iff_ker_eq_bot, Submodule.tensorEquivSpan_apply_tmul, Polynomial.iterate_derivative_zero, IsBaseChange.basis_repr_comp_apply, RingHom.ker_eq_top_of_subsingleton, LinearMap.baseChange_comp, MeasureTheory.FiniteMeasure.toWeakDualBCNN_apply, Ideal.smul_top_eq_map, LinearEquiv.coe_baseChange, Module.DualBases.basis_repr_symm_apply, Module.Basis.toDual_linearCombination_left, MvPowerSeries.mem_hasEvalIdeal_iff, PowerSeries.catalanSeries_coeff, Ideal.rank_prime_pow_ramificationIdx, Polynomial.card_rootSet_le_derivative, PowerSeries.coeff_X_mul_largeSchroderSeries, realPart_ofReal, geometric_hahn_banach_open_point, LinearMap.BilinForm.toMatrixAux_eq, PrimeSpectrum.zeroLocus_eq_univ_iff, CliffordAlgebra.ι_mul_ι_mul_invOf_ι, Ideal.exists_nonzero_mem_of_ne_bot, HomogeneousLocalization.val_nsmul, PowerSeries.coeff_trunc, StrongDual.polarSubmodule_eq_setOf, Algebra.Extension.H1Cotangent.equiv_apply, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul', LieModule.trace_toEnd_eq_zero_of_mem_lcs, Algebra.Generators.cotangentSpaceBasis_apply, Polynomial.derivative_mul, IsAdjoinRootMonic.coeff_apply, PowerSeries.trunc_mul_trunc, Module.Relations.Solution.surjective_π_iff_span_eq_top, Matrix.toLinearMap₂'_comp, AffineMap.pi_lineMap_apply, Finset.weightedVSubOfPoint_insert, Submodule.mul_annihilator, MeasureTheory.OuterMeasure.comap_map, Polynomial.Chebyshev.one_sub_X_sq_mul_derivative_derivative_U_eq_poly_in_U, Quaternion.rank_eq_four, MvPolynomial.IsWeightedHomogeneous.pderiv, ExteriorAlgebra.lift_symm_apply, Rep.coindToInd_of_support_subset_orbit, finrank_quotient_span_eq_natDegree_norm, PowerSeries.zero_weierstrassDiv, RootPairing.root_coroot_two, ModuleCat.extendScalarsId_hom_app_one_tmul, WithIdealFilter.mem_nhds_zero_iff, MvPolynomial.X_pow_eq_monomial, CommSemiring.finrank_self, LinearMap.toMatrix_apply', Polynomial.opRingEquiv_op_monomial, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sSup, Polynomial.natDegree_monomial_le, span_singleton_dvd_span_singleton_iff_dvd, Ideal.le_span_singleton_mul_iff, Polynomial.Sequence.span, mem_pairSelfAdjointMatricesSubmodule, LinearMap.toMatrix₂_mul, Quaternion.finrank_eq_four, LinearMap.BilinForm.smul_left_of_tower, dist_sq_lineMap_lineMap_of_inner_eq_zero, Rep.leftRegularHom_hom, Ring.coe_jacobson_quotient, Ideal.mem_leadingCoeffNth_zero, LieAlgebra.LoopAlgebra.twoCochainOfBilinear_apply_apply, IdealFilter.isPFilter_gabrielComposition, Module.Basis.dualBasis_coord_toDualEquiv_apply, PowerSeries.trunc_X_pow_self_mul, Matrix.repr_toLin, PrimeSpectrum.equivSubtype_apply_coe, AdicCompletion.of_ofAlgEquiv_symm, LieModule.traceForm_eq_sum_genWeightSpaceOf, WeierstrassCurve.Projective.nonsingularLift_some, Lagrange.derivative_nodal, DividedPowers.IsDPMorphism.ideal_comp, PowerSeries.IsWeierstrassDivisorAt.div_coe_eq_zero, LocalizedModule.restrictScalars_map_eq, LinearMap.polar_eq_iInter, ProbabilityTheory.covarianceBilinDual_apply, MvPowerSeries.coeff_pow, Module.forall_dual_apply_eq_zero_iff, Polynomial.monomial_mem_lifts, Convex.lineMap_mem, KaehlerDifferential.mvPolynomialBasis_repr_D_X, PowerSeries.zero_weierstrassMod, ProbabilityTheory.IsGaussian.map_eq_gaussianReal, exteriorPower.coe_basis, Polynomial.taylor_mul, Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero, Finsupp.linearCombination_id_surjective, LinearMap.det_mulLeft, ProjectiveSpectrum.zeroLocus_mul_ideal, Module.rankAtStalk_eq, Subspace.dualLift_of_subtype, Submodule.mem_colon', PowerSeries.eq_mul_weierstrassDiv_add_weierstrassMod, Module.Invertible.free_iff_linearEquiv, AlgebraicGeometry.StructureSheaf.const_self, RootPairing.EmbeddedG2.shortAddLongRoot_shortRoot, PadicInt.appr_spec, LieAlgebra.IsKilling.isCompl_ker_weight_span_coroot, Polynomial.sum_bernoulli, Finset.affineCombination_sdiff_sub, exteriorPower.ιMulti_family_linearIndependent_field, minimalPrimes_eq_minimals, Ring.isField_iff_isSimpleOrder_ideal, Ideal.sum_eq_sup, Orientation.volumeForm_robust_neg, CliffordAlgebra.isUnit_of_isUnit_ι, Polynomial.C_mul_X_eq_monomial, skewAdjointLieSubalgebraEquiv_apply, CliffordAlgebraQuaternion.toQuaternion_star, Ideal.one_eq_top, IsLocalization.AtPrime.mk'_mem_maximal_iff, Pi.counit_coe_dFinsupp, idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical, traceForm_nondegenerate_tfae, PowerSeries.derivative_exp, TensorProduct.piScalarRight_symm_algebraMap, LinearMap.BilinForm.toMatrix_basisFun, Algebra.trace_eq_of_ringEquiv, Subspace.dualRestrict_comp_dualLift, Ideal.mem_colon_span_singleton, PowerSeries.coeff_of_lt_order, Ideal.IsHomogeneous.sInf, MvPowerSeries.constantCoeff_subst, LinearMap.BilinForm.congr_apply, range_comap_of_surjective, LieModule.traceForm_eq_sum_finrank_nsmul', eval₂_minpolyDiv_of_eval₂_eq_zero, MvPolynomial.divMonomial_add_modMonomial, Ideal.under_top, LinearMap.BilinForm.toMatrix_compRight, PrimeSpectrum.isRetrocompact_zeroLocus_compl_of_fg, RootPairing.rootSpan_dualAnnihilator_le_ker_rootForm, Ideal.height_top, Ideal.natCast_eq_top, TensorProduct.AlgebraTensorModule.rid_tmul, KaehlerDifferential.tensorKaehlerEquiv_symm_D_tmul, Polynomial.hasseDeriv_one', Polynomial.derivative_comp_one_sub_X, Ideal.smul_closure, ringKrullDim_eq_one_iff_of_isLocalRing_isDomain, Submodule.dualQuotEquivDualAnnihilator_apply, NumberField.mixedEmbedding.fundamentalCone.integerSetEquivNorm_apply_fst, ContinuousLinearMap.toBilinForm_apply, WittVector.IsocrystalHom.frob_equivariant, MvPolynomial.eval₂Hom_monomial, Polynomial.contentIdeal_one, Coalgebra.sum_map_tmul_counit_eq, IsDedekindDomain.differentIdeal_eq_differentIdeal_mul_differentIdeal_of_isCoprime, Algebra.IsPushout.cancelBaseChange_tmul, Polynomial.derivative_pow_eq_zero, ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm, Ideal.instCanLiftTwoSidedIdealCoeOrderHomAsIdealIsTwoSided, LinearMap.BilinForm.IsNonneg.nonneg, LieAlgebra.IsKilling.instIsReducedSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, QuadraticModuleCat.toIsometry_comp, PowerSeries.coeff_succ_mul_X, LinearMap.IsPosSemidef.add, Module.exists_dual_forall_apply_eq_one, PolyEquivTensor.right_inv, LinearIndependent.linearCombination_repr, exteriorPower.basis_repr_ne, IsLocalizedModule.exists_subsingleton_away, PolynomialModule.aeval_equivPolynomial, PowerSeries.spanFinrank_le_spanFinrank_map_constantCoeff_add_one_of_isPrime, Module.length_of_free, MvPolynomial.le_vanishingIdeal_zeroLocus, Algebra.Presentation.differentials.comm₂₃, Coalgebra.sum_tmul_counit_eq, rank_finsupp_self, RingPreordering.coe_support, MvPolynomial.homogeneousComponent_mem, Representation.ofCoinvariantsTprodLeftRegular_mk_tmul_single, Orientation.volumeForm_neg_orientation, sbtw_iff_mem_image_Ioo_and_ne, MvPowerSeries.coeff_monomial_mul, PrimeSpectrum.equivSubtype_symm_apply_asIdeal, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_left, LinearMap.toMatrix_one, Polynomial.degreeLT_mono, Finsupp.weight_single_one_apply, Module.isTorsionBySet_quotient_ideal_smul, DualNumber.range_lift, QuaternionAlgebra.imIₗ_apply, LinearMap.IsReflective.smul_coroot, QuotSMulTop.equivTensorQuot_naturality, IntermediateField.relfinrank_mul_finrank_top, Ideal.isHomogeneous_iff_forall_subset, LieAlgebra.IsKilling.rootSystem_coroot_apply, Module.Basis.mk_repr, Polynomial.roots_monomial, Prod.counit_comp_inr, Module.Flat.finsupp, MvPowerSeries.coeff_X_pow, Valuation.mem_ltIdeal_iff, MvPolynomial.support_monomial, CliffordAlgebraDualNumber.equiv_symm_eps, LinearMap.IsReflective.apply_self_mul_coroot_apply, MeasureTheory.OuterMeasure.restrict_trim, ProbabilityTheory.uncenteredCovarianceBilin_apply, Finset.weightedVSubOfPoint_congr, ExteriorAlgebra.ιMulti_span_fixedDegree, PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical, Wbtw.right_mem_image_Ici_of_left_ne, Polynomial.aeval_iterate_derivative_of_ge, MvPolynomial.monomial_finsupp_sum_index, KaehlerDifferential.cotangentComplexBaseChange_tmul, MvPolynomial.weightedHomogeneousSubmodule_mul, Module.Invertible.instTensorProduct_1, Module.Relations.Solution.injective_fromQuotient_iff_ker_π_eq_span, cross_cross, Matrix.separatingRight_toBilin_iff, Ideal.mem_pointwise_smul_iff_inv_smul_mem, Ideal.span_pow_eq_map_homogeneousSubmodule, Polynomial.derivative_expand, Polynomial.derivative_sum, Polynomial.isCoveringMapOn_eval, IsBaseChange.of_fintype_basis_eq, IsDiscreteValuationRing.instIsHausdorffMaximalIdeal, MvPolynomial.mul_def, Matrix.spectrum_toLpLin, ExteriorAlgebra.ι_eq_zero_iff, Ideal.span_singleton_mul_left_injective, Submodule.IsPrincipal.map_ringHom, Complex.reLm_coe, Polynomial.derivative_pow_succ, AdicCompletion.instIsScalarTower_1, CliffordAlgebra.equivOfIsometry_trans, Valuation.leSubmodule_monotone, Polynomial.Sequence.basis_eq_self, uniqueDiffOn_Iic, PadicInt.sub_zmodRepr_mem, Module.Basis.repr_isUnitSMul, Submodule.ideal_span_singleton_smul, AffineMap.lineMap_rev_mem_affineSpan_pair, RootPairing.InvariantForm.pairing_mul_eq_pairing_mul_swap, DividedPowers.SubDPIdeal.sInf_carrier_def, sum_smul_minpolyDiv_eq_X_pow, Polynomial.derivative_ofNat, Submodule.top_eq_ofList_cons_smul_iff, BoxIntegral.unitPartition.mem_smul_span_iff, Submodule.quotDualCoannihilatorToDual_apply, CliffordAlgebra.equivOfIsometry_refl, AlternatingMap.smulRight_apply, Ideal.exists_smul_eq_of_isGaloisGroup, InnerProductSpace.toDual_symm_apply, Module.Dual.eval_comp_comp_evalEquiv_eq, ContinuousMap.notMem_idealOfSet, aleph0_le_rank_of_isEmpty_oreSet, OreLocalization.universalHom_apply, Polynomial.monomial_one_one_eq_X, OreLocalization.nsmul_eq_nsmul, LinearMap.BilinForm.toMatrix'_symm, OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation, ExteriorAlgebra.map_comp_map, Matrix.cramer_row_self, Span.repr_def, PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical, Module.Basis.det_reindex', Polynomial.Splits.eval_derivative, Submodule.dualPairing_apply, ValuationRing.le_total_ideal, Matrix.toLin_mul_apply, RingHom.liftOfSurjective_comp, AlgebraicGeometry.Scheme.zeroLocus_mul, Polynomial.derivative_smul, toMatrix_distrib_mul_action_toLinearMap, PrimeSpectrum.gc, PowerSeries.gaussNorm_monomial, Module.dualMap_dualMap_eq_iff, MvPowerSeries.X_pow_eq, exists_spanRank_le_and_le_height_of_le_height, LinearMap.baseChange_baseChange, AdicCompletion.val_smul, DividedPowers.isSubDPIdeal_iInf, Polynomial.Splits.monomial, PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply_comp_toReal, Valuation.Integers.isPrincipal_iff_exists_isGreatest, PiToModule.fromEnd_apply, ModuleCat.CoextendScalars.smul_apply', TruncatedWittVector.iInf_ker_truncate, Ideal.subset_pointwise_smul_iff, Ideal.IsPrime.prod_le, isUnit_iff_notMem_of_isAdicComplete_maximal, MeasureTheory.charFun_toDual_symm_eq_charFunDual, Module.Basis.isUnit_det, TensorPower.gMul_eq_coe_linearMap, Module.FaithfullyFlat.finsupp, RatFunc.laurent_algebraMap, AlgebraicIndependent.repr_ker, Polynomial.iterate_derivative_eq_factorial_smul_sum, Module.FinitePresentation.exists_notMem_bijective, LinearMap.BilinForm.IsRefl.groupSMul, LinearMap.BilinForm.toMatrix'_apply, Submodule.mem_smul_span_singleton, Module.Basis.dual_rank_eq, Localization.localRingHom_mk', WittVector.factorPowSucc_comp_fontaineThetaModPPow, Polynomial.iterate_derivative_X_add_pow, Algebra.Extension.formallySmooth_iff_split_injection, exposed_point_def, linearIndependent_iff_injective_finsuppLinearCombination, Ideal.bot_lt_of_maximal, range_specComap_of_surjective, lintegral_fderiv_lineMap_eq_edist, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, AdicCompletion.incl_apply, Module.mem_support_iff_of_finite, AdicCompletion.map_ext'_iff, Submodule.exists_le_ker_of_lt_top, Ideal.quotientInfToPiQuotient_inj, AlgCat.hom_inv_rightUnitor, Unitization.starAlgHom_ext_iff, AdicCompletion.ofTensorProduct_surjective_of_finite, MeasureTheory.Measure.comapₗ_eq_comap, LinearMap.nondegenerate_toLinearMap₂'_of_det_ne_zero', MvPolynomial.scalarRTensor_apply_X_tmul_apply, HomogeneousIdeal.toIdeal_add, DividedPowers.SubDPIdeal.le_iff, Algebra.IsSmoothAt.exists_notMem_isStandardSmooth, LinearEquiv.congrRight₂_apply, OrthonormalBasis.det_eq_neg_det_of_opposite_orientation, Subfield.relfinrank_mul_finrank_top, cross_self, eval_minpolyDiv_of_aeval_eq_zero, Module.Basis.mk_coord_apply_eq, Polynomial.derivative_X, lineMap_le_map_iff_slope_le_slope_right, CliffordAlgebra.contractRight_mul_ι, Polynomial.sumIDeriv_map, AdicCompletion.val_smul_eq_evalₐ_smul, Matrix.Nondegenerate.toLinearMap₂, MvPolynomial.smul_monomial, nilradical_le_prime, OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation, Affine.Simplex.wSameSide_affineSpan_faceOpposite_iff, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_apply, contractLeft_assoc_coevaluation, Module.Basis.prod_repr_inl, Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_right_iff, Lagrange.eval_interpolate_not_at_node, AffineBasis.linear_eq_sumCoords, LinearMap.toContinuousLinearMap₁_coe, RootPairing.rootForm_anisotropic, PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal, ValuationRing.TFAE, Quaternion.snd_re_dualNumberEquiv_symm, Ideal.notMem_of_isUnit, LinearMap.injective_compr₂_of_injective, Module.Basis.flag_le_ker_dual, LieModule.lowerCentralSeries_one_inf_center_le_ker_traceForm, Polynomial.mirror_monomial, Orientation.eq_or_eq_neg, Algebra.PreSubmersivePresentation.cotangentComplexAux_zero_iff, Bialgebra.TensorProduct.counitAlgHom_def, Ideal.isPrime_iff_bot_or_prime, MvPolynomial.IsWeightedHomogeneous.weightedHomogeneousComponent_same, Matrix.intrinsicStar_toLin', Ideal.span_singleton_eq_bot, IntermediateField.relrank_mul_rank_top, Polynomial.dvd_iterate_derivative_pow, QuaternionAlgebra.imKₗ_apply, IsGalois.card_fixingSubgroup_eq_finrank, Subfield.relfinrank_eq_finrank_of_le, Ideal.range_cotangentToQuotientSquare, exteriorPower.ιMulti_family_span_fixedDegree_of_span, Ideal.Filtration.smul_le, isLinearTopology_iff_hasBasis_open_twoSidedIdeal, Polynomial.coeffs_monomial, KaehlerDifferential.polynomialEquiv_D, IsLocalization.mem_span_map, Ideal.mem_toTwoSided, Module.supportDim_add_length_eq_supportDim_of_isRegular, Ideal.submodule_span_eq, Polynomial.ofFn_eq_sum_monomial, hasStrictFDerivAt_exp_smul_const_of_mem_ball', Matrix.coe_mulVecLin, TwoSidedIdeal.instIsTwoSidedCoeOrderHomIdealAsIdeal, Ideal.ne_top_iff_one, LinearEquiv.dualMap_trans, Affine.Simplex.sign_signedInfDist_lineMap_incenter_touchpoint, Module.Basis.coe_dualBasis, Matrix.mulVecLin_apply, uniqueDiffOn_Ioc, LieAlgebra.IsKilling.corootSpace_eq_bot_iff, PowerSeries.coeff_ne_zero_C, IsLocalizedModule.iso_symm_apply, HomogeneousIdeal.toIdeal_iInf, continuous_equivFun_basis, ExteriorAlgebra.liftAlternating_ι_mul, Module.Relations.Solution.surjective_fromQuotient_iff_surjective_π, Module.rank_baseChange, Affine.Simplex.ExcenterExists.affineCombination_eq_excenter_iff, Algebra.Extension.CotangentSpace.map_id, Algebra.IsPushout.cancelBaseChange_symm_tmul, RootPairing.coPolarization_apply_eq_zero_iff, Lagrange.interpolate_eq_nodalWeight_mul_nodal_div_X_sub_C, Polynomial.abc, Fintype.linearIndependent_iff', exists_le_isAssociatedPrime_of_isNoetherianRing, AffineSubspace.sOppSide_lineMap_left, finrank_quotient_span_eq_natDegree', HahnSeries.embDomainLinearMap_apply, Module.le_rank_iff_exists_linearMap, LinearMap.intrinsicStar_toSpanSingleton, Ideal.quotientInfToPiQuotient_surj, instIsScalarTowerAtPrimeFractionRing_1, Finsupp.linearCombination_zero_apply, Lagrange.interpolate_eq_sum_interpolate_insert_sdiff, QuadraticMap.toMatrix'_comp, LinearMap.trace_mul_cycle, ExteriorAlgebra.toTrivSqZeroExt_comp_map, IsSemisimpleModule.jacobson_le_annihilator, ContinuousMultilinearMap.mkPiRing_apply_one_eq_self, CliffordAlgebra.toProd_comp_ofProd, Module.exists_smul_eq_zero_and_mk_eq, LinearMap.separatingRight_iff_det_ne_zero, algebraMapCLM_coe, TensorProduct.quotTensorEquivQuotSMul_comp_mkQ_rTensor, Matrix.GeneralLinearGroup.coe_toLin, Complex.instFiniteDimensionalReal, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_comap_ideal, SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix, Ideal.map_spanIntNorm, ContinuousMultilinearMap.mkPiRing_apply, Ideal.LiesOver.smul, lineMap_inv_two, Finset.weightedVSub_apply_const, CliffordAlgebra.equivBaseChange_symm_apply, Module.FaithfullyFlat.instTensorProduct, Module.Basis.repr_self_apply, RootPairing.coroot'In_rootSpanMem_eq_pairingIn, Ideal.IsPrime.pow_mem_iff_mem, Polynomial.Splits.eval_root_derivative, exteriorPower.alternatingMapLinearEquiv_ιMulti, Algebra.normalizedTrace_comp_algHom, ContinuousLinearMap.smulRight_one_pow, Ideal.minimalPrimes_top, IsLocalizedModule.iso_apply_mk, KaehlerDifferential.kerToTensor_apply, Quaternion.imJ_fst_dualNumberEquiv, RatFunc.laurent_div, Subspace.forall_mem_dualAnnihilator_apply_eq_zero_iff, Finsupp.mem_span_iff_linearCombination, star_dotProduct_toMatrix₂_mulVec, RingHom.ker_equiv, isStablyFiniteRing_iff_injective_of_surjective, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, RootPairing.root_coroot_eq_pairing, CliffordAlgebraQuaternion.toQuaternion_ofQuaternion, IsLocalFrameOn.coeff_apply_of_mem, TensorProduct.AlgebraTensorModule.lTensor_comp_cancelBaseChange, QuadraticModuleCat.forget₂_map_associator_inv, Polynomial.degreeLT.basis_val, Subspace.dualAnnihilator_dualAnnihilator_eq, Finset.weightedVSubOfPoint_indicator_subset, setBasisOfLinearIndependentOfCardEqFinrank_repr_apply, IsSemisimpleRing.exists_linearEquiv_ideal_of_isSimpleModule, ModuleCat.RestrictionCoextensionAdj.HomEquiv.toRestriction_hom_apply, MvPowerSeries.monomial_mem_nonzeroDivisors, Algebra.Presentation.differentials.hom₁_single, PowerSeries.derivative_inv', LocalizedModule.divBy_apply, RingOfIntegers.exponent_eq_sInf, instIsPrincipalSpanSingletonSet, RingHom.SurjectiveOnStalks.exists_mul_eq_tmul, LieModule.exists_nontrivial_weightSpace_of_lieIdeal, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_right, LinearMap.BilinForm.linMulLin_comp, Ideal.iInf_mul, skewAdjointLieSubalgebraEquiv_symm_apply, Algebra.zariskisMainProperty_iff, PrimeSpectrum.denseRange_comap_iff_minimalPrimes, Module.Basis.det_isEmpty, Algebra.Generators.exists_presentation_of_basis_cotangent, LinearMap.nondegenerate_of_det_ne_zero, range_vecMulLinear, Matrix.SpecialLinearGroup.center_equiv_rootsOfUnity'_symm_apply_coe_coe, Bialgebra.counitBialgHom_apply, zero_prodMap_dualTensorHom, LinearMap.sum_repr_mul_repr_mulₛₗ, Ideal.map_under_le_under_map, Polynomial.derivative_monomial, cfcL_apply, Ideal.mem_toCotangent_ker, Algebra.TensorProduct.basis_apply, RingHom.liftOfSurjective_comp_apply, hasFDerivWithinAt_inv, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, LinearMap.ringLmapEquivSelf_symm_apply, MvPolynomial.pderiv_one, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mul, LinearMap.liftBaseChange_one_tmul, Module.Invertible.inst, Ideal.ideal_prod_prime, MvPolynomial.pderiv_mul, RootPairing.self_comp_coPolarization_eq_corootForm, ExteriorAlgebra.algebraMap_inj, Polynomial.aeval_sumIDeriv, Representation.ofMulAction_apply, ContinuousAffineMap.coe_lineMap_eq, geometric_hahn_banach_compact_closed, Matrix.toBilin_symm, LinearMap.spectrum_toMatrix', Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd, LieModule.trace_toEnd_genWeightSpace, Polynomial.sumIDeriv_C, Polynomial.Chebyshev.T_derivative_eq_U, Real.smul_map_diagonal_volume_pi, Algebra.traceMatrix_apply, Ideal.comap_finsetInf, MvPowerSeries.coeff_mul_right_one_sub_of_lt_weightedOrder, ModuleCat.extendRestrictScalarsAdj_homEquiv_apply, LinearMap.smulRight_id, Polynomial.coeff_taylor_natDegree, RootPairing.span_coroot'_eq_top, Submodule.colon_finsetInf, CliffordAlgebraComplex.ofComplex_toComplex, IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal, toMatrix_rotation, AffineBasis.coord_apply_ne, LinearMap.BilinForm.sum_right, Algebra.TensorProduct.piScalarRight_tmul, Ideal.Quotient.tower_quotient_map_quotient, ContinuousLinearMap.toSpanSingleton_apply, FreeNonUnitalNonAssocAlgebra.of_comp_lift, Module.Basis.det_adjustToOrientation, Module.eval_apply_injective, AlgHom.mulLeftRightMatrix.inv_comp, Submodule.dualAnnihilator_map_dualMap_le, nilradical_eq_zero, LieAlgebra.IsKilling.ker_traceForm_eq_bot_of_isCartanSubalgebra, Submodule.dualCopairing_eq, AffineIndependent.affineCombination_eq_iff_eq, Nat.exists_mem_span_nat_finset_of_ge, CliffordAlgebraComplex.toComplex_comp_ofComplex, Ideal.map_eq_top_or_isMaximal_of_surjective, LinearMap.polar_singleton, ringKrullDim_quotient, Ideal.liesOver_iff_dvd_map, Ideal.Quotient.subsingleton_iff, Ideal.finsuppTotal_apply_eq_of_fintype, WittVector.factorPowSucc_fontaineThetaModPPow_eq, Polynomial.aeval_iterate_derivative_self, NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_dvd_norm_le, KaehlerDifferential.quotKerTotalEquiv_symm_comp_D, strongRankCondition_iff_forall_not_injective, PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical, IntermediateField.LinearDisjoint.algebraMap_basisOfBasisRight_repr_apply, RootPairing.RootPositiveForm.zero_lt_posForm_iff, QuotSMulTop.equivTensorQuot_naturality_mk, RootPairing.toLinearMap_apply_CoPolarization, IsSimpleRing.tfae, LinearMap.BilinForm.ext_iff_of_isSymm, IsLocalRing.map_mkQ_eq, dot_cross_self, Subalgebra.instIsTorsionFree', Convex.mapsTo_lineMap, ContinuousLinearMap.toSpanSingleton_apply_one, NormedSpace.Dual.toWeakDual_continuous, Polynomial.Chebyshev.iterate_derivative_T_eval_one_recurrence, WeierstrassCurve.Projective.Point.zero_point, Ideal.jacobson_bot, Matrix.toLpLin_mul_same, Polynomial.toFinsupp_monomial, Submodule.quotDualCoannihilatorToDual_nondegenerate, CliffordAlgebra.toProd_ι_tmul_one, cfcₙ_eq_cfcₙL_mkD, QuadraticModuleCat.ofIso_hom, Ideal.bot_pow, RootPairing.Hom.weight_coweight_transpose, Polynomial.coe_monomial, Ideal.mem_iff_of_associated, Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity, FixedPoints.rank_le_card, finsuppTensorFinsupp'_symm_single_eq_tmul_single_one, Algebra.PreSubmersivePresentation.isUnit_jacobian_iff_aevalDifferential_bijective, Module.DualBases.coe_dualBasis, CliffordAlgebra.changeForm_self_apply, PowerSeries.aeval_eq_sum, Algebra.normalizedTrace_algebraMap, Rep.finsuppToCoinvariantsTensorFree_single, Module.Relations.Solution.IsPresentation.linearEquiv_symm_var, Submodule.mem_dualAnnihilator, maximalIdeal_isPrincipal_of_isDedekindDomain, PolyEquivTensor.toFunAlgHom_apply_tmul, Submodule.mem_ideal_smul_span_iff_exists_sum', QuadraticAlgebra.linearEquivTuple_apply, HahnSeries.SummableFamily.lsum_apply, QuadraticForm.equivalent_signType_weighted_sum_squared, LinearMap.toMatrix_baseChange, Ideal.FG.pow, curveIntegralFun_segment, TwoSidedIdeal.bot_asIdeal, Coalgebra.sum_counit_tmul_eq, Finset.weightedVSub_map, IdealFilter.isTorsionQuot_inter_left_iff, MvPolynomial.pow_idealOfVars, LieAlgebra.IsKilling.le_chainBotCoeff_of_rootSpace_ne_top, SemimoduleCat.hom_inv_rightUnitor, IsLocalRing.maximalIdeal_le_jacobson, RCLike.geometric_hahn_banach_closed_point, RootPairing.toPerfPair_comp_root, Submodule.dualAnnihilator_eq_top_iff, MeasureTheory.OuterMeasure.comap_iInf, LinearMap.IsContPerfPair.bijective_left, Matrix.linearIndependent_rows_of_invertible, Algebra.trace_eq_of_algEquiv, Rep.coinvariantsTensorFreeLEquiv_symm_apply, linearIndependent_iff_ker, TensorProduct.dualDistrib_dualDistribInvOfBasis_right_inverse, FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal, CliffordAlgebra.contractRight_comm, adjoin_monomial_eq_reesAlgebra, Ring.jacobson_smul_top_le, Orientation.map_eq_neg_iff_det_neg, Rep.standardComplex.d_eq, instIsTorsionFree, Matrix.coe_vecMulLinear, Module.presentationFinsupp_G, CliffordAlgebra.EquivEven.reverse_e0, IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_mem, biUnion_associatedPrimes_eq_compl_regular, ContinuousMap.mem_idealOfSet, Pi.counit_comp_finsuppLcoeFun, Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange, MvPolynomial.monomial_zero, ExteriorAlgebra.algebraMap_eq_one_iff, Polynomial.monomial_mul_C, LinearMap.trace_eq_matrix_trace_of_finset, AddMonoidAlgebra.mapDomainBialgHom_id, Polynomial.X_pow_eq_monomial, Finset.affineCombination_apply_const, Matrix.SeparatingLeft.toLinearMap₂', DirectSum.decompose_eq_mul_idempotent, QuadraticAlgebra.instFinite, Polynomial.Bivariate.Polynomial.Bivariate.pderiv_one_equivMvPolynomial, irreducible_pow_sup_of_ge, ClassGroup.mk0_eq_one_iff, TensorProduct.dualDistrib_apply_comm, LinearMap.BilinForm.IsAlt.eq_of_add_add_eq_zero, Submodule.coe_dualCoannihilator_span, Finsupp.llift_apply, idealFactorsFunOfQuotHom_comp, ExteriorAlgebra.liftAlternating_comp, HasDerivWithinAt.hasFDerivWithinAt, LinearMap.IsSymm.ortho_comm, Algebra.isIntegral_trace, ContinuousAffineMap.lineMap_toAffineMap, LDL.lowerInv_eq_gramSchmidtBasis, wbtw_lineMap_iff, Ring.DimensionLEOne.not_lt_lt, SeparatingDual.exists_eq_one, Quaternion.instFree, nndist_right_lineMap, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_single, Matrix.sum_cramer, Module.Basis.det_isUnitSMul, coe_algebraMapCLM, Finsupp.linearCombination_smul, finrank_algHom, ModuleCat.exteriorPower.iso₀_hom_naturality, mem_vectorSpan_iff_eq_weightedVSub, Module.Basis.toDualEquiv_apply, CliffordAlgebra.changeFormAux_changeFormAux, Fintype.bilinearCombination_apply_single, Submodule.pow_smul_top_le, LinearMap.BilinForm.nondegenerate_toMatrix_iff, Matrix.derivative_det_one_add_X_smul, exteriorPower.ιMulti_span_of_span, LinearMap.toMatrixₛₗ₂'_symm, LieAlgebra.IsKilling.apply_coroot_eq_cast', Ideal.range_mul', LinearMap.BilinForm.exists_bilinForm_self_ne_zero, CliffordAlgebra.ι_mul_ι_comm, MonoidAlgebra.of_mem_span_of_iff, IsLocalRing.residue_eq_zero_iff, Module.equiv_directSum_of_isTorsion, PowerSeries.binomialSeries_coeff, PointedCone.subset_dual_dual, LinearMap.isSymm_iff_eq_flip, sbtw_iff_left_ne_and_right_mem_image_Ioi, QuadraticMap.discr_smul, RootPairing.root'_coroot_eq_pairing, Polynomial.mem_iff_eq_smul_annIdealGenerator, dotProductEquiv_apply_apply, LinearMap.rank_diagonal, Polynomial.derivative_rootMultiplicity_of_root, RootPairing.injOn_dualMap_subtype_span_root_coroot, ModuleCat.ExtendRestrictScalarsAdj.HomEquiv.toRestrictScalars_hom_apply, PrimeSpectrum.vanishingIdeal_anti_mono_iff, MvPolynomial.mem_pow_idealOfVars_iff, AffineEquiv.apply_lineMap, Ideal.finsuppTotal_apply, RootPairing.disjoint_corootSpan_ker_corootForm, AlternatingMap.map_basis_eq_zero_iff, LinearMap.separatingRight_toMatrix₂'_iff, Quaternion.imI_fst_dualNumberEquiv, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, Polynomial.taylor_injective, LieAlgebra.IsKilling.rootSpace_neg_nsmul_add_chainTop_of_lt, Polynomial.iterate_derivative_mul_X, Algebra.toMatrix_lmul_eq, NormedSpace.isClosed_polar, LieModule.traceForm_eq_zero_of_isNilpotent, MeasureTheory.charFunDual_eq_charFun_map_one, TensorProduct.tensorQuotEquivQuotSMul_comp_mkQ_lTensor, LinearMap.toMatrixAlgEquiv'_id, isAdjointPair_toLinearMap₂, Ideal.subset_union_prime', OreLocalization.mul_zero, Ideal.jacobson_bot_polynomial_le_sInf_map_maximal, MvPowerSeries.monomial_mem_nonzeroDivisorsLeft, MvPowerSeries.smul_eq_C_mul, CliffordAlgebra.ι_sq_scalar, Ideal.quotientEquivAlgOfEq_coe_eq_factor, Algebra.Generators.liftBaseChange_injective_of_isLocalizationAway, Rep.trivial_projective_of_subsingleton, LinearMap.BilinForm.sub_left, Algebra.intTrace_eq_of_isLocalization, TensorProduct.equivFinsuppOfBasisLeft_apply_tmul_apply, ContinuousLinearMapWOT.le_nhds_iff_forall_dual_apply_le_nhds, Finsupp.lmapDomain_linearCombination, exteriorPower.alternatingMapLinearEquiv_symm_map, LieIdeal.coe_killingCompl_top, Ideal.relNorm_top, PowerSeries.coeff_subst, Polynomial.sumIDeriv_eq_self_add, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, Algebra.trace_algebraMap, Polynomial.self_sub_monomial_natDegree_leadingCoeff, Module.Basis.repr_algebraMap, LinearMap.det_baseChange, exteriorPower.instFree, NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_norm_eq_mul_torsion, QuadraticForm.tmul_tensorMap_apply, DividedPowers.mem_dpEqualizer_iff, PrimeSpectrum.union_zeroLocus, Polynomial.iterate_derivative_X, ModN.basis_apply_eq_mkQ, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, HomogeneousIdeal.eq_bot_iff, AlgebraicGeometry.Scheme.zeroLocus_biInf, ExteriorAlgebra.ιMulti_succ_apply, PowerSeries.le_order_smul, nndist_left_lineMap, IsPrecomplete.bot, Algebra.Generators.cotangentSpaceBasis_repr_one_tmul, Polynomial.eval₂_monomial, Ideal.IsPrincipal.of_finite_maximals_of_isUnit, Ideal.toTwoSided_asIdeal, Polynomial.Splits.eval_derivative_eq_eval_mul_sum, MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent, MvPolynomial.pderiv_rename, PowerSeries.coeff_coe, AlgebraicGeometry.eq_bot_of_comp_quotientMk_eq_sigmaSpec, IntermediateField.finiteDimensional_right, AffineBasis.coord_apply, Polynomial.divByMonic_add_X_sub_C_mul_derivative_divByMonic_eq_derivative, Ideal.relNorm_eq_pow_of_isPrime_isGalois, LocalizedModule.divBy_mul_by, Rep.coindFunctorIso_hom_app_hom_hom_apply_hom_hom_apply, Finsupp.basis_repr, LinearMap.toMatrix_symm, Matrix.toBilin_apply, IsLocalRing.basisQuotient_repr, AdjoinRoot.evalEval_apply, LinearMap.toMatrix₂'_compl₁₂, Ideal.comap_iInf, Module.Relations.surjective_toQuotient, MvPolynomial.monomial_eq_monomial_iff, Subfield.relrank_mul_rank_top, LinearEquiv.det_baseChange, Ideal.mul_comm, RootPairing.flip_comp_polarization_eq_rootForm, DualNumber.lift_apply_eps, cfcL_integrable, Ideal.absNorm_mem, Polynomial.derivative'_apply, MeasureTheory.charFunDual_pi', AlternatingMap.ext_ring_iff, Polynomial.mem_support_derivative, MvPowerSeries.coeff_X, BoxIntegral.unitPartition.integralSum_eq_tsum_div, Ideal.IsMaximal.mem_associatedPrimes_of_isFractionRing, Ideal.map_mul, PowerSeries.eq_shift_mul_X_pow_add_trunc, Polynomial.monomial_pow, Representation.ofMulAction_single, Module.Basis.toLin_toMatrix, MvPolynomial.eval₂_mul_monomial, Submodule.map_smul'', MvPolynomial.weightedHomogeneousComponent_zero, LinearMap.flip_injective_iff₁, Finsupp.linearCombination_one_tmul, ExteriorAlgebra.map_comp_ι, Matrix.toLinearEquivRight'OfInv_symm_apply, WeierstrassCurve.Projective.addMap_of_Z_eq_zero_left, ExteriorAlgebra.ιMulti_span, Ideal.prod_bot_bot, Ideal.isIdempotentElem_iff_eq_bot_or_top, Ideal.mul_bot, Pi.counit_eq_adjoint, LinearMap.BilinForm.coeFnAddMonoidHom_apply, IsSimpleRing.exists_ringEquiv_matrix_end_mulOpposite, LocalizedModule.map_surjective, Ideal.sup_eq_top_iff_isCoprime, IsLocalizedModule.map_linearMap_of_isLocalization, MvPolynomial.homogeneousComponent_C_mul, Ideal.le_radical, Module.Basis.map_repr, CliffordAlgebra.EquivEven.ι_eq_v_add_smul_e0, AlgEquiv.quotientBot_mk, AdicCompletion.pi_apply_coe, Matrix.toLin'_mul_apply, Finset.sum_smul_vsub_eq_weightedVSubOfPoint_sub, PreValuationRing.iff_ideal_total, Polynomial.toLaurent_C_mul_T, MeasureTheory.OuterMeasure.map_apply, Ideal.smul_sup, Polynomial.card_roots_le_derivative, Matrix.toLinearMapₛₗ₂'_single, CliffordAlgebra.ofBaseChange_toBaseChange, Ideal.comap_le_map_of_inverse, IsLinearMap.isLinearMap_smul', Algebra.Extension.Cotangent.val_mk, Matrix.nondegenerate_toBilin'_iff, Ideal.smul_eq_mul, Ideal.Quotient.stabilizerHom_surjective, IsHausdorff.of_isDomain, LinearMap.trace_eq_contract_of_basis', PowerSeries.IsWeierstrassDivisorAt.coeff_seq_succ_sub_seq_mem, RingEquiv.quotientBot_symm_mk, DifferentialAlgebra.deriv_algebraMap, QuadraticForm.tmul_comp_tensorComm, LinearMap.BilinForm.congr_comp, Ideal.isPrime_ideal_prod_top', LinearMap.toMatrixAlgEquiv_apply, QuadraticForm.polarBilin_baseChange, Lagrange.eq_interpolate_iff, cfcₙ_apply_mkD, RingTheory.Sequence.isWeaklyRegular_iff_Fin, Polynomial.derivative_eval, Ideal.isPrimary_finsetInf, FreeNonUnitalNonAssocAlgebra.lift_unique, Algebra.traceForm_apply, PrimeSpectrum.coe_preimageHomeomorphFiber_apply_asIdeal, Algebra.IsUnramifiedAt.exists_hasStandardEtaleSurjectionOn, MvPolynomial.instFree, Algebra.TensorProduct.basis_repr_tmul, List.SortedLE.wbtw, cfcₙHom_id, LinearMap.BilinForm.apply_dualBasis_left, ContinuousLinearMapWOT.inducingFn_apply, AdicCompletion.map_injective, Ideal.factors_decreasing, Basis.linearEquiv_dual_iff_finiteDimensional, linearIndependent_monoidHom, Orientation.areaForm_neg_orientation, LinearMap.toMatrix'_toLin', PrimeSpectrum.iSup_basicOpen_eq_top_iff, AdicCompletion.val_add_apply, tendsto_card_div_pow_atTop_volume, affineSpan_eq_affineSpan_lineMap_units, PolynomialModule.monomial_smul_single, Submodule.annihilator_iSup, lineMap_mono_endpoints, IntermediateField.LinearDisjoint.trace_algebraMap, Pi.counit_comp_dFinsuppCoeFnLinearMap, MvPolynomial.weightedDecomposition.decompose'_apply, DividedPowers.SubDPIdeal.isSubideal, Module.mapEvalEquiv_apply, LinearMap.IsSymm.domRestrict, CliffordAlgebra.changeForm.zero_proof, RootPairing.ker_copolarization_eq_ker_corootForm, ContinuousMultilinearMap.ext_ring_iff, IsLocalRing.exists_maximalIdeal_pow_le_of_finite_quotient, LinearMap.bijective_compr₂_of_equiv, Valuation.leIdeal_map_algebraMap_eq_leSubmodule_min, QuadraticModuleCat.forget₂_map, TensorProduct.toMatrix_assoc, Subspace.finrank_dualCoannihilator_eq, NumberField.mixedEmbedding.stdBasis_apply_isComplex_snd, Matrix.toLinearMap₂'_apply', ExteriorAlgebra.map_comp_ιMulti, Submodule.iInf_colon_iSup, TrivSqZeroExt.lift_comp_inrHom, RootPairing.exists_ge_zero_eq_rootForm, Orientation.reindex_neg, AdicCompletion.smul_eval, Affine.Simplex.excenter_eq_affineCombination, exteriorPower.ιMultiDual_apply_diag, Module.DualBases.eval_of_ne, IntermediateField.LinearDisjoint.rank_right_mul_adjoin_rank_eq_of_isAlgebraic, MvPolynomial.expand_monomial, Ideal.pointwise_smul_subset_iff, MvPowerSeries.substAlgHom_eq_aeval, Polynomial.X_pow_smul_rTensor_monomial, Ideal.isPrime_int_iff, Finset.weightedVSubOfPoint_vadd, PiToModule.fromEnd_injective, IsLocalRing.mem_maximalIdeal, PadicInt.exists_mem_range, Module.Basis.mk_coord_apply_ne, KaehlerDifferential.mapBaseChange_tmul, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable, ExteriorAlgebra.algebraMap_eq_zero_iff, SkewPolynomial.monomial_def, MvPolynomial.monomial_mul_mem_coeffsIn, MonoidAlgebra.mapDomainBialgHom_comp, is_bot_adic_iff, MvPowerSeries.coeff_monomial_same, Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_eq, LinearMap.toMatrix_id, Polynomial.smul_monomial, uniqueDiffWithinAt_iff_accPt, InnerProductSpace.toDualMap_apply_apply, NNRealRMK.lintegral_rieszMeasure, ExteriorAlgebra.lift_ι_apply, ExteriorAlgebra.ιMulti_zero_apply, LinearMap.BilinForm.apply_eq_dotProduct_toMatrix_mulVec, KaehlerDifferential.kerTotal_eq, ContinuousLinearEquiv.unitsEquivAut_apply, Ideal.toCotangent_range, LinearMap.BilinForm.isNonneg_def, Finsupp.span_image_eq_map_linearCombination, MatrixEquivTensor.invFun_add, MvPowerSeries.hasSum_aeval, ModuleCat.ExtendRestrictScalarsAdj.Counit.map_hom_apply, LinearMap.baseChange_one, FractionalIdeal.mem_coeSubmodule, PrimeSpectrum.zeroLocus_subset_zeroLocus_iff, RootPairing.RootPositiveForm.posForm_apply_root_root_le_zero_iff, AlgebraicGeometry.StructureSheaf.comap_const, Module.Invertible.exists_linearEquiv_ideal, OreLocalization.zero_smul, Ideal.instCovariantClassHSMulLe, Matrix.ker_mulVecLin_conjTranspose_mul_self, ProperCone.mem_dual, Submodule.colon_union, Subspace.finiteDimensional_quot_dualCoannihilator_iff, Matrix.toAlgEquiv_kroneckerStarAlgEquiv, MvPowerSeries.weightedOrder_monomial_of_ne_zero, TensorPower.algebraMap₀_mul, Polynomial.iterate_derivative_X_sub_pow, ContinuousLinearMap.one_smulRight_eq_toSpanSingleton, Ideal.prod_le_inf, LinearIndependent.repr_eq_single, PowerSeries.coeff_inv_aux, MvPowerSeries.X_pow_dvd_iff, MeasureTheory.OuterMeasure.restrict_iInf, MeasureTheory.OuterMeasure.comap_apply, IsLocalization.AtPrime.coe_primeSpectrumOrderIso_apply_coe_asIdeal, WittVector.mk_pow_fontaineTheta, MvPolynomial.monomial_one_dvd_iff_modMonomial_eq_zero, PowerSeries.degree_trunc_lt, IsAdjoinRootMonic.basis_repr, MeasureTheory.FiniteMeasure.injective_toWeakDualBCNN, LinearMap.toMatrix_transpose, Localization.exists_finite_awayMapₐ_of_surjective_awayMapₐ, WeakDual.toStrongDual_inj, Representation.leftRegular_norm_eq_zero_iff, PowerSeries.smul_weierstrassDiv, Finsupp.supported_eq_span_single, Ideal.ker_Pi_Quotient_mk, FirstOrder.Field.lift_genericMonicPoly, Submodule.instSMulCommClass, ModuleCat.RestrictionCoextensionAdj.counit'_app, Ideal.adic_module_basis, MvPowerSeries.coeff_mul_prod_one_sub_of_lt_order, Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective, Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange_residueField, Submodule.IsPrincipal.contentIdeal_le_span_iff_dvd, PowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto, IsLocalizedModule.map_surjective_iff_localizedModuleMap_surjective, LinearMap.extendTo𝕜'_apply, LieIdeal.mem_killingCompl, CliffordAlgebra.toBaseChange_comp_ofBaseChange, AffineMap.lineMap_injective, PrimeSpectrum.coe_preimageOrderIsoFiber_symm_apply_coe_asIdeal, Ideal.isCoprime_iff_gcd, HopfAlgebra.sum_mul_antipode_eq_smul, LinearMap.isSymm_dualProd, Algebra.Extension.subsingleton_h1Cotangent, Matrix.mulVecLin_add, Matrix.linearIndependent_cols_of_isUnit, LinearMap.trace_smulRight, Algebra.TensorProduct.basisAux_map_smul, Pi.basis_repr_single, LinearMap.convOne_apply, Module.Relations.Solution.fromQuotient_toQuotient, AddMonoidAlgebra.counit_single, CompactlySupportedContinuousMap.toRealLinearMap_apply_apply, Pi.intrinsicStar_comul_commSemiring, LocalizedModule.mem_ker_mkLinearMap_iff, LinearMap.isIdempotentElem_map_one_iff, Algebra.Generators.exists_presentation_of_free_cotangent, Algebra.IsInvariant.exists_smul_of_under_eq, Ideal.iSup_eq_span, TensorProduct.equivFinsuppOfBasisRight_apply_tmul, MvPolynomial.X_mul_pderiv_monomial, AddSubgroup.index_eq_natAbs_det, Polynomial.isPrimitive_iff_contentIdeal_eq_top, Module.Basis.ofEquivFun_repr_apply, AlgebraicGeometry.instIsScalarTowerObjOppositeOpensCarrierTopValStructureSheafInType, QuadraticAlgebra.imₗ_apply, Ideal.jacobson_eq_top_iff, LinearMap.det_mulRight, Submodule.IsPrincipal.contentIdeal_generator_dvd_coeff, MvPowerSeries.coeff_index_single_X, PowerSeries.WithPiTopology.uniformContinuous_coeff, MeasureTheory.OuterMeasure.restrict_ofFunction, Module.one_le_rank_iff, IsLocalizedModule.fromLocalizedModule.bij, Lagrange.interpolate_eq_sum, Matrix.SpecialLinearGroup.toLin'_symm_to_linearMap, Module.Basis.sumQuot_repr_left, Algebra.Extension.CotangentSpace.map_tmul, mem_skewAdjointMatricesSubmodule', Matrix.toLinearMap₂'Aux_single, MvPowerSeries.LinearTopology.mem_basis_iff, MvPowerSeries.subst_monomial, Representation.ofMulAction_self_smul_eq_mul, Module.isTorsionBySet_annihilator, Module.Basis.reindexRange_repr', cross_anticomm, PrimeSpectrum.zeroLocus_span, RingEquiv.moduleEndSelf_symm_apply, ConcaveOn.antitoneOn_slope_gt, IsAdic.isPrecomplete_iff, Polynomial.degreeLT.addLinearEquiv_castAdd, KaehlerDifferential.polynomialEquiv_comp_D, LocalizedModule.equivTensorProduct_symm_apply_tmul_one, map_lt_lineMap_iff_slope_lt_slope_right, Ideal.powQuotSuccInclusion_apply_coe, LinearMap.dualMap_surjective_of_injective, LinearEquiv.toSpanNonzeroSingleton_homothety, RootPairing.Base.exists_mem_span_pairingIn_ne_zero_and_pairwise_ne, Module.Basis.reindexRange_repr, IsBaseChange.endHom_toMatrix, MeasureTheory.OuterMeasure.map_iInf_comap, Ideal.range_mul, Ideal.map_inf_comap_of_surjective, range_cfcₙHom, LocalizedModule.equivTensorProduct_apply_mk, Algebra.TensorProduct.lid_tmul, cross_cross_eq_smul_sub_smul, Real.map_matrix_volume_pi_eq_smul_volume_pi, affineCombination_mem_affineSpan_pair, TrivSqZeroExt.algHom_ext'_iff, MeasureTheory.FiniteMeasure.tendsto_iff_weakDual_tendsto, Submodule.dualAnnihilator_anti, PowerSeries.IsWeierstrassDivision.unique, Algebra.discr_powerBasis_eq_norm, LinearMap.BilinForm.sum_left, LinearMap.surjective_iff_ne_zero, HopfAlgebra.sum_mul_antipode_eq_algebraMap_counit, TensorProduct.equivFinsuppOfBasisLeft_apply_tmul, IsLocalization.mapFrameHom_apply, MvPolynomial.weightedHomogeneousComponent_eq_zero', PowerSeries.coeff_zero_one, QuadraticForm.polarBilin_tmul, PiTensorProduct.ofFinsuppEquiv'_tprod_single, TensorProduct.lidIsometry_apply, Ideal.isIdempotentElem_iff_of_fg, AlgebraicGeometry.StructureSheaf.comap_apply, Finset.weightedVSubOfPoint_filter_of_ne, MvPowerSeries.coeff_of_lt_order, LocalizedModule.map_injective, RootPairing.toLinearMap_apply_apply_Polarization, Algebra.Extension.H1Cotangent.val_zero, nilradical_le_jacobson, Nat.subset_span_setGcd, StrongDual.toWeakDual_inj, RootPairing.exist_set_root_not_disjoint_and_le_ker_coroot'_of_invtSubmodule, LinearMap.BilinForm.Nondegenerate.congr, IsAdic.hasBasis_nhds, Polynomial.map_monomial, rieszContentAux_image_nonempty, QuadraticAlgebra.instFree, HomogeneousIdeal.toAddSubmonoid_irrelevant_le, AlgebraicGeometry.Scheme.IdealSheafData.ideal_pow, isGroupLikeElem_iff, PowerSeries.coeff_X_mul_largeSchroderSeriesSeries_sq, CommSemiring.rank_self, ProperCone.subset_dual_dual, TensorProduct.comm_trans_lid, LinearMap.flip_surjective_iff₁, TensorAlgebra.toExterior_ι, IsLocalization.map_inf, Module.Basis.coe_toOrthonormalBasis_repr, IntermediateField.relfinrank_top_right, ExteriorAlgebra.liftAlternatingEquiv_symm_apply, Matrix.toLinearMap₂'_apply, PrimeSpectrum.isClosed_iff_zeroLocus_radical_ideal, matPolyEquiv_coeff_apply_aux_1, WeakDual.coe_toStrongDual, NormedSpace.eq_zero_iff_forall_dual_eq_zero, Affine.Simplex.affineCombination_mem_interior_face_iff_mem_Ioo, Polynomial.degreeLT.addLinearEquiv_apply_fst, LinearMap.BilinForm.separatingLeft_toMatrix'_iff, Matrix.linfty_opNNNorm_eq_opNNNorm, TensorProduct.finsuppScalarRight_apply, Algebra.FormallySmooth.iff_injective_lTensor_residueField, ModularGroup.lcRow0Extend_symm_apply, rank_lt_rank_dual', AdicCompletion.sumInv_apply, IsLocalization.isMaximal_iff_isMaximal_disjoint, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, Polynomial.derivative_X_sub_C_sq, FreeLieAlgebra.liftAux_map_smul, Module.dual_finite, MvPolynomial.homogeneousComponent_isHomogeneous, MonoidAlgebra.mapDomainBialgHom_mapDomainBialgHom, Module.Relations.Solution.ofQuotient_var, Finset.affineCombination_congr, LinearMap.trace_mul_comm, LinearMap.trace_eq_contract, CategoryTheory.Iso.toIsometryEquiv_toFun, AddMonoidAlgebra.mapDomainBialgHom_mapDomainBialgHom, NumberField.mixedEmbedding.fundamentalCone.integerSetEquiv_apply_fst, FiniteField.algebraMap_trace_eq_sum_pow, IsLocalRing.maximalIdeal_eq_bot, ModuleCat.CoextendScalars.smul_apply, CliffordAlgebra.changeFormAux_apply_apply, LocalizedModule.equivTensorProduct_symm_apply_tmul, AddValuation.self_le_supp_comap, Module.comap_freeLocus_le, ContinuousLinearMap.norm_toSpanSingleton, CliffordAlgebra.toBaseChange_comp_involute, AdicCompletion.sum_lof, TensorProduct.finsuppScalarRight_apply_tmul, Polynomial.smeval.linearMap_apply, LinearMap.trace_eq_sum_inner, Polynomial.eval_derivative_of_splits, Polygon.edgeSet_eq_image_edgePath, MvPolynomial.sum_homogeneousComponent, LinearMap.BilinForm.not_nondegenerate_zero, Ideal.under_smul, Polynomial.isAlt_wronskianBilin, Module.FinitePresentation.linearEquivMap_symm_apply, exists_basis_of_pairing_ne_zero, HomogeneousIdeal.toIdeal_bot, Affine.Simplex.circumcenter_eq_affineCombination_of_pointsWithCircumcenter, Ideal.IsPrime.inf_le, Submodule.annihilator_mul, Subspace.dualPairing_nondegenerate, Matrix.isNilpotent_toLin'_iff, Finset.affineCombination_eq_linear_combination, Module.supportDim_self_eq_ringKrullDim, divRadical_dvd_derivative, Bialgebra.TensorProduct.counit_eq_algHom_toLinearMap, Ideal.map_relNorm, Coalgebra.TensorProduct.lid_tmul, MvPolynomial.degreesLE_nsmul, MvPolynomial.coe_monomial, LinearMap.trace_one, Module.Basis.constrL_apply, ProbabilityTheory.uncenteredCovarianceBilinDual_of_not_memLp, Ideal.Quotient.factorₐ_refl, Polynomial.aeval_add_of_sq_eq_zero, MvPowerSeries.X_def, Polynomial.mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero, Polynomial.derivativeFinsupp_apply_toFun, ExteriorAlgebra.GradedAlgebra.ι_sq_zero, MvPolynomial.C_apply, Ideal.eq_bot_or_top, ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal, Polynomial.deriv_aeval, hasGradientWithinAt_iff_hasFDerivWithinAt, IsModuleTopology.continuous_bilinear_of_pi_fintype, Ideal.localized'_eq_map, MeasureTheory.Measure.comapₗ_apply, Ideal.IsPrime.inf_le', Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero, MeasureTheory.charFunDual_prod, Submodule.mul_mem_smul_iff, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff_of_mem, ExteriorAlgebra.lift_comp_ι, Ideal.span_singleton_mul_eq_span_singleton_mul, HomogeneousIdeal.toIdeal_iSup₂, AddMonoidAlgebra.decomposeAux_coe, AffineMap.apply_lineMap, MeasureTheory.OuterMeasure.map_comap_le, Polynomial.ofFinsupp_single, Matrix.toLinearEquiv'_symm_apply, MvPowerSeries.ext_iff, LinearMap.BilinForm.IsSymm.eq, IsAdic.isAdicComplete_iff, PiTensorProduct.constantBaseRingEquiv_symm, LinearMap.BilinForm.mul_toMatrix'_mul, Module.Basis.repr_smul, Matrix.toLin_symm, Ideal.comap_sInf', Ideal.inf_mul, AffineMap.hasDerivAt_lineMap, Ideal.isTorsionBySet_cotangent, Polynomial.le_trailingDegree_monomial, WeakDual.CharacterSpace.instNonUnitalAlgHomClass, Ideal.isRadical_iInf, Polynomial.iterate_derivative_derivative_mul_X_sq, Hausdorffification.instIsHausdorff, HomogeneousLocalization.val_zsmul, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, ProjectiveSpectrum.union_zeroLocus, MvPolynomial.single_eq_monomial, AddMonoidAlgebra.gradeBy.isInternal, finsuppTensorFinsupp'_single_tmul_single, AddMonoidAlgebra.grade_eq_lsingle_range, Finset.weightedVSub_empty, Rep.standardComplex.εToSingle₀_comp_eq, Derivation.comp_aeval_eq, Coalgebra.subsingleton_to_ring, Module.Basis.orientation_eq_iff_det_pos, CliffordAlgebraQuaternion.k_quaternionBasis, Submodule.traceDual_top', MeasureTheory.Measure.mapₗ_congr, WeakDual.CharacterSpace.continuousMapEval_bijective, Module.Flat.iff_rTensor_injective, LinearMap.baseChange_pow, CompactlySupportedContinuousMap.coe_toRealLinearMap, AlgebraicGeometry.Scheme.ΓSpecIso_inv, SeparatingDual.dualMap_surjective_iff, NumberField.canonicalEmbedding.integralBasis_repr_apply, InnerProductGeometry.norm_toLp_symm_crossProduct, dualTensorHom_apply, BilinForm.mul_toMatrix_mul, isNoetherian_linearMap_pi, ZSpan.measureReal_fundamentalDomain, Ideal.sup_pow_add_le_pow_sup_pow, Polynomial.modByMonic_eq_zero_iff_quotient_eq_zero, PrimeSpectrum.vanishingIdeal_eq_top_iff, QuadraticForm.tensorRId_symm_apply, Ideal.relNorm_le_comap, IsNoetherianRing.isClosed_ideal, Finset.sum_smul_const_vsub_eq_vsub_affineCombination, Ideal.bot_mul, Ideal.map_includeRight_eq, Quaternion.fst_imK_dualNumberEquiv_symm, MeasureTheory.OuterMeasure.map_le_restrict_range, LinearMap.trace_comp_eq_mul_of_commute_of_isNilpotent, PowerSeries.trunc_trunc_mul_trunc, Module.Basis.linearMap_apply, AdicCompletion.mkₐ_apply_coe, PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, Ideal.span_singleton_one, Submodule.dualCoannihilator_iSup_eq, Module.comap_eval_surjective, NormedSpace.inclusionInDoubleDual_norm_eq, AffineMap.left_vsub_lineMap, PowerSeries.coeff_invOfUnit, instPosMulStrictMonoIdeal, Matrix.toLinearMap₂_compl₁₂, Affine.Simplex.affineCombination_mem_closedInterior_face_iff_nonneg, LinearMap.dualMap_id, Ideal.span_eq_top_iff_finite, Ideal.radical_inf, Submodule.coe_span_smul, Ideal.pow_succ_lt_pow, AffineMap.hasDerivWithinAt_lineMap, Module.Finite.exists_nat_not_surjective, Polynomial.C_mul_X_pow_eq_monomial, Submodule.exists_le_ker_of_notMem, basisOfLinearIndependentOfCardEqFinrank_repr_apply, mem_pairSelfAdjointMatricesSubmodule', SkewMonoidAlgebra.nonUnitalAlgHom_ext'_iff, ExteriorAlgebra.liftAlternating_ιMulti, MonomialOrder.leadingTerm_monomial, IsDedekindDomain.primesOverEquivPrimesOver_inertiagDeg_eq, MvPowerSeries.WithPiTopology.continuous_coeff, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', cfcₙHom_integral, Ideal.map_mk_comap_factorPow, Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomial, AdicCompletion.map_comp_apply, IsCyclotomicExtension.Rat.p_mem_span_zeta_sub_one, MvPowerSeries.monomial_def, PowerSeries.trunc_mul_C, ContinuousLinearMap.nnnorm_toSpanSingleton, QuaternionAlgebra.instIsTorsionFree, IsLocalizedModule.iso_symm_comp, LieModule.traceForm_lieSubalgebra_mk_right, MonoidAlgebra.scalarTensorEquiv_tmul, Matrix.cramer_apply, tendsto_integral_exp_smul_cocompact, mem_affineSpan_iff_eq_affineCombination, QuadraticForm.tensorAssoc_toLinearEquiv, FGModuleCat.FGModuleCatEvaluation_apply, Ring.DimensionLeOne.prime_le_prime_iff_eq, Ideal.span_single_eq_top, Ideal.Quotient.span_singleton_one, Matrix.toLinearMapₛₗ₂'_symm, Module.Relations.range_map, Module.Basis.forall_coord_eq_zero_iff, Polynomial.taylor_zero', curveIntegral_segment, Matrix.cRank_toNat_eq_finrank, ContinuousMap.idealOfSet_closed, Submodule.mem_annihilator', DualNumber.fst_eq_zero_iff_eps_dvd, IsIdempotentElem.ker_toSpanSingleton_eq_span, Ideal.IsHomogeneous.bot, AdicCompletion.isAdicCauchy_iff, LinearMap.polyCharpoly_baseChange, CliffordAlgebra.changeForm_self, LinearMap.isAlt_iff_eq_neg_flip, RootPairing.RootPositiveForm.exists_eq, RootPairing.corootSpan_map_flip_toPerfPair, LinearMap.toLinearMap_toContPerfPair, convexHull_basis_eq_stdSimplex, Module.Basis.trace_traceDual_mul, IntermediateField.relrank_top_right, PowerSeries.hasSum_of_monomials_self, Polynomial.monomial_eq_zero_iff, Polynomial.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ, Ideal.disjoint_powers_iff_notMem, coevaluation_apply_one, CliffordAlgebraQuaternion.i_quaternionBasis, MatrixEquivTensor.toFunBilinear_apply, MonoidAlgebra.basis_apply, Module.Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul, ValuationSubring.idealOfLE_le_of_le, AffineBasis.toMatrix_apply, Ideal.IsPrime.multiset_prod_map_le, ContinuousLinearMap.norm_smulRightL, Subfield.finrank_comap, Submodule.dualRestrict_ker_eq_dualAnnihilator, IsBaseChange.transvection, Ideal.mul_left_self_sup, MvPowerSeries.le_lexOrder_iff, LinearMap.toBilin'Aux_toMatrixAux, MeasureTheory.FiniteMeasure.self_eq_mass_smul_normalize, LieAlgebra.IsKilling.invtSubmoduleToLieIdeal_top, Finset.weightedVSub_weightedVSubVSubWeights, Matrix.linearIndependent_cols_of_det_ne_zero, PrimeSpectrum.sup_vanishingIdeal_le, sub_mem_pNilradical_iff_pow_expChar_pow_eq, AffineMap.lineMap_vsub_right, Module.Dual.eval_apply, MvPolynomial.IsHomogeneous.HomogeneousSubmodule.gcommSemiring, TwoSidedIdeal.orderIsoIsTwoSided_symm_apply, Representation.coinvariantsTprodLeftRegularLEquiv_apply, AdicCompletion.val_sum, LinearMap.toMatrix₂_compl₂, MvPolynomial.degreesLE_add, finsuppTensorFinsuppLid_symm_single_smul, Finset.weightedVSubOfPoint_sdiff, HasDerivAt.hasFDerivAt, Ideal.eq_jacobson_iff_sInf_maximal', Pi.ker_ringHom, InnerProductSpace.trace_rankOne, MonomialOrder.span_leadingTerm_eq_span_monomial, Ideal.pi_mkQ_surjective, LinearMap.trace_eq_contract_apply, PiTensorProduct.constantBaseRingEquiv_tprod, TensorPower.algebraMap₀_one, ExteriorAlgebra.liftAlternatingEquiv_apply, QuaternionAlgebra.instFree, dualTensorHomEquivOfBasis_symm_cancel_right, WithIdealFilter.instIsLinearTopology, MonomialOrder.span_leadingTerm_eq_span_monomial₀, Algebra.linearMap_self, NormedSpace.eq_iff_forall_dual_eq, Matrix.toLpLin_symm_pow, Ideal.card_stabilizer_eq, Ideal.comap_eq_top_iff, Polynomial.mkDerivation_one_eq_derivative', MvPolynomial.scalarRTensor_apply_tmul, Polynomial.derivative_X_add_C, AffineMap.lineMap_continuous, LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton', RootPairing.rootForm_root_self, MvPowerSeries.monomial_mem_nonzeroDivisorsRight, Polynomial.hermite_succ, Algebra.TensorProduct.map_ker, KaehlerDifferential.kerTotal_mkQ_single_algebraMap_one, finsuppTensorFinsupp'_symm_single_mul, Submodule.mem_smul_iff_inv_mul_mem, Finset.affineCombination_of_eq_one_of_eq_zero, LinearMap.polyCharpolyAux_map_eval, isJacobsonRing_iff_sInf_maximal, affineCombination_mem_affineSpan, Matrix.toLinearMapₛₗ₂'_aux_eq, CommRing.Pic.inv_eq_dual, ZSpan.repr_ceil_apply, stdSimplex.map_coe, LinearMap.BilinForm.isRefl_zero, PowerSeries.coe_hasEvalIdeal, MvPolynomial.C_mul_X_eq_monomial, MatrixEquivTensor.invFun_algebraMap, PowerSeries.coeff_derivativeFun, Algebra.TensorProduct.instSMulCommClassTensorProduct_1, LinearMap.IsPerfectCompl.isCompl_right, RingHom.mem_ker, Ideal.span_singleton_le_span_singleton, Submodule.surjective_tensorToSpan, LinearMap.BilinForm.tensorDistribEquiv_toLinearMap, PolyEquivTensor.toFunLinear_mul_tmul_mul, PrimeSpectrum.mem_compl_zeroLocus_iff_notMem, PrimeSpectrum.zeroLocus_radical, Polynomial.hermite_eq_iterate, DualNumber.maximalIdeal_eq_span_singleton_eps, PrimeSpectrum.zeroLocus_minimalPrimes, Ideal.FiniteHeight.eq_top_or_height_ne_top, Polynomial.hasseDeriv_comp, Ideal.IsPrimary.inf, Matrix.ker_mulVecLin_eq_bot_iff, MvPowerSeries.coeff_index_single_self_X, dist_affineCombination_lt_of_strictConvexSpace, SemimoduleCat.hom_hom_rightUnitor, Ideal.mem_span_range_self, IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff', Matrix.linfty_opNNNorm_toMatrix, dist_lineMap_left, MvPowerSeries.WithPiTopology.as_tsum, MeasureTheory.OuterMeasure.isometry_comap_mkMetric, KaehlerDifferential.tensorKaehlerEquivBase_tmul, MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real, ModuleCat.exteriorPower.iso₀_hom_apply, MeasureTheory.OuterMeasure.map_comap, IsHausdorff.iInf_pow_smul, strongRankCondition_iff_forall_rank_lt_aleph0, PrimeSpectrum.isIrreducible_zeroLocus_iff, MvPolynomial.bind₂_monomial, IsLocalizedModule.fromLocalizedModule.inj, Module.Basis.addSubgroupOfClosure_repr_apply, Module.Invertible.bijective_curry, Algebra.traceForm_toMatrix, KaehlerDifferential.tensorKaehlerEquivBase_symm_apply, IsLocalization.coeSubmodule_bot, LinearMap.trace_eq_zero_of_mapsTo_ne, DualNumber.lift_inlAlgHom_eps, MvPolynomial.monomialOneHom_apply, Module.Basis.coe_sumCoords_of_fintype, Module.Basis.repr_range, IsDedekindDomain.exists_forall_sub_mem_ideal, IsDedekindDomain.HeightOneSpectrum.valuation_def, basis_toMatrix_basisFun_mul, cross_anticomm', Affine.Simplex.mongePoint_eq_affineCombination_of_pointsWithCircumcenter, Matrix.toLin_scalar, Ideal.isPrimary_finset_inf, ValuativeRel.supp_def, AdicCompletion.map_comp, Polynomial.iterate_derivative_natCast_mul, hasStrictFDerivAt_exp_smul_const', lineMap_lt_right_iff_lt_one, MonoidAlgebra.mem_span_support, instFinitePresentationFinsupp, Ideal.span_singleton_mul_right_inj, IntermediateField.LinearDisjoint.rank_right_mul_adjoin_rank_eq_of_isAlgebraic_right, Ideal.IsPrime.smul_iff, toBilin'Aux_toMatrixAux, NumberField.det_basisOfFractionalIdeal_eq_absNorm, IsSimpleModule.toSpanSingleton_surjective, ZSpan.measure_fundamentalDomain, MeasureTheory.FiniteMeasure.map_fst_prod, MvPolynomial.pderiv_sumToIter, MvPolynomial.coeff_homogeneousComponent, Ideal.FG.mul, Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_left_iff, FixedPoints.finrank_le_card, ContinuousLinearMap.toSpanSingleton_apply_map_one, lineMap_le_lineMap_iff_of_lt', BilinForm.dotProduct_toMatrix_mulVec, LinearMap.BilinForm.tensorDistrib_tmul, AdjoinRoot.mk_leftInverse, algebraMap.coe_deriv, CliffordAlgebra.coe_toEven_reverse_involute, Finset.eq_weightedVSubOfPoint_subset_iff_eq_weightedVSubOfPoint_subtype, RingEquiv.quotientBot_mk, QuadraticForm.baseChange_tmul, LinearMap.trace_transpose', Ideal.image_subset_nonunits_valuationSubring, Polynomial.natTrailingDegree_monomial, MvPowerSeries.WithPiTopology.uniformContinuous_coeff, lineMap_strict_mono_endpoints, ExteriorAlgebra.ι_comp_lift, bernsteinPolynomial.iterate_derivative_at_1_eq_zero_of_lt, Projectivization.cross_mk_of_cross_ne_zero, Submodule.mem_annihilator, finrank_quotient_span_eq_natDegree, Matrix.toLpLin_one, Module.Basis.trace_mul_traceDual, CliffordAlgebra.ι_mul_ι_mul_ι, finsuppTensorFinsupp'_apply_apply, Polynomial.eval_monomial_one_add_sub, ContinuousOn.lineMap, Ideal.disjoint_map_primeCompl_iff_comap_le, IsAdicComplete.bot, RootPairing.InvariantForm.exists_apply_eq_or, bddAbove_slope_gt_of_mem_interior, MeasureTheory.charFunDual_eq_prod_iff, Rep.coinvariantsTensorFreeLEquiv_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_bot, LinearRecurrence.solSpace_rank, MvPolynomial.monomial_left_inj, Module.Dual.transpose_apply, Module.Basis.reindexRange_repr_self, Sbtw.right_mem_image_Ioi, Module.Invertible.instDual, Polynomial.map_taylor, LinearMap.isPosSemidef_iff_posSemidef_toMatrix, MvPolynomial.X_dvd_monomial, CliffordAlgebra.EvenHom.compr₂_bilin, RootPairing.zero_le_rootForm, Module.Basis.det_unitsSMul, nonPrincipals_eq_empty_iff, Polynomial.map_under_lt_comap_of_weaklyQuasiFiniteAt, Ideal.index_pow_le, TannakaDuality.FiniteGroup.sumSMulInv_apply, hasDerivWithinAt_iff_hasFDerivWithinAt, right_lt_lineMap_iff_lt, Algebra.PreSubmersivePresentation.aevalDifferential_toMatrix'_eq_mapMatrix_jacobiMatrix, Derivation.mapCoeffs_monomial, Ideal.IsMaximal.isClosed, Algebra.PreSubmersivePresentation.jacobiMatrix_apply, FractionalIdeal.coeIdeal_inf, IntermediateField.LinearDisjoint.basisOfBasisLeft_apply, Module.rankAtStalk_baseChange, MvPowerSeries.monomial_zero_eq_C, MvPowerSeries.coeff_inv, KaehlerDifferential.tensorKaehlerEquivOfFormallyEtale_apply, exteriorPower.finrank_eq, StrongDual.toLp_of_not_memLp, Module.Basis.smul_det, Ideal.iInf_span_singleton_natCast, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff, Finsupp.linearCombination_linear_comp, Submodule.comap_dualAnnihilator, Polynomial.contentIdeal_eq_top_iff_forall_gaussNorm_eq_one, linearIndependent_algHom_toLinearMap, MeasureTheory.OuterMeasure.restrict_univ, Module.Basis.repr_eq_iff, lineMap_mem_openSegment, Polynomial.degreeLT.addLinearEquiv_apply_snd, IsLocalization.mk'_mem_iff, PowerSeries.eval₂_eq_tsum, AlgebraicClosure.instIsAlgClosureOfIsAlgebraic, MvPolynomial.C_mul_monomial, Algebra.normalizedTrace_intermediateField, LinearMap.BilinForm.smul_right_of_tower, TensorProduct.inner_lid_lid, TensorProduct.tmul_eq_smul_one_tmul, LinearMap.BilinForm.tensorDistribEquiv_apply, ClassGroup.mkMMem_surjective, Finset.attach_affineCombination_coe, linearIndependent_toLinearMap, WeierstrassCurve.Projective.smul_eq, Algebra.Generators.H1Cotangent.exact_map_δ, Ideal.homogeneousHull.gc, Matrix.piLp_ofLp_toEuclideanLin, eq_affineCombination_of_mem_affineSpan_image, Algebra.Generators.H1Cotangent.δAux_monomial, LinearMap.BilinForm.toMatrix_apply, Module.Basis.sumQuot_repr_inl, Ideal.IsRadical.inf, MvPolynomial.vars_monomial, Polynomial.modByMonicHom_apply, Algebra.trace_isNilpotent_of_isNilpotent, MvPolynomial.pow_idealOfVars_eq_span, DualNumber.isNilpotent_iff_eps_dvd, LieAlgebra.killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero, TrivSqZeroExt.map_comp_inrHom, RootPairing.reflectionPerm_coroot, NormedSpace.inclusionInDoubleDual_norm_le, Polynomial.addHom_ext'_iff, AddMonoidAlgebra.mem_ideal_span_of'_image, LieSubmodule.traceForm_eq_zero_of_isTrivial, LinearMap.dualProd_apply_apply, lieEquivMatrix'_apply, Rep.barResolution_complex, Algebra.Generators.CotangentSpace.compEquiv_symm_zero, PowerSeries.trunc_X_of, TensorProduct.dualDistribEquivOfBasis_symm_apply, BilinForm.toMatrix_toBilin, Module.dual_free, DualNumber.isMaximal_span_singleton_eps, WeakDual.isCompact_closedBall, LinearMap.IsPerfectCompl.flip_iff, Polynomial.derivative_X_pow_succ, LinearMap.isSymm_def, spinGroup.conjAct_smul_range_ι, Matrix.toPerfectPairing_apply_apply, CliffordAlgebra.EquivEven.neg_e0_mul_v, Matrix.mulVecLin_zero, WittVector.mem_span_p_pow_iff_le_coeff_eq_zero, KaehlerDifferential.kerTotal_mkQ_single_add, IsBaseChange.of_basis, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux, RCLike.reCLM_coe, LinearMap.BilinForm.IsRefl.smul, MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported, Finset.map_affineCombination, Differential.coeff_mapCoeffs, AdicCompletion.range_eval, InnerProductSpace.toDual_apply, LinearMap.BilinForm.IsNonneg.add, MeasureTheory.Measure.map_def, Nat.finite_setOf_setGcd_dvd_and_mem_span, Ideal.Quotient.smul_top, Algebra.Generators.cotangentSpaceBasis_repr_tmul, instFinitePresentationForall, right_le_lineMap_iff_le, Ideal.IsPrime.one_notMem, LinearMap.ker_toSpanSingleton, Ideal.span_singleton_zero, LinearMap.IsPerfPair.compl₁₂, NumberField.mixedEmbedding.fundamentalCone.prod_deriv_expMap_single, NonUnitalAlgHom.toAlgHom_zero, Ideal.comp_quotientMap_eq_of_comp_eq, LinearMap.minpoly_toMatrix', Ideal.Quotient.eq_zero_iff_dvd, Ideal.mem_of_liesOver, Ideal.instIsScalarTowerQuotientHPowKerRingHomAlgebraMapOfNat, LinearMap.BilinForm.IsPosSemidef.smul, List.exists_map_eq_of_sorted_nonempty_iff_sbtw, MvPolynomial.bind₂_monomial_one, Module.isTorsionBySet_span_singleton_iff, AdicCompletion.val_neg_apply, Polynomial.gaussNorm_lt_one_iff_contentIdeal_le, Module.Basis.flag_le_ker_coord, Polynomial.degree_monomial_le, LinearMap.trace_prodMap, Submodule.iSup_dualAnnihilator_le_iInf, ProbabilityTheory.covarianceBilinDual_eq_covariance, smoothSheafCommRing.isUnit_stalk_iff, Polynomial.pow_sub_one_dvd_derivative_of_pow_dvd, Ideal.eq_bot_of_minimalPrimes_eq_empty, ExteriorAlgebra.invertibleAlgebraMapEquiv_apply_invOf, Matrix.linearIndependent_rows_of_isUnit, LinearMap.liftBaseChangeEquiv_symm_apply, Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes, LieAlgebra.IsKilling.instIsCrystallographicSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, IntermediateField.finrank_lt_of_gt, ContinuousLinearMap.toSpanSingleton_pow, QuadraticForm.dualProd_apply, Fintype.bilinearCombination_apply, Algebra.WeaklyQuasiFiniteAt.finite_locoalization, Polynomial.coeff_monomial_of_ne, Algebra.TensorProduct.equivFinsuppOfBasis_apply, CompactlySupportedContinuousMap.toRealLinearMap_apply, Polynomial.derivative_X_sub_C_pow, LinearMap.toMatrix₂_comp, AlgebraicGeometry.StructureSheaf.algebraMap_self_map, exteriorPower.map_apply_ιMulti_family, AffineMap.lineMap_lineMap_right, ConcaveOn.slope_anti, Lagrange.degree_interpolate_erase_lt, Module.Quotient.mk_smul_mk, Algebra.isNilpotent_trace_of_isNilpotent, RingPreordering.mem_support, isometry_cfcₙHom, ExteriorAlgebra.ιMulti_range, Ideal.isMaximal_iff_of_bijective, differentialIdeal_le_iff, Affine.Simplex.wOppSide_affineSpan_faceOpposite_iff, TrivSqZeroExt.sndCLM_apply, IntermediateField.LinearDisjoint.basisOfBasisRight_apply, LinearMap.lsmul_flip_apply, ProbabilityTheory.covarianceBilinDual_zero, Ideal.primesOver_bot, Bialgebra.counit_surjective, Matrix.SeparatingRight.toLinearMap₂', Algebra.Extension.Hom.sub_one_tmul, Module.Basis.toDual_ker, killingForm_eq_zero_of_mem_zeroRoot_mem_posFitting, Ideal.ramificationIdx_bot, IsLinearTopology.hasBasis_open_ideal, LinearMap.BilinForm.IsSymm.baseChange, RatFunc.ofFractionRing_add, NormedSpace.polar_ball_subset_closedBall_div, DualNumber.lift_smul, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff, Module.Basis.repr_linearCombination, Module.Basis.dualBasis_apply_self, Ideal.isTwoSided_iff, PowerSeries.order_monomial, Coalgebra.sum_counit_smul, PointedCone.mem_dual, WittVector.IsocrystalEquiv.frob_equivariant, Quaternion.imK_fst_dualNumberEquiv, Ideal.Factors.isScalarTower, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff, Algebra.TensorProduct.equivFinsuppOfBasis_symm_apply, QuadraticForm.tensorRId_apply, InnerProductSpace.toDualMap_apply, geometric_hahn_banach_point_closed, MvPolynomial.rTensor_apply_X_tmul, Ideal.isRadical_bot, QuadraticForm.comp_tensorLId_eq, Algebra.SubmersivePresentation.linearIndependent_aeval_val_pderiv_relation, Ideal.span_singleton_generator, top_isPrincipal, MvPolynomial.pderiv_X, Algebra.zariskisMainProperty_iff', MvPowerSeries.coeff_homogeneousComponent, Algebra.TensorProduct.lmulEquiv_eq_lidOfCompatibleSMul, KaehlerDifferential.exact_mapBaseChange_map, PowerSeries.monomial_mul_monomial, LinearEquiv.isAdjointPair_symm_iff, Ideal.Quotient.index_eq_zero, LinearMap.BilinForm.add_left, Ideal.radical_bot_of_noZeroDivisors, PiLp.basisFun_repr, LinearMap.toSpanSingleton_apply_one, LinearMap.tensorProductEnd_apply, MvPolynomial.restrictSupport_univ, lift_rank_lt_rank_dual', Matrix.toLpLin_pow, ValuationSubring.primeSpectrumOrderEquiv_apply_coe_carrier, KaehlerDifferential.tensorKaehlerEquiv_tmul, RingPreordering.supportAddSubgroup_eq, LinearMap.trace_eq_contract', RootPairing.toLinearMap_apply_PolarizationIn, Algebra.Generators.H1Cotangent.δAux_toAlgHom, Matrix.kroneckerAlgEquiv_apply, IsDiscreteValuationRing.isOpen_iff, Finsupp.linearCombination_single_index, exteriorPower.zeroEquiv_naturality, Polynomial.derivWithin, LinearEquiv.symm_flip, instNonUnitalSubsemiringClassIdeal, Ideal.height_le_spanRank_toENat_of_mem_minimal_primes, KaehlerDifferential.D_apply, MeasureTheory.OuterMeasure.map_iInf_le, CliffordAlgebra.EquivEven.e0_mul_v_mul_e0, CliffordAlgebraComplex.ofComplex_I, LinearMap.BilinForm.separatingRight_toMatrix'_iff, LinearMap.extendTo𝕜_apply, Matrix.isUnit_toLin'_iff, Ideal.IsHomogeneous.iSup, DividedPowers.isDPMorphism_iff, Module.piEquiv_apply_apply, Submodule.span_algebraMap_image, isSimpleModule_self_iff_isUnit, Polynomial.Chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T, RootPairing.EmbeddedG2.long_eq_three_mul_short, LinearMap.BilinForm.smul_left, IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_dvd, Finsupp.lift_apply, LinearIndependent.linearCombinationEquiv_apply_coe, LinearMap.range_dualMap_eq_dualAnnihilator_ker, Submodule.dualCoannihilator_map_linearEquiv_flip, RootPairing.rootSpan_map_toPerfPair, Polynomial.adjSylvester_comp_sylveserMap, Polynomial.exists_iterate_derivative_eq_factorial_smul, AffineMap.lineMap_mem, IsLocalization.mk'_eq_mk', IsLocalization.algebraMap_mem_map_algebraMap_iff, Finset.weightedVSub_smul, Module.Relations.Solution.range_π, Ideal.Quotient.mk_singleton_self, RootPairing.GeckConstruction.trace_toEnd_eq_zero, ContinuousLinearMap.toSpanSingleton_smul, Polynomial.monomial_injective, Matrix.toLin'_mul, Polynomial.taylor_eq_zero, Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_right_iff, ModularGroup.lcRow0Extend_apply, AffineMap.lineMap_mono, LinearMap.isNilpotent_toMatrix_iff, AddMonoidAlgebra.mapDomainBialgHom_apply, LinearMap.BilinForm.lieInvariant_iff, cfcₙHom_predicate, Ideal.eq_span_singleton_mul, TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply, LinearMap.BilinForm.dotProduct_toMatrix_mulVec, Lagrange.degree_interpolate_lt, CliffordAlgebraQuaternion.equiv_apply, AffineMap.right_vsub_lineMap, Quaternion.re_fst_dualNumberEquiv, LinearMap.det_eq_det_toMatrix_of_finset, LieModule.traceForm_comm, LieSubmodule.trace_eq_trace_restrict_of_le_idealizer, Algebra.linearMap_apply, mem_conductor_iff, LinearMap.toMatrix_reindexRange, ExteriorAlgebra.liftAlternating_apply_ιMulti, MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict, QuadraticAlgebra.reₗ_apply, IsDedekindDomain.primesOverEquivPrimesOver_apply, Wbtw.left_mem_image_Ici_of_right_ne, ModuleCat.MonoidalCategory.rightUnitor_def, exteriorPower.ιMulti_span_fixedDegree_of_span_eq_top, AdicCompletion.transitionMap_comp_eval, Associates.le_singleton_iff, CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe, CommRing.Pic.mapAlgebra_apply, RatFunc.laurentAux_div, Matrix.ofLp_toLpLin, traceForm_dualSubmodule_adjoin, TensorPower.galgebra_toFun_def, Module.range_piEquiv, polyEquivTensor_symm_apply_tmul, RootPairing.coroot'_apply_apply_mem_of_mem_span, LinearIndependent.linearCombination_comp_repr, Matrix.toMatrix₂Aux_toLinearMap₂'Aux, MvPowerSeries.weightedHomogeneousComponent_of_lt_weightedOrder_eq_zero, Ideal.mem_map_of_equiv, Matrix.toBilin'_symm, Module.Basis.dualBasis_equivFun, AlgebraicGeometry.Scheme.IdealSheafData.le_ofIdeals_iff, HomogeneousIdeal.toIdeal_sSup, MvPowerSeries.WithPiTopology.hasSum_iff_hasSum_coeff, PowerSeries.eq_X_mul_shift_add_const, RootSystem.rootForm_anisotropic, RootPairing.corootSpan_dualAnnihilator_map_eq, AffineSubspace.wOppSide_lineMap_left, NumberField.FinitePlace.norm_lt_one_iff_mem, MvPowerSeries.coeff_subst, CoalgHomClass.counit_comp_apply, Module.support_of_algebra, MeasureTheory.OuterMeasure.comap_top, LinearMap.toMatrix_smulBasis_left, Polynomial.degreeLT.instFiniteSubtypeMemSubmodule, Module.Basis.coe_toOrthonormalBasis_repr_symm, IsAdjoinRootMonic.modByMonicHom_map, Ideal.mul_sup, KaehlerDifferential.one_smul_sub_smul_one_mem_ideal, PowerSeries.IsWeierstrassDivisorAt.mod_coe_eq_self, AdicCompletion.map_id, RootPairing.EmbeddedG2.allRoots_eq_map_allCoeffs, Module.Basis.orientation_eq_or_eq_neg, Ideal.finrank_prime_pow_ramificationIdx, Module.Dual.transpose_comp, CoalgHom.counit_comp, Polynomial.derivativeFinsupp_apply_apply, IsSMulRegular.notMem_of_mem_minimalPrimes, Algebra.Extension.CotangentSpace.map_toInfinitesimal_bijective, Ideal.le_pow_ramificationIdx, LinearMap.BilinForm.isSymm_iff_flip, AlgebraicGeometry.AffineSpace.spec_le_iff, MvPolynomial.algebraTensorAlgEquiv_symm_comp_aeval, MvPolynomial.finitePresentation_universalFactorizationMap, RootPairing.corootForm_apply_apply, LinearMap.flip_bijective_iff₁, Polynomial.taylor_apply, InnerProductSpace.toDual_apply_apply, KaehlerDifferential.range_mapBaseChange, LinearMap.flip_bijective_iff₂, Algebra.SubmersivePresentation.sectionCotangent_eq_iff, Matrix.mulVecLin_one, Ideal.Quotient.stabilizerHom_surjective_of_profinite, Algebra.Extension.Cotangent.mk_eq_zero_iff, Rat.valuation_le_one_iff_den, ContinuousLinearMap.adjointAux_apply, Polynomial.eval_derivative_eq_eval_mul_sum_of_splits, LinearMap.BilinForm.mem_dualSubmodule, LinearMap.range_dualMap_dual_eq_span_singleton, LinearMap.det_toLin', CompactlySupportedContinuousMap.toRealPositiveLinear_apply, MvPolynomial.map_restrict_dom_evalₗ, Polynomial.iterate_derivative_comp_one_sub_X, Algebra.Presentation.differentials.surjective_hom₁, Module.FinitePresentation.exists_fin, LinearMap.range_toSpanSingleton, MvPolynomial.tensorEquivSum_one_tmul_C, Algebra.QuasiFiniteAt.exists_basicOpen_eq_singleton, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, LaurentSeries.powerSeriesPart_coeff, PowerBasis.trace_gen_eq_nextCoeff_minpoly, Ideal.pi_tensorProductMk_quotient_surjective, Matrix.rank_eq_finrank_range_toLin, MvPolynomial.rTensor_apply_tmul, PowerSeries.IsWeierstrassDivisorAt.mod'_mk_eq_mod, CliffordAlgebra.invOf_ι_mul_ι_mul_ι, Matrix.toLinearMapRight'_mul_apply, CommRing.Pic.mk_eq_one_iff, Submodule.annihilator_smul, Submodule.jacobson_smul_lt_top, CliffordAlgebraQuaternion.equiv_symm_apply, PrimeSpectrum.mem_zeroLocus, Polynomial.degree_derivative_lt, BilinForm.toMatrix_comp, Matrix.eRank_toNat_eq_finrank, MvPolynomial.IsWeightedHomogeneous.weightedHomogeneousComponent_ne, AlgebraicGeometry.StructureSheaf.const_mul_cancel, CharacterModule.intSpanEquivQuotAddOrderOf_apply, Ideal.inertia_le_stabilizer, PowerBasis.exists_smodEq, OreLocalization.numeratorRingHom_apply, CliffordAlgebra.EvenHom.contract, Polynomial.derivative_natCast_mul, AlgHom.ker_coe_equiv, Module.Basis.linearCombination_dualBasis, FiniteDimensional.basisSingleton_repr_apply, sbtw_iff_right_ne_and_left_mem_image_Ioi, NumberField.RingOfIntegers.ker_algebraMap_eq_bot, DFinsupp.counit_comp_lsingle, IsLocalization.isPrime_iff_isPrime_disjoint, Polynomial.fderiv_aeval, Polynomial.Splits.eval_derivative_div_eval_of_ne_zero, Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one, transpose_dualTensorHom, Algebra.Generators.sq_ker_comp_le_ker_compLocalizationAwayAlgHom, PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff, ContinuousLinearMap.norm_smulRight_apply, Valuation.pow_Uniformizer_is_pow_generator, MvPolynomial.coeff_monomial_mul, HomogeneousIdeal.irrelevant_le, Matrix.toLin_mul, Rep.indResAdjunction_counit_app_hom_hom, CliffordAlgebra.map_apply_ι, exteriorPower.ιMulti_apply_coe, FGModuleCat.FGModuleCatDual_obj, Localization.mk_sum, Matrix.toLin_pow, Submodule.coe_tensorSpanEquivSpan_apply_tmul, Module.Basis.equivFunL_symm_apply_repr, LinearEquiv.baseChange_mul, PowerSeries.isWeierstrassDivision_weierstrassDiv_weierstrassMod, LinearMap.smulRight_eq_comp, PrimeSpectrum.subset_vanishingIdeal_zeroLocus, Algebra.Generators.H1Cotangent.exact_δ_map, lineMap_le_left_iff_nonpos, Ideal.Quotient.mk_bijective_iff_eq_bot, PowerSeries.coeff_invUnitsSub, PowerBasis.repr_gen_pow_isIntegral, ContinuousLinearMap.det_toSpanSingleton, Module.Basis.det_reindex_symm, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_comp_inl, SkewPolynomial.monomial_zero_one, IsDedekindDomain.map_differentIdeal_dvd_differentIdeal, Matrix.toLin_self, Rep.coindToInd_apply, CliffordAlgebra.ι_mul_ι_add_swap, IsLocalization.AtPrime.to_map_mem_maximal_iff, Submodule.dualCoannihilator_bot, LinearMap.BilinForm.toQuadraticMap_isOrtho, AffineMap.vadd_lineMap, Polynomial.valuation_inv_monomial_eq_valuation_X_zpow, mem_affineSpan_pair_iff_exists_lineMap_rev_eq, Polynomial.sum_monomial_index, Polynomial.derivativeFinsupp_derivative, AdicCompletion.sumInv_comp_sum, Subalgebra.LinearDisjoint.algebraMap_basisOfBasisRight_repr_apply, QuadraticAlgebra.rank_eq_two, KaehlerDifferential.ker_map, MvPolynomial.dvd_monomial_iff_exists, LinearMap.BilinForm.neg_left, QuadraticForm.equivalent_one_zero_neg_one_weighted_sum_squared, mem_span_C_coeff, AlgHom.toKerIsLocalization_apply, LinearMap.BilinForm.sum_repr_mul_repr_mul, Module.Basis.mulOpposite_repr_op, Orientation.kahler_neg_orientation, QuadraticForm.dualProdIsometry_invFun, PowerSeries.prod_monomial, AdicCompletion.coe_eval, LinearMap.separatingLeft_toMatrix₂'_iff, Module.Basis.smulTower'_repr_mk, CliffordAlgebraQuaternion.ofQuaternion_star, KaehlerDifferential.mapBaseChange_surjective, IntermediateField.LinearDisjoint.finrank_right_eq_finrank, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, PowerSeries.coeff_succ_C, LinearMap.nondegenerate_iff_det_ne_zero, Polynomial.sum_monomial_eq, QuadraticForm.equivalent_weightedSumSquares, Ideal.mem_sInf, ContinuousLinearMap.toSpanSingleton_comp, ModuleCat.ExtendRestrictScalarsAdj.HomEquiv.fromExtendScalars_hom_apply, TannakaDuality.FiniteGroup.leftRegular_apply, Ideal.instIsTwoSidedBot, Nat.mem_maximalIdeal_iff, IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite, AdicCompletion.coe_ofTensorProductEquivOfFiniteNoetherian, CliffordAlgebra.contractLeft_contractLeft, trace_eq_finrank_mul_minpoly_nextCoeff, isNoetherianRing_iff, PowerSeries.trunc_trunc_mul, Matrix.cramer_zero, Submodule.dualCoannihilator_top, hasEigenvector_toLin'_diagonal, MvPolynomial.DirectSum.coeAddMonoidHom_eq_support_sum, Matrix.GeneralLinearGroup.toLin'_apply, MvPolynomial.mkDerivationₗ_monomial, Finsupp.counit_comp_lsingle, Ideal.IsPrime.prod_mem_iff, LinearMap.toMatrixAlgEquiv'_apply, Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin, PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal, AlgebraicGeometry.Scheme.zeroLocus_iInf_of_nonempty, Module.Dual.instIsReflecive, TensorProduct.finsuppScalarRight_smul, LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict, LinearMap.toMvPolynomial_baseChange, IsLocalization.bot_lt_comap_prime, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections_mem_grothendieckTopology, LinearMap.BilinForm.IsAlt.self_eq_zero, Ideal.pi_le_pi_iff, Polynomial.degree_derivative_eq, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, Polynomial.C_mul_monomial, Orientation.map_eq_det_inv_smul, AdicCompletion.factorₐ_evalₐ_one, Algebra.FormallySmooth.iff_of_surjective, Polynomial.natDegree_hasseDeriv_le, TensorProduct.lid_symm_apply, PrimeSpectrum.vanishingIdeal_anti_mono, Unitization.inrNonUnitalStarAlgHom_apply, Matrix.vecMulLinear_apply, Matrix.toLinearMapₛₗ₂'_toMatrix', LinearMap.continuous_of_isContPerfPair, Affine.Simplex.affineCombination_mem_setInterior_face_iff_mem, LinearMap.toMatrix'_algebraMap, StrongDual.polarSubmodule_eq_polar, MvPolynomial.IsWeightedHomogeneous.sum_weight_X_mul_pderiv, Algebra.IsInvariant.orbit_eq_primesOver, TensorProduct.tensorQuotEquivQuotSMul_tmul_mk, LinearMap.flip.instIsPerfPair, Matrix.cramer_reindex, AlgCat.hom_hom_rightUnitor, instIsScalarTowerAtPrimeFractionRing, TensorProduct.counit_tmul, ConcaveOn.antitoneOn_slope_lt, PiToModule.fromMatrix_apply_single_one, LocalizedModule.coe_map_eq, coe_topologicalNilradical, MvPolynomial.sum_weightedHomogeneousComponent, RCLike.geometric_hahn_banach_point_point, Algebra.Extension.CotangentSpace.map_sub_map, exteriorPower.oneEquiv_naturality, HomogeneousLocalization.val_sub, IntermediateField.rank_top, LinearMap.dualMap_injective_of_surjective, Valuation.ideal_isPrincipal, ExteriorAlgebra.ι_range_disjoint_one, Ideal.exists_radical_pow_le_of_fg, LinearMap.span_singleton_sup_ker_eq_top, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, MvPowerSeries.HasSubst.coeff_zero, LinearMap.dualMap_injective_iff, Polynomial.leval_eq_smeval.linearMap, IsHausdorff.map_algebraMap_iff, Algebra.normalizedTrace_eq_of_fininteDimensional, LinearEquiv.instIsPerfPair, FreeLieAlgebra.liftAux_map_add, MvPolynomial.pderiv_def, MeasureTheory.OuterMeasure.map_map, MultilinearMap.mkPiRing_zero, Polynomial.as_sum_range, FractionalIdeal.coeIdeal_le_coeIdeal', Module.Basis.eval_range, hasFDerivAt_exp_smul_const', SkewPolynomial.monomial_zero_right, AdicCompletion.val_mul, MvPolynomial.mem_pow_idealOfVars_iff', Module.Basis.repr_reindex, BilinearForm.toMatrixAux_eq, CliffordAlgebra.contractLeft_ι_mul, LinearMap.BilinForm.exists_orthogonal_basis, TensorProduct.lid_tmul, Commute.cfcₙHom, CliffordAlgebraQuaternion.toQuaternion_comp_ofQuaternion, RootPairing.rootForm_reflection_reflection_apply, Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem, LinearMap.dualPairing_nondegenerate, Submodule.localized₀_smul, Submodule.annihilator_bot, CliffordAlgebra.comp_ι_sq_scalar, RootPairing.posRootForm_posForm_pos_of_ne_zero, Matrix.SeparatingLeft.toBilin, Ideal.Quotient.mk_span_range, SemimoduleCat.hom_inv_leftUnitor, FreeNonUnitalNonAssocAlgebra.lift_symm_apply, Matrix.charpoly_toLin, Module.subsingletonEquiv_symm_apply, FGModuleCat.FGModuleCatCoevaluation_apply_one, Module.Basis.coe_sumCoords_eq_finsum, Affine.Simplex.ExcenterExists.sign_signedInfDist_lineMap_excenter_touchpoint, LinearMap.toMatrixOrthonormal_symm_apply, biUnion_associatedPrimes_eq_zero_divisors, IsLocalizedModule.map_linearCombination, LinearMap.BilinMap.baseChange_tmul, rank_finsupp_self', Module.Basis.algebraMapCoeffs_repr_apply_toFun, IsBaseChange.of_fintype_basis, Subspace.dualRestrict_leftInverse, QuadraticForm.tmul_tensorAssoc_apply, IsDomain.minimalPrimes_eq_singleton_bot, StrongDual.dualPairing_apply, HopfAlgebra.sum_antipode_mul_eq_smul, LinearMap.BilinForm.IsPosSemidef.add, LinearEquiv.piRing_symm_apply, MeasureTheory.OuterMeasure.restrict_empty, Ideal.mem_jacobson_iff, trace_eq_trace_adjoin, Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge, LinearMap.toMatrix_mulVec_repr, OreLocalization.zsmul_eq_zsmul, LinearMap.BilinForm.zero_left, isFiniteLength_quotient_span_singleton, LinearMap.toMatrix_innerₛₗ_apply, Module.instFiniteSubtypeMemIdealOfIsNoetherian, MonoidAlgebra.mapDomainBialgHom_apply, AdicCompletion.ofTensorProductEquivOfFiniteNoetherian_symm_of, ProperCone.hyperplane_separation_point, RingOfIntegers.not_dvd_exponent_iff, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul, Ideal.sup_mul_right_self, Matrix.toLin_apply_eq_zero_iff, MvPolynomial.modMonomial_add_divMonomial, TwoSidedIdeal.asIdeal_jacobson, MonomialOrder.span_leadingTerm_eq_span_monomial', WeakDual.CharacterSpace.coe_toNonUnitalAlgHom, LinearMap.extendTo𝕜'_apply_re, Algebra.PreSubmersivePresentation.jacobian_eq_det_aevalDifferential, Matrix.toBilin_basisFun, Matrix.ker_mulVecLin_transpose_mul_self, MeasureTheory.Measure.toSignedMeasure_smul, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul, Polynomial.det_taylorLinearEquiv_toLinearMap, List.exists_map_eq_of_sorted_nonempty_iff_wbtw, Matrix.nondegenerate_toBilin_iff, Coalgebra.sum_counit_tmul_map_eq, Ideal.map_eq_bot_iff_of_injective, AffineMap.vsub_lineMap, AdicCompletion.mk_ofAlgEquiv_symm, linearDepOn_iff, Ideal.exact_mulQuot_quotOfMul, MvPolynomial.decompose'_apply, LinearMap.BilinForm.isSymm_neg, AdicCompletion.mk_smul_mk, TensorProduct.AlgebraTensorModule.cancelBaseChange_tmul, lineMap_le_right_iff_le, Submodule.top_colon, dist_lineMap_right, Matrix.toLin'_reindex, Ideal.mul_le, IsFractionRing.stabilizerHom_surjective, Module.Basis.orientation_ne_iff_eq_neg, Submodule.mem_colon_iff_le, LocalizedModule.subsingleton_iff_ker_eq_top, PowerSeries.smul_weierstrassMod, MeasureTheory.Measure.mapₗ_mk_apply_of_aemeasurable, LinearMap.transvection.apply, PowerSeries.IsRestricted.isRestricted_iff, LinearMap.BilinForm.toDual_def, LinearMap.ker_toSpanSingleton_eq_bot_iff, CliffordAlgebra.contractRight_mul_algebraMap, Submodule.baseChange_bot, MeasureTheory.OuterMeasure.map_ofFunction, TwoSidedIdeal.mem_fromIdeal, IsSemiprimaryRing.isNilpotent, Ideal.span_singleton_eq_top, LieAlgebra.IsKilling.coroot_mem_corootSpace, AffineBasis.toMatrix_vecMul_coords, Module.Basis.algebraMapCoeffs_repr_apply_apply, Algebra.QuasiFinite.finite_fiber, Matrix.PosSemidef.toLinearMap₂'_zero_iff, LinearMap.BilinForm.toMatrix'_compLeft, Matrix.toLinearMapₛₗ₂_apply, IsLocalizedModule.fromLocalizedModule_mk, Polynomial.monomial_mul_X, MeasureTheory.OuterMeasure.map_comap_of_surjective, MultilinearMap.freeDFinsuppEquiv_single, MeasureTheory.Measure.addHaar_parallelepiped, Polynomial.rootSet_derivative_subset_convexHull_rootSet, Polynomial.mapAlgHom_monomial, Module.FaithfullyFlat.iff_flat_and_ideal_smul_eq_top, WeakDual.isSeqCompact_closedBall, LinearMap.BilinForm.dualBasis_repr_apply, AlgebraicGeometry.Scheme.zeroLocus_radical, Polynomial.Chebyshev.iterate_derivative_T_eval_zero_recurrence, MvPowerSeries.coeff_one, Polynomial.iterate_derivative_X_pow_eq_natCast_mul, LinearMap.lsmul_injective, Module.finrank_pi, instIsLocalizedModuleLinearMapOfIsLocalization, Ideal.quotientMulEquivQuotientProd_snd, Module.IsReflexive.bijective_dual_eval', PiLp.basisFun_apply, WittVector.mem_ker_truncate, AdicCompletion.ofTensorProduct_bijective_of_finite_of_isNoetherian, LocalSubring.map_maximalIdeal_eq_top_of_isMax, Polynomial.natDegree_taylor, Orientation.rotation_neg_orientation_eq_neg, Unitization.sndHom_apply, Polynomial.degree_taylor, span_range_eq_top_iff_surjective_fintypeLinearCombination, ExteriorAlgebra.liftAlternating_one, Complex.selfAdjointEquiv_apply, Subspace.flip_quotDualCoannihilatorToDual_bijective, PrimeSpectrum.zeroLocus_ideal_mem_irreducibleComponents, Subspace.dual_finrank_eq, linearIndepOn_iff_disjoint, Algebra.TensorProduct.instFree, map_lt_lineMap_iff_slope_lt_slope_left, MvPolynomial.monomial_one_mul_cancel_left_iff, MeasureTheory.hausdorffMeasure_lineMap_image, Ideal.dvd_bot, Module.Basis.sumQuot_repr_inl_of_mem, Module.Basis.end_apply, Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors, TensorPower.algebraMap₀_eq_smul_one, LinearMap.separatingLeft_iff_det_ne_zero, LinearMap.toSpanSingleton_smul, CoalgCat.tensorUnit_instCoalgebra, PowerSeries.coeff_mk, ModuleCat.hom_hom_leftUnitor, Polynomial.annIdealGenerator_mem, Polynomial.coe_smul, MvPolynomial.coe_basisMonomials, Polynomial.hasFDerivAt, isPrincipalIdealRing_iff, Ideal.symm_apply_mem_of_equiv_iff, IsDedekindDomain.HeightOneSpectrum.intAdicAbv_lt_one_iff, wbtw_zero_one_iff, WeierstrassCurve.Affine.derivative_addPolynomial_slope, Module.rankAtStalk_eq_finrank_tensorProduct, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, RingHom.toKerIsLocalization_apply, Matrix.Nondegenerate.toBilin, LieModule.exists_nontrivial_weightSpace_of_isNilpotent, WeakDual.CharacterSpace.toAlgHom_apply, PowerSeries.sub_const_eq_X_mul_shift, TensorProduct.finsuppScalarRight_apply_tmul_apply, LinearMap.BilinForm.nondegenerate_toBilin'_iff_det_ne_zero, Ideal.comap_bot_of_injective, LinearMap.toMatrix'_apply, IntermediateField.finrank_top, LinearMap.mem_span_iff_continuous_of_finite, LinearEquiv.baseChange_tmul, MvPolynomial.monomial_mem_pow_idealOfVars_iff, Ideal.Quotient.stabilizerHom_apply, Subalgebra.LinearDisjoint.algebraMap_basisOfBasisRight_apply, QuadraticForm.tmul_tensorComm_apply, Ideal.mem_leadingCoeff, Module.Basis.repr_symm_single, Matrix.toLpLin_toLp, PowerSeries.coeff_largeSchroderSeries, MvPowerSeries.coeff_mul_right_one_sub_of_lt_order, RingEquiv.moduleEndSelf_apply, apply_eq_dotProduct_toMatrix₂_mulVec, IsIdempotentElem.ker_toSpanSingleton_one_sub_eq_span, Ideal.fst_comp_quotientMulEquivQuotientProd, Polynomial.degreeLT.addLinearEquiv_apply, PrimeSpectrum.zeroLocus_iSup, Ideal.ResidueField.exists_smul_eq_tmul_one, DividedPowers.RatAlgebra.dpow_apply, kroneckerLinearEquiv_symm_kronecker, minpoly.ker_aeval_eq_span_minpoly, Ideal.span_mul_span', Matrix.rank_eq_finrank_span_row, LinearMap.BilinForm.separatingRight_toMatrix_iff, Module.mem_support_iff_exists_annihilator, RootPairing.prod_rootForm_smul_coroot_mem_range_domRestrict, AffineMap.lineMap_apply_module', MultilinearMap.freeDFinsuppEquiv_def, IsLocalRing.isField_iff_maximalIdeal_eq, ModuleCat.hom_hom_rightUnitor, ContinuousLinearMap.toSpanSingletonLE_apply, LinearMap.ext_ring_iff, MvPolynomial.irreducible_sumSMulX, ContinuousAffineMap.apply_lineMap', Lagrange.eq_interpolate, Polynomial.mem_annIdeal_iff_aeval_eq_zero, finsuppTensorFinsuppRid_symm_single_smul, Subalgebra.LinearDisjoint.trace_algebraMap, MvPowerSeries.coeff_expand_smul, finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply, Ideal.finiteHeight_iff, Module.Basis.toDual_range, Complex.equivRealProdLm_apply, PiTensorProduct.dualDistribInvOfBasis_apply, AffineSubspace.wSameSide_lineMap_left, Ideal.homogeneousHull_eq_iSup, MeasureTheory.FiniteMeasure.instContinuousSMulNNReal, isSimpleRing_iff_isTwoSided_imp, ContinuousWithinAt.lineMap, Finsupp.degree_eq_weight_one, QuadraticForm.tensorLId_toLinearEquiv, FractionalIdeal.num_zero_eq, LieAlgebra.IsKilling.apply_coroot_eq_cast, slope_pos_iff, signedDist_lineMap_lineMap, Polynomial.separable_def', IsAdic.hasBasis_nhds_zero, Module.Basis.restrictScalars_repr_apply, MvPowerSeries.le_weightedOrder_smul, Orthonormal.inner_finsupp_eq_sum_right, AffineBasis.sum_coord_apply_eq_one, Matrix.ker_toLin_eq_bot, Polynomial.derivative_of_natDegree_zero, TensorProduct.AlgebraTensorModule.dualDistrib_apply, IntermediateField.LinearDisjoint.finrank_left_eq_finrank, LinearMap.annihilator_le_of_injective, WeierstrassCurve.Projective.addMap_eq, AlgebraicGeometry.Scheme.Hom.ideal_ker_le, Finsupp.span_eq_range_linearCombination, Polynomial.toFinsuppIsoLinear_symm_apply_toFinsupp, Module.Basis.equivFunL_apply, Algebra.Generators.snd_comp_cotangentCompLocalizationAwayEquiv, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, Submodule.dualAnnihilator_sup_eq, Ring.KrullDimLE.mem_minimalPrimes_iff_le_of_isPrime, Finsupp.isCompl_range_lmapDomain_span, Ideal.isRadical_bot_iff, ContDiff.smulRight, Fintype.linearIndependent_iff'ₛ, QuadraticForm.tmul_tensorRId_apply, BoundedContinuousFunction.probCharDual_zero, AlgebraicGeometry.Scheme.zeroLocus_inf, AdicCompletion.kerProj_surjective, WeakDual.CharacterSpace.coe_toCLM, HahnSeries.ofPowerSeries_apply_coeff, MvPowerSeries.eval₂_eq_tsum, HasGradientAt.hasFDerivAt, RingTheory.Sequence.IsWeaklyRegular.regular_mod_prev, ideal_eq_bot_of_localization, image_comap_zeroLocus_eq_zeroLocus_comap, Module.FinitePresentation.linearEquivMap_apply, MvPowerSeries.homogeneousComponent_of_lt_order_eq_zero, MvPolynomial.combinatorial_nullstellensatz_exists_linearCombination, MvPolynomial.finite_universalFactorizationMap, AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono, Module.Relations.map_single, Module.presentationFinsupp_R, Polynomial.Splits.taylor, MvPolynomial.rTensorAlgHom_toLinearMap, Module.annihilator_eq_top_iff, Subfield.relfinrank_top_right, range_cfcₙ_eq_range_cfcₙHom, CliffordAlgebra.toEven_comp_ofEven, Finset.affineCombination_map, coe_polyEquivTensor'_symm, ContinuousMap.AlgHom.closure_ker_inter, baseChange_ext_iff, Ideal.Quotient.torsionBy_eq_span_singleton, Submodule.dualCoannihilator_sup_eq, AffineBasis.det_smul_coords_eq_cramer_coords, LinearMap.BilinForm.zero_apply, reesAlgebra.monomial_mem, FreeLieAlgebra.liftAux_map_mul, PrimeSpectrum.zeroLocus_eq_iff, Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative, Algebra.Extension.CotangentSpace.map_cotangentComplex, QuadraticModuleCat.forget₂_map_associator_hom, MvPolynomial.homogeneousSubmodule_zero, ModuleCat.exteriorPower.iso₀_hom_naturality_assoc, MvPolynomial.coeff_sumSMulX, exists_max_ideal_of_mem_nonunits, Ideal.IsHomogeneous.iInf, Complex.restrictScalars_one_smulRight, LieAlgebra.IsKilling.instIsRootSystemSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, Ideal.pow_le_self, QuotSMulTop.equivQuotTensor_naturality, LinearMap.smulRightₗ_apply, LinearMap.IsNonneg.smul, Ideal.relNorm_algebraMap', PowerSeries.derivativeFun_coe, AddSubgroup.relIndex_eq_abs_det, MvPowerSeries.le_weightedOrder_subst, LinearMap.toMatrix'_symm, Polynomial.coeff_monomial, eq_affineCombination_of_mem_affineSpan_of_fintype, MvPolynomial.universalFactorizationMapPresentation_map, Module.symm_dualMap_evalEquiv, Localization.finite_of_primesOver_eq_singleton, StrongDual.dualPairing_separatingLeft, AffineBasis.coord_apply_combination_of_notMem, Finsupp.linearCombinationOn_range, ProperCone.dual_singleton, Algebra.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex, CliffordAlgebra.EquivEven.Q'_apply, Module.Basis.coe_singleton, Algebra.TensorProduct.basisAux_tmul, Ideal.isCompact_of_fg, MvPowerSeries.monomial_zero_eq_C_apply, Module.Basis.singleton_repr, RootPairing.algebraMap_rootFormIn, MvPolynomial.restrictSupport_mono, KaehlerDifferential.derivationQuotKerTotal_apply, InnerProductGeometry.norm_ofLp_crossProduct, AdicCompletion.transitionMap_comp_reduceModIdeal, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', Ideal.mem_matrix, TwoSidedIdeal.mem_asIdealOpposite, FractionalIdeal.coe_one_eq_coeSubmodule_top, RingPreordering.one_notMem_support, MvPolynomial.algebraTensorAlgEquiv_symm_X, Ideal.torsionOf_zero, Differential.mapCoeffs_monomial, MvPolynomial.mem_restrictDegree, ExteriorAlgebra.ι_sq_zero, Submodule.exists_dual_map_eq_bot_of_lt_top, Polynomial.taylor_coeff_zero, MonomialOrder.monic_monomial, PowerSeries.monomial_zero_eq_C, AdicCompletion.instIsScalarTowerQuotientIdealHSMulTopSubmodule, LinearMap.polyCharpolyAux_map_eq_charpoly, FractionalIdeal.mem_coeIdeal, HomogeneousIdeal.toIdeal_sInf, Ideal.map_quotient_self, Matrix.range_mulVecLin, Submodule.baseChange_eq_span, linearMap_toMatrix_mul_basis_toMatrix, tensorKaehlerQuotKerSqEquiv_symm_tmul_D, CliffordAlgebra.isUnit_ι_iff, Submodule.dualCopairing_apply, LinearEquiv.toSpanNonzeroSingleton_apply, IsLocalization.mk'_mem_map_algebraMap_iff, Module.dualPairing_apply, Polynomial.mul_eq_sum_sum, MulOpposite.counit_def, PowerSeries.monomial_pow, Ideal.ofNat_eq_top, finite_iff_ideal_quotient, Ideal.coe_restrictScalars, coe_polyEquivTensor', AlgebraicClosure.instIsAlgClosure, LinearMap.polyCharpolyAux_eval_eq_toMatrix_charpoly_coeff, LinearMap.BilinForm.isNonneg_zero, Submodule.span_algebraMap_image_of_tower, Ideal.mk_mem_cotangentIdeal, AffineMap.lineMap_vsub, RootPairing.toPerfPair_flip_comp_coroot, QuadraticModuleCat.toIsometry_whiskerRight, Coalgebra.rTensor_counit_comp_comul, Algebra.trace_prod, LinearMap.toMatrix_transpose_apply, MvPolynomial.indicator_mem_restrictDegree, LinearMap.BilinForm.IsSymm.sub, Pi.basis_repr, MvPowerSeries.coeff_comp_monomial, IntermediateField.finrank_eq_one_iff_eq_top, PowerSeries.eq_X_pow_mul_shift_add_trunc, FractionalIdeal.coe_inv_of_ne_zero, Polynomial.iterate_derivative_C, SkewMonoidAlgebra.instFree, toMatrix_dualTensorHom, PowerSeries.trunc_trunc_of_le, exists_dual_vector, IsDedekindDomain.primesOverEquivPrimesOver_ramificationIdx_eq, conductor_subset_adjoin, NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm, LinearMap.BilinForm.nondegenerate_iff_ker_eq_bot, MonomialOrder.sPolynomial_def, exteriorPower.alternatingMapLinearEquiv_apply_ιMulti, OrthonormalBasis.det_to_matrix_orthonormalBasis, Algebra.Generators.H1Cotangent.δAux_ofComp, MvPolynomial.restrictSupport_zero, Polynomial.taylor_X_pow, RootPairing.isCompl_rootSpan_ker_rootForm, Module.Relations.Quotient.linearMap_ext_iff, RootPairing.reflectionPerm_root, Finset.affineCombination_indicator_subset, ProbabilityTheory.IsGaussian.charFunDual_eq, Matrix.range_toLin', LinearIsometryEquiv.toSpanUnitSingleton_apply, ContinuousMapZero.nonUnitalStarAlgHom_precomp_apply, Ideal.isOpen_pow_of_isMaximal, LinearIndependent.span_repr_eq, associatedPrimes.eq_singleton_of_isPrimary, ProbabilityTheory.covarianceBilin_eq_covarianceBilinDual, MeasureTheory.OuterMeasure.map_iSup, Differential.mapCoeffs_C, AffineMap.lineMap_eq_lineMap_iff, QuaternionAlgebra.rank_eq_four, IsLocalization.coeSubmodule_le_coeSubmodule, LinearMap.pi_apply_eq_sum_univ, RootPairing.rootForm_self_eq_zero_iff, Algebra.IsInvariant.exists_smul_of_under_eq_of_profinite, Ideal.iSup_mul, ModularFormClass.qExpansion_coeff, Rep.coindFunctorIso_inv_app_hom_hom_apply_coe, Quaternion.fst_imJ_dualNumberEquiv_symm, Submodule.injective_tensorToSpan, MvPolynomial.as_sum, LinearMap.dualMap_apply', PadicInt.mahlerTerm_one, Ideal.mem_leadingCoeffNth, AlgCat.hom_inv_leftUnitor, Ideal.spanRank_map_le, LinearMap.mul_toMatrix', Ideal.IsPrime.multiset_prod_le, CoalgCat.MonoidalCategoryAux.leftUnitor_hom_toLinearMap, HasStrictDerivAt.hasStrictFDerivAt, LinearMap.toMatrix₂_basisFun, AdicCompletion.ofTensorProduct_bijective_of_pi_of_fintype, MonomialOrder.leadingCoeff_monomial, Algebra.Presentation.differentials.comm₂₃', Ideal.spanFinrank_map_le_of_fg, inr_comp_cfcₙHom_eq_cfcₙAux, Submodule.finite_quotient_smul, exteriorPower.ιMultiDual_apply_ιMulti, InnerProductSpace.gramSchmidt_triangular, MvPolynomial.monomial_add_single, Algebra.normalizedTrace_algebraMap_apply_eq_self, Ideal.setOf_isPrincipal_wellFoundedOn_gt, QuadraticAlgebra.basis_repr_apply, IsLocalizedModule.lift_comp_iso, CliffordAlgebra.EquivEven.neg_v_mul_e0, PiTensorProduct.dualDistribEquivOfBasis_apply_apply, LinearMap.dualMap_bijective_iff, Algebra.zariskisMainProperty_iff_exists_saturation_eq_top, TensorPower.multilinearMapToDual_apply_tprod, Polynomial.toFn_zero, Module.Basis.sum_dual_apply_smul_coord, Ideal.Filtration.Stable.exists_pow_smul_eq, Polynomial.ofFn_zero', TrivSqZeroExt.inrCLM_apply, HomogeneousLocalization.val_add, Ideal.exists_not_mem_forall_mem_of_ne_of_liesOver, Matrix.mul_adjugate_apply, ContinuousLinearMap.range_smulRight_apply, Polynomial.derivative_C_mul_X, IsHausdorff.bot, MvPolynomial.pointToPoint_zeroLocus_le, LinearMap.toMatrix_smulBasis_right, Matrix.toLpLin_symm_id, lineMap_lt_left_iff_lt, Polynomial.eval_iterate_derivative_rootMultiplicity, OreLocalization.right_distrib, Polynomial.lcoeff_apply, Polynomial.iterate_derivative_derivative_mul_X, Module.Basis.toDual_eq_repr, Polynomial.Chebyshev.add_one_mul_self_mul_T_eq_poly_in_T, Matrix.range_toLin_eq_top, AdicCompletion.smul_mk, List.exists_map_eq_of_sorted_iff_sbtw, Unitization.inrHom_apply, Module.Basis.baseChange_end, Complex.rank_real_complex', TensorProduct.lid'_apply_tmul, Matrix.toLin_finTwoProd_apply, FractionalIdeal.coeIdeal_eq_zero', MeasureTheory.OuterMeasure.restrict_iSup, KaehlerDifferential.instSMulCommClassTensorProduct_1, Localization.mkAddMonoidHom_apply, Sbtw.of_lt_of_lt, Ideal.isMaximal_iff, Ideal.Quotient.stabilizerQuotientInertiaEquiv_mk, PowerSeries.coeff_exp, LinearPMap.adjointDomainMkCLM_apply, Module.Basis.sum_repr_mul_repr, LinearMap.BilinForm.dualBasis_eq_iff, map_le_lineMap_iff_slope_le_slope_right, Finsupp.linearCombination_apply, FractionalIdeal.coeIdeal_mul, LocalizedModule.exists_subsingleton_away, Module.Basis.baseChange_apply, Polynomial.iterate_derivative_eq_zero, Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent, iInter_halfSpaces_eq, Polynomial.hasseDeriv_coeff, NumberField.integralBasis_repr_apply, Matrix.toBilin'Aux_single, IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_mem, LieModule.lie_traceForm_eq_zero, geometric_hahn_banach_closed_compact, Module.Basis.det_map', exteriorPower.ιMulti_family_eq_coe_comp, exteriorPower.pairingDual_ιMulti_ιMulti, Polynomial.derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors, Polynomial.monomial_zero_one, Polynomial.hasseDeriv_mul, Valuation.ltSubmodule_monotone, ExteriorAlgebra.map_apply_ι, LinearMap.mul_toMatrix₂, Ideal.isRadical_bot_of_noZeroDivisors, Ideal.isPrime_bot, mem_skewAdjointMatricesLieSubalgebra, PowerSeries.trunc_one_left, MonomialOrder.monic_monomial_one, MvPowerSeries.isHomogeneous_iff_eq_homogeneousComponent, Submodule.mulRightMap_eq_mulMap_comp, Algebra.TensorProduct.basis_repr_symm_apply', LieAlgebra.IsKilling.corootForm_rootSystem_eq_killing, AffineMap.image_uIcc, ExteriorAlgebra.ι_leftInverse, MvPolynomial.zeroLocus_bot, LinearMap.surjective_compr₂_of_equiv, Algebra.trace_comp_trace_of_basis, QuadraticMap.discr_comp, Module.Basis.linearMap_repr_apply, Ideal.matrix_top, AffineMap.lineMap_vadd_apply, Affine.Simplex.affineCombination_mem_interior_iff, exteriorPower.alternatingMapLinearEquiv_comp_ιMulti, MvPolynomial.monomial_eq_zero, differentialIdeal_le_fractionalIdeal_iff, PowerSeries.IsWeierstrassDivisorAt.mod_add, LinearMap.lsmul_eq_DistribMulAction_toLinearMap, toLinearMap_algebraMapCLM, Module.eval_apply_eq_zero_iff, Int.absNorm_under_mem, MvPolynomial.eval_monomial, LinearEquiv.baseChange_zpow, MvPolynomial.coeff_coe, Module.Basis.det_reindex, isLinearTopology_iff_hasBasis_ideal, AdicCompletion.map_mk, PadicInt.zmodRepr_spec, PrimeSpectrum.vanishingIdeal_union, MultilinearMap.freeFinsuppEquiv_def, localizedModuleIsLocalizedModule, ProbabilityTheory.indepFun_iff_charFunDual_prod', MvPolynomial.coeff_monomial_mul', Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner_naturality, LieModule.trace_comp_toEnd_genWeightSpace_eq, MvPolynomial.monomial_mem_coeffsIn, WittVector.ker_map_le_ker_mk_comp_ghostComponent, Bialgebra.TensorProduct.rid_toAlgEquiv, Ideal.le_comap_of_ramificationIdx_ne_zero, Matrix.sum_cramer_apply, uniqueDiffWithinAt_Ioo, ProbabilityTheory.covarianceBilinDual_self_eq_variance, Ideal.span_eq, Polynomial.eraseLead_add_monomial_natDegree_leadingCoeff, RatFunc.laurentAux_algebraMap, QuadraticForm.tensorComm_toLinearEquiv, MvPowerSeries.coeff_trunc', CliffordAlgebra.contractLeft_comm, IsAdjoinRoot.repr_zero_mem_span, CompactlySupportedContinuousMap.integralLinearMap_apply, coe_subset_nonunits, PowerSeries.sub_const_eq_shift_mul_X, geometric_hahn_banach_open_open, LaurentSeries.intValuation_le_iff_coeff_lt_eq_zero, AdjoinRoot.modByMonicHom_mk, Matrix.UnitaryGroup.coe_toGL, FractionalIdeal.mem_dual, Matrix.linearIndependent_cols_iff_isUnit, Ideal.isSimpleOrder, Matrix.dualNumberEquiv_symm_apply, Finset.weightedVSub_const_smul, LinearMap.toMatrix'_mul, Ideal.zero_eq_bot, ContinuousLinearMap.toLinearMap_toSpanSingleton, MvPowerSeries.coeff_zero_mul_X, tfae_of_isNoetherianRing_of_isLocalRing_of_isDomain, Algebra.trace_self_apply, Ideal.localized₀_eq_restrictScalars_map, Ideal.span_singleton_le_iff_mem, CliffordAlgebra.ofBaseChange_tmul_ι, mul_basis_toMatrix, Affine.Simplex.affineCombination_eq_touchpoint_iff, Module.Basis.toDual_toDual, DualNumber.exp_eps, Finsupp.linearCombination_embDomain, WittVector.fontaineThetaModPPow_teichmuller, Polynomial.iterate_derivative_X_pow_eq_smul, KaehlerDifferential.tensorKaehlerEquiv_left_inv, Configuration.ofField.instNondegenerateProjectivizationForallFinOfNatNat, Module.dualProdDualEquivDual_apply_apply, MvPowerSeries.coeff_inv_aux, QuadraticModuleCat.toIsometry_hom_leftUnitor, trace_eq_sum_roots, Algebra.TensorProduct.rid_tmul, RootPairing.zero_le_posForm, IsBezout.iff_span_pair_isPrincipal, ContinuousLinearEquiv.toSpanNonzeroSingleton_symm_apply, Polynomial.Chebyshev.derivative_U_eval_one, LinearMap.BilinForm.sub_apply, Submodule.annihilator_top_inter_nonZeroDivisors, Ideal.comap_sInf, LinearMap.toSpanSingleton_apply, AlgHom.IsArithFrobAt.le_comap, killingForm_apply_apply, Ring.jacobson_le_of_isMaximal, LinearMap.trace_id, LinearMap.toMatrix_algebraMap, QuadraticModuleCat.toIsometry_hom_rightUnitor, Ideal.le_spanNorm_spanNorm, Algebra.trace_trace, LinearMap.spectrum_toMatrix, AdicCompletion.ofAlgEquiv_apply, Polynomial.monomial_one_eq_iff, LaurentPolynomial.leval_apply, Ideal.hasBasis_nhds_zero_adic, LinearMap.BilinForm.congr_refl, Ideal.fg_top, StandardEtalePresentation.toSubmersivePresentation_jacobian, Finset.sum_smul_const_vsub_eq_neg_weightedVSub, IsBaseChange.equiv_symm_apply, Algebra.norm_eq_norm_adjoin, Ideal.idealProdEquiv_symm_apply, Ideal.add_eq_one_iff, MvPowerSeries.coeff_weightedHomogeneousComponent, Ideal.homogeneousHull_mono, Matrix.isUnit_toLin_iff, Polynomial.degreeLT.addLinearEquiv_apply', Ideal.isCancelMulZero, RootPairing.InvariantForm.apply_weylGroup_smul, Algebra.TensorProduct.cancelBaseChange_tmul, Polynomial.X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic, Representation.ofMulActionSelfAsModuleEquiv_apply, Subspace.quotAnnihilatorEquiv_apply, PowerSeries.derivative_invOf, Polynomial.hasseDeriv_zero, tendsto_integral_exp_smul_cocompact_of_inner_product, PowerSeries.HasSubst.monomial', Projectivization.mk_eq_mk_iff_crossProduct_eq_zero, ModuleCat.RestrictionCoextensionAdj.unit'_app, Ideal.mem_span_singleton_sup, Ideal.finset_inf_span_singleton, Matrix.diagonal_toLin', Module.isTorsionBySet_iff_subseteq_ker_lsmul, Submodule.instIsScalarTowerQuotientSpanSingletonSetSubtypeMemTorsionBy, KaehlerDifferential.kerTotal_map, Polynomial.coeff_monomial_zero_mul, pNilradical_eq_bot, FractionalIdeal.exists_mem_algebraMap_eq, InnerProductSpace.nullSubmodule_le_ker_toDualMap_left, Bialgebra.counit_natCast, exteriorPower.oneEquiv_symm_apply, PowerSeries.IsWeierstrassDivisorAt.isUnit_shift, QuadraticForm.tensorLId_symm_apply, RingEquiv.idealComapOrderIso_symm_apply, RingTheory.Sequence.isWeaklyRegular_append_iff', Coalgebra.counitCoalgHom_toLinearMap, IsAdjoinRootMonic.ext_elem_iff, Matrix.charpoly_toLin', Quaternion.fst_imI_dualNumberEquiv_symm, LinearMap.BilinForm.toMatrix_symm, Lagrange.iterate_derivative_interpolate, Rep.ofMulActionSubsingletonIsoTrivial_inv_hom, Matrix.toLinearMap₂'_toMatrix', Polynomial.pow_sub_dvd_iterate_derivative_pow, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, Polynomial.coeff_mul_invOneSubPow_eq_hilbertPoly_eval, Ideal.map_le_comap_of_inverse, rank_dual_eq_card_dual_of_aleph0_le_rank', MvPolynomial.le_zeroLocus_iff_le_vanishingIdeal, nndist_lineMap_right, Matrix.toLinearMapₛₗ₂'_apply, Module.Basis.orientation_comp_linearEquiv_eq_neg_iff_det_neg, Polynomial.hasFDerivAt_aeval, TensorProduct.finsuppScalarRight_symm_apply_single, AlgCat.hom_hom_leftUnitor, Ideal.relNorm_algebraMap, Rat.HeightOneSpectrum.natGenerator_dvd_iff, Ideal.spanNorm_mul, exists_dual_vector'', Submodule.smul_le_right, Module.Flat.iff_lTensor_injective, Matrix.toLin'_pow, IsCoprime.sup_eq, Algebra.trace_eq_zero_of_not_exists_basis, Module.Basis.mem_span_image, ProbabilityTheory.IsGaussian.memLp_dual, Finset.affineCombination_linear, CliffordAlgebra.ofBaseChangeAux_ι, Ideal.ker_tensorProductMk_quotient, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_notMem, WittVector.inv_pair₂, FiniteDimensional.finiteDimensional_self, MvPowerSeries.coeff_apply, Unitary.toAlgEquiv_conjStarAlgAut, rank_dual_eq_card_dual_of_aleph0_le_rank, Finset.affineCombination_apply, Submodule.annihilator_span_singleton, irreducible_pow_sup, instIsLocalizedModuleQuotientSubmoduleLocalizedModuleLocalizationLocalizedToLocalizedQuotient, ProbabilityTheory.covarianceBilinDual_self_nonneg, Ideal.IsRadical.radical_le_iff, NumberField.mixedEmbedding.stdBasis_apply_isComplex_fst, Ideal.natCast_mem_of_charP_quotient, PowerSeries.constantCoeff_divXPowOrder, Module.Basis.localizationLocalization_repr_algebraMap, Ideal.Quotient.eq_zero_iff_mem, exteriorPower.map_comp_ιMulti, Ideal.span_mul_span, Ideal.span_one, PrimeSpectrum.isClosed_iff_zeroLocus_ideal, Matrix.toLinAlgEquiv'_apply, PowerSeries.monomial_eq_mk, Complex.equivRealProdLm_symm_apply, Polynomial.natDegree_derivative_le, cfcₙHom_isClosedEmbedding, MeasureTheory.charFunDual_map_const_add, RootPairing.rootForm_apply_apply, LieModule.traceForm_eq_zero_of_isTrivial, Module.Basis.dualBasis_apply, Ideal.basisSpanSingleton_apply, LinearMap.apply_toPerfPair_flip, Submodule.colon_eq_top_iff_subset, MeasureTheory.charFunDual_map_add_const, MvPowerSeries.coeff_trunc, Complex.equivRealProdLm_symm_apply_im, HahnSeries.toPowerSeriesAlg_symm_apply_coeff, apply_eq_star_dotProduct_toMatrix₂_mulVec, tensorKaehlerQuotKerSqEquiv_tmul_D, MvPolynomial.mkDerivation_monomial, HopfAlgebra.mul_antipode_rTensor_comul, LinearMap.toMatrix_mul, Complex.coe_basisOneI_repr, Finsupp.linearCombination_single, LinearMap.polar_gc, Module.Presentation.CokernelData.π_lift, TwoSidedIdeal.subtype_injective, LinearMap.toMatrix₂_compl₁₂, LinearMap.BilinForm.IsSymm.smul, Ideal.coe_smul_primesOver_mk, uniqueDiffWithinAt_Ioi, ExteriorAlgebra.instGradedMonoidNatSubmoduleExteriorPower, Polynomial.iterate_derivative_eq_sum, Matrix.minpoly_toLin', Algebra.Presentation.relation_mem_ker, Ideal.comap_le_comap_iff_of_surjective, WeierstrassCurve.Projective.Point.fromAffine_some, Module.Basis.reindexFinsetRange_repr_self, QuadraticForm.tensorDistrib_tmul, affineCombination_mem_affineSpan_image, signedDist_lineMap_left, MvPolynomial.mem_map_C_iff, ExteriorAlgebra.ι_add_mul_swap, AffineMap.lineMap_apply_zero, MvPolynomial.rename_monomial, RootPairing.iInf_ker_root'_eq, lift_rank_lt_rank_dual, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, MeasureTheory.FiniteMeasure.smul_testAgainstNN_apply, Polynomial.Chebyshev.iterate_derivative_U_eval_zero_recurrence, WittVector.ghostComponentModPPow_teichmuller_coeff, LinearMap.toSpanSingleton_eq_zero_iff, QuadraticModuleCat.Hom.toIsometry_injective, BilinForm.toMatrix_compRight, LinearMap.BilinForm.nondegenerate_toMatrix'_iff, Finset.affineCombination_affineCombinationLineMapWeights, Polynomial.wronskianBilin_apply, Submodule.annihilator_mono, MvPolynomial.restrictScalars_restrictSupportIdeal, MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictDegreeOfFinite, LinearMap.detAux_def', Orthonormal.inner_right_finsupp, PolyEquivTensor.invFun_add, LinearMap.toMatrix_transpose_apply', MvPowerSeries.eq_zero_iff_forall_coeff_eq_zero_and, Submodule.IsPrincipal.generator_map_dvd_of_mem, Ideal.iSup_iInf_eq_top_iff_pairwise, Wbtw.of_le_of_le, Submodule.colon_singleton_zero, Ideal.isCoprime_tfae, LinearMap.norm_extendTo𝕜'_apply_sq, RingHom.ker_eq_bot_iff_eq_zero, Ideal.map_sup_comap_of_surjective, MvPolynomial.mem_restrictSupport_iff, Ideal.map_radical_le, Module.Basis.repr_symm_single_one, Ideal.quotEquivOfEq_eq_factor, Rep.diagonalOneIsoLeftRegular_inv_hom, Ideal.relNorm_eq_bot_iff, LinearMap.toMatrix'_intrinsicStar, IsAdicComplete.le_jacobson_bot, MvPolynomial.degreesLE_zero, hasStrictFDerivAt_inv, DividedPowers.SubDPIdeal.inf_carrier_def, LinearMap.BilinForm.congr_congr, DividedPowers.IsSubDPIdeal.isSubideal, AddMonoidAlgebra.GradesBy.decompose_single, WeierstrassCurve.Projective.negMap_of_Z_ne_zero, ModuleCat.hom_inv_rightUnitor, MvPolynomial.universalFactorizationMapPresentation_relation, AffineMap.fst_lineMap, TrivSqZeroExt.inrHom_apply, LinearMap.isPosSemidef_zero, InnerProductSpace.gramSchmidtOrthonormalBasis_det, ModuleCat.ExtendScalars.smul_tmul, ConvexOn.slope_mono, Ideal.map_bot, QuadraticForm.tensorComm_apply, PowerSeries.trunc_C, Ideal.le_toIdeal_homogeneousHull, Algebra.PreSubmersivePresentation.aevalDifferential_single, Module.Flat.linearIndependent_one_tmul, cross_apply, MvPolynomial.isWeightedHomogeneous_monomial, cfcₙHom_le_iff, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem, IsBaseChange.toDual_comp_apply, IsBaseChange.linearMapLeftRightHom_toMatrix, Polynomial.sumIDeriv_apply, RootPairing.PolarizationIn_apply, LinearMap.BilinForm.isAlt_zero, AlternatingMap.eq_smul_basis_det, OnePoint.equivProjectivization_symm_apply_mk, LinearMap.BilinForm.toMatrixAux_apply, Polynomial.sylvesterDeriv_updateRow, AlgebraicGeometry.StructureSheaf.algebraMap_obj_top_bijective, MeasureTheory.OuterMeasure.comap_mono, Finset.affineCombination_filter_of_ne, LinearMap.BilinForm.isAdjointPair_iff_eq_of_nondegenerate, DividedPowers.SubDPIdeal.sSup_carrier_def, Polynomial.dvd_derivative_iff, Polynomial.coeff_mul_monomial_zero, OrthonormalBasis.det_adjustToOrientation, Matrix.SpecialLinearGroup.toLin_equiv.symm_toLinearMap_eq, PowerSeries.weierstrassMod_zero_left, LinearEquiv.dualMap_symm, Ideal.exists_forall_sub_mem_ideal, WeierstrassCurve.Projective.nonsingularLift_iff, PiTensorProduct.dualDistrib_apply, Fintype.range_linearCombination, CliffordAlgebraQuaternion.ofQuaternion_mk, Algebra.Generators.Cotangent.exact, Pi.basisFun_det, IsFractionRing.coeSubmodule_le_coeSubmodule, Matrix.toLinOfInv_symm_apply, LinearMap.toMatrixAlgEquiv'_toLinAlgEquiv', LinearMap.toMatrix_smulRight, ModularFormClass.qExpansion_coeff_zero, Module.Relations.solutionFinsupp_var, MvPowerSeries.LinearTopology.instIsLinearTopologyOfMulOpposite, Polynomial.resultant_taylor, Matrix.represents_iff, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact, QuadraticModuleCat.forget₂_obj, Ideal.is_fg_degreeLE, MvPolynomial.monomial_zero', RingHom.liftOfRightInverse_comp_apply, Polynomial.degreeLT.addLinearEquiv_symm_apply_inl, pinGroup.conjAct_smul_range_ι, MvPowerSeries.coeff_truncFun', LinearMap.lsmul_apply, Polynomial.degreeLT.addLinearEquiv_symm_apply_inr_basis, Algebra.Smooth.exists_span_eq_top_isStandardSmooth, FreeLieAlgebra.liftAux_spec, Finsupp.counit_single, MvPolynomial.weightedHomogeneousComponent_directSum, PrimeSpectrum.exists_primeSpectrum_prod_le, Polynomial.natDegree_hasseDeriv, NumberField.mixedEmbedding.stdBasis_repr_eq_matrixToStdBasis_mul, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply, QuadraticForm.tmul_comp_tensorMap, Ideal.Quotient.algebraMap_quotient_pow_ramificationIdx, isAdjointPair_toLinearMap₂', Module.Dual.baseChange_apply_tmul, uniqueDiffOn_Icc_zero_one, Matrix.toLin'_apply', LinearMap.rTensor_baseChange, AddMonoidAlgebra.mem_grade_iff', RootPairing.Base.toCoweightBasis_repr_coroot, Ideal.coe_toTwoSided, AdjoinRoot.powerBasisAux'_repr_symm_apply, Polynomial.Bivariate.swap_monomial_monomial, MvPowerSeries.coeff_prod, Configuration.ofField.crossProduct_eq_zero_of_dotProduct_eq_zero, Matrix.Represents.congr_fun, ContinuousLinearMap.norm_extendTo𝕜'_bound, slope_def_field, Matrix.ker_toLin'_eq_bot_iff, LinearMap.isIdempotentElem_apply_one_iff, IsAzumaya.coe_tensorEquivEnd, cardQuot_pow_of_prime, uniqueDiffWithinAt_Iio, eq_lineMap_of_dist_eq_mul_of_dist_eq_mul, DualNumber.lift_apply_apply, AdicCompletion.Ideal.mk_eq_mk, hasDerivAt_iff_hasFDerivAt, LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg, Ideal.iInf_span_singleton, LinearMap.toMatrix'_toLinearMap₂', Ideal.IsPrime.mul_le, Module.isTorsionBySet_iff_is_torsion_by_span, Polynomial.jacobson_bot_of_integral_localization, Polynomial.eval_derivative_div_eval_of_ne_zero_of_splits, Ideal.map_sup, Polynomial.Chebyshev.one_sub_X_sq_mul_iterate_derivative_U_eval, Polynomial.Monic.leadingCoeff_notMem, Basis.multilinearMap_apply_apply, LinearMap.BilinForm.toLin'Flip_apply, TensorProduct.tensorQuotEquivQuotSMul_symm_mk, LinearMap.reduceModIdeal_apply, Valuation.Integers.maximalIdeal_eq_setOf_le_v_algebraMap, Finset.weightedVSubOfPoint_smul, RingTheory.Sequence.isWeaklyRegular_append_iff, LocalizedModule.lift_comp, cfc_eq_cfcL_mkD, PolynomialModule.equivPolynomial_single, dvd_differentIdeal_iff, MvPolynomial.homogeneousComponent_apply, instIsTorsionFreeLocalizationAlgebraMapSubmonoidPrimeCompl_1, BilinForm.toMatrix_mul_basis_toMatrix, LinearMap.polar_weak_closed, Algebra.IsSmoothAt.exists_notMem_smooth, Polynomial.toFn_comp_ofFn_eq_id, Submodule.instIsPrincipalSpanSingletonSet, WeakDual.isClosed_closedBall, Ideal.IsTwoSided.instHPowNat, Rep.standardComplex.d_of, Module.Basis.coe_finTwoProd_repr, Polynomial.supNorm_monomial, QuaternionAlgebra.instFinite, QuadraticForm.dualProdIsometry_toFun, Polynomial.derivative_comp, LinearMap.BilinForm.mul_toMatrix', MvPowerSeries.coeff_C_mul, MvPolynomial.divMonomial_mul_monomial, transcendental_iff_ker_eq_bot, QuadraticForm.tensorAssoc_symm_apply, AffineMap.lineMap_apply, Finsupp.linearCombination_unique, PiTensorProduct.isEmptyEquiv_symm_apply, Module.mem_freeLocus_iff_tensor, AffineBasis.coord_apply_eq, Ideal.comap_map_quotientMk, LinearMap.map_mul_iff, AffineBasis.coord_apply_centroid, PowerSeries.X_eq, Localization.exists_awayMap_bijective_of_residueField_surjective, Algebra.Presentation.tensorModelOfHasCoeffsEquiv_tmul, TensorProduct.Algebra.linearMap_comp_mul', LinearMap.toSpanSingleton_homothety, ContinuousLinearEquiv.coord_toSpanNonzeroSingleton, WittVector.isAdicCompleteIdealSpanP, MvPolynomial.weightedHomogeneousComponent_finsupp, PrimeSpectrum.vanishingIdeal_empty, Polynomial.gaussNorm_monomial, ExteriorAlgebra.toTrivSqZeroExt_ι, left_le_lineMap_iff_le, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, Finsupp.linearCombination_mapDomain, Ideal.prod_eq_top_iff, sbtw_mul_sub_add_iff, HomogeneousIdeal.toIdeal_top, Span.finsupp_linearCombination_repr, InnerProductSpace.rankOne_def', Polynomial.taylor_taylor, IntermediateField.LinearDisjoint.lift_rank_right_mul_lift_adjoin_rank_eq_of_isAlgebraic_right, MeasureTheory.OuterMeasure.le_comap_map, PrimeSpectrum.zeroLocus_bot, LinearMap.toMatrixRight'_id, CliffordAlgebra.toBaseChange_ofBaseChange, leibniz_cross, MvPolynomial.linearIndependent_X, CliffordAlgebraComplex.equiv_symm_apply, Polynomial.expand_monomial, ProbabilityTheory.iIndepFun_iff_charFunDual_pi, MvPolynomial.prod_X_pow_eq_monomial, TensorProduct.rid_tmul, Submodule.colon_univ, ProbabilityTheory.norm_uncenteredCovarianceBilinDual_le, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_mul, Submodule.mem_dualCoannihilator, IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite, LinearMap.minpoly_toMatrix, DividedPowers.DPMorphism.ideal_comp, LinearMap.det_dualMap, PiToModule.fromEnd_apply_single_one, Ideal.pow_le_pow_right, MvPowerSeries.weightedHomogeneousComponent_mul_of_le_weightedOrder, Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one, Submodule.tmul_mem_baseChange_of_mem, matrixEquivTensor_apply_single, Ideal.radical_eq_sInf, LinearMap.BilinForm.IsRefl.neg, WittVector.inv_pair₁, MvPolynomial.constantCoeff_monomial, RootPairing.coroot_root_two, CategoryTheory.Iso.toIsometryEquiv_refl, QuadraticModuleCat.toIsometry_inv_rightUnitor, PiTensorProduct.piTensorHomMapFun₂_smul, Matrix.nondegenerate_toLinearMap₂'_iff_nondegenerate_toLinearMap₂, DividedPowers.isDPMorphism_def, FractionalIdeal.coeIdeal_eq_one, Algebra.TensorProduct.cancelBaseChange_symm_tmul, MvPolynomial.universalFactorizationMapPresentation_algebra_algebraMap, ModuleCat.ExtendScalars.map_tmul, LinearMap.toMatrixOrthonormal_apply, ContinuousLinearEquiv.unitsEquivAut_symm_apply, Ideal.Quotient.eq, cfcₙHom_eq_cfcₙ_extend, Ideal.IsHomogeneous.isPrime_iff, LieAlgebra.IsKilling.biSup_corootSpace_eq_top, LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg_aux, Matrix.charpoly_mulVecLin, diffToIdealOfQuotientCompEq_apply, Module.Basis.toDual_injective, KaehlerDifferential.instSMulCommClassTensorProduct, IsAdjoinRootMonic.coeff_apply_lt, Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge, mem_affineSpan_pair_iff_exists_lineMap_eq, LieModule.traceForm_eq_zero_if_mem_lcs_of_mem_ucs, Algebra.FormallyEtale.Algebra.FormallyEtale.iff_of_surjective, ZLattice.covolume_eq_det_mul_measureReal, openSegment_eq_image_lineMap, Ideal.eq_jacobson_iff_notMem, Affine.Simplex.sSameSide_affineSpan_faceOpposite_iff, RootPairing.iInf_ker_coroot'_eq, Subspace.finrank_add_finrank_dualCoannihilator_eq, Affine.Simplex.sSameSide_affineSpan_faceOpposite_of_sign_eq, CliffordAlgebra.map_comp_ι, MvPolynomial.mem_weightedHomogeneousSubmodule, RingHom.surjectiveOnStalks_iff_forall_maximal', HopfAlgebra.counit_comp_antipode, PowerSeries.coeff_comp_monomial, Ideal.top_liesOver_top, LieAlgebra.IsKilling.disjoint_ker_weight_corootSpace, DirectSum.IsInternal.collectedBasis_repr_of_mem_ne, Module.dualProdDualEquivDual_apply, List.Sorted.sbtw, MvPolynomial.divMonomial_monomial, Ideal.IsPrime.multiset_prod_mem_iff_exists_mem, Module.Basis.toMatrix_apply, IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_dvd, Module.Flat.baseChange, StrongDual.toLpₗ_apply, wbtw_iff_right_eq_or_left_mem_image_Ici, Ideal.span_singleton_mul_span_singleton, RootPairing.Hom.weight_coweight_transpose_apply, Polynomial.derivative_sq, KaehlerDifferential.mvPolynomialBasis_repr_apply, LinearMap.BilinForm.IsSymm.add, MonoidAlgebra.nonUnitalAlgHom_ext'_iff, Differential.mapCoeffs_X, Polynomial.Monic.finite_quotient, IntermediateField.AdjoinSimple.trace_gen_eq_zero, AlgebraicGeometry.Scheme.IdealSheafData.ideal_top, Module.DualBases.eval_same, MvPowerSeries.coeff_zero_one, QuadraticModuleCat.toIsometry_whiskerLeft, MeasureTheory.FiniteMeasure.normalize_eq_inv_mass_smul_of_nonzero, Ideal.map_eq_top_iff, Valued.integer.coe_span_singleton_eq_closedBall, MatrixEquivTensor.invFun_smul, QuadraticForm.toDualProd_apply, tendsto_iff_forall_eval_tendsto_topDualPairing, Ideal.IsPrime.mem_or_mem_of_mul_eq_zero, LinearMap.IsNonneg.nonneg, Rep.diagonalSuccIsoTensorTrivial_hom_hom_single, PadicInt.mem_span_pow_iff_le_valuation, RootPairing.CoPolarization_apply, finsuppTensorFinsuppRid_single_tmul_single, AddMonoidAlgebra.nonUnitalAlgHom_ext'_iff, AdicCompletion.sum_of, instLiesOverResidueFieldBotIdeal, LinearIndependent.linearCombinationEquiv_symm_apply, eq_affineCombination_of_mem_affineSpan, Localization.mk_list_sum, Module.DualBases.basis_repr_apply, Algebra.Extension.Hom.mapKer_apply_coe, Module.preReflection_apply, NumberField.discr_mem_differentIdeal, Algebra.toMatrix_lmul', mem_skewAdjointMatricesSubmodule, PrimeSpectrum.range_comap_snd, idealFactorsEquivOfQuotEquiv_symm, Polynomial.hasseDeriv_monomial, Algebra.Presentation.tensorModelOfHasCoeffsHom_tmul, PowerSeries.trunc_trunc_pow, PowerSeries.coeff_mul, ExteriorAlgebra.isUnit_algebraMap, Module.Relations.Solution.isPresentation_iff, Ideal.sup_mul_inf, Ideal.map_iSup_comap_of_surjective, Affine.Simplex.affineCombination_mem_interior_face_iff_pos, ContinuousLinearMap.toSpanSingleton_one, Module.DualBases.dual_lc, wbtw_iff_left_eq_or_right_mem_image_Ici, ExteriorAlgebra.map_injective_field, Ideal.ker_le_comap, Algebra.Extension.contangentEquiv_tmul, IntermediateField.relrank_dvd_rank_top_of_le, LinearMap.BilinForm.isAlt_neg, WeierstrassCurve.Affine.CoordinateRing.XYIdeal_neg_mul, Polynomial.Bivariate.pderiv_zero_equivMvPolynomial, MvPolynomial.weightedHomogeneousComponent_of_isWeightedHomogeneous_ne, Polynomial.derivativeFinsupp_map, Ideal.quotOfMul_surjective, AlgebraicClosure.le_maxIdeal, WeierstrassCurve.Projective.addMap_of_Y_eq, IsPRadical.ker_le, LinearMap.BilinForm.ext_iff, Finsupp.llift_symm_apply, lineMap_lt_map_iff_slope_lt_slope, IsPrecomplete.top, TensorProduct.finsuppScalarLeft_apply_tmul_apply, Matrix.ker_diagonal_toLin', left_lt_lineMap_iff_lt, Polynomial.monomial_add, Algebra.Generators.cMulXSubOneCotangent_eq, ValuationSubring.coe_mem_nonunits_iff, CliffordAlgebraComplex.toComplex_ι, Rep.linearization_obj_ρ, AffineBasis.toMatrix_inv_vecMul_toMatrix, RootPairing.RootPositiveForm.two_mul_apply_root_root, MvPolynomial.isHomogeneous_monomial, Ideal.pointwise_smul_eq_comap, ContinuousLinearEquiv.toSpanNonzeroSingleton_coord, Polynomial.one_lt_rootMultiplicity_iff_isRoot_gcd, Ideal.map_le_iff_le_comap, Ideal.toCotangent_apply, LinearMap.polyCharpolyAux_baseChange, Polynomial.hasDerivWithinAt_aeval, Ideal.snd_comp_quotientMulEquivQuotientProd, ContinuousAlternatingMap.smulRight_toContinuousMultilinearMap, Polynomial.isEisensteinAt_iff, HahnSeries.coeff_toPowerSeries, ContinuousLinearMap.extendTo𝕜_apply, Ideal.Quotient.isScalarTower_of_liesOver, MvPolynomial.pderiv_C_mul, LinearMap.baseChange_sub, Module.Basis.det_comp, Module.Basis.symmetricAlgebra_repr_apply, LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective, Coalgebra.lift_lsmul_comp_counit_comp_comul, LinearMap.convOne_def, PowerSeries.IsWeierstrassDivisorAt.coeff_seq_mem, PowerSeries.coeff_smul, Polynomial.coeff_monomial_same, Ideal.ramificationIdx_bot', Ideal.homogeneousCore'_le, ProbabilityTheory.IsGaussian.integral_dual, Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq, matrixEquivTensor_apply_symm, cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt, Bialgebra.counit_mul, LieModule.traceForm_eq_sum_finrank_nsmul, ZLattice.exists_forall_abs_repr_le_norm, Matrix.toBilin'_apply, Polynomial.hasStrictDerivAt_aeval, LinearMap.flip_injective_iff₂, Algebra.IsEtaleAt.exists_isStandardEtale, ValuationSubring.mem_nonunits_iff_exists_mem_maximalIdeal, AffineMap.lineMap_anti, QuadraticForm.tensorRId_toLinearEquiv, MvPowerSeries.coeff_add_mul_monomial, Matrix.adjugate_def, PrimeSpectrum.zeroLocus_mul, RootPairing.prod_rootFormIn_smul_coroot_mem_range_PolarizationIn, Ideal.comap_map_of_surjective, Ideal.mem_normalizedFactors_iff, Polynomial.span_of_finite_le_degreeLT, IsIntegralClosure.range_le_span_dualBasis, Ideal.matrix_jacobson_le, Ideal.eq_top_iff_one, ContinuousMapZero.toContinuousMapHom_toNNReal, StrongDual.polar_univ, isSimpleModule_iff_quot_maximal, AlgebraicGeometry.ProjectiveSpectrum.Proj.mk_mem_toSpec_base_apply, Ideal.annihilator_quotient, LinearMap.toMatrix_prodMap, Matrix.toLin_apply, Ideal.absNorm_top, Matrix.PosDef.toQuadraticForm', bernsteinPolynomial.iterate_derivative_succ_at_0_eq_zero, Matrix.toLin'_one, ExteriorAlgebra.map_injective, AddMonoidAlgebra.single_mem_grade, Polynomial.iterate_derivative_eq_zero_of_degree_lt, Ideal.absNorm_apply, ContinuousLinearEquiv.unitsEquivAut_apply_symm, MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_apply, Module.Finite.self, Polynomial.toFinsuppIsoLinear_apply, MvPowerSeries.monomial_pow, ProjectiveSpectrum.zeroLocus_bot, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, RootPairing.reflection_dualMap_eq_coreflection, DualNumber.coe_lift_symm_apply, Algebra.Extension.h1Cotangentι_ext_iff, Ideal.coe_comap, Complex.restrictScalars_toSpanSingleton, MultilinearMap.freeFinsuppEquiv_apply, Algebra.Smooth.exists_subalgebra_finiteType, MatrixEquivTensor.left_inv, Ideal.quotientInfEquivQuotientProd_snd, Ideal.toCotangent_eq_zero, Finsupp.linearCombination_range, Module.Basis.constr_apply_fintype, exteriorPower.ιMulti_family_span_of_span, Ideal.span_singleton_toAddSubgroup_eq_zmultiples, instIsSimpleModule, hasGradientAt_iff_hasFDerivAt, algebraMapCLM_apply, AddMonoidAlgebra.liftMagma_apply_apply, IsLocalRing.finrank_cotangentSpace_le_one_iff, Matrix.cramer_submatrix_equiv, Ideal.Filtration.pow_smul_le_pow_smul, Ideal.mul_unit_mem_iff_mem, MvPolynomial.decomposition.decompose'_apply, MvPolynomial.restrictSupport_nsmul, Matrix.mulVecLin_submatrix, bernsteinPolynomial.linearIndependent_aux, Algebra.Generators.toKaehler_tmul_D, Quaternion.imJ_snd_dualNumberEquiv, Polynomial.isRoot_iterate_derivative_of_lt_rootMultiplicity, cfcₙHom_map_quasispectrum, PowerSeries.coeff_monomial_same, MvPolynomial.mem_degreesLE, TensorProduct.isBaseChange, CoalgCat.counit_def, Polynomial.iterate_derivative_prod_X_sub_C, doublyStochastic_eq_convexHull_permMatrix, LinearEquiv.algEquivOfRing_symm_apply, Module.Basis.SmithNormalForm.le_ker_coord_of_notMem_range, Module.isTorsionBy_iff_mem_ker_lsmul, MvPolynomial.weightedHomogeneousComponent_isWeightedHomogeneous, mem_affineSpan_iff_eq_weightedVSubOfPoint_vadd, OreLocalization.zero_mul, AffineSubspace.sSameSide_lineMap_right, WeierstrassCurve.Projective.negMap_eq, Ideal.mul_assoc, IsDedekindDomain.inf_prime_pow_eq_prod, Module.finrank_fin_fun, MeasureTheory.Measure.mapₗ_apply_of_measurable, instFlatAtPrime, MultilinearMap.smulRight_apply, RootPairing.rootForm_self_sum_of_squares, Polynomial.opRingEquiv_symm_monomial, Polynomial.toAddCircle_monomial_eq_smul_fourier, Matrix.toLin_conjTranspose, Ideal.zero_mem, CliffordAlgebra.toBaseChange_comp_reverseOp, Orthonormal.inner_finsupp_eq_sum_left, Ideal.isCoprime_iff_exists, MvPolynomial.map_monomial, Ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot, HopfAlgebra.mul_antipode_rTensor_comul_apply, Ideal.mem_primeCompl_iff, RatFunc.ofFractionRing_sub, MvPolynomial.monomial_mem_restrictSupport, LinearMap.BilinForm.isRefl_neg, LieAlgebra.IsKilling.span_weight_isNonZero_eq_top, LinearIndependent.finsuppLinearCombination_injective, Ideal.cotangentEquivIdeal_apply, KaehlerDifferential.exact_kerCotangentToTensor_mapBaseChange, Subalgebra.LinearDisjoint.linearIndependent_right_of_flat, Representation.ker_leftRegular_norm_eq, Ideal.instIsTwoSidedIInf, exteriorPower.map_injective_field, MvPowerSeries.coeff_zero_X_mul, AdicCompletion.congr_symm_apply, Subfield.relrank_dvd_rank_top_of_le, Polynomial.not_ker_le_map_C_of_surjective_of_weaklyQuasiFiniteAt, Algebra.Etale.exists_subalgebra_fg, continuous_barycentric_coord, AlgebraicGeometry.Scheme.Hom.iInf_ker_openCover_map_comp_apply, LieAlgebra.IsKilling.chainTopCoeff_add_chainBotCoeff, exists_maximalIdeal_pow_eq_of_principal, Algebra.Extension.H1Cotangent.val_add, PolynomialModule.equivPolynomialSelf_apply_eq, AffineMap.lineMap_same_apply, Subspace.dualLift_of_mem, ExteriorAlgebra.ιMulti_apply, Polynomial.degreeLT.addLinearEquiv_symm_apply, coord_norm', RootPairing.algebraMap_posRootForm_posForm, CliffordAlgebra.EquivEven.v_sq_scalar, IsAdjoinRootMonic.coeff_injective, PadicInt.norm_le_pow_iff_mem_span_pow, ContinuousLinearMapWOT.ext_dual_iff, FractionalIdeal.coeIdeal_bot, Ideal.sum_pow_mem_span_pow, Ideal.apply_mem_of_equiv_iff, Polynomial.Chebyshev.iterate_derivative_U_eval_one_recurrence, MeasureTheory.ProbabilityMeasure.coe_toWeakDualBCNN, Quaternion.snd_imK_dualNumberEquiv_symm, MvPolynomial.homogeneousComponent_eq_zero, Submodule.dualAnnihilator_gc, Basis.mem_ideal_iff', Polynomial.coe_basisMonomials, MvPowerSeries.map_monomial, Rep.ofMulActionSubsingletonIsoTrivial_hom_hom, PowerSeries.trunc_apply, KaehlerDifferential.submodule_span_range_eq_ideal, Algebra.idealMap_isLocalizedModule, isArtinianRing_iff, LinearMap.baseChange_neg, IsDedekindDomain.HeightOneSpectrum.intValuation_eq_one_iff, RootPairing.span_root'_eq_top, PrimeSpectrum.closure_image_comap_zeroLocus, instInvertibleReprₛ, RingHom.one_notMem_ker, Module.isTrivialRelation_iff_vanishesTrivially, Bialgebra.TensorProduct.coalgebra_rid_eq_algebra_rid_apply, Ideal.IsHomogeneous.sSup, bddBelow_slope_lt_of_mem_interior, Ideal.associatesEquivIsPrincipal_symm_apply, LinearMap.BilinForm.nondegenerate_congr_iff, NNRealRMK.le_rieszMeasure_of_isCompact_tsupport_subset, RatFunc.ofFractionRing_neg, exteriorPower.alternatingMapLinearEquiv_comp, Matrix.trace_toLin'_eq, MvPowerSeries.le_order_smul, Finset.weightedVSub_filter_of_ne, wbtw_one_zero_iff, Ideal.adic_basis, Matrix.toLin'OfInv_symm_apply, Orientation.rotation_eq_matrix_toLin, Finsupp.linearCombination_restrict, Ideal.stableFiltration_N, Algebra.Extension.H1Cotangent.val_smul, AffineMap.lineMap_apply_ring, LinearMap.BilinForm.flip_apply, TrivSqZeroExt.inlCLM_apply, Algebra.Presentation.differentials.comm₁₂, RootPairing.orthogonal_corootSpan_eq, AffineSubspace.sOppSide_lineMap_right, HasDerivAt.hasFDerivAt_equiv, QuadraticForm.posDef_of_toMatrix', Module.Basis.sumQuot_repr_inr_of_mem, MonomialOrder.sPolynomial_monomial_mul, Module.eval_ker, Finsupp.coe_basisSingleOne, Module.Basis.sumCoords_self_apply, MvPolynomial.mem_homogeneousSubmodule, Algebra.trace_localization, Finset.weightedVSub_vadd, LinearMap.isNilpotent_trace_of_isNilpotent, RootPairing.ker_rootForm_eq_dualAnnihilator, Differential.algHom_deriv', Rep.linearizationTrivialIso_inv_hom, LinearMap.BilinForm.compRight_apply, QuadraticAlgebra.linearEquivTuple_symm_apply, exteriorPower.basis_coord, Submodule.sup_dualAnnihilator_le_inf, Module.Basis.repr_injective, Module.FinitePresentation.linearEquivMapExtendScalars_apply, Polynomial.isNilpotent_monomial_iff, IsLocalizedModule.iso_symm_apply_aux, Polynomial.as_sum_range', IsBaseChange.endHom_apply, LinearIndependent.fintypeLinearCombination_injective, StrongDual.norm_toLpₗ_le, MvPolynomial.homogeneousComponent_zero, LinearIndependent.repr_range, PowerSeries.coeff_monomial, Algebra.Extension.Cotangent.mk_eq_mk_iff_sub_mem, LinearMap.mul''_apply, Module.Basis.parallelepiped_eq_map, Polynomial.natDegree_monomial, IsDedekindDomain.HeightOneSpectrum.intAdicAbv_eq_one_iff, Polynomial.mem_iff_annIdealGenerator_dvd, TensorProduct.piScalarRight_symm_single, associated_norm_prod_smith, MvPolynomial.ker_eval₂Hom_universalFactorizationMap, PolyEquivTensor.toFunAlgHom_apply_tmul_eq_smul, Module.Basis.repr_apply_eq, Polynomial.factorial_mul_shiftedLegendre_eq, MvPolynomial.monomial_left_injective, Polynomial.Chebyshev.one_sub_X_sq_mul_derivative_derivative_T_eq_poly_in_T, RingHom.pNilradical_le_ker_of_perfectRing, TensorPower.algebraMap₀_mul_algebraMap₀, Ideal.constr_basisSpanSingleton, IsAdjoinRootMonic.map_modByMonicHom, TrivSqZeroExt.sndHom_comp_map, MeasureTheory.Measure.map_toOuterMeasure, QuadraticModuleCat.ofIso_trans, Affine.Simplex.incenter_eq_affineCombination, Module.isTorsionBySet_annihilator_top, LinearMap.mem_polar_singleton, MvPolynomial.monomial_sum_index, LinearMap.baseChange_zero, Finsupp.linearCombination_comp, LinearEquiv.isUnit_det, Localization.exists_awayMap_injective_of_localRingHom_injective, LinearMap.IsSymm.eq, Localization.instIsTorsionFreeAtPrimeAlgebraMapSubmonoidPrimeComplOfIsDomain, Ideal.apply_coe_mem_map, FiniteDimensional.finiteDimensional_pi, Polynomial.separable_def, CliffordAlgebra.contractLeft_ι, exteriorPower.presentation_relation, LinearMap.polyCharpolyAux_map_aeval, MvPolynomial.monic_monomial_eq, MvPowerSeries.coeff_zero_iff, dvd_differentIdeal_of_not_isSeparable, IsDedekindRing.toIsNoetherian, PrimeSpectrum.closure_singleton, Pi.counit_coe_finsupp, Polynomial.iterate_derivative_smul, Ideal.exists_maximal_not_isPrincipal, Finset.weightedVSubOfPoint_apply_const, Ideal.mem_map_C_iff, PowerSeries.coeff_X, Subspace.dualAnnihilator_iInf_eq, MvPolynomial.pderiv_monomial_single, Algebra.trace_eq_matrix_trace, Sbtw.left_mem_image_Ioi, Ideal.finite_mulSupport, LinearMap.liftBaseChange_tmul, mem_selfAdjointMatricesSubmodule, PowerSeries.IsWeierstrassDivisorAt.mod_smul, Polynomial.sumIDeriv_X, Algebra.traceMatrix_of_basis, TensorProduct.quotTensorEquivQuotSMul_symm_mk, Algebra.trace_algebraMap_of_basis, AdicCompletion.kerProj_of, LieModule.traceForm_lieSubalgebra_mk_left, NumberField.inverse_basisMatrix_mulVec_eq_repr, MonoidAlgebra.liftMagma_symm_apply, Finsupp.range_linearCombination, Polynomial.contentIdeal_monomial, AddMonoidAlgebra.liftMagma_symm_apply, linearIndependent_iff_injective_fintypeLinearCombination, HahnSeries.coeff_toPowerSeries_symm, Ideal.iInf_pow_eq_bot_of_isLocalRing, Polynomial.injective_ofFn, Matrix.spectrum_toLin, AddMonoidAlgebra.gradeBy.gradedMonoid, geometric_hahn_banach_point_open, MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictTotalDegreeOfFinite, Submodule.quotDualCoannihilatorToDual_injective, Module.Basis.eval_injective, Module.Basis.coePiBasisFun.toMatrix_eq_transpose, PowerSeries.IsWeierstrassDivisorAt.coeff_div, PolyEquivTensor.toFunBilinear_apply_apply, LinearMap.BilinForm.IsSymm.neg, Ideal.IsPrime.smul, LinearMap.toMatrix'_comp, Matrix.toLinearEquiv'_apply, LinearMap.isUnit_toMatrix'_iff, DividedPowers.isSubDPIdeal_ker, PrimeSpectrum.coe_primesOverOrderIsoFiber_apply_asIdeal, ContinuousMultilinearMap.mkPiRing_zero, LinearMap.BilinForm.add_right, Ideal.span_singleton_mul_left_mono, Matrix.toBilin_toMatrix, IsBaseChange.toDual_apply, LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker, ZLattice.volume_image_eq_volume_div_covolume, Module.Relations.ker_toQuotient, Valuation.ltSubmodule_le_leSubmodule, PolyEquivTensor.toFunLinear_tmul_apply, Finsupp.bilinearCombination_apply, Ideal.IsHomogeneous.inf, UniqueFactorizationMonoid.iff_exists_prime_mem_of_isPrime, Valuation.leIdeal_mono, MvPolynomial.DirectSum.coeLinearMap_eq_finsum, Matrix.nondegenerate_toLinearMap₂_iff, Polynomial.coeff_iterate_derivative, Finsupp.apply_linearCombination, StrongDual.toLpₗ_of_not_memLp, LinearMap.BilinForm.comp_congr, Module.Relations.Solution.IsPresentation.surjective_π, Matrix.cramer_subsingleton_apply, AffineMap.coe_lineMap, Matrix.toBilin'_comp, MvPolynomial.support_monomial_subset, Subspace.dualRestrict_surjective, cfcₙHomSuperset_id, MvPolynomial.monomial_mul, Module.compHom.toLinearMap_apply, RootPairing.InvariantForm.apply_eq_or_aux, KaehlerDifferential.kerTotal_mkQ_single_mul, MvPowerSeries.coeff_mul_left_one_sub_of_lt_weightedOrder, Rep.standardComplex.quasiIso_forget₂_εToSingle₀, LinearMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal', uniqueDiffOn_Iio, AdicCompletion.evalOneₐ_of, Ideal.mem_iInf, MultilinearMap.freeFinsuppEquiv_single, Algebra.Generators.H1Cotangent.exact_map_δ', WeakBilin.coeFn_continuous, Lagrange.degree_interpolate_le, Polynomial.bernoulli_def, AlgebraicGeometry.Scheme.default_asIdeal, Polynomial.derivative_eval₂_C, Valuation.leIdeal_zero, MvPowerSeries.weightedOrder_eq_nat, Ideal.Quotient.mk_out, Ideal.eq_bot_of_prime, PiLp.basisFun_eq_pi_basisFun, LinearMap.lift_lsmul_mul_eq_lsmul_lift_lsmul, StrongDual.mem_polarSubmodule, exists_dual_vector', lineMap_le_map_iff_slope_le_slope, MeasureTheory.measureReal_abs_dual_gt_le_integral_charFunDual, Matrix.cramer_is_linear, ProjectiveSpectrum.gc_ideal, CliffordAlgebra.ofEven_comp_toEven, ClassGroup.equivPic_symm_apply, LocalizedModule.lift_mk, PowerSeries.as_tsum, CliffordAlgebra.invOf_ι, CliffordAlgebra.prodEquiv_symm_apply, AffineMap.lineMap_eq_right_iff, Ideal.IsTwoSided.mul_one, HomogeneousIdeal.toIdeal_sup, Module.Basis.end_repr_symm_apply, MvPowerSeries.expand_monomial, Ideal.restrictScalars_mul, Module.evalEquiv_apply, Ideal.jacobson_eq_iff_jacobson_quotient_eq_bot, Polynomial.lhom_ext'_iff, LinearMap.IsReflective.dvd_two_mul, CliffordAlgebra.contractRight_contractRight, InnerProductSpace.toMatrix_rankOne, QuadraticForm.tmul_tensorLId_apply, Affine.Simplex.affineCombination_mem_affineSpan_faceOpposite_iff, separate_convex_open_set, LieAlgebra.InvariantForm.mem_orthogonal, MvPolynomial.rTensor_symm_apply_single, PowerSeries.derivative_coe, rank_matrix', Module.FinitePresentation.exists_lift_equiv_of_isLocalizedModule, Complex.equivRealProdLm_symm_apply_re, ContinuousAffineMap.apply_lineMap, MvPolynomial.pderiv_X_of_ne, LinearMap.IsPerfPair.bijective_right, AdicCompletion.val_add, PowerSeries.coeff_zero_eq_constantCoeff_apply, Ideal.not_isPrime_iff, Polynomial.Bivariate.Polynomial.Bivariate.pderiv_zero_equivMvPolynomial, Polynomial.X_pow_mul_monomial, exteriorPower.instFinite, FractionalIdeal.den_mul_self_eq_num, QuadraticForm.tensorLId_apply, slope_pos_iff_gt, Representation.free_single_single, RCLike.geometric_hahn_banach_open, LinearMap.mul_toMatrix₂_mul, Polynomial.aeval_derivative_of_splits, LinearEquiv.coord_self, Module.Flat.self, Ideal.subset_union_prime_finite, MeasureTheory.OuterMeasure.isometryEquiv_map_mkMetric, MvPolynomial.pderiv_monomial, Algebra.trace_quotient_mk, ContinuousLinearMap.toSpanSingleton_comp_toSpanSingleton, WeierstrassCurve.Projective.addMap_of_Z_eq_zero_right, Module.Basis.apply_eq_iff, Module.Flat.iff_forall_exists_factorization, instIsTorsionFreeLocalizationAlgebraMapSubmonoidPrimeCompl, Orientation.map_neg, exteriorPower.alternatingMapToDual_apply_ιMulti, Module.Basis.mk_coord_apply, Ideal.IsMaximal.out, IsNoetherianRing.isNilpotent_nilradical, AffineBasis.coords_apply, LinearMap.BilinForm.tensorDistribEquiv_tmul, instFiniteAtPrimeLocalizationAlgebraMapSubmonoidPrimeCompl, Algebra.trace_surjective, HahnSeries.ofPowerSeriesAlg_apply_coeff, Module.Dual.lie_apply, RingHom.liftOfRightInverse_comp, RingHom.injective_iff_ker_eq_bot, LinearMap.trace_tensorProduct', Algebra.Generators.toComp_toAlgHom_monomial, Module.Relations.Solution.π_comp_map, LinearMap.polar_eq_biInter_preimage, Submodule.annihilator_span, Module.Finite.exists_comp_eq_id_of_projective, LinearMap.IsReflective.coroot_apply_self, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mk_mem_carrier, LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff, KaehlerDifferential.D_tensorProductTo, Rep.standardComplex.d_comp_ε, Ideal.mul_le_left, Polynomial.IsDistinguishedAt.algEquivQuotient_symm_apply, Ring.KrullDimLE.isNilpotent_iff_mem_maximalIdeal, Polynomial.derivative_pow, Ideal.mem_span, Ideal.IsPrime.pow_le_iff, Matrix.stdBasis_eq_single, Algebra.Extension.toKaehler_surjective, ContinuousMap.ideal_gc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iInf, IsLocalizedModule.map_bijective_iff_localizedModuleMap_bijective, Module.Basis.abs_det_adjustToOrientation, Ideal.mem_span_range_iff_exists_fun, QuotSMulTop.equivQuotTensor_naturality_mk, Finset.weightedVSubOfPoint_eq_of_weights_eq, MvPolynomial.monomial_mem_homogeneousSubmodule_pow_degree, Module.dual_rank_eq, Ideal.radical_le_radical_iff, CompactlySupportedContinuousMap.toNNRealLinear_apply, Polynomial.derivative_prod, Matrix.toLpLinAlgEquiv_apply_apply_ofLp, ValuationRing.iff_ideal_total, LinearMap.toMatrix'_id, Module.Basis.det_self, setBasisOfTopLeSpanOfCardEqFinrank_repr_apply, LinearMap.polyCharpolyAux_coeff_eval, RingEquiv.toSemilinearEquiv_symm_apply, Ideal.isPrime_nat_iff, WStarAlgebra.exists_predual, LinearEquiv.baseChange_inv, QuaternionAlgebra.finrank_eq_four, weightedVSub_mem_vectorSpan, LocalizedModule.mul_by_divBy, EuclideanGeometry.inner_weightedVSub, AffineMap.lineMap_apply_one, LindemannWeierstrass.integral_exp_mul_eval, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, Ideal.associatesEquivIsPrincipal_apply, Derivation.tensorProductTo_mul, nnnorm_cfcₙHom, ModularFormClass.hasSum_qExpansion_of_abs_lt, PrincipalIdealRing.isMaximal_of_irreducible, PowerSeries.coeff_map, contractLeft_apply, Algebra.Generators.disjoint_ker_toKaehler_of_linearIndependent, Algebra.Extension.Cotangent.mk_surjective, Polynomial.valuation_monomial_eq_valuation_X_pow, Algebra.exists_formallyUnramified_of_isUnramifiedAt, Projectivization.orthogonal_mk, AffineMap.lineMap_apply', Polynomial.contentIdeal_eq_bot_iff, Projectivization.cross_mk, AdicCompletion.mk_smul_top_ofAlgEquiv_symm, LinearMap.BilinForm.mul_toMatrix_mul, Polynomial.eval_sumIDeriv_of_pos, LinearMap.trace_transpose, isLinearTopology_iff_hasBasis_open_ideal, MvPolynomial.finrank_eq_zero, MeasureTheory.charFunDual_eq_pi_iff, MvPolynomial.dvd_monomial_mul_iff_exists, TwoSidedIdeal.top_asIdeal, AdicCompletion.val_one, IsBaseChange.map_id_lsmul_eq_lsmul_algebraMap, Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_aeval_zero_of_monic_mem_map, PrimeSpectrum.gc_set, mem_pNilradical, IsLocalization.orderIsoOfPrime_symm_apply_coe, Ideal.bot_isMaximal, Module.rank_self, Finsupp.linearCombination_comp_addSingleEquiv, IsBezout.isPrincipal_of_FG, AddMonoidAlgebra.mem_grade_iff, TwoSidedIdeal.mem_asIdeal, QuadraticForm.comp_tensorRId_eq, LinearMap.tripolar_eq_polar, LinearMap.BilinForm.toMatrix_mul_basis_toMatrix, Matrix.toLinAlgEquiv_apply, Submodule.mem_smul_top_iff, Module.Basis.coe_ofRepr, AdicCompletion.tensor_map_id_left_eq_map, ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, MultilinearMap.mkPiRing_apply_one_eq_self, SubmodulesRingBasis.leftMul, LinearMap.nondegenerate_toMatrix₂'_iff, QuadraticMap.weightedSumSquares_apply, LinearMap.BilinForm.toMatrix'_comp, Module.equiv_free_prod_directSum, Submodule.map_map₂, DualNumber.range_inlAlgHom_sup_adjoin_eps, Submodule.torsionBySet_span_singleton_eq, Submodule.smul_one_eq_span, ContinuousMap.setOfTop_eq_univ, LieModule.exists_nontrivial_weightSpace_of_isSolvable, Algebra.Generators.CotangentSpace.exact, MonomialOrder.div_set, CliffordAlgebra.prodEquiv_apply, AddMonoidAlgebra.grade.decompose_single, LinearMap.ker_dualMap_eq_dualCoannihilator_range, MvPowerSeries.coeff_rescale, TensorProduct.equivFinsuppOfBasisRight_apply, Submodule.mem_annihilator_span, IsArtinianRing.isNilpotent_jacobson_bot, right_le_lineMap_iff_one_le, lineMap_le_lineMap_iff_of_lt, MultilinearMap.ext_ring_iff, Submodule.mulLeftMap_eq_mulMap_comp, IsExposed.eq_inter_halfSpace, MvPowerSeries.WithPiTopology.hasSum_of_monomials_self, MonoidAlgebra.scalarTensorEquiv_symm_single, WittVector.StandardOneDimIsocrystal.frobenius_apply, triple_product_eq_det, LinearEquiv.dualMap_apply, TensorProduct.counit_def, Matrix.toLinearMap₂_basisFun, ProbabilityTheory.isPosSemidef_covarianceBilinDual, LinearMap.IsNonneg.add, List.exists_map_eq_of_sorted_iff_wbtw, LinearMap.finrank_range_dualMap_eq_finrank_range, LocalizedModule.map_exact, PowerSeries.add_weierstrassMod, LinearMap.BilinForm.IsAlt.neg, Submodule.IsPrincipal.dvd_generator_span_iff, Ideal.asIdeal_toTwoSided, Algebra.TensorProduct.quotIdealMapEquivTensorQuot_symm_tmul, Module.Basis.det_basis, KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination, RingHom.IsIntegral.quotient, instFiniteFractionRingLocalizationAlgebraMapSubmonoidNonZeroDivisors, Module.bijective_dual_eval, Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_root_of_monic_mem, PowerSeries.coeff_one, Ideal.map_eq_submodule_map, Affine.Simplex.sOppSide_affineSpan_faceOpposite_of_pos_of_neg, HopfAlgebra.counit_antipode, Ideal.span_singleton_mul_right_mono, PowerSeries.eq_shift_mul_X_add_const, Algebra.Extension.Hom.subToKer_apply_coe, nonPrincipals_def, Polynomial.Chebyshev.iterate_derivative_U_eval_one, Matrix.toLinearMapRight'_apply, Module.annihilator_eq_bot, LinearMap.BilinForm.toMatrix_mul, rank_matrix, PowerSeries.coeff_prod, Algebra.Generators.ofComp_kerCompPreimage, IntermediateField.LinearDisjoint.linearIndependent_right, Finset.weightedVSub_congr, AlgHom.coe_toContinuousLinearMap, Ideal.isCompactElement_top, dividedPowersBot_dpow_eq, Module.Flat.exists_factorization_of_apply_eq_zero_of_free, Polynomial.iterate_derivative_C_mul, Subfield.relrank_top_right, IsLocalization.disjoint_comap_iff, NumberField.linearDisjoint_of_isGalois_isCoprime_discr, Ideal.finite_factors, MvPolynomial.mem_restrictDegree_iff_sup, Continuous.matrix_cramer, PowerSeries.coeff_mul_X_pow', PolyEquivTensor.invFun_monomial, ContinuousLinearMap.smulRight_comp_smulRight, Polynomial.contentIdeal_zero, Matrix.toLinearMap₂_apply, Ideal.Factors.finrank_pow_ramificationIdx, Module.Basis.det_inv, PerfectRing.pNilradical_eq_bot, ExteriorAlgebra.map_surjective_iff, MvPolynomial.weightedHomogeneousComponent_eq_zero_of_notMem, Ideal.mem_inf, ZLattice.abs_repr_lt_of_norm_lt, PrimeSpectrum.nilradical_eq_iInf, Polynomial.mkDerivation_one_eq_derivative, Complex.coe_selfAdjointEquiv, Ideal.quotientMulEquivQuotientProd_fst, QuadraticModuleCat.toIsometry_id, SeparatingDual.exists_eq_one_ne_zero_of_ne_zero_pair, MultilinearMap.freeDFinsuppEquiv_apply, MeasureTheory.OuterMeasure.restrict_biInf, CliffordAlgebra.forall_mul_self_eq_iff, CliffordAlgebra.even.lift_symm_apply_bilin, AlgebraicGeometry.StructureSheaf.algebraMap_germ_assoc, RootPairing.coroot_root_eq_pairing, CliffordAlgebra.evenEquivEvenNeg_apply, Rep.coindVEquiv_apply_hom, CoalgHomClass.counit_comp, Matrix.linearIndependent_rows_of_det_ne_zero, Ideal.instLiesOverBotOfIsPrime, Ring.KrullDimLE.mem_minimalPrimes_iff, AdicCompletion.ofTensorProduct_tmul, isClosedEmbedding_cfcₙHom_of_cfcHom, Polynomial.degreeLT.addLinearEquiv_symm_apply', LieAlgebra.LoopAlgebra.residuePairing_apply_apply, LinearEquiv.toSpanNonzeroSingleton_symm_apply_smul, NormedSpace.polar_ball, Polynomial.coeff_mem_radical_span_coeff_of_dvd, MonoidAlgebra.mem_ideal_span_of_image, KaehlerDifferential.kerTotal_mkQ_single_smul, LinearMap.nondegenerate_toLinearMap₂'_iff_det_ne_zero, MvPolynomial.rTensor_apply_monomial_tmul, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_injective, CliffordAlgebraRing.ι_eq_zero, Polynomial.iterate_derivative_mul_X_pow, PowerSeries.coeff_zero_C, IntermediateField.finrank_fixedField_eq_card, Algebra.Generators.repr_CotangentSpaceMap, Module.Basis.dualBasis_repr, Algebra.normalizedTrace_minpoly, IsAdjoinRootMonic.modByMonicHom_root_pow, Ideal.mem_iInf_smul_pow_eq_bot_iff, lieEquivMatrix'_symm_apply, InnerProductSpace.toContinuousLinearMap_toDualMap, Valuation.Integers.coe_span_singleton_eq_setOf_le_v_algebraMap, MvPowerSeries.eq_zero_iff_forall_coeff_zero, IntermediateField.finrank_comap, Matrix.proj_diagonal, imaginaryPart_ofReal, Module.Dual.baseChange_baseChange, separatingDual_iff_injective, TensorProduct.dualDistribInvOfBasis_apply, Ideal.Filtration.pow_smul_le, Polynomial.taylor_coeff_one, Ideal.isPrime_iff_of_isPrincipalIdealRing_of_noZeroDivisors, isVisible_iff_lineMap, Hausdorffification.lift_comp_of, isNilpotent_tensor_residueField_iff, MvPolynomial.pderiv_pow, LinearMap.trace_eq_sum_trace_restrict, RootPairing.root'_apply_apply_mem_of_mem_span, IsSelfAdjoint.commute_cfcₙHom, PowerSeries.IsWeierstrassDivisorAt.div_add, PowerSeries.forall_coeff_eq_zero, Polynomial.fderivWithin_aeval, NNRealRMK.le_rieszMeasure_of_tsupport_subset, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, dualTensorHom_prodMap_zero, MvPowerSeries.coeff_truncFun, Ideal.rank_pow_quot_aux, PolyEquivTensor.toFunLinear_one_tmul_one, lineMap_lt_map_iff_slope_lt_slope_left, CliffordAlgebraComplex.equiv_apply, LinearMap.range_liftBaseChange, NumberField.mixedEmbedding.stdBasis_apply_isReal, Polynomial.fderivWithin, LinearMap.IsContPerfPair.bijective_right, RingHom.ker_eq_comap_bot, PowerSeries.coeff_C, lineMap_le_right_iff_le_one, Module.annihilator_pi, HahnSeries.coeff_toMvPowerSeries_symm, DividedPowers.isSubDPIdeal_iSup, Polynomial.monomial_neg, LinearMap.isUnit_toMatrix_iff, PadicInt.instIsAdicCompleteMaximalIdeal, Polynomial.hasDerivAt, Subfield.relrank_eq_rank_of_le, Subspace.dualEquivDual_def, PrimeSpectrum.coe_preimageOrderIsoFiber_apply_asIdeal, Module.Basis.linearCombination_repr, List.Sorted.wbtw, MvPolynomial.mkDerivationₗ_C, Ring.map_jacobson_le, LocalizedModule.map_id, ContinuousLinearMap.toSpanSingleton_add, MvPowerSeries.isWeightedHomogeneous_weightedHomogeneousComponent, union_support_maximal_linearIndependent_eq_range_basis, LinearMap.BilinForm.toMatrix_compLeft, Polynomial.monomial_zero_right, Submodule.colon_mono, LinearMap.trace_eq_matrix_trace, FractionalIdeal.spanSingleton_mul_coeIdeal_eq_coeIdeal, RootPairing.polarizationEquiv_symm_apply_coroot, Module.FinitePresentation.out, BilinForm.toMatrix_apply, WeakDual.CharacterSpace.norm_le_norm_one, AdicCompletion.map_exact, Algebra.SubmersivePresentation.cotangentComplexAux_surjective, LinearMap.continuous_on_pi, Matrix.linfty_opNorm_toMatrix, cfcL_integral, exteriorPower.zeroEquiv_ιMulti, not_dvd_differentIdeal_iff, RingHom.toSemilinearMap_apply, MvPolynomial.ker_evalₗ, Polynomial.iterate_derivative_sub, Matrix.toBilin'_apply', Affine.Simplex.point_eq_affineCombination_of_pointsWithCircumcenter, Ideal.ofList_append, Module.Basis.linearCombination_coord, LinearMap.BilinForm.IsSymm.polarization, Polynomial.signVariations_monomial, Ideal.homogeneousCore'_eq_sSup, Representation.FiniteCyclicGroup.coinvariantsKer_leftRegular_eq_ker, InnerProductSpace.toDual_apply_eq_toDualMap_apply, Derivation.map_aeval, Matrix.toLinearMap₂'_single, Ideal.smul_restrictScalars, ExteriorAlgebra.liftAlternating_comp_ιMulti, IntermediateField.LinearDisjoint.lift_rank_right_mul_lift_adjoin_rank_eq_of_isAlgebraic_left, CliffordAlgebra.toEven_ι, Algebra.Extension.H1Cotangent.map_apply_coe, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, LinearMap.separatingRight_toLinearMap₂'_of_det_ne_zero', FractionalIdeal.coe_inv_of_nonzero, TensorProduct.dualDistrib_dualDistribInvOfBasis_left_inverse, PolyEquivTensor.left_inv, CliffordAlgebra.equivBaseChange_apply, MvPowerSeries.coe_hasEvalIdeal, Polynomial.monomial_mul_X_pow, Module.Basis.toDual_apply, PowerSeries.coeff_inv, LinearMap.BilinForm.mul_toMatrix, instNonUnitalSubringClassIdeal, MvPowerSeries.coeff_expand_of_not_dvd, Module.Basis.tensorAlgebra_repr_apply, aeval_derivative_mem_differentIdeal, Algebra.trace_eq_zero_of_not_isSeparable, Module.projective_def', AlgebraicGeometry.Scheme.IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, Algebra.Generators.toKaehler_cotangentSpaceBasis, AccPt.uniqueDiffWithinAt, Submodule.mem_smul_span, Polynomial.iterate_derivative_X_pow_eq_C_mul, Algebra.normalizedTrace_eq_of_finiteDimensional_apply, AdicCompletion.surjective_evalₐ, IsAdjoinRootMonic.coeff_algebraMap, Ideal.ne_top_iff_exists_maximal, Subspace.dualLift_rightInverse, AffineBasis.surjective_coord, SemimoduleCat.MonoidalCategory.leftUnitor_naturality, ContinuousMap.Homotopy.affine_apply, DualNumber.algHom_ext_iff, MvPowerSeries.coeff_zero_C, QuadraticMap.associated_linMulLin, MeasureTheory.JordanDecomposition.toSignedMeasure_smul, Representation.coinvariantsTprodLeftRegularLEquiv_symm_apply, Polynomial.derivative_X_pow, Module.Basis.equivFun_apply, Ideal.natAbs_det_equiv, dualTensorHomEquivOfBasis_toLinearMap, QuadraticForm.associated_isSymm, Submodule.torsion_gc, IsBaseChange.dual, Polynomial.sumIDeriv_derivative, Finset.eq_affineCombination_subset_iff_eq_affineCombination_subtype, Module.apply_evalEquiv_symm_apply, TensorProduct.finsuppScalarLeft_apply_tmul, MvPowerSeries.coeff_monomial, Submodule.coe_dualAnnihilator_span, MvPowerSeries.X_dvd_iff, MvPolynomial.vanishingIdeal_empty, Ideal.torsionOf_eq_top_iff, Polynomial.X_mul_monomial, MvPowerSeries.coeff_C, ProbabilityTheory.iIndepFun_iff_charFunDual_pi', Lagrange.interpolate_eq_add_interpolate_erase, CliffordAlgebraRing.involute_eq_id, Ideal.FG.isNilpotent_iff_le_nilradical, LinearMap.trace_conj', ProbabilityTheory.covarianceBilinDual_comm, AffineMap.lineMap_eq_left_iff, StandardEtalePair.HasMap.isUnit_derivative_f, Matrix.SpecialLinearGroup.toLin'_to_linearMap, TwoSidedIdeal.coe_subtype, ModularFormClass.qExpansion_coeff_eq_circleIntegral, ProbabilityTheory.isGaussian_iff_charFunDual_eq, Module.Basis.mulOpposite_repr_eq, MvPolynomial.tensorEquivSum_X_tmul_X, WittVector.quotientPEquiv_mk, Module.Basis.coe_repr_symm, Module.Basis.smulTower'_repr, Ideal.map_algebraMap_eq_finset_prod_pow, isLocalizedModule_iff_isLocalization', PreTilt.exists_smodEq_untiltAux, TensorProduct.norm_lid, LinearMap.mul_toMatrix₂'_mul, AlgebraicGeometry.Scheme.zeroLocus_biInf_of_nonempty, Rep.coinvariantsTensor_hom_ext_iff, Module.Relations.toQuotient_map_apply, LinearMap.toMatrix₂'_mul, Finset.eq_weightedVSub_subset_iff_eq_weightedVSub_subtype, Ideal.absNorm_eq_one_iff, MeasureTheory.charFun_eq_charFunDual_toDualMap, Polynomial.aeval_sumIDeriv_eq_eval, AdicCompletion.transitionMap_map_pow, PowerSeries.coeff_expand_of_not_dvd, IsLocalRing.le_maximalIdeal, Module.Projective.exists_dual_eq_one, IsCoprime.wronskian_eq_zero_iff, LinearMap.BilinForm.flip_flip, OnePoint.smul_infty_def, ZLattice.volume_image_eq_volume_div_covolume', Algebra.trace_complex_apply, MvPowerSeries.homogeneousComponent_mul_of_le_order, Algebra.toMatrix_lsmul, AlgEquiv.moduleEndSelf_apply_apply, Submodule.fg_iff_exists_fin_linearMap, exteriorPower.map_injective, toMatrix_innerSL_apply, Finset.weightedVSub_eq_linear_combination, Module.Basis.map_orientation_eq_det_inv_smul, isAssociatedPrime_iff_exists_injective_linearMap, isGroupLikeElem_self, MvPolynomial.coeff_rTensorAlgHom_monomial_tmul, Finsupp.linearCombination_fin_zero, Ideal.Quotient.ker_stabilizerHom, KaehlerDifferential.polynomial_D_apply, Ideal.rank_pow_quot, AlgebraicGeometry.StructureSheaf.algebraMap_germ, Ideal.subtype_isoBaseOfIsPrincipal_eq_mul, MvPolynomial.support_sum_monomial_coeff, Matrix.separatingLeft_toLinearMap₂'_iff, Module.Relations.Solution.π_relation, LinearEquiv.dilatransvection.baseChange, Orientation.rightAngleRotation_trans_neg_orientation, Orientation.volumeForm_robust', AdicCompletion.mk_ofAlgEquiv_symm_eq_evalOneₐ, exteriorPower.basis_repr_apply, triple_product_permutation, LinearMap.separatingLeft_toLinearMap₂'_of_det_ne_zero', Polynomial.leval_apply, MvPolynomial.algebraTensorAlgEquiv_symm_map, TensorProduct.rid_symm_apply, IsAssociatedPrime.annihilator_le, Module.Basis.toMatrix_update, LinearMap.intrinsicStar_smulRight, Bialgebra.TensorProduct.lid_toAlgEquiv, PiLp.basis_toMatrix_basisFun_mul, PiTensorProduct.dualDistrib_dualDistribInvOfBasis_right_inverse, NormedSpace.Dual.dual_norm_topology_le_weak_dual_topology, exists_derivative_mul_eq_and_isIntegral_coeff, Module.End.baseChangeHom_apply_apply, dist_left_lineMap, Polynomial.ofFn_degree_lt, Polynomial.taylor_monomial, MvPolynomial.weightedDecomposition.decompose'_eq, ModP.preVal_zero, LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpace, Ideal.instIsTwoSidedTop, Finsupp.linearCombination_equivMapDomain, NumberField.mixedEmbedding.fundamentalCone.idealSetEquiv_symm_apply, BilinForm.toMatrix_mul, Subspace.dualCopairing_nondegenerate, Pi.basisFun_apply, Module.finite_dual_iff, Lagrange.interpolate_apply, exteriorPower.presentation_R, affineCombination_mem_convexHull, Algebra.Generators.CotangentSpace.compEquiv_symm_inr, exists_lt_rieszContentAux_add_pos, isArtinianRing_iff_isNilpotent_maximalIdeal, LinearMap.toMatrix_comp, ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus, Polynomial.valuation_aeval_monomial_eq_valuation_pow, Algebra.normalizedTrace_algebraMap_apply, MvPolynomial.tensorEquivSum_C_tmul_one, CliffordAlgebra.ofProd_comp_toProd, Algebra.normalizedTrace_def, Lagrange.interpolate_eq_iff_values_eq_on, crossProduct_ne_zero_iff_linearIndependent, Module.dual_projective, PowerSeries.coeff_zero_eq_constantCoeff, Polynomial.factorial_smul_hasseDeriv, RootPairing.coreflection_apply, Submodule.exists_isInternal_prime_power_torsion, NormedSpace.sInter_polar_eq_closedBall, Module.Basis.coe_toDual_self, TensorProduct.lid_eq_rid, instIsPrincipalMapRingHom, Submodule.mul_smul, RootPairing.RootPositiveForm.exists_pos_eq, Rep.FiniteCyclicGroup.resolution_complex, Polynomial.mem_ker_modByMonic, Matrix.separatingLeft_toBilin'_iff, Module.Basis.baseChange_repr_tmul, Matrix.kroneckerAlgEquiv_symm_apply, Submodule.dualAnnihilator_top, Matrix.trace_toLin_eq, AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal, lineMap_lt_right_iff_lt, LinearMap.toMatrix₂_mul_basis_toMatrix, ZSpan.repr_fract_apply, IsArtinianRing.exists_not_mem_forall_mem_of_ne, map_le_lineMap_iff_slope_le_slope, Basis.mem_ideal_iff, MeasureTheory.charFunDual_prod', Polynomial.Chebyshev.one_sub_X_sq_mul_iterate_derivative_T_eq_poly_in_T, AffineMap.coe_homothetyAffine, LieAlgebra.IsKilling.rootSystem_pairing_apply, Module.Basis.toMatrix_eq_toMatrix_constr, Matrix.det_reindexLinearEquiv_self, Finset.weightedVSub_vadd_affineCombination, Rep.leftRegularTensorTrivialIsoFree_inv_hom, LieAlgebra.IsKilling.chainTopCoeff_zero_right, Algebra.TensorProduct.basis_repr_symm_apply, Polynomial.hasDerivAt_aeval, LinearMap.polyCharpoly_map_eq_charpoly, AffineMap.snd_lineMap, Ideal.pi_mkQ_rTensor, MvPolynomial.coeffsIn_eq_span_monomial, Ideal.le_comap_map, AlgebraicGeometry.StructureSheaf.const_mul_cancel', Polynomial.aeval_root_derivative_of_splits, ContinuousMapZero.toContinuousMapHom_apply, Orthonormal.inner_finsupp_eq_zero, Algebra.Generators.ofComp_toAlgHom_monomial_sumElim, Finsupp.single_mem_span_single, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_subset_zeroLocus, KaehlerDifferential.quotKerTotalEquiv_symm_apply, LaurentSeries.coeff_zero_of_lt_intValuation, Polynomial.iterate_derivative_neg, support_of_supportDim_eq_zero, finsuppTensorFinsuppLid_apply_apply, PreTilt.untiltAux_smodEq_untiltFun, polyEquivTensor_apply, lineMap_one_half, AlgebraicGeometry.StructureSheaf.Localizations.comapFun_mk, Matrix.mulVec_cramer, LinearIndependent.repr_ker, Matrix.cramerMap_is_linear, CliffordAlgebra.contractRight_one, span_singleton_inf_span_singleton, affineCombination_mem_affineSpan_of_nonempty, PolyEquivTensor.toFunBilinear_apply_eq_smul, CliffordAlgebraDualNumber.equiv_ι, Algebra.Presentation.algebraTensorAlgEquiv_symm_relation, WeierstrassCurve.Projective.Point.zero_def, MvPowerSeries.WithPiTopology.summable_iff_summable_coeff, MeasureTheory.OuterMeasure.restrict_sInf_eq_sInf_restrict, Polynomial.instFree, Polynomial.sum_taylor_eq, LinearMap.BilinForm.isSymm_def, ModuleCat.MonoidalCategory.tensorUnit_isModule, Matrix.toLinOfInv_apply, Polynomial.taylor_eval_sub, Ideal.mem_quotient_iff_mem_sup, Ideal.mem_torsionOf_iff, Polynomial.derivative_prod_finset, AdicCompletion.val_sub, IsHausdorff.eq_iff_smodEq, dualTensorHomEquivOfBasis_symm_cancel_left, Ideal.span_eq_bot, List.SortedLT.sbtw, MvPolynomial.monomial_eq, Ideal.smul_bot, ProbabilityTheory.uncenteredCovarianceBilin_of_not_memLp, LinearMap.IsPerfPair.id, LinearMap.transvection.det, ContinuousLinearMap.det_one_smulRight, Module.Invertible.rTensorEquiv_symm_apply_apply, LinearMap.surjective_compr₂_of_exists_rightInverse, Complex.selfAdjointEquiv_symm_apply, QuaternionAlgebra.coe_basisOneIJK_repr, LaurentPolynomial.counit_C_mul_T, MvPolynomial.homogeneousSubmodule_eq_finsupp_supported, Valuation.Integers.isPrincipal_iff_exists_eq_setOf_valuation_le, cfcₙAux_mem_range_inr, Ideal.Quotient.norm_mk_lt, Module.Flat.iff_lift_lsmul_comp_subtype_injective, Algebra.exists_etale_of_isEtaleAt, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun, Ideal.radical_le_jacobson, Ideal.range_finsuppTotal, Differential.deriv_aeval_eq, strongRankCondition_iff_forall_zero_lt_finrank, LieAlgebra.IsKilling.orthogonal_span_coroot_eq_ker, Affine.Simplex.centroid_eq_affineCombination, Ideal.span_zero, Subalgebra.LinearDisjoint.basisOfBasisLeft_apply, Polynomial.hasseDeriv_C, AlgEquiv.quotientBot_symm_mk, Ideal.ofList_nil, Coalgebra.TensorProduct.lid_toLinearEquiv, Submodule.IsPrincipal.prime_generator_of_isPrime, IsDiscreteValuationRing.TFAE, AddMonoidAlgebra.basis_apply, MvPolynomial.pderiv_map, Polynomial.coeff_list_sum, biUnion_associatedPrimes_eq_compl_nonZeroDivisors, IsSemisimpleRing.exists_ringEquiv_pi_matrix_end_mulOpposite, HomogeneousIdeal.toIdeal_iInf₂, HomogeneousLocalization.val_neg, TensorProduct.AlgebraTensorModule.cancelBaseChange_symm_tmul, Finset.weightedVSub_apply, Polynomial.smeval_monomial, PowerSeries.IsWeierstrassDivisorAt.mod_zero, ContinuousLinearMap.nnnorm_smulRight_apply, Subfield.relfinrank_dvd_finrank_top_of_le, IsCoprime.exists, PowerSeries.X_pow_eq, PowerBasis.trace_gen_eq_sum_roots, ContinuousAlternatingMap.ext_ring_iff, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_right_iff, RingTheory.Sequence.IsRegular.quot_ofList_smul_nontrivial, Matrix.toBilin'Aux_eq, MvPowerSeries.coeff_invOfUnit, LieAlgebra.IsKilling.cartanEquivDual_apply_apply, rank_fin_fun, MvPolynomial.esymm_eq_sum_monomial, Polynomial.homogenize_monomial_of_lt, MeasureTheory.OuterMeasure.comap_boundedBy, Finset.weightedVSubOfPoint_eq_of_sum_eq_zero, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', Matrix.toLin_finTwoProd_toContinuousLinearMap, LinearMap.IsSymm.isRefl, Ideal.IsMaximal.coprime_of_ne, MvPowerSeries.coeff_map, AddMonoidAlgebra.grade.isInternal, NumberField.FinitePlace.norm_eq_one_iff_notMem, Complex.finrank_real_complex, Ideal.unit_mul_mem_iff_mem, Module.Free.self, PowerSeries.monomial_eq_C_mul_X_pow, trace_quotient_eq_trace_localization_quotient, Localization.add_mk, StandardEtalePair.cond, LinearMap.toContinuousLinearMap₁_apply, Ideal.top_mul, Ideal.Quotient.mk_eq_mk_iff_sub_mem, FractionalIdeal.coeIdeal_eq_zero, AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_supportSet, Affine.Simplex.affineCombination_touchpointWeights, Ideal.add_eq_sup, MvPolynomial.HomogeneousSubmodule.gradedMonoid, Orientation.volumeForm_zero_neg, Submodule.mem_traceDual, LinearMap.toMatrixAlgEquiv'_symm, Ideal.coe_closure, Ideal.inertiaDeg_bot, MeasureTheory.FiniteMeasure.map_smul, MvPowerSeries.coeff_mul_eq_coeff_trunc'_mul_trunc', RootPairing.flip_toFun_apply, Algebra.Extension.Hom.sub_aux, Algebra.normalizedTrace_trans_apply, FractionalIdeal.instPosMulReflectLEIdealOfIsDedekindDomain, Algebra.IsAlgebraic.rank_fractionRing, Finset.weightedVSubOfPoint_const_smul, Matrix.finiteDimensional, AdicCompletion.val_neg, IsGroupLikeElem.counit_eq_one, Polynomial.derivative_sub, Bialgebra.counit_algebraMap, Ideal.ringJacobson_le_jacobson, Rep.linearization_δ_hom, Matrix.toPerfectPairing, MeasureTheory.OuterMeasure.map_id, instFiniteLocalizationAlgebraMapSubmonoidPrimeCompl, Polynomial.IsEisensteinAt.mem, KaehlerDifferential.derivationTensorProduct_algebraMap, Ideal.isUnit_iff, Polynomial.hasStrictDerivAt, MvPolynomial.instFiniteOfIsEmpty, Ideal.powQuotSuccInclusion_injective, LinearMap.IsProj.trace, IsLocalRing.CotangentSpace.map_eq_top_iff, MvPolynomial.weightedHomogeneousComponent_eq_zero, Submodule.smul_comap_le_comap_smul, DualNumber.instIsLocalRing, MvPolynomial.zeroLocus_vanishingIdeal_galoisConnection, Ideal.pointwise_smul_le_pointwise_smul_iff, AdicCompletion.mul_apply, Subspace.map_le_dualAnnihilator_dualAnnihilator, Complex.finrank_real_complex_fact, LinearMap.surjective, AdicCompletion.ofTensorProductEquivOfFiniteNoetherian_apply, Ideal.radical_iInf_le, AffineSubspace.sSameSide_lineMap_left, LinearMap.BilinForm.isSymm_zero, lineMap_le_left_iff_le, MvPolynomial.restrictSupport_add, IsLocalRing.isOpen_maximalIdeal_pow, Ideal.le_comap_pow, Algebra.norm_one_add_smul, AffineMap.lineMap_lineMap_left, LinearMap.BilinForm.orthogonal_top_eq_ker, QuadraticMap.apply_linearCombination', Ideal.associatesEquivIsPrincipal_mul, KaehlerDifferential.tensorProductTo_surjective, LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff_mem, MvPolynomial.totalDegree_monomial_le, Submodule.isInternal_prime_power_torsion, RootPairing.rootSpan_dualAnnihilator_map_eq_iInf_ker_root', RingEquiv.toSemilinearEquiv_apply, PowerSeries.coeff_divXPowOrder, ExteriorAlgebra.algebraMap_leftInverse, Polynomial.aroots_monomial, IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime, Coalgebra.rTensor_counit_comul, Ideal.map_inf_le, Rep.standardComplex.instQuasiIsoNatεToSingle₀, MeasureTheory.OuterMeasure.isometry_map_mkMetric, Polynomial.det_taylorLinearEquiv, PowerSeries.trunc_one, Ideal.iInf_pow_eq_bot_of_isDomain, LinearMap.trace_tensorProduct, Ideal.mem_map_iff_of_surjective, Polynomial.derivative_monomial_succ, comp_dualTensorHom, Polynomial.resultant_deriv, LocalizedModule.lift_mk_one, Algebra.trace_quotient_eq_of_isDedekindDomain, QuadraticForm.dualProdProdIsometry_toFun, Algebra.Extension.h1Cotangentι_apply, Finsupp.linearCombination_linearCombination, Matrix.toLinearEquivRight'OfInv_apply, HahnSeries.SummableFamily.powerSeriesFamily_of_orderTop_pos, IntermediateField.relfinrank_dvd_finrank_top_of_le, Module.Relations.Solution.IsPresentation.desc_comp_π, hasEigenvalue_toLin'_diagonal_iff, StrongDual.zero_mem_polar, IsLocalizedModule.mk_eq_mk', IsLocalizedModule.map_injective_iff_localizedModuleMap_injective, PowerSeries.ext_iff, LinearEquiv.smul_id_of_finrank_eq_one_apply, MvPolynomial.pderiv_inr_universalFactorizationMap_X, LinearMap.BilinForm.coe_injective, RootPairing.algebraMap_coroot'In_apply, Algebra.Extension.cotangentComplex_injective_iff, LinearMap.toMatrix'_toLinearMapₛₗ₂', Module.Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod, AdicCompletion.transitionMap_ideal_mk, LinearMap.BilinForm.smul_right, idealFactorsFunOfQuotHom_id, PowerSeries.coeff_X_pow_self, BilinForm.toMatrix_basisFun, Ideal.mul_le_inf, PowerSeries.trunc_succ, IsCyclic.mulAutMulEquiv_symm_apply_symm_apply, Matrix.toLpLin_mul, instMulPosStrictMonoIdeal, MeasureTheory.charFunDual_eq_pi_iff', MvPowerSeries.coeff_smul, LinearPMap.adjointDomainMkCLMExtend_apply, uniqueDiffOn_Ioi, MvPolynomial.weightedHomogeneousComponent_C_mul, Matrix.toLin'_submatrix, QuaternionAlgebra.imJₗ_apply, Algebra.Presentation.tensorModelOfHasCoeffsEquiv_symm_tmul, IsAdjoinRootMonic.coeff_one, Polynomial.monomial_one_right_eq_X_pow, Ideal.isCoprime_biInf, ClassGroup.exists_mk0_eq_mk0, RootPairing.rootForm_restrict_nondegenerate_of_isAnisotropic, Derivation.compAEval_eq, Ideal.Quotient.smulCommClass', Ideal.spanNorm_bot, Matrix.separatingLeft_toLinearMap₂_iff, RingHom.ker_coe_equiv, instIsSimpleOrderIdeal, Ideal.map_iInf_comap_of_surjective, stdSimplex.image_linearMap, Derivation.tensorProductTo_tmul, AdicCompletion.one_apply, IntermediateField.LinearDisjoint.rank_right_mul_adjoin_rank_eq_of_isAlgebraic_left, instIsScalarTowerQuotientIdealResidueField, Ideal.card_stabilizer_eq_card_inertia_mul_finrank, Polynomial.hasseDeriv_apply_one, Finsupp.lift_symm_apply, AdicCompletion.transitionMap_map_mul, CommSemiring.instIsCocomm, Differential.logDeriv_eq_zero, MvPowerSeries.coeff_zero_X, WeierstrassCurve.Projective.negMap_of_Z_eq_zero, Ideal.span_mono, LinearMap.ringLmapEquivSelf_apply, Matrix.toLin_toMatrix, LieModule.traceForm_genWeightSpace_eq, AffineMap.lineMap_same, AddMonoidAlgebra.mapDomainBialgHom_comp, NumberField.mixedEmbedding.latticeBasis_repr_apply, Affine.Simplex.mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter, LinearMap.toSpanSingleton_convMul_toSpanSingleton, exteriorPower.map_comp_ιMulti_family, ModularFormClass.qExpansion_isBigO, Valued.integer.finite_quotient_maximalIdeal_pow_of_finite_residueField, LieModule.traceForm_apply_apply, cross_dot_cross, MvPolynomial.mul_monomial_mem_coeffsIn, AdicCompletion.sumEquivOfFintype_apply, CategoryTheory.Iso.toIsometryEquiv_invFun, AddMonoidAlgebra.scalarTensorEquiv_symm_single, hasFDerivWithinAt_iff_hasGradientWithinAt, IntermediateField.finrank_le_of_le_left, IsLocalRing.exists_maximalIdeal_pow_le_of_isArtinianRing_quotient, RootPairing.rootFormIn_self_smul_coroot, LinearMap.IsSymmetric.trace_eq_sum_eigenvalues, MvPolynomial.C_mem_pow_idealOfVars_iff, IntermediateField.LinearDisjoint.lift_rank_right_mul_lift_adjoin_rank_eq_of_isAlgebraic, Module.Basis.ext_elem_iff, cfcₙHom_continuous, ContinuousLinearMap.comp_lsmul_flip_apply, Continuous.lineMap, Submodule.tensorToSpan_apply_tmul, Rep.FiniteCyclicGroup.resolution_quasiIso, Valuation.integer.coe_span_singleton_eq_setOf_le_v_coe, Ideal.quotientInfToPiQuotient_mk', exteriorPower.map_id, Rep.FiniteCyclicGroup.homResolutionIso_hom_f_hom_apply, RootPairing.corootForm_self_smul_root, nilpotent_iff_mem_prime, mem_fintypeAffineCoords_iff_sum, PiToModule.fromMatrix_apply, RootPairing.exists_form_eq_form_and_form_ne_zero, Polynomial.surjective_toFn, signedDist_right_lineMap, KaehlerDifferential.mvPolynomialBasis_repr_D, IsDedekindDomain.primesOverEquivPrimesOver_symm_apply, QuadraticForm.tensorAssoc_apply, ProbabilityTheory.uncenteredCovarianceBilin_zero, RootPairing.coroot'_reflection, LinearMap.toContPerfPair_apply, MonomialOrder.sPolynomial_monomial_mul', LieModule.trace_toEnd_genWeightSpaceChain_eq_zero, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mono, nilradical_eq_bot_iff, HahnSeries.coeff_toMvPowerSeries, Ideal.ofList_cons_smul, Ideal.Quotient.mk_eq_mk, LinearMap.dualMap_apply, Algebra.mem_ideal_map_adjoin, Lagrange.interpolate_empty, WeierstrassCurve.Projective.addMap_of_Z_ne_zero, image_specComap_zeroLocus_eq_zeroLocus_comap, AlgebraicGeometry.ΓSpec.toSpecΓ_of, Rep.FiniteCyclicGroup.homResolutionIso_inv_f_hom_apply_hom_hom_apply, MvPolynomial.pderiv_eq_zero_of_notMem_vars, MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_continuous, bernsteinPolynomial.derivative_succ, LinearMap.dualMap_surjective_iff, AlgebraicGeometry.Scheme.zeroLocus_span, Finset.attach_affineCombination_of_injective, Ideal.nonempty_inter_nonZeroDivisors_of_faithfulSMul, AlgebraicGeometry.StructureSheaf.toOpen_germ, Polynomial.pow_sub_dvd_iterate_derivative_of_pow_dvd, AddAut.mulLeft_apply_apply, Algebra.SubmersivePresentation.aevalDifferentialEquiv_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iSup, cfcₙHomSuperset_continuous, RatFunc.neg_def, Submodule.iInf_colon, finsuppTensorFinsuppRid_apply_apply, Module.Basis.repr_sum_self, uniqueDiffOn_Ico, TensorProduct.toLinearMap_symm_lid, Matrix.linfty_opNorm_eq_opNorm, Subalgebra.LinearDisjoint.linearIndependent_left_of_flat_of_commute, RootPairing.Base.toWeightBasis_repr_root, DividedPowers.SubDPIdeal.memCarrier, MvPolynomial.C_mul_X_pow_eq_monomial, Finsupp.linearCombination_option, Ideal.mul_iSup, LieAlgebra.IsKilling.chainBotCoeff_of_eq_zsmul_add, Polynomial.iterate_derivative_X_sub_pow_self, IsCyclotomicExtension.Rat.map_eq_span_zeta_sub_one_pow, QuadraticMap.associated_sq, Polynomial.derivativeFinsupp_one, LinearMap.dualMap_def, Algebra.leftMulMatrix_mulVec_repr, span_flip_eq_top_iff_linearIndependent, MvPowerSeries.order_monomial, ExteriorAlgebra.map_apply_ιMulti, Polynomial.degreeLT_eq_span_X_pow, RootPairing.toPerfPair_conj_reflection, Finsupp.linearCombination_eq_fintype_linearCombination, MvPolynomial.weightedHomogeneousComponent_mem, segment_eq_image_lineMap, KaehlerDifferential.ker_map_of_surjective, Finsupp.basisSingleOne_repr, AffineMap.lineMap_continuous_uncurry, Dual.apply_one_mul_eq, max_aleph0_card_le_rank_fun_nat, Ideal.disjoint_primeCompl_of_liesOver, Lagrange.eq_interpolate_of_eval_eq, LinearEquiv.toSpanNonzeroSingleton_one, Ideal.IsPrime.exists_mem_prime_of_ne_bot, Ideal.colon_span, cfcₙ_apply_pi, CliffordAlgebra.EquivEven.e0_mul_e0, Matrix.cramer_one, MeasureTheory.charFunDual_dirac, CliffordAlgebra.contractLeft_mul_algebraMap, map_dualTensorHom, Module.Basis.smulTower_repr_mk, Submodule.range_dualMap_mkQ_eq, Module.Basis.repr_support_subset_of_mem_span, Polynomial.taylor_zero, Subspace.dualLift_injective, Algebra.Generators.cotangentRestrict_mk, Polynomial.toMatrix_sylvesterMap, PowerSeries.coeff_mul_eq_coeff_trunc_mul_trunc, LinearMap.smulRightₗ_apply_apply, Ideal.coe_piOrderIso_symm_apply, Module.DualBases.finite, MvPolynomial.optionEquivLeft_monomial, IsAdjoinRootMonic.modByMonicHom_root, Algebra.TensorProduct.equivPiOfFiniteBasis_symm_apply, nndist_lineMap_lineMap, Irreducible.maximalIdeal_eq_closedBall, IntermediateField.finrank_bot', Submodule.IsPrincipal.associated_generator_span_self, Algebra.Generators.cotangentRestrict_bijective_of_basis_kaehlerDifferential, PrimeSpectrum.coe_vanishingIdeal, Matrix.dualNumberEquiv_apply, CliffordAlgebra.toProd_one_tmul_ι, Module.finrank_linearMap_self, Matrix.separatingRight_toBilin'_iff, Module.FaithfullyFlat.self, ModuleCat.MonoidalCategory.leftUnitor_def, AlgEquiv.moduleEndSelf_symm_apply, MvPowerSeries.substAlgHom_monomial, Polynomial.derivative_neg, Ideal.jacobson_top, RootPairing.InvariantForm.apply_root_root_zero_iff, LinearMap.transvection.of_left_eq_zero, Matrix.toLin_finTwoProd, Ideal.sup_height_eq_ringKrullDim, stdSimplex.barycenter_eq_centerMass, Polynomial.deriv, KaehlerDifferential.tensorKaehlerEquivOfFormallyEtale_symm_D_algebraMap, Unitization.inrRangeEquiv_apply_coe, Coalgebra.counitCoalgHom_apply, LinearMap.BilinForm.apply_toDual_symm_apply, Module.Basis.flag_le_ker_coord_iff, Matrix.SpecialLinearGroup.toLin_equiv.toLinearMap_eq, Ideal.IsHomogeneous.top, Algebra.TensorProduct.quotIdealMapEquivTensorQuot_mk, ContDiffWithinAt.smulRight, qExpansion_mul_coeff_zero, Ideal.radicalInfTopHom_apply, CliffordAlgebra.toBaseChange_involute, Ideal.prod_eq_bot_iff, Module.Basis.linearMap_repr_symm_apply, Matrix.toLinAlgEquiv'_toMatrixAlgEquiv', MeasureTheory.Measure.mapₗ_eq_zero_iff, Hausdorffification.lift_of, KaehlerDifferential.mvPolynomialBasis_repr_comp_D, Submodule.iInf_colon_iUnion, LinearMap.BilinForm.neg_apply, RootPairing.rootForm_pos_of_ne_zero, IsBaseChange.equiv_tmul, TwoSidedIdeal.gc, PowerSeries.trunc_coe_eq_self, instCharPLinearMapSubtypeMemSubringCenterId, LinearMap.tensorProduct_apply, Ideal.pow_right_strictAnti, LinearMap.BilinForm.isPosSemidef_zero, Polynomial.hasseDeriv_natDegree_eq_C, Polynomial.hasFDerivWithinAt_aeval, DividedPowers.IsSubDPIdeal.dpow_eq, Ideal.mem_span_pair, Algebra.Generators.snd_cotangentCompLocalizationAwayEquiv, QuaternionAlgebra.coe_linearEquivTuple, WeakBilin.eval_continuous, finsuppTensorFinsuppLid_single_tmul_single, Ideal.isRadical_iff_pow_one_lt, isStablyFiniteRing_iff_isDedekindFiniteMonoid_moduleEnd, Subfield.lift_rank_comap, LieModule.traceForm_apply_lie_apply, Submodule.IsPrincipal.contentIdeal_generator_dvd, Ideal.pointwise_central_scalar, Ideal.span_empty, Projectivization.cross_mk_of_ne, Algebra.tensorH1CotangentOfIsLocalization_toLinearMap, Ideal.quotientInfToPiQuotient_mk, Ideal.exists_mem_pow_notMem_pow_succ, Ideal.finiteHeight_iff_lt, Ideal.spanNorm_le_comap, Rep.indMap_hom, Module.Basis.finTwoProd_zero, MvPolynomial.algebraTensorAlgEquiv_tmul, PowerSeries.coeff_derivative, ExteriorAlgebra.lift_unique, Module.support_eq_zeroLocus, LieAlgebra.IsKilling.chainBotCoeff_le_chainLength, mem_reesAlgebra_iff, Polynomial.derivative_C_mul_X_pow, Polynomial.derivative_C, Ideal.span_singleton_mul_left_inj, FractionalIdeal.abs_det_basis_change, MvPolynomial.homogeneousSubmodule_one_eq_span_X, Unitization.inrNonUnitalAlgHom_apply, LinearMap.nondegenerate_toMatrix₂_iff, Ideal.span_singleton_lt_span_singleton, Polynomial.derivative_bernoulli_add_one, span_range_eq_top_iff_surjective_finsuppLinearCombination, Ideal.homogeneousCore'_mono, AffineMap.comp_lineMap, ModuleCat.extendScalarsComp_hom_app_one_tmul, Polynomial.hasseDeriv_apply, LinearMap.polar_isClosed, Ideal.matrix_strictMono_of_nonempty, Algebra.norm_eq_prod_roots, Affine.Simplex.affineCombination_mem_closedInterior_face_iff_mem_Icc, Pi.linearIndependent_single_one, Polynomial.addSubmonoid_closure_setOf_eq_monomial, ContinuousLinearMap.det_smulRight, Polynomial.map_under_lt_comap_of_quasiFiniteAt, Ideal.mem_isPrincipalSubmonoid_iff, Matrix.toLinAlgEquiv'_one, RootPairing.flip_toLinearMap, Polynomial.eval₂_derivative_of_splits, trace_eq_sum_embeddings, Matrix.toLin_one, Ideal.map_sSup, CliffordAlgebra.contractLeft_algebraMap_mul, PowerSeries.coeff_subst_finite, Ideal.exists_le_maximal, liftOfDerivationToSquareZero_mk_apply', LinearMap.BilinForm.compLeft_apply, Submodule.FG.jacobson_smul_lt, isFullyInvariant_iff_isTwoSided, IsDedekindDomain.HeightOneSpectrum.maxPowDividing_eq_pow_multiset_count, Ideal.quotientInfEquivQuotientProd_fst, Quaternion.re_snd_dualNumberEquiv, contractRight_apply, cfc_eq_cfcL, LinearMap.trace_eq_contract_of_basis, LinearMap.range_dualMap_le_dualAnnihilator_ker, PowerSeries.coeff_X_pow, Matrix.toLin_transpose, Ideal.univ_eq_iUnion_image_add, Ideal.pointwise_smul_toAddSubgroup, AffineBasis.coord_vadd, Submodule.span_singleton_mul, PrimeSpectrum.basicOpen_le_basicOpen_iff, nilradical_eq_sInf, Complex.toMatrix_conjAe, StrongDual.toLp_apply, Ideal.mem_mul_span_singleton, MvPowerSeries.isHomogeneous_homogeneousComponent, LaurentPolynomial.counit_T, ContinuousMap.mem_setOfIdeal, ContinuousAt.lineMap, Submodule.finite_dualAnnihilator_iff, Representation.leftRegular_norm_apply, BilinForm.mul_toMatrix, Algebra.FormallySmooth.exists_adicCompletionEvalOneₐ_comp_eq, Ideal.IsTwoSided.pow_add, ModularFormClass.hasSum_qExpansion_of_norm_lt, RootPairing.RootPositiveForm.algebraMap_apply_eq_form_iff, CompactlySupportedContinuousMap.eq_toRealPositiveLinear_toReal, Rep.standardComplex.forget₂ToModuleCatHomotopyEquiv_f_0_eq, Coalgebra.TensorProduct.rid_symm_apply, MatrixEquivTensor.right_inv, Polynomial.sum_modByMonic_coeff, Module.Relations.Solution.fromQuotient_comp_toQuotient, ClassGroup.exists_min, Algebra.Generators.CotangentSpace.map_toComp_injective, AdicCompletion.evalₐ_of, Module.Basis.det_mul_det, Polynomial.Chebyshev.iterate_derivative_U_eval_one_eq_div, PowerSeries.expand_monomial, Matrix.toLpLin_symm_comp, Module.Basis.SmithNormalForm.repr_comp_embedding_eq_smul, Polynomial.aeval_monomial, MvPowerSeries.aeval_eq_sum, Representation.ind_mk, Module.Basis.toDual_linearCombination_right, PrimeSpectrum.mem_basicOpen, Algebra.IsUnramifiedAt.exists_notMem_forall_ne_mem_and_adjoin_eq_top, IsLocalRing.CotangentSpace.span_image_eq_top_iff, LaurentPolynomial.counit_C, qExpansion_coeff_unique, isPrecomplete_iff, ContinuousMultilinearMap.smulRight_toMultilinearMap, norm_sub_le_mul_volume_of_norm_fderiv_le, ModuleCat.preservesFiniteLimits_tensorLeft_of_ringHomFlat, Polynomial.Chebyshev.iterate_derivative_T_eval_one, LinearEquiv.coord_apply_smul, PiTensorProduct.ofFinsuppEquiv'_apply_apply, IsLocalRing.notMem_maximalIdeal, CliffordAlgebraRing.reverse_apply, Ideal.isPrime_iff, Lagrange.eval_interpolate_at_node, RootPairing.restrictScalars_toLinearMap_apply_apply, MvPolynomial.divMonomial_monomial_mul, sbtw_lineMap_iff, ExteriorAlgebra.hom_ext_iff, DFinsupp.counit_single, Polynomial.isWeaklyEisensteinAt_iff, ExteriorAlgebra.liftAlternating_algebraMap, AlgebraicGeometry.Scheme.IdealSheafData.ideal_inf, NumberField.mixedEmbedding.fundamentalCone.preimage_of_IdealSetMap, Ideal.coe_prod, RCLike.imCLM_coe, MeasureTheory.FiniteMeasure.toWeakDualBCNN_continuous, Algebra.TensorProduct.rid_toLinearEquiv, PowerSeries.coeff_subst', Polynomial.sylvesterMap_apply_coe, Bialgebra.TensorProduct.lid_toCoalgEquiv, LinearMap.toMatrix_toSpanSingleton, RootPairing.toPerfPair_flip_conj_coreflection, QuadraticAlgebra.finrank_eq_two, Differential.implicitDeriv_X, MvPolynomial.sum_monomial_eq, Module.Invertible.bijective, Polynomial.degreeLT.addLinearEquiv_natAdd, ExteriorAlgebra.map_id, TensorProduct.lidOfCompatibleSMul_tmul, linearDepOn_iffₛ, TrivSqZeroExt.sndHom_apply, MvPolynomial.degrees_monomial_eq, MvPolynomial.zeroLocus_top, PrimeSpectrum.subset_zeroLocus_vanishingIdeal, Coalgebra.TensorProduct.rid_toLinearEquiv, Valuation.mem_ltSubmodule_iff, AffineSubspace.wSameSide_lineMap_right, Ideal.lcm_eq_inf, Ideal.neg_mem_iff, Module.Flat.iff_flat_tensorProduct, HopfAlgebra.mul_antipode_lTensor_comul_apply, Module.Basis.singleton_apply, CoalgCat.rightUnitor_def, Module.finrank_finsupp_self, Ideal.isCoprime_iff_add, MvPolynomial.pUnitAlgEquiv_monomial, Algebra.Generators.compLocalizationAwayAlgHom_relation_eq_zero, Matrix.toLin_eq_toLin', conductor_mul_differentIdeal, WeakDual.CharacterSpace.continuousMapEval_apply_apply, Localization.neg_mk, Polynomial.support_derivativeFinsupp_subset_range, MvPolynomial.coeff_monomial, MvPolynomial.coeff_mul_monomial, CliffordAlgebraDualNumber.ι_mul_ι, Polynomial.ofFn_comp_toFn_eq_id_of_natDegree_lt, Polynomial.monomial_zero_left, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_apply, Bialgebra.TensorProduct.rid_symm_apply, HomogeneousIdeal.irrelevant_eq_iSup, Ideal.Quotient.mk_eq_one_iff_sub_mem, exteriorPower.basis_repr, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_inter, exists_extension_norm_eq, MvPolynomial.mem_ideal_span_monomial_image_iff_dvd, LinearMap.liftBaseChange_comp, Ideal.eq_prime_pow_mul_coprime, ProjectiveSpectrum.zeroLocus_sup_ideal, IsSMulRegular.subsingleton_linearMap_iff, CliffordAlgebra.EquivEven.involute_e0, MatrixEquivTensor.invFun_zero, Subspace.dualAnnihilator_le_dualAnnihilator_iff, LinearMap.separatingRight_of_det_ne_zero, Bialgebra.counit_one, IsRegular.monomial, Module.finrank_self, Module.Basis.orientation_unitsSMul, RootPairing.InvariantForm.apply_eq_or, Module.annihilator_dfinsupp, Ideal.radical_top, Matrix.Nondegenerate.toBilin', Lagrange.interpolate_poly_eq_self, Module.Relations.Solution.span_relation_le_ker_π, ContinuousLinearMap.toSpanSingletonLE_symm_apply, VectorFourier.fourierIntegral_comp_add_right, NormedSpace.closedBall_inv_subset_polar_closedBall, IsSemisimpleRing.jacobson_eq_bot, LinearMap.BilinForm.congr_trans, Polynomial.taylor_coeff, LinearMap.toMatrix_singleton, LinearMap.isNonneg_def, Localization.mk_multiset_sum, LinearMap.BilinForm.toMatrix'_mul, ProbabilityTheory.IsGaussian.integral_dual_conv_map_neg_eq_zero, Algebra.trace_prod_apply, Submodule.mem_colon_singleton, Ideal.comap_top, signedDist_lineMap_right, ContinuousLinearMap.norm_smulRightL_le, ConvexOn.monotoneOn_slope_gt, MvPolynomial.monomial_pow, CliffordAlgebraQuaternion.ofQuaternion_comp_toQuaternion, Algebra.trace_trace_of_basis, AffineMap.lineMap_symm, PowerSeries.derivative_subst, Polynomial.taylor_inj, LinearMap.trace_restrict_eq_of_forall_mem, Polynomial.derivative_zero, Algebra.traceMatrix_of_basis_mulVec, IsBaseChange.basis_repr_comp, MvPowerSeries.monomial_zero_one, Module.Dual.finrank_ker_add_one_of_ne_zero, MvPowerSeries.monomial_one_eq, Algebra.Extension.Hom.sub_tmul, Rep.diagonalOneIsoLeftRegular_hom_hom, PowerSeries.IsRestricted.monomial, exteriorPower.alternatingMapLinearEquiv_symm_apply, OrthonormalBasis.coe_toBasis_repr_apply, ContinuousMap.idealOfEmpty_eq_bot, MvPolynomial.eval₂_monomial, TensorProduct.lid'_symm_apply, SkewPolynomial.monomial_mul_monomial, PowerSeries.coeff_expand, Orientation.reindex_apply, affineCombination_eq_centerMass, Ideal.quotientMap_injective, Submodule.colon_top, Rep.coinvariantsTensorIndHom_mk_tmul_indVMk, DistribMulActionHom.ext_ring_iff, instIsStablyFiniteRingEndForallOfFinite, AdicCompletion.evalOneₐ_surjective, Ideal.prod_top_top, ZLattice.normBound_spec, Algebra.map_intTraceAux, Ideal.map_top, QuadraticMap.apply_linearCombination, Submodule.top_smul, LinearMap.toMatrix₂'_compl₂, CliffordAlgebra.evenEquivEvenNeg_symm_apply, Ideal.eq_top_iff_of_liesOver, Module.isTorsionBySet_iff_subset_annihilator, DualNumber.exp_smul_eps, Ideal.minimalPrimes_eq_empty_iff, TensorPower.mul_algebraMap₀, ContinuousMultilinearMap.smulRight_apply, instFreeSubtypeMemIdealOfFiniteOfIsTorsionFree, Basis.multilinearMap_apply, HopfAlgCat.leftUnitor_def, AddValuation.mem_supp_iff, Matrix.toLinAlgEquiv'_symm, AffineIndependent.injOn_affineCombination_fintypeAffineCoords, Quaternion.snd_imI_dualNumberEquiv_symm, CliffordAlgebraComplex.toComplex_involute, Ideal.mul_iInf, MvPolynomial.homogeneousSubmodule_one_pow, LinearMap.IsSymm.add, MeasureTheory.OuterMeasure.map_top_of_surjective, Submodule.le_dualAnnihilator_iff_le_dualCoannihilator, Ideal.spanNorm_mul_spanNorm_le, MvPolynomial.universalFactorizationMapPresentation_algebra_smul, Ideal.mem_span_insert, Ideal.isPrimary_iff, IsDedekindDomainDvr.toIsNoetherian, MvPolynomial.IsHomogeneous.sum_X_mul_pderiv, AdjoinRoot.powerBasisAux'_repr_apply_to_fun, LieAlgebra.IsKilling.traceForm_coroot, Matrix.toLinearEquiv_kroneckerAlgEquiv, IsCyclic.mulAutMulEquiv_symm_apply_apply, PrimeSpectrum.coe_preimageHomeomorphFiber_symm_apply_coe_asIdeal, Polynomial.succ_signVariations_X_sub_C_mul_monomial, MeasureTheory.charFunDual_map, IsAdjoinRootMonic.coeff_apply_coe, Polynomial.derivative_X_add_C_sq, Ideal.Factors.finiteDimensional_quotient_pow, Submodule.span_smul_eq, Module.Basis.mapCoeffs_repr, Polynomial.one_lt_rootMultiplicity_iff_isRoot_iterate_derivative, Polynomial.span_le_degreeLE_of_finite, CliffordAlgebraComplex.toComplex_ofComplex, LinearMap.IsScalarTower.compatibleSMul', uniqueDiffWithinAt_Ici, LinearMap.BilinForm.comp_apply, ZSpan.discreteTopology_pi_basisFun, AlgebraicGeometry.Scheme.IdealSheafData.ideal_ofIdeals_le, LinearMap.rid_comp_lTensor, isOpenMap_barycentric_coord, Polynomial.as_sum_support, ExteriorAlgebra.ιInv_comp_map, MvPolynomial.aeval_sumElim_pderiv_inl, FDRep.char_dual, Affine.Simplex.signedInfDist_affineCombination, MvPolynomial.monomial_one_dvd_monomial_one, Submodule.mem_colon, MeasureTheory.OuterMeasure.restrict_apply, Module.Basis.traceDual_repr_apply, Polynomial.taylorLinearEquiv_symm, Ideal.Quotient.smulCommClass, MvPolynomial.mul_monomial_modMonomial, MvPolynomial.tensorEquivSum_one_tmul_X, Matrix.nondegenerate_toLinearMap₂'_iff, Algebra.SubmersivePresentation.sectionCotangent_comp, LinearMap.separatingRight_toMatrix₂_iff, isSimpleRing_isArtinianRing_iff, FGModuleCat.FGModuleCatEvaluation_apply', Polynomial.natTrailingDegree_monomial_le, LieModule.traceForm_eq_sum_finrank_nsmul_mul, Matrix.SpecialLinearGroup.toLin'_injective, pNilradical_one, ContinuousLinearMap.smulRight_id, TensorProduct.piScalarRight_apply, DividedPowers.SubDPIdeal.sup_carrier_def, Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_aux, Ideal.instNoZeroDivisors, Coalgebra.lTensor_counit_comp_comul, Ideal.eq_inf_of_isPrime_inf, PowerSeries.eval₂_trunc_eq_sum_range, Differential.algEquiv_deriv', Ideal.mem_radical_iff, LinearMap.det_ring, AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff, Polynomial.degreeLT.addLinearEquiv_symm_apply_inl_basis, ProbabilityTheory.IsGaussian.charFunDual_eq_of_integral_eq_zero, ModuleCat.hom_inv_leftUnitor, CliffordAlgebra.changeForm.associated_neg_proof, Matrix.mulVecLin_reindex, PowerSeries.trunc_one_X, Module.Relations.toQuotient_relation, Polynomial.leval_coe_eq_smeval, conductor_eq_top_iff_adjoin_eq_top, LinearEquiv.baseChange_pow, LinearMap.BilinForm.neg_right, RootPairing.RootPositiveForm.zero_lt_apply_root_root_iff, ProbabilityTheory.variance_dual_prod, CliffordAlgebra.ofEven_ι, exteriorPower.map_apply_ιMulti, Module.Flat.exists_factorization_of_comp_eq_zero_of_free, MvPolynomial.mkDerivationₗ_X, LinearMap.BilinForm.IsNonneg.smul, Ideal.le_pow_of_le_ramificationIdx, Algebra.H1Cotangent.exact_map_δ, EuclideanGeometry.dist_affineCombination, CliffordAlgebra.ofBaseChange_tmul_one, Module.Basis.ofZLatticeComap_repr_apply, MvPowerSeries.monomial_eq, LinearMap.BilinForm.linMulLin_compLeft, Ideal.spanNorm_eq_bot_iff, MvPolynomial.universalFactorizationMap_freeMonic, uniqueDiffOn_Ioo, Coalgebra.TensorProduct.rid_tmul, Algebra.Generators.Hom.toAlgHom_monomial, Finset.sum_smul_vsub_const_eq_weightedVSub, Algebra.normalizedTrace_algebraMap_of_lifts, HopfAlgebra.sum_antipode_mul_eq_algebraMap_counit, Finset.sum_smul_vsub_const_eq_affineCombination_vsub, Ideal.associatesEquivIsPrincipal_map_zero, Ideal.span_singleton_mul_right_injective, Algebra.Generators.H1Cotangent.δ_comp_equiv, Matrix.cramer_smul, LieAlgebra.IsKilling.chainBotCoeff_zero_right, RootPairing.posRootForm_posForm_anisotropic, MeasureTheory.Measure.restrictₗ_apply, Ideal.pow_lt_self, MvPolynomial.universalFactorizationMapPresentation_val, Module.projective_def, Submodule.dualAnnihilator_bot, DualNumber.lift_apply_inl, Ideal.absNorm_eq_zero_iff, Localization.le_comap_primeCompl_iff, CliffordAlgebraComplex.Q_apply, PrimeSpectrum.zeroLocus_inf, Finsupp.mem_ideal_span_range_iff_exists_finsupp, Matrix.UnitaryGroup.toLin'_mul, ContinuousLinearMap.extendTo𝕜'_apply, AddMonoidAlgebra.mem_span_support', Module.supportDim_quotient_eq_ringKrullDim, LinearMap.BilinForm.linMulLin_apply, PrimeSpectrum.mem_vanishingIdeal, Orientation.map_apply, MvPolynomial.universalFactorizationMapPresentation_σ', PolynomialModule.monomial_smul_apply, IsBaseChange.linearMap, AdicCompletion.eval_of, RootPairing.corootSpan_dualAnnihilator_le_ker_rootForm, Valuation.mem_leIdeal_iff, PowerSeries.coeff_succ_X_mul, PowerSeries.mem_hasEvalIdeal_iff, Ideal.span_univ, LinearMap.toMvPolynomial_eval_eq_apply, ProbabilityTheory.norm_uncenteredCovarianceBilin_le, Module.Dual.eval_naturality, slope_fun_def_field, Submodule.torsionBySet_eq_torsionBySet_span, LinearMap.BilinForm.comp_symmCompOfNondegenerate_apply, KaehlerDifferential.mvPolynomialBasis_repr_symm_single, MvPowerSeries.coeff_monomial_ne, Ideal.exists_comap_eq_of_mem_minimalPrimes, PowerSeries.IsWeierstrassDivisorAt.mk_mod'_eq_self, AdicCompletion.map_zero, Ideal.subset_union_prime, ringKrullDim_le_ringKrullDim_add_spanFinrank, Polynomial.logMahlerMeasure_monomial, Function.ExtendByZero.linearMap_apply, PowerSeries.trunc_trunc, Module.Basis.toMatrix_transpose_apply, LinearOrderedSemiring.toLocallyConvexSpace, ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id, Module.finite_finsupp_self_iff, instFinitePresentation, idealFactorsFunOfQuotHom_coe_coe, AddMonoidAlgebra.decomposeAux_single, IdealFilter.IsTorsionQuot.inf, ContinuousLinearMap.smulRight_comp, cfcₙ_eq_cfcₙL, AdicCompletion.component_sumInv, Algebra.normalizedTrace_trans, DualNumber.lift_op_smul, cyclotomic_comp_X_add_one_isEisensteinAt, LinearMap.trace_conj, Bialgebra.TensorProduct.rid_tmul, Submodule.map_range_rTensor_subtype_lid, QuadraticMap.Isometry.tmul_apply, isSMulRegular_on_quot_iff_lsmul_comap_eq, MeasureTheory.FiniteMeasure.isEmbedding_toWeakDualBCNN, IsLinearTopology.hasBasis_ideal, Rep.freeLift_hom, Polynomial.rootSet_monomial, ContinuousLinearEquiv.coord_norm', PowerSeries.substAlgHom_eq_aeval, Ideal.coe_torsionOf, LinearEquiv.baseChange_symm, ContinuousLinearMap.smulRight_one_one, exists_bijective_map_powers, Polynomial.monomial_left_inj, RatFunc.taylor_mem_nonZeroDivisors, LinearMap.toSpanSingleton_zero, IsLocalizedModule.iso_localizedModule_eq_refl, LieAlgebra.IsKilling.chainTopCoeff_of_eq_zsmul_add, IsLocalization.map_linearMap_eq_toLinearMap_mapₐ, LinearMap.BilinForm.congr_fun, jacobi_cross, AffineMap.lineMap_apply_ring', CliffordAlgebra.toBaseChange_ι, Polynomial.newtonMap_apply, LinearMap.separatingLeft_toMatrix₂_iff, isArtinianRing_iff_isFiniteLength, Algebra.IsAlgebraic.rank_fractionRing_mvPolynomial, Derivation.apply_aeval_eq', Subalgebra.LinearDisjoint.linearIndependent_left_of_flat, Module.map_eval_injective, basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix, AdicCompletion.evalₐ_mk, Module.Basis.toDualFlip_apply, LinearMap.toSpanSingleton_one, RootPairing.EmbeddedG2.twoShortAddLongRoot_shortRoot, Bialgebra.TensorProduct.lid_tmul, cfcₙL_apply, Set.Finite.convexHull_eq_image, AdicCompletion.piEquivOfFintype_apply, MvPolynomial.monomial_sum_one, Algebra.algebraMap_intTrace, hasFDerivAt_exp_smul_const_of_mem_ball', Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_right_iff, LieAlgebra.IsKilling.chainBotCoeff_add_chainTopCoeff, TensorProduct.dualDistrib_apply, LinearMap.BilinForm.toLin_restrict_ker_eq_inf_orthogonal, Module.Basis.ofZLatticeBasis_repr_apply, Ideal.KerLift.map_smul, tendsto_card_div_pow_atTop_volume', MeasureTheory.charFunDual_pi, HopfAlgebra.mul_antipode_lTensor_comul, CliffordAlgebraComplex.reverse_eq_id, LinearMap.BilinForm.IsAlt.sub, LinearMap.BilinForm.toMatrix'_toBilin', Fintype.linearCombination_apply_single, map_le_lineMap_iff_slope_le_slope_left, Ideal.multiset_prod_le_inf, isSimpleModule_iff_toSpanSingleton_surjective, PrimeSpectrum.coe_primesOverOrderIsoFiber_symm_apply_coe, Fintype.linearCombination_apply, neg_cross, Ideal.sInf_minimalPrimes, Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors, LocalizedModule.map_mk, TwoSidedIdeal.orderIsoIsTwoSided_apply_coe, PiTensorProduct.map_update_smul, Submodule.coe_torsion_eq_annihilator_ne_bot, TwoSidedIdeal.asIdeal_matrix, Polynomial.natDegree_iterate_derivative, Module.exists_surjective_quotient_of_finite, FaithfulSMul.ker_algebraMap_eq_bot, instIsTorsionFreeLocalizationAlgebraMapSubmonoidPrimeCompl_2, finsuppTensorFinsuppRid_self, ModuleCat.ofHom₂_compr₂, AdicCompletion.evalₐ_mkₐ, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff_of_mem, Ideal.le_comap_sup, IsLocalization.coeSubmodule_strictMono, RootPairing.four_smul_rootForm_sq_eq_coxeterWeight_smul, Irreducible.maximalIdeal_pow_eq_closedBall_pow, PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal, StrongDual.coe_toWeakDual, AddAut.mulLeft_apply_symm_apply, LinearMap.det_toMatrix', Matrix.separatingLeft_toBilin_iff, conductor_eq_top_of_powerBasis, WeakDual.toStrongDual_apply, convexHull_rangle_single_eq_stdSimplex, Matrix.mulVec_injective_iff, Polynomial.eval_add_of_sq_eq_zero, IsFractionRing.coeSubmodule_isPrincipal, CharacterModule.intSpanEquivQuotAddOrderOf_apply_self, Polynomial.derivative_X_add_C_pow, RatFunc.sub_def, Module.Finite.exists_fin', HomogeneousIdeal.irrelevant_eq_closure, LieAlgebra.IsKilling.ker_restrict_eq_bot_of_isCartanSubalgebra, MeasureTheory.OuterMeasure.map_top, MultilinearMap.mkPiRing_apply, Polynomial.contentIdeal_le_span_content, LinearMap.toMatrixAlgEquiv_apply', trace_eq_sum_automorphisms, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_inter, MvPolynomial.mem_vanishingIdeal_iff, WithIdealFilter.mem_nhds_iff, is_ideal_adic_pow, MvPolynomial.mem_vanishingIdeal_singleton_iff, Matrix.toLin'OfInv_apply, MvPowerSeries.coeff_subst_finite, MonomialOrder.div, Basis.equivFun_symm_single, isSemiprimaryRing_iff, Ideal.ofList_cons, dist_right_lineMap, PowerSeries.IsWeierstrassDivisorAt.div_zero, AffineBasis.affineCombination_coord_eq_self, AlgHom.toKerIsLocalization_isLocalizedModule, Finsupp.linearCombination_surjective, TensorProduct.tensorQuotEquivQuotSMul_comp_mk, geometric_hahn_banach_closed_point, AdicCompletion.ofTensorProduct_naturality, LinearMap.BilinForm.isOrtho_def, CommRing.Pic.mk_self, IntermediateField.LinearDisjoint.basisOfBasisLeft_repr_apply, MvPolynomial.scalarRTensor_symm_apply_single, FractionalIdeal.equivNum_apply, Irreducible.maximalIdeal_pow_eq_setOf_le_v_coe_pow, IsTopologicallyNilpotent.mem_topologicalNilradical_iff, Polynomial.natDegree_monomial_eq, Module.finrank_fintype_fun_eq_card, Ideal.homogeneousCore_eq_sSup, Ideal.iUnion_minimalPrimes, ContinuousLinearMap.toSpanSingleton_smul', PowerSeries.IsWeierstrassDivisorAt.isWeierstrassDivisionAt_div_mod, ExteriorAlgebra.lhom_ext_iff, DividedPowers.isSubDPIdeal_inf_iff, Ideal.ideal_prod_prime_aux, RatFunc.add_def, TensorProduct.equivFinsuppOfBasisLeft_apply, ContinuousMap.mem_idealOfSet_compl_singleton, Module.Basis.prod_repr_inr, AddSubgroup.relIndex_eq_natAbs_det, instExpCharLinearMapSubtypeMemSubringCenterId, Polynomial.iterate_derivative_mul, Finsupp.range_lmapDomain, QuadraticModuleCat.toIsometry_inv_leftUnitor, Polynomial.aeval_iterate_derivative_of_lt, RootPairing.rootSpan_dualAnnihilator_map_eq, Ideal.absNorm_bot, PrimeSpectrum.asIdeal_lt_asIdeal, Ideal.Quotient.quotient_ring_saturate, Rep.leftRegularTensorTrivialIsoFree_hom_hom_single_tmul_single, AffineBasis.coe_coord_of_subsingleton_eq_one, map_lt_lineMap_iff_slope_lt_slope, Ideal.comap_map_of_surjective', Submodule.instSMulCommClass_1, Ideal.relNorm_apply, LinearMap.isSymm_iff_basis, Ideal.comap_fiberIsoOfBijectiveResidueField_symm, Polynomial.derivative_map, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_inl, Finsupp.linearCombination_comp_lmapDomain, LinearMap.toMatrix_basis_equiv, Matrix.UnitaryGroup.toGL_one, Ideal.top_pow, Finsupp.mem_span_image_iff_linearCombination, Algebra.Generators.cotangentCompLocalizationAwayEquiv_symm_inr, ExteriorAlgebra.ι_range_map_map, Submodule.baseChange_top, IntermediateField.rank_bot', AdicCompletion.val_zero, IsDedekindDomain.isOpen_of_ne_bot, Valuation.self_le_supp_comap, ProjectiveSpectrum.zeroLocus_span, Module.evalEquiv_toLinearMap, MvPowerSeries.order_eq_nat, exteriorPower.map_comp, MvPolynomial.irreducible_sumSMulXSMulY, isJacobsonRing_iff_sInf_maximal', LieModule.traceForm_apply_eq_zero_of_mem_lcs_of_mem_center, Finset.sum_smul_vsub_eq_affineCombination_vsub, topDualPairing_apply, NumberField.mixedEmbedding.fundamentalCone.mem_idealSet, DualNumber.lift_comp_inlHom, LinearMap.trace_prodMap', Ideal.mem_pi, SemimoduleCat.hom_hom_leftUnitor, MvPowerSeries.IsWeightedHomogeneous.coeff_eq_zero, SemimoduleCat.MonoidalCategory.tensorUnit_isModule, PowerSeries.coeff_C_mul, PowerSeries.X_pow_dvd_iff, exteriorPower.ιMultiDual_apply_nondiag, LinearMap.annihilator_le_of_surjective, MvPolynomial.mem_ideal_span_monomial_image, Polynomial.iterate_derivative_intCast_mul, Polynomial.taylor_mem_degreeLT, Ideal.mem_span_singleton_mul, Polynomial.leadingCoeff_monomial, AddMonoidAlgebra.decomposeAux_eq_decompose, IsLocalizedModule.fromLocalizedModule.surj, Ideal.exists_spanRank_eq_and_height_eq, Lagrange.eval_nodal_derivative_eval_node_eq, LinearMap.dualMap_comp_dualMap, Submodule.exists_dual_map_eq_bot_of_notMem, Finset.weightedVSub_subtype_eq_filter, ValuationSubring.coe_primeSpectrumOrderEquiv_symm_apply_asIdeal, Matrix.SeparatingRight.toBilin', RCLike.iInter_halfSpaces_eq, exteriorPower.presentation_var, irreducible_pow_sup_of_le, IsAdjoinRootMonic.coeff_apply_le, LinearMap.compQuadraticMap_polarBilin, Module.Basis.traceDual_powerBasis_eq, Algebra.SubmersivePresentation.sectionCotangent_zero_of_notMem_range, CliffordAlgebra.EquivEven.reverse_v, Polynomial.card_support_le_one_iff_monomial, Algebra.injective_lift_lsmul, MvPowerSeries.coeff_mul_left_one_sub_of_lt_order, BilinForm.apply_eq_dotProduct_toMatrix_mulVec, Polynomial.coeff_mul_monomial, Algebra.algebraMap_intTrace_fractionRing, TensorProduct.lift_compr₂, Ring.isField_iff_maximal_bot, KaehlerDifferential.range_kerCotangentToTensor, Ideal.homogeneousCore_mono, Ideal.mul_top, LaurentPolynomial.trunc_C_mul_T, ProperCone.hyperplane_separation, Ideal.isIdempotentElem_iff_eq_bot_or_top_of_isLocalRing, DividedPowers.SubDPIdeal.lt_iff, ModuleCat.extendRestrictScalarsAdj_unit_app_apply, LinearMap.compBilinForm_apply_apply, LinearMap.lid_comp_rTensor, DualNumber.algHom_ext'_iff, Ideal.pow_eq_top_iff, Matrix.range_diagonal, AdicCompletion.mk_apply_coe, Prod.counit_apply, Module.Relations.toQuotient_map, Submodule.torsionBySet.mk_smul, PowerSeries.derivative_C, LinearMap.BilinForm.Nondegenerate.ker_eq_bot, Polynomial.derivative_natCast, QuadraticModuleCat.ofIso_refl, LinearMap.IsSymmetric.re_trace_eq_sum_eigenvalues, StrongDual.mem_polar_iff, Ideal.IsPrime.prod_mem_iff_exists_mem, DualNumber.ideal_trichotomy, NumberField.mixedEmbedding.fundamentalCone.idealSetEquiv_apply, PowerSeries.IsWeierstrassDivisorAt.div_smul, RootPairing.Base.cartanMatrixIn_mul_diagonal_eq, Submodule.mem_annihilator_span_singleton, Finset.sum_smul_const_vsub_eq_sub_weightedVSubOfPoint, LinearMap.dualMap_eq_lcomp, Module.Basis.constr_def, ContinuousLinearEquiv.coord_self, CliffordAlgebraComplex.ofComplex_comp_toComplex, IsLocalRing.map_mkQ_eq_top, isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule, cardinalMk_algHom, LinearMap.trace_comp_comm, RingHom.sub_mem_ker_iff, Polynomial.hasseDeriv_X, Ideal.inf_eq_mul_of_isCoprime, RCLike.geometric_hahn_banach_compact_closed, Derivation.liftKaehlerDifferential_apply, Algebra.Generators.instFreeCotangentSpaceToExtension, rank_fun', LinearMap.range_smulRight_apply, MvPolynomial.coeff_weightedHomogeneousComponent, CliffordAlgebra.equivOfIsometry_symm, IsAdic.isHausdorff_iff, LinearMap.trace_eq_sum_trace_restrict', BilinForm.toMatrix_compLeft, PowerSeries.order_eq, Matrix.maxGenEigenspace_toLin_diagonal_eq_eigenspace, Ideal.iInf_pow_smul_eq_bot_of_isTorsionFree, Polynomial.derivative_bernoulli, AdicCompletion.congr_apply, Module.Basis.toDual_apply_left, LinearMap.toPerfPair_apply, ExteriorAlgebra.comp_ι_sq_zero, exteriorPower.map_surjective, KaehlerDifferential.DLinearMap_apply, Ideal.matrix_bot, ContDiffOn.smulRight, QuadraticMap.sum_repr_sq_add_sum_repr_mul_polar, Ideal.mem_span_singleton, Algebra.TensorProduct.lmul''_eq_lid_comp_mapOfCompatibleSMul, LinearMap.toMatrix₂'_comp, Ideal.radical_eq_top, Ideal.multiset_prod_eq_bot, MvPolynomial.rTensor_apply, Polynomial.ofFn_coeff_eq_val_of_lt, Finset.weightedVSubOfPoint_erase, IsAdjoinRootMonic.coeff_root, MvPolynomial.disjoint_support_monomial, CliffordAlgebra.ofProd_ι_mk, IsLocalization.mem_coeSubmodule, RCLike.geometric_hahn_banach_point_closed, MvPowerSeries.monomial_mul_monomial, Module.DualBases.lc_def, ContinuousMultilinearMap.mkPiRing_eq_zero_iff, AffineBasis.coord_smul, Module.exists_ker_toSpanSingleton_eq_annihilator, AffineBasis.coord_apply_combination_of_mem, PolynomialModule.monomial_smul_lsingle, Algebra.normalizedTrace_eq_of_fininteDimensional_apply, IsLocalRing.le_maximalIdeal_of_isPrime, HomogeneousIdeal.toIdeal_irrelevant_le, IntermediateField.rank_comap, Ideal.mul_inf, PowerSeries.coeff_zero_X_mul, LieAlgebra.IsKilling.chainTopCoeff_le_chainLength, RingTheory.Sequence.map_first_exact_on_four_term_right_exact_of_isSMulRegular_last, LinearMap.toMatrix_eq_toMatrix', LinearMap.toMatrix_pow, MvPolynomial.aeval_monomial, RingTheory.Sequence.isWeaklyRegular_iff, RootPairing.disjoint_rootSpan_ker_rootForm, Ideal.relNorm_bot, CommSemiring.counit_apply, mem_selfAdjointMatricesSubmodule', NNRealRMK.integral_rieszMeasure, Module.Relations.Solution.IsPresentation.exact, Module.Basis.orientation_neg_single, ExteriorAlgebra.ι_mul_prod_list, Ideal.map_comap_le, Ideal.fst_comp_quotientInfEquivQuotientProd, IsBaseChange.linearMapLeftRightHom_apply, Algebra.trace_apply, CoalgCat.tensorUnit_isModule, Module.Finite.base_change, Ideal.span_union, NormedSpace.polar_closedBall, CompactlySupportedContinuousMap.monotone_of_nnreal, LinearMap.BilinForm.toMatrix_comp, ExteriorAlgebra.ιMulti_succ_curryLeft, Matrix.SpecialLinearGroup.toLin'_symm_apply, ClassGroup.mk0_eq_mk0_inv_iff, NumberField.canonicalEmbedding_eq_basisMatrix_mulVec, AlgebraicGeometry.Scheme.IdealSheafData.ideal_biInf, Module.Relations.solutionFinsupp_isPresentation, Finset.weightedVSub_sdiff_sub, Ideal.matrix_monotone, CharP.quotient_iff_le_ker_natCast, IsExposed.eq_inter_halfSpace', QuadraticAlgebra.det_toLinearMap_eq_norm, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_map_one, ExteriorAlgebra.leftInverse_map_iff, PrimeSpectrum.zeroLocus_sup, Module.Basis.reindexFinsetRange_repr, ProjectiveSpectrum.zeroLocus_iSup_ideal, Polynomial.degreeLT.basisProd_natAdd, ProbabilityTheory.covarianceBilinDual_of_not_memLp, exteriorPower.presentation_G, Algebra.Generators.CotangentSpace.fst_compEquiv_apply, Module.Basis.continuous_coe_repr, AlgebraicGeometry.structurePresheafInCommRingCat_obj_carrier, TensorProduct.finsuppScalarLeft_apply, MonomialOrder.degree_monomial_le, LinearMap.trace_comp_cycle, Module.Basis.det_unitsSMul_self, PowerSeries.WithPiTopology.tendsto_trunc_atTop, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_coe, Module.Basis.smulTower_repr, Module.Basis.toDual_apply_right, Polynomial.Sequence.linearIndependent, MeasureTheory.charFun_map_eq_charFunDual_smul, PrimeSpectrum.mem_image_comap_zeroLocus_sdiff, LinearMap.trace_comp_cycle', IsDedekindDomain.exists_representative_mod_finset, Pi.basisFun_equivFun, MvPolynomial.coeff_mul_monomial', Valuation.ltIdeal_mono, Submodule.dualAnnihilator_map_linearEquiv_flip_symm, LinearMap.toMatrix'_one, Representation.dual_apply, Algebra.FormallySmooth.exists_kerProj_comp_eq_id, Localization.flat, Ideal.IsHomogeneous.sup, Polynomial.monomial_natDegree_leadingCoeff_eq_self, Polynomial.contentIdeal_mul_le_mul_contentIdeal, AdicCompletion.transitionMap_comp_eval_apply, Finsupp.linearCombination_comapDomain, AlgebraicGeometry.StructureSheaf.instAwayObjOppositeOpensCarrierTopValStructureSheafInTypeOpBasicOpen, Polynomial.degreeLE_mono, NormedSpace.double_dual_bound, Module.DualBases.coeffs_apply, Submodule.comap_smul'', ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id, Submodule.baseChange_span, MvPolynomial.pderiv_X_self, ModP.preVal_eq_zero, LinearMap.BilinForm.toLin_restrict_range_dualCoannihilator_eq_orthogonal, KaehlerDifferential.tensorKaehlerEquiv_tmul_D, Module.Basis.det_smul_mk_coord_eq_det_update, Ideal.height_le_spanRank_toENat, Module.surjective_piEquiv_apply_iff, LinearMap.transvection.congr, Real.zero_at_infty_vector_fourierIntegral, MvPolynomial.pderiv_C, PowerSeries.coeff_C_mul_X_pow, Submodule.dualAnnihilator_iSup_eq, PiLp.basisFun_map, BialgCat.leftUnitor_def, sbtw_zero_one_iff, MvPolynomial.universalFactorizationMapPresentation_jacobiMatrix, Submodule.dualAnnihilator_eq_bot_iff, TensorProduct.quotTensorEquivQuotSMul_comp_mk, AffineIndependent.affineCombination_eq_lineMap_iff_weight_lineMap, Bialgebra.TensorProduct.lid_symm_apply, RatFunc.toFractionRingRingEquiv_symm_eq, Bialgebra.TensorProduct.rid_toCoalgEquiv, ProbabilityTheory.uncenteredCovarianceBilinDual_apply, Algebra.PreSubmersivePresentation.ofHasCoeffs_algebra_smul, AddMonoidAlgebra.scalarTensorEquiv_tmul, RingHom.coe_toSemilinearMap, bernsteinPolynomial.linearIndependent, isPRadical_iff, bernsteinPolynomial.derivative_zero, OnePoint.equivProjectivization_apply_coe, ZSpan.repr_floor_apply, PointedCone.mem_maxTensorProduct, MvRatFunc.rank_eq_max_lift, TannakaDuality.FiniteGroup.map_mul_toRightFDRepComp, Polynomial.iterate_derivative_map, Matrix.iSup_eigenspace_toLin_diagonal_eq_top, Polynomial.sumIDeriv_apply_of_le, MvPolynomial.homogeneousSubmodule_mul, LinearEquiv.dualMap_refl, Configuration.ofField.mem_iff, Ideal.IsHomogeneous.mul, lineMap_lt_lineMap_iff_of_lt', LinearMap.BilinForm.IsAlt.add, cross_cross_eq_smul_sub_smul', rank_lt_rank_dual, SemimoduleCat.MonoidalCategory.rightUnitor_naturality, Module.Basis.repr_symm_apply, Finsupp.apply_linearCombination_id, QuadraticForm.tmul_comp_tensorAssoc, Polynomial.Bivariate.pderiv_one_equivMvPolynomial, Module.Dual.congr_apply_apply, LieAlgebra.IsKilling.span_weight_eq_top, Algebra.coe_linearMap, Polynomial.support_monomial', Localization.exists_awayMap_bijective_of_localRingHom_bijective, WittVector.mem_span_p_iff_coeff_zero_eq_zero, basis_toMatrix_mul_linearMap_toMatrix, IntermediateField.lift_rank_comap, Polynomial.mem_degreeLE, LieModule.traceForm_flip, Path.segment_apply, lipschitzWith_lineMap, Submodule.span_smul_span, isIntegral_quotientMap_iff, LinearMap.toMatrixₛₗ₂'_apply, Representation.dualTensorHom_comm, LinearMap.baseChange_tmul, IsLocalization.mem_map_algebraMap_iff, MvPolynomial.lcoeff_apply, LinearMap.transvection.baseChange, LinearMap.polar_mem_iff, LinearMap.IsContPerfPair.continuous_uncurry, Module.Basis.tensorProduct_repr_tmul_apply, MvPolynomial.algebraTensorAlgEquiv_symm_monomial, exteriorPower.ιMulti_family_linearIndependent_ofBasis, Module.Basis.coe_ofEquivFun, Ideal.mem_span_insert', PrimeSpectrum.localization_comap_range, PolynomialLaw.ground_apply, Matrix.toLinearMapRight'_one, MvPolynomial.restrictSupport_eq_span, Module.length_of_free_of_finite, Ideal.IsPrincipal.of_comap, RingHom.ker_eq, cfcₙL_integrable, IsLocalization.orderIsoOfPrime_apply_coe, isSMulRegular_iff_ker_lsmul_eq_bot, Matrix.maxGenEigenspace_toLin'_diagonal_eq_eigenspace, bernsteinPolynomial.iterate_derivative_at_0_eq_zero_of_lt, AffineMap.lineMap_apply_one_sub, Ideal.IsHomogeneous.radical_eq, Polynomial.taylor_X, PointedCone.dual_flip_dual_dual_flip, QuadraticForm.associated_tmul, pNilradical_eq_bot_of_frobenius_inj, ModN.instModuleFinite, AffineBasis.linear_combination_coord_eq_self, Representation.free_asModule_free, Affine.Simplex.interior_eq_image_Ioo, Polynomial.contentIdeal_le_contentIdeal_of_dvd, bernsteinPolynomial.iterate_derivative_at_0, Submodule.one_eq_range, Rep.linearizationTrivialIso_hom_hom, MvPolynomial.rank_eq, PowerSeries.coeff_X_pow_mul', Ideal.ext_iff, Basis.piTensorProduct_repr_tprod_apply, RootPairing.toLinearMap_apply_apply_mem_range_algebraMap, Matrix.cramer_transpose_apply, Finset.sum_smul_vsub_const_eq_weightedVSubOfPoint_sub, ClassGroup.equivPic_apply, MvPowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto, Polynomial.Chebyshev.derivative_U_eval_one_eq_div, AdicCompletion.val_zero_apply, Finset.weightedVSub_indicator_subset, Module.Relations.Solution.π_comp_map_apply, Ring.le_comap_jacobson, Submodule.dualRestrict_apply, Polynomial.hasFDerivWithinAt, Ideal.span_prod_le, TrivSqZeroExt.liftEquiv_symm_apply_coe, Valuation.leSubmodule_zero, HomogeneousIdeal.eq_top_iff, Complex.isometryOfOrthonormal_symm_apply, Ideal.mem_comap, IsLocalizedModule.map_iso_commute, Module.Basis.sum_equivFun, Pi.counit_comp_single, Algebra.coe_trace_int, SymmetricAlgebra.IsSymmetricAlgebra.mvPolynomial, Submodule.index_smul_le, LieModule.traceForm_apply_lie_apply', TensorProduct.finsuppScalarLeft_symm_apply_single, MvPowerSeries.isWeightedHomogeneous_iff_eq_weightedHomogeneousComponent, Module.Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix, MeasureTheory.OuterMeasure.restrict_mono, Module.Finite.exists_fin_quot_equiv, Submodule.IsPrincipal.mem_iff_generator_dvd, TensorProduct.piScalarRightHom_tmul, Int.absNorm_under_eq_sInf, QuadraticForm.tensorComm_symm, Ideal.dvdNotUnit_iff_lt, Ideal.IsLocal.mem_jacobson_or_exists_inv, RatFunc.toFractionRingRingEquiv_apply, Ideal.algebraMap_residueField_eq_zero, Submodule.colon_iUnion, LinearMap.flip_surjective_iff₂, Module.Basis.repr_smul', AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec, RootPairing.RootPositiveForm.algebraMap_posForm, Bialgebra.counit_pow, Ideal.radical_finset_inf, Bialgebra.mul_compr₂_comul, Algebra.FormallyEtale.iff_of_surjective, Algebra.TensorProduct.equivPiOfFiniteBasis_apply, Algebra.SubmersivePresentation.cotangentComplex_injective, PrimeSpectrum.range_comap_fst, IsGaloisGroup.card_fixingSubgroup_eq_finrank, Module.Projective.out, LinearMap.span_singleton_eq_range, hasStrictDerivAt_iff_hasStrictFDerivAt, Polynomial.degreeLE_eq_span_X_pow, Submodule.dualAnnihilator_eq_bot_iff', LinearMap.ker_lsmul, Module.Basis.repr_unitsSMul, Module.compHom.toLinearEquiv_symm_apply, RootPairing.reflection_apply, Matrix.toLpLin_apply, Representation.ind_apply, ProbabilityTheory.IsGaussian.charFunDual_eq', lineMap_mono_left, Ideal.dvd_span_singleton, Polynomial.iterate_derivative_one, LinearEquiv.baseChange_one, Algebra.TensorProduct.piScalarRight_tmul_apply, OrthonormalBasis.det_to_matrix_orthonormalBasis_real, Ideal.Quotient.factor_eq, PowerSeries.coeff_pow, trace_eq_sum_embeddings_gen, Matrix.toLin'_apply, CliffordAlgebra.contractRight_ι, IsGaloisGroup.finrank_fixedPoints_eq_card_subgroup, ModuleCat.extendScalarsId_inv_app_apply, Polynomial.natDegree_derivative_lt, Ideal.map_toCotangent_ker, isSMulRegular_on_quot_iff_lsmul_comap_le, Ideal.absNorm_algebraMap, PrimeSpectrum.zeroLocus_empty_iff_eq_top, Submodule.inf_colon, Ideal.isCoprime_iff_codisjoint, lineMap_le_map_iff_slope_le_slope_left, Submodule.map_dualCoannihilator_linearEquiv_flip, ProjectiveSpectrum.zeroLocus_inf, IsLocalization.coeSubmodule_fg, Module.Basis.constr_apply, MvPolynomial.finrank_eq_one, Pi.mem_span_range_single_inl_iff, IdealFilter.isTorsionQuot_def, AlgebraicGeometry.StructureSheaf.comapₗ_eq_localRingHom, Polynomial.Chebyshev.one_sub_X_sq_mul_iterate_derivative_U_eq_poly_in_U, PowerSeries.coeff_one_pow, LieModule.Weight.toLinear_neg, Ideal.map_pointwise_smul, StrongDual.polar_singleton, Ideal.smul_mem_pointwise_smul_iff, RatFunc.algebraMap_monomial, instFiniteDimensionalQuotientPolynomialIdealSpanSingletonSetSmithCoeffs, MvPowerSeries.monomial_smul_const, IsPRadical.ker_le', PowerSeries.coeff_mul_X_pow, MeasureTheory.FiniteMeasure.map_snd_prod, exteriorPower.ιMulti_family_span, FreeNonUnitalNonAssocAlgebra.lift_of_apply, Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_inv_f_hom_apply, TensorProduct.toLinearMap_symm_rid, Submodule.smul_top_le_comap_smul_top, AlgebraicGeometry.StructureSheaf.toOpen_res, HasStrictDerivAt.hasStrictFDerivAt_equiv, Submodule.set_smul_top_eq_span, exteriorPower.ιMulti_span_fixedDegree, CoalgCat.MonoidalCategoryAux.rightUnitor_hom_toLinearMap, Ideal.sup_mul, Polynomial.taylor_pow, Module.Dual.instLieModule, Finsupp.linearCombination_eq_fintype_linearCombination_apply, LinearMap.toMatrix_adjoint, PowerSeries.smul_eq_C_mul, Affine.Simplex.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter, AffineIndependent.units_lineMap, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec_assoc, MvPolynomial.tensorEquivSum_X_tmul_one, Polynomial.smul_X_eq_monomial, Ideal.Quotient.norm_mk_le, Submodule.dualQuotEquivDualAnnihilator_symm_apply_mk, Ideal.snd_comp_quotientInfEquivQuotientProd, Ideal.quotTorsionOfEquivSpanSingleton_apply_mk, Polynomial.Sequence.basis_degree_strictMono, Submodule.smul_le_span, SpecialLinearGroup.baseChange_apply_coe, CoalgCat.leftUnitor_def, LinearMap.diag_toMatrix_directSum_collectedBasis_eq_zero_of_mapsTo_ne, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_spec, Subspace.map_dualCoannihilator, Orientation.eq_or_eq_neg_of_isEmpty, Polynomial.one_lt_rootMultiplicity_iff_isRoot, lineMap_slope_lineMap_slope_lineMap, Int.cast_mem_ideal_iff, sbtw_one_zero_iff, Polynomial.monomial_sub, LinearMap.separatingLeft_toLinearMap₂'_iff_det_ne_zero, Polynomial.coeff_monomial_succ, Orientation.oangle_neg_orientation_eq_neg, Module.Basis.dvd_coord_smul, Subalgebra.LinearDisjoint.basisOfBasisLeft_repr_apply, CliffordAlgebraRing.reverse_eq_id, Ideal.toIdeal_homogeneousCore_le, RingHom.surjectiveOnStalks_iff_forall_ideal, cfcₙHom_of_cfcHom_map_quasispectrum, Ideal.isCoprime_iff_sup_eq, Ideal.IsPrime.radical_le_iff, ExteriorAlgebra.GradedAlgebra.ι_apply, ModularFormClass.qExpansionFormalMultilinearSeries_coeff, PowerSeries.trunc_sub, ExteriorAlgebra.invertibleAlgebraMapEquiv_symm_apply_invOf_toQuot, Algebra.SubmersivePresentation.basisCotangent_apply, MvPowerSeries.hasSum_eval₂, RootPairing.posRootForm_rootFormIn_posDef, LinearMap.BilinForm.Nondegenerate.toMatrix, AdicCompletion.tensor_map_id_left_injective_of_injective, Ideal.IsTwoSided.instHMul, MultilinearMap.mkPiRing_eq_zero_iff, EuclideanGeometry.Sphere.secondInter_eq_lineMap, Module.rankAtStalk_self, RootPairing.isCompl_corootSpan_ker_corootForm, Ideal.le_comap_mul, Ideal.mem_span_pow_iff_exists_isHomogeneous, Polynomial.annIdealGenerator_eq_zero_iff, ContinuousLinearMap.isOpenMap_of_ne_zero, Finset.weightedVSubOfPoint_subtype_eq_filter, lineMap_strict_mono_left, LinearMap.dualCoannihilator_range_eq_ker_flip, TensorProduct.nnnorm_lid, AddMonoidAlgebra.single_mem_gradeBy, PowerSeries.IsWeierstrassDivisorAt.seq_one, PrimeSpectrum.iSup_basicOpen_eq_top_iff', weightedVSub_mem_vectorSpan_pair, Algebra.trace_eq_of_equiv_equiv, LinearMap.BilinForm.congr_symm, Nat.Partition.coeff_genFun, Polynomial.degree_monomial, strongRankCondition_iff_succ, mem_finsuppAffineCoords_iff_linearCombination, Algebra.kerTensorProductMapIdToAlgHomEquiv_symm_apply, exteriorPower.oneEquiv_ιMulti, Ideal.span_pair_mul_span_pair, IsArtinianRing.isNilpotent_nilradical, MeasureTheory.OuterMeasure.comap_iSup, matrixEquivTensor_apply, Ideal.bot_quotient_isMaximal_iff, Matrix.toEuclideanLin_toLp, Ideal.toIdeal_homogeneousHull_eq_iSup, TensorProduct.lift_mk_compr₂, ContinuousLinearMapWOT.continuous_dual_apply, Algebra.Extension.Cotangent.map_mk, RootPairing.CoPolarizationIn_apply, Ideal.height_bot, Module.isTorsionBy_iff_mem_annihilator, PowerSeries.trunc_derivativeFun, MatrixEquivTensor.toFunAlgHom_apply, LinearMap.BilinForm.baseChange_tmul, Module.Basis.repr_eq_iff', linearDepOn_iff', StrongDual.mem_polar_singleton, IsArtinianRing.isSemisimpleRing_iff_jacobson, Algebra.Extension.CotangentSpace.map_comp, LinearMap.toLinearMap_toPerfPair, AdicCompletion.eval_surjective, CliffordAlgebra.GradedAlgebra.ι_sq_scalar, cardQuot_mul_of_coprime, Polynomial.sylveserMap_comp_adjSylvester, LieAlgebra.IsKilling.ker_killingForm_eq_bot, RootPairing.corootSpan_dualAnnihilator_map_eq_iInf_ker_coroot', isAdic_iff, LinearMap.toSpanSingleton_injective, SkewMonoidAlgebra.mem_span_support, LinearMap.BilinForm.isSymm_iff_basis, Submodule.FG.smul, exteriorPower.ιMulti_span, Ideal.coe_smul_primesOver, Ideal.iInf_pow_smul_eq_bot_of_noZeroSMulDivisors, RootPairing.invtSubmodule_reflection_of_invtSubmodule_coreflection, cfcₙHomSuperset_apply, PowerSeries.coeff_subst_finite', IsLocalization.coeSubmodule_isPrincipal, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.num_mem_carrier_iff, Real.exists_extension_norm_eq, PowerSeries.HasSubst.monomial, TannakaDuality.FiniteGroup.ofRightFDRep_hom, RootPairing.root_coroot'_eq_pairing, CliffordAlgebra.EvenHom.contract_mid, Valuation.leSubmodule_comap_algebraMap_eq_leIdeal, antilipschitzWith_lineMap, LinearMap.baseChange_mul, cfcₙ_apply, LinearMap.det_toMatrix, exteriorPower.toTensorPower_apply_ιMulti, Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOver, Algebra.traceForm_toMatrix_powerBasis, Ideal.homogeneousHull_eq_sInf, algebraMapCLM_toLinearMap, Matrix.minpoly_toLin, IsLocalization.coeSubmodule_top, basis_toMatrix_mul, PowerSeries.trunc_map, MeasureTheory.OuterMeasure.map_biInf_comap, Matrix.linearIndependent_cols_of_invertible, Rep.coinvariantsTensorIndInv_mk_tmul_indMk, Polynomial.ker_modByMonicHom, Ideal.span_singleton_absNorm_le, LinearMap.BilinForm.isAdjointPairLeftAdjointOfNondegenerate, Ideal.bot_liesOver_bot, Ideal.exists_map_eq_of_isGalois, Ideal.Factors.piQuotientEquiv_mk, conductor_eq_top_of_adjoin_eq_top, Matrix.toLin'_toMatrix', Ideal.sup_mul_left_self, LinearMap.det_toLin, RootPairing.coroot'_reflectionPerm, lineMap_slope_slope_sub_div_sub, Module.Basis.det_map, ProbabilityTheory.covarianceBilinDual_of_not_memLp', LinearMap.posSemidef_toMatrix_iff, PowerSeries.coeff_coe_trunc_of_lt, Ideal.finprod_not_dvd, Affine.Simplex.affineCombination_mem_setInterior_iff, Prod.counit_comp_inl, Submodule.localized'_smul, Algebra.Extension.exact_cotangentComplex_toKaehler, Ideal.natAbs_det_basis_change, instIsLocalizedModuleLinearMapIdLocalizationLocalizedModuleMapOfFinitePresentation, LinearMap.trace_mul_cycle', Polynomial.isMonicOfDegree_monomial_one, MonoidAlgebra.counit_single, Algebra.TensorProduct.lid_toLinearEquiv, PowerSeries.trunc_zero', Module.Basis.norm_repr_le_norm, PadicInt.coe_dpow_eq, MeasureTheory.FiniteMeasure.coe_toWeakDualBCNN, Ideal.bot_prime, Module.Flat.iff_rTensor_injective', LinearMap.baseChange_id, LinearMap.BilinForm.nondegenerate_toBilin'_of_det_ne_zero', Module.mem_support_iff_of_span_eq_top, Module.Basis.coord_unitsSMul, FractionalIdeal.trace_mem_dual_one, LinearMap.BilinForm.sum_apply, MvPolynomial.scalarRTensor_apply_tmul_apply, AddMonoidAlgebra.grade.gradedMonoid, Ideal.instNormalSubtypeMemSubgroupStabilizerSubgroupOfInertia, Module.Basis.mem_span_repr_support, Pi.basisFun_repr, PowerSeries.coeff_mul_prod_one_sub_of_lt_order, IsLocalRing.finite_quotient_iff, Ideal.leadingCoeffNth_mono, Module.Basis.SmithNormalForm.toAddSubgroup_index_eq_ite, RootPairing.InvariantForm.apply_eq_or_of_apply_ne, isLinearTopology_iff_hasBasis_twoSidedIdeal, Algebra.leftMulMatrix_eq_repr_mul, Algebra.trace_comp_trace, Polynomial.hasseDeriv_eq_zero_of_lt_natDegree, Polynomial.ofFn_coeff_eq_zero_of_ge, IsPrincipalIdealRing.principal, Module.finrank_baseChange, OrthonormalBasis.abs_det_adjustToOrientation, AdicCompletion.transitionMap_map_one, RootPairing.InvariantForm.two_mul_apply_root_root, HasGradientWithinAt.hasFDerivWithinAt, Module.Basis.toDual_eq_equivFun, MeasureTheory.OuterMeasure.restrict_iInf_restrict, Polynomial.degreeLT.basisProd_castAdd, LinearMap.isNonneg_zero, AlgebraicGeometry.Scheme.IdealSheafData.strictMono_ideal, Matrix.derivative_det_one_add_X_smul_aux, Polynomial.IsEisensteinAt.leading, MvPolynomial.coeffsIn_le, IsLocalization.map_algebraMap_ne_top_iff_disjoint, TensorProduct.toMatrix_comm, Module.Basis.sum_repr, Bialgebra.mul_compr₂_counit, Polynomial.trailingDegree_monomial, ContinuousMapZero.coe_toContinuousMapHom, MonomialOrder.C_mul_leadingCoeff_monomial_degree, EuclideanGeometry.inversion_eq_lineMap, Algebra.TensorProduct.lid_symm_apply, Polynomial.hasDerivWithinAt, instFreeCarrierX₂ModuleCatProjectiveShortComplex, ZSpan.mem_fundamentalDomain, dualTensorHomEquivOfBasis_apply, NormedSpace.Dual.coe_toWeakDual, LinearMap.polar_mem, Ideal.Quotient.isScalarTower_right, LinearMap.toMatrix₂'_apply, Matrix.vecMul_injective_iff, Ideal.mul_right_self_sup, AffineMap.lineMap_mem_affineSpan_pair, Ideal.spanNorm_top, IsAdjoinRootMonic.liftPolyₗ_apply, HahnSeries.toPowerSeries_symm_apply_coeff, Module.Basis.toMatrix_mulVec_repr, IsLocalRing.local_hom_TFAE, Ideal.one_notMem, PowerSeries.trunc_derivative, Subspace.quotDualCoannihilatorToDual_bijective, Algebra.Generators.comp_σ, HenselianRing.jac, LinearMap.ker_tensorProductMk, CliffordAlgebraQuaternion.toQuaternion_ι, AdicCompletion.sumEquivOfFintype_symm_apply, LinearMap.IsPerfectCompl.flip, DualNumber.instIsPrincipalIdealRing, Algebra.Generators.CotangentSpace.map_ofComp_surjective, bernsteinPolynomial.derivative_succ_aux, MvPowerSeries.monomial_smul_eq, LinearMap.ker_dualMap_eq_dualAnnihilator_range, LaurentSeries.coeff_coe_powerSeries, Ideal.toCotangent_surjective, PiTensorProduct.dualDistrib_dualDistribInvOfBasis_left_inverse, Matrix.kroneckerStarAlgEquiv_apply, Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂, LinearEquiv.trans_dualMap_symm_flip, FractionalIdeal.num_eq_zero_iff, Ideal.coe_piOrderIso_apply, Ideal.cotangentIdeal_square, Polynomial.mul_contentIdeal_le_radical_contentIdeal_mul, dist_sq_lineMap_of_inner_eq_zero, LinearMap.BilinForm.separatingLeft_toMatrix_iff, MvPolynomial.tensorEquivSum_C_tmul_C, LinearMap.trace_comp_comm', Polynomial.derivative_X_sub_C, Quaternion.instFinite, MvPolynomial.finsum_weightedHomogeneousComponent, signedDist_left_lineMap, OreLocalization.add_smul, MvPowerSeries.order_monomial_of_ne_zero, hasFDerivAt_inv, IntermediateField.finrank_dvd_of_le_left, exteriorPower.zeroEquiv_symm_apply, finsetBasisOfLinearIndependentOfCardEqFinrank_repr_apply, Module.torsion_by_prime_power_decomposition, Ideal.IsHomogeneous.iSup₂, Polynomial.IsWeaklyEisensteinAt.mem, QuadraticForm.associated_baseChange, pNilradical_eq_bot', Real.volume_preserving_transvectionStruct, ContinuousLinearMap.smulRightL_apply_apply, KaehlerDifferential.map_liftBaseChange_smul, ModularFormClass.hasSum_qExpansion, PowerSeries.weierstrassDiv_zero_left, MeasureTheory.OuterMeasure.map_ofFunction_le, ContinuousLinearMap.toSpanSingletonCLE_apply_apply, Matrix.toLpLinAlgEquiv_symm_apply, AddValuation.supp_quot_supp, wbtw_iff_of_le, MvPowerSeries.IsHomogeneous.coeff_eq_zero, Matrix.toBilin'_toMatrix', qExpansion_coeff_isBigO_of_norm_isBigO, Sbtw.mem_image_Ioo, Module.Basis.algebraMapCoeffs_repr, Polynomial.coeff_mem_contentIdeal, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, Algebra.Generators.H1Cotangent.map_comp_cotangentComplex_baseChange, Ideal.mem_span_singleton_self, AdicCompletion.eval_apply, Ideal.mul_eq_bot, LinearMap.trace_baseChange, Ideal.map_pow, CliffordAlgebra.contractLeftAux_apply_apply, RingHom.toKerIsLocalization_isLocalizedModule, PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain, PowerSeries.coeff_one_mul, Algebra.idealMap_apply_coe, LinearMap.BilinForm.IsAlt.smul, LaurentPolynomial.comul_C_mul_T_self, PrimeSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal, CliffordAlgebra.ofBaseChange_comp_toBaseChange, HomogeneousIdeal.toIdeal_inf, Module.Basis.coord_apply, CliffordAlgebra.contractLeft_one, Matrix.rank_eq_finrank_span_cols, LinearMap.IsPerfPair.flip, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq, ContDiffAt.smulRight, LinearMap.toMatrix'_mulVec, RootPairing.EmbeddedG2.threeShortAddLongRoot_longRoot, LinearMap.trace_eq_sum_trace_restrict_of_eq_biSup, IsLocalRing.isOpen_iff_finite_quotient, RootPairing.ker_corootForm_eq_dualAnnihilator, Unitization.inrRangeEquiv_symm_apply, MvPowerSeries.coeff_eq_zero_of_lt_lexOrder, Nat.one_mem_span_iff, Submodule.le_traceDual_iff_map_le_one, Subspace.dualAnnihilator_inf_eq, Module.Basis.mem_span_iff_repr_mem, PowerSeries.add_weierstrassDiv, PowerSeries.monomial_zero_eq_C_apply, ModuleCat.CoextendScalars.map'_hom_apply_apply, Module.Basis.SmithNormalForm.repr_eq_zero_of_notMem_range, IsBaseChange.toDualBaseChange_tmul, PrimeSpectrum.vanishingIdeal_iUnion, IsLocalizedModule.map_LocalizedModules, Matrix.SeparatingLeft.toLinearMap₂, Module.Relations.Solution.IsPresentation.ker_π, KaehlerDifferential.quotKerTotalEquiv_apply, AffineMap.lineMap_vadd, Polynomial.card_roots_toFinset_le_derivative, PrimeSpectrum.isCompact_isOpen_iff_ideal, instFiniteDimensionalFractionRingOfFinite, MvPowerSeries.coeff_mul_of_add_lexOrder, KaehlerDifferential.kerTotal_map', ModuleCat.RestrictionCoextensionAdj.HomEquiv.fromRestriction_hom_apply_apply, IsDedekindDomain.quotientEquivPiFactors_mk, Finset.centroid_def, IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_eq_one_iff, hasEigenvector_toLin_diagonal, Rep.leftRegularTensorTrivialIsoFree_hom_hom, LinearMap.separatingRight_toLinearMap₂'_iff_det_ne_zero, cfcₙHom_nonneg_iff, ContinuousLinearMap.toSpanSingletonCLE_symm_apply, Module.Basis.coord_equivFun_symm, Module.rank_linearMap_self, Polynomial.iterate_derivative_sum, pNilradical_le_nilradical, Polynomial.Chebyshev.one_sub_X_sq_mul_iterate_derivative_T_eval, TensorProduct.quotTensorEquivQuotSMul_symm_comp_mkQ, Submodule.annihilator_eq_top_iff, IsArithFrobAt.conj, Ring.jacobson_lt_top, dotProduct_toMatrix₂_mulVec, Algebra.Generators.CotangentSpace.fst_compEquiv, lineMap_lt_map_iff_slope_lt_slope_right, Ring.jacobson_quotient_jacobson, Algebra.Extension.CotangentSpace.map_comp_apply, MvPolynomial.linearMap_ext_iff, Affine.Simplex.sOppSide_affineSpan_faceOpposite_iff, AdicCompletion.flat_of_isNoetherian, Ideal.map_includeLeft_eq, LieModule.range_traceForm_le_span_weight, Ideal.map_iSup, RatFunc.laurentAux_ofFractionRing_mk, MeasureTheory.OuterMeasure.map_sup, CliffordAlgebra.equivOfIsometry_apply, Matrix.diagonal_comp_single, RCLike.geometric_hahn_banach_closed_compact, FDRep.dualTensorIsoLinHom_hom_hom, MeasureTheory.charFunDual_eq_prod_iff', ContinuousLinearMapWOT.tendsto_iff_forall_dual_apply_tendsto, FreeNonUnitalNonAssocAlgebra.hom_ext_iff, Algebra.Generators.H1Cotangent.equiv_apply, ModuleCat.ExtendScalars.hom_ext_iff, LieAlgebra.IsKilling.corootSpace_zero_eq_bot, Ring.jacobson_eq_sInf_isMaximal, MvPolynomial.monomial_single_add, CliffordAlgebra.map_comp_map, Algebra.TensorProduct.rid_symm_apply, LinearMap.IsPerfectCompl.isCompl_left, Algebra.Extension.CotangentSpace.map_comp_cotangentComplex, LinearMap.toMatrixAlgEquiv_transpose_apply, HomogeneousIdeal.toIdeal_iSup, Coalgebra.TensorProduct.lid_symm_apply, Module.annihilator_le_of_mem_support, OreLocalization.universalHom_commutes, Ideal.IsHomogeneous.iInf₂, LinearRecurrence.is_sol_iff_mem_solSpace, Module.Basis.repr_self, LinearMap.toSpanSingleton_isIdempotentElem_iff, AdicCompletion.piEquivFin_apply, Polynomial.coeff_coe, Algebra.IsAlgebraic.rank_fractionRing_polynomial, Orientation.rightAngleRotation_neg_orientation, MvPolynomial.monomial_one_mul_cancel_right_iff, Polynomial.sumIDeriv_apply_of_lt, Submodule.map_dualAnnihilator_linearEquiv_flip_symm, Submodule.colon_bot, LinearMap.separatingLeft_of_det_ne_zero, Polynomial.erase_monomial, Quaternion.fst_re_dualNumberEquiv_symm, PowerSeries.derivative_inv, Module.Basis.is_basis_iff_det, NormedSpace.Dual.toWeakDual_inj, MvPowerSeries.coeff_eq_zero_of_lt_weightedOrder, IsLocalizedModule.lift_iso, SpecialLinearGroup.coe_dualMap, QuadraticMap.basisRepr_apply, PrimeSpectrum.zeroLocus_pow, Polynomial.homogenizeLM_apply, IsBezout.span_pair_isPrincipal, Polynomial.degree_derivative_le, Ideal.associatesEquivIsPrincipal_map_one, Module.Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord, MvPolynomial.degreeOf_monomial_eq, Affine.Simplex.centroid_eq_affineCombination_of_pointsWithCircumcenter, Finsupp.linearCombination_zero, Polynomial.Chebyshev.iterate_derivative_T_eval_one_eq_div, Polynomial.isRoot_of_isRoot_iff_dvd_derivative_mul, Module.Basis.det_apply, Quaternion.imI_snd_dualNumberEquiv, Ideal.span_insert, Rep.linearization_ε_hom, Polynomial.derivative_one, Ideal.Quotient.zero_eq_one_iff, Polynomial.taylorLinearEquiv_apply_coe, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, MonoidAlgebra.mapDomainBialgHom_id, PowerSeries.natDegree_trunc_lt, Finsupp.lcomapDomain_eq_linearProjOfIsCompl, MeasureTheory.OuterMeasure.comap_ofFunction, LinearMap.traceAux_def, Module.Flat.iff_lTensor_injective', SkewPolynomial.smul_monomial, Ideal.Fiber.exists_smul_eq_one_tmul, Algebra.leftMulMatrix_apply, left_lt_lineMap_iff_pos, Complex.imLm_coe, Polynomial.Chebyshev.derivative_T_eval_one, LinearMap.charpoly_baseChange, Submodule.IsPrincipal.isPrimitive_iff_contentIdeal_eq_top, AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology_iff_sectionsOfPresieve, Polynomial.taylor_C, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, RCLike.re_extendTo𝕜'ₗ, Subspace.isCompl_dualAnnihilator, LinearMap.BilinForm.toMatrix'_compRight, Ideal.hasBasis_nhds_adic, isBoundedBilinearMap_smulRight, Polynomial.mem_degreeLT, Ideal.relNorm_eq_pow_of_isMaximal, LocalizedModule.mkLinearMap_apply, Polynomial.derivative_apply, IsDedekindDomain.HeightOneSpectrum.ideal_ne_top_iff_exists, PowerSeries.trunc_X, Module.subsingletonEquiv_apply, Localization.sub_mk, Algebra.Extension.Cotangent.ker_mk, Submodule.IsPrimary.mem_or_mem, Algebra.normalizedTrace_self_apply, ProbabilityTheory.uncenteredCovarianceBilinDual_zero, Polynomial.normalizedFactors_cyclotomic_card, AdicCompletion.val_sum_apply, Polynomial.degreeLT.basis_repr, Ideal.pointwise_smul_toAddSubmonoid, Rep.FiniteCyclicGroup.resolution_π, Module.Basis.eval_ker, Affine.Simplex.faceOppositeCentroid_eq_affineCombination, polyEquivTensor_symm_apply_tmul_eq_smul, WeierstrassCurve.Projective.nonsingularLift_zero, Pi.counit_single, IsLocalization.coe_primeSpectrumOrderIso_apply_coe_asIdeal, ExteriorAlgebra.liftAlternating_ι, AddMonoidAlgebra.mem_gradeBy_iff, Ideal.prime_generator_of_prime, trace_adjoinSimpleGen, PowerSeries.derivative_X, Differential.algEquiv_deriv, TensorProduct.comm_trans_rid, Polynomial.monomial_comp, rank_matrix'', Matrix.SeparatingRight.toBilin, Ideal.span_singleton_pow, TwoSidedIdeal.subtype_apply, Polynomial.IsEisensteinAt.notMem, NumberField.Ideal.primesOverSpanEquivMonicFactorsMod_symm_apply, Module.Basis.repr_reindex_apply, Finset.affineCombination_subtype_eq_filter, Ideal.le_jacobson, lineMap_strict_mono_right, CliffordAlgebra.contractLeftAux_contractLeftAux, wbtw_mul_sub_add_iff, HomogeneousIdeal.mem_iff, lipschitzGroup.conjAct_smul_range_ι, RingPreordering.support_eq_bot, MvPowerSeries.monomial_eq', Subspace.comap_dualAnnihilator_dualAnnihilator, LinearMap.toMatrix_apply, RootPairing.ker_polarization_eq_ker_rootForm, PowerSeries.IsWeierstrassDivisorAt.coeff_div_sub_seq_mem, linearDepOn_iff'ₛ, Ideal.gc_map_comap, LinearMap.baseChange_add, NumberField.house.basis_repr_norm_le_const_mul_house, exteriorPower.ιMulti_family_apply_coe, PrimeSpectrum.le_vanishingIdeal_zeroLocus, Polynomial.ofFn_natDegree_lt, MvPolynomial.weightedHomogeneousComponent_apply, CliffordAlgebra.EquivEven.involute_v, Module.erange_coe, TensorProduct.toMatrix_map, Matrix.rank_vecMulVec, LinearMap.continuous_uncurry_of_isContPerfPair, Submodule.mem_traceDual_iff_isIntegral, LieAlgebra.IsKilling.rootSystem_root_apply, AdicCompletion.map_of, MeasureTheory.SignedMeasure.toJordanDecomposition_smul, PadicInt.existsUnique_mem_range, Rep.coinvariantsTensorFreeToFinsupp_mk_tmul_single, PowerSeries.coeff_X_pow_mul, RootPairing.polarization_apply_eq_zero_iff, Submodule.le_dualCoannihilator_dualAnnihilator, MvPolynomial.range_evalᵢ, ContinuousLinearMap.ext_ring_iff, Nat.coe_maximalIdeal, Polynomial.monomial_mul_monomial, IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_lt_one_iff, ContinuousLinearMap.norm_smulRightL_apply, Submodule.torsionBy.mk_smul, AddMonoidAlgebra.mem_span_support, PowerSeries.coeff_zero_mul_X, RootPairing.EmbeddedG2.threeShortAddTwoLongRoot_longRoot, TannakaDuality.FiniteGroup.rightRegular_apply, MvPowerSeries.coeff_mul_C, Valuation.Integers.maximalIdeal_pow_eq_setOf_le_v_algebraMap_pow, Polynomial.IsEisensteinAt.coeff_mem, CliffordAlgebraQuaternion.Q_apply, LinearMap.BilinForm.restrict_apply, AffineMap.homothety_eq_lineMap, MvPolynomial.rank_eq_lift, Algebra.H1Cotangent.exact_δ_mapBaseChange, Finsupp.linearIndependent_single_one, LinearMap.map_finsupp_linearCombination, PadicInt.toZMod_spec, Module.FinitePresentation.exists_basis_localizedModule_powers, KaehlerDifferential.instIsScalarTowerTensorProduct_2, Rep.FiniteCyclicGroup.resolution.π_f, Algebra.SubmersivePresentation.cotangentEquiv_apply, IsSimpleModule.ker_toSpanSingleton_isMaximal, PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure, LinearMap.toMatrix_id_eq_basis_toMatrix, contractLeft_assoc_coevaluation', Rep.FiniteCyclicGroup.leftRegular.range_applyAsHom_sub_eq_ker_linearCombination, FractionalIdeal.coeIdeal_pow, RCLike.imCLM_apply, LinearMap.toMatrixAlgEquiv_transpose_apply', TensorProduct.dualDistribEquivOfBasis_apply_apply, QuadraticModuleCat.ofIso_inv, Polynomial.mkDerivation_apply, Polynomial.not_finite, LinearMap.baseChangeHom_apply, Module.Relations.Solution.π_single, Module.Basis.finTwoProd_one, Ideal.mem_inv_pointwise_smul_iff, Polynomial.coeff_monomial_mul, ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_one, linearIndepOn_iff_linearCombinationOnₛ, LinearMap.toMatrixRight'_comp, TwoSidedIdeal.coe_asIdeal, Matrix.kroneckerStarAlgEquiv_symm_apply, CommSemiring.comul_apply, UniformSpace.inseparableSetoid_ring, Localization.add_mk_self, Algebra.Generators.H1Cotangent.δ_map, Ideal.radical_mul, Module.dualProdDualEquivDual_symm_apply, AffineMap.lineMap_vadd_lineMap, convexHull_range_eq_exists_affineCombination, MvPolynomial.IsHomogeneous.pderiv, Ideal.IsPrime.mul_mem_iff_mem_or_mem, LinearMap.trace_map, MvPolynomial.mem_ideal_span_X_image, AdicCompletion.of_apply, Ideal.pointwise_smul_def, WittVector.ghostComponentModPPow_map_mk, MvPolynomial.mem_restrictTotalDegree, Module.presentationFinsupp_var, uniqueDiffOn_Ici, NormedSpace.dual_def, QuadraticMap.toMatrix'_smul, Rep.linearization_map_hom, RCLike.geometric_hahn_banach_open_open, IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_mem, Matrix.toLpLin_eq_toLin, MvPolynomial.restrictTotalDegree_le_restrictDegree, QuadraticModuleCat.ofIso_symm, LinearMap.dualProd.toQuadraticForm, Ideal.comap_jacobson, Subspace.finrank_add_finrank_dualAnnihilator_eq, Algebra.SubmersivePresentation.basisDeriv_apply, MvPolynomial.bind₁_monomial, Lagrange.interpolate_singleton, IsDedekindDomain.isOpen_iff, MeasureTheory.OuterMeasure.map_mono, Algebra.Extension.Cotangent.map_sub_map, Matrix.toBilin_comp, TensorProduct.lid_tensor, TensorProduct.coe_finsuppScalarRight', LinearMap.polyCharpoly_coeff_eval, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj, lineMap_lt_lineMap_iff_of_lt, MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto, LinearMap.BilinForm.apply_smul_sub_smul_sub_eq, SimpleGraph.mem_ker_toLin'_lapMatrix_of_connectedComponent, MvPolynomial.induction_on_monomial, lsum_comp_mapRange_toSpanSingleton, MeasureTheory.OuterMeasure.restrict_le_self, RootPairing.orthogonal_rootSpan_eq, Module.Basis.repr_unop_eq_mulOpposite_repr, RootPairing.algebraMap_root'In_apply, LinearMap.det_toMatrix_eq_det_toMatrix, MvPolynomial.mem_image_comap_C_basicOpen, Matrix.SpecialLinearGroup.toLin'_apply, PowerSeries.order_monomial_of_ne_zero, DualNumber.isNilpotent_eps, Matrix.GeneralLinearGroup.toLin_apply, PowerSeries.trunc_C_mul, Polynomial.Chebyshev.add_one_mul_T_eq_poly_in_U, Matrix.mulVecLin_mul, Algebra.Smooth.exists_subalgebra_fg, IsLocalRing.isOpen_maximalIdeal, Algebra.Presentation.mem_ker_naive, linearIndepOn_iff_linearCombinationOn, Polynomial.coeffList_monomial, ClassGroup.isPrincipal_of_isUnit_coeIdeal, MvPowerSeries.coeff_mul, Ideal.isRadical_iff_quotient_reduced, lineMap_mono_right, MvPolynomial.vars_monomial_single, MvPolynomial.evalₗ_apply, Complex.rank_real_complex, AlgebraicGeometry.Scheme.IdealSheafData.le_def, ContinuousAlternatingMap.smulRight_apply, finsuppTensorFinsuppLid_self, Ideal.instIsCoatomic, Ideal.IsOka.top, Representation.IndV.hom_ext_iff, dist_lineMap_lineMap, InnerProductSpace.nullSubmodule_le_ker_toDualMap_right, Module.Relations.Solution.IsPresentation.π_desc_apply, Polynomial.derivative_C_mul_X_sq, LinearMap.BilinForm.add_apply, PowerSeries.WithPiTopology.summable_iff_summable_coeff, Polynomial.content_monomial, LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton, AffineMap.lineMap_vsub_lineMap, exteriorPower.basis_repr_self, Matrix.linearIndependent_rows_iff_isUnit, Ideal.comap_le_iff_le_map, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, PointedCone.dual_dual_flip_dual, Ideal.mem_prod, Ideal.exists_pow_inf_eq_pow_smul, Path.eqOn_extend_segment, MvPowerSeries.coeff_mul_prod_one_sub_of_lt_weightedOrder, Polynomial.derivativeFinsupp_C, Submodule.flip_quotDualCoannihilatorToDual_injective, MvPolynomial.monomial_modMonomial, Algebra.Extension.cotangentComplex_mk, ContinuousLinearMap.comp_toSpanSingleton, AdicCompletion.map_surjective, Rep.indResHomEquiv_symm_apply_hom, MvPolynomial.WeightedHomogeneousSubmodule.gradedMonoid, Polynomial.taylor_one, RingHom.surjective_localRingHom_iff, Subspace.dualPairing_eq, WeierstrassCurve.Affine.CoordinateRing.XYIdeal_mul_XYIdeal, IdealFilter.isUniform_iff_exists_ringFilterBasis, QuadraticForm.dualProdProdIsometry_invFun, Pi.basisFun_det_apply, nndist_lineMap_left, AlternatingMap.coe_smulRight, Matrix.Nondegenerate.toLinearMap₂', Polynomial.derivativeFinsupp_X, AlgHom.mulLeftRightMatrix.comp_inv, AdicCompletion.eval_comp_of, Finset.sum_smul_vsub_eq_weightedVSub_sub, Ideal.to_quotient_square_comp_toCotangent, Module.Flat.exists_factorization_of_isFinitelyPresented, AdicCompletion.sum_comp_sumInv, Ideal.IsHomogeneous.mem_iff, Ideal.map_eq_bot_iff_le_ker, CategoryTheory.Iso.toIsometryEquiv_symm, Finset.weightedVSubOfPoint_sdiff_sub, KaehlerDifferential.linearCombination_surjective, Submodule.restrictScalars_localized'_smul, Matrix.separatingRight_toLinearMap₂'_iff_separatingRight_toLinearMap₂, Module.Basis.sumQuot_repr_inr, AdicCompletion.val_smul_apply, Module.Basis.coord_toDualEquiv_symm_apply, Module.annihilator_prod, Ideal.le_comap_pow_ramificationIdx, Finsupp.linearCombination_onFinset, Module.Basis.baseChange_linearMap, QuaternionAlgebra.reₗ_apply, MvPowerSeries.coeff_add_monomial_mul, IsAdjoinRootMonic.coeff_root_pow, Ideal.mem_jacobson_bot, LinearEquiv.baseChange_trans, LinearMap.BilinForm.sub_right, Module.Basis.coord_repr_symm, Module.Relations.Solution.linearCombination_var_relation, Ideal.span_iUnion, LinearMap.toMatrixAlgEquiv'_comp, BoxIntegral.unitPartition.tag_mem_smul_span, IsCoprime.codisjoint, Ideal.Factors.piQuotientEquiv_map, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_eq_iInter_zeroLocus, Module.Dual.range_eq_top_of_ne_zero, CategoryTheory.Iso.toIsometryEquiv_trans, PowerSeries.hasSum_eval₂, left_le_lineMap_iff_nonneg, Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top, Algebra.IsStandardSmoothOfRelativeDimension.exists_subalgebra_fg, Polynomial.eval_multiset_prod_X_sub_C_derivative, MonomialOrder.sPolynomial_mul_monomial, lineMap_lt_left_iff_neg, BoundedContinuousFunction.probCharDual_apply, Module.mapEvalEquiv_symm_apply, mem_nilradical, TensorProduct.quotTensorEquivQuotSMul_mk_tmul, Subspace.dualEquivDual_apply, LinearMap.BilinForm.iIsOrtho_def, MvPowerSeries.prod_monomial, OnePoint.equivProjectivization_apply_infinity, Matrix.cramer_eq_adjugate_mulVec, Ideal.isoBaseOfIsPrincipal_apply, Submodule.map_le_smul_top, LinearMap.BilinForm.apply_dualBasis_right, Rep.leftRegularTensorTrivialIsoFree_inv_hom_single_single, Ideal.homogeneousCore.gc, Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors', ZLattice.abs_repr_le, TensorProduct.tensorQuotEquivQuotSMul_symm_comp_mkQ, Module.Dual.congr_symm_apply_apply, DirectSum.IsInternal.collectedBasis_repr_of_mem, Module.Basis.coe_sumCoords, Matrix.det_smul_inv_vecMul_eq_cramer_transpose, ModuleCat.isSeparator, MvPolynomial.decomposition.decompose'_eq, IsCoprime.add_eq, Algebra.Generators.C_mul_X_sub_one_mem_ker, MvPolynomial.universalFactorizationMapPresentation_jacobian, PointedCone.dual_singleton, Ideal.eq_bot_of_liesOver_bot, MvPowerSeries.prod_smul_X_eq_smul_monomial_one, PowerSeries.coeff_one_X, TensorPower.pairingDual_tprod_tprod, LinearMap.sum_repr_mul_repr_mul, MonomialOrder.sPolynomial_leadingTerm_mul, Polynomial.UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap, bernsteinPolynomial.iterate_derivative_at_1, Submodule.instIsScalarTower, Ideal.Quotient.noZeroDivisors, LinearMap.IsPerfPair.dualEval, Lagrange.nodalWeight_eq_eval_derivative_nodal, Polynomial.derivative_intCast, IsLocalizedModule.iso_mk_one, rieszContentAux_le, QuadraticModuleCat.hom_hom_associator, LinearMap.charpoly_toMatrix, Polynomial.leadingCoeff_taylor, LieAlgebra.IsKilling.instIsIrreducibleSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystemOfIsSimple, LinearMap.toLinearMap₂'Aux_toMatrix₂Aux, Polynomial.homogenize_monomial, Polynomial.derivative_add, LinearMap.isSymm_iff_isHermitian_toMatrix, Ideal.isOpen_of_isMaximal, Ideal.span_singleton_mul_le_iff, LinearMap.trace_lie, LinearMap.flip.instIsContPerfPair, Module.Basis.ofIsLocalizedModule_repr_apply, Valuation.HasExtension.algebraMap_mem_maximalIdeal_iff, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff, LinearMap.lsmul_eq_distribSMultoLinearMap, Polynomial.UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap', Matrix.toBilin'_single, Rep.linearization_μ_hom, Polynomial.monomial_add_erase, Submodule.natAbs_det_basis_change, NumberField.mixedEmbedding.fundamentalCone.intNorm_idealSetEquiv_apply, Module.Relations.Solution.ofQuotient_π, Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply, Ideal.le_colon, Module.FinitePresentation.linearEquivMapExtendScalars_symm_apply, ProbabilityTheory.indepFun_iff_charFunDual_prod, LinearMap.separatingLeft_dualProd, Submodule.spanRank_bot, LinearMap.toMatrix_toLin, WeakDual.CharacterSpace.instContinuousLinearMapClass, TensorProduct.includeRight_lid, Polynomial.eval_monomial, MeasureTheory.OuterMeasure.isometryEquiv_comap_mkMetric, FreeNonUnitalNonAssocAlgebra.lift_comp_of, Ideal.isOka_isPrincipal, Complex.restrictScalars_toSpanSingleton', CliffordAlgebraComplex.reverse_apply, FractionalIdeal.coeIdeal_le_coeIdeal, MvPowerSeries.commute_monomial, LinearMap.toSpanSingleton_add, Quaternion.imK_snd_dualNumberEquiv, ModuleCat.CoextendScalars.map_apply, Module.Basis.traceDual_eq_iff, MvPolynomial.monomial_dvd_monomial, Module.compHom.toLinearEquiv_apply, PowerSeries.derivative_pow, Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd', PowerSeries.order_eq_nat, instFinitePresentationTensorProduct, RCLike.geometric_hahn_banach_open_point, Polynomial.aeval_sumIDeriv_of_pos, PrimeSpectrum.asIdeal_le_asIdeal, MvPolynomial.monomial_mul_modMonomial, LinearMap.polyCharpolyAux_map_eq_toMatrix_charpoly, Representation.ofMulAction_def, CliffordAlgebraQuaternion.ofQuaternion_toQuaternion, MvPolynomial.pderiv_inl_universalFactorizationMap_X, Matrix.toLinearEquiv_apply, LinearEquiv.piRing_apply, QuadraticModuleCat.hom_inv_associator, hasEigenvalue_toLin_diagonal_iff, PrimeSpectrum.zeroLocus_eq_singleton, Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_left_iff, PowerSeries.coeff_zero_X, CliffordAlgebra.contractLeft_algebraMap, Matrix.spectrum_toLin', LinearMap.dualAnnihilator_ker_eq_range_flip, AdicCompletion.ofAlgEquiv_symm_of, Derivation.apply_aeval_eq, Module.notMem_support_iff', Ideal.isMaximal_def, Algebra.ZariskisMainProperty.exists_fg_and_exists_notMem_and_awayMap_bijective, PowerSeries.IsWeierstrassDivisionAt.coeff_f_sub_r_mem, Algebra.Presentation.tensorModelOfHasCoeffsInv_aeval_val, PowerSeries.trunc_derivative', Ideal.cotangent_subsingleton_iff, Ideal.mulQuot_injective, KaehlerDifferential.kerTotal_mkQ_single_algebraMap, kroneckerLinearEquiv_tmul, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, LieAlgebra.killingForm_of_equiv_apply, MvPolynomial.rTensorAlgHom_apply_eq

(root)

Definitions

Name	Category	Theorems
instDistribSMul 📖	CompOp	191 math math: Submodule.isScalarTower, Nat.floorDiv_eq_div, Subring.center.smulCommClass_right, MvPolynomial.rTensor_apply_tmul_apply, QuadraticAlgebra.coe_mul_eq_smul, Matrix.single_vecMul, Matrix.mul_mul_left, AddMonoidAlgebra.isScalarTower_self, Algebra.to_smulCommClass, Matrix.Semiring.smulCommClass, descPochhammer_succ_comp_X_sub_one, HahnSeries.embDomain_smul, SkewMonoidAlgebra.smul_of, QuotSMulTop.equivTensorQuot_naturality, Submodule.top_eq_ofList_cons_smul_iff, Module.Basis.isScalarTower_of_nonempty, Matrix.vecMul_vecMulVec, IsScalarTower.right, Polynomial.C_mul', Matrix.smul_one_eq_diagonal, KaehlerDifferential.quotKerTotalEquiv_symm_comp_D, QuotSMulTop.equivTensorQuot_naturality_mk, Algebra.TensorProduct.sMulCommClass_right, QuotSMulTop.map_comp_mkQ, Polynomial.monic_of_isUnit_leadingCoeff_inv_smul, NonarchimedeanRing.left_mul_subset, DualNumber.inv_eps, MonoidAlgebra.smulCommClass_self, Algebra.TensorProduct.basis_repr_tmul, Algebra.TensorProduct.basisAux_map_smul, Polynomial.sum_smul_index, gc_floorDiv_mul, QuotSMulTop.map_surjective, Polynomial.smul_eq_C_mul, exists_basis_of_pairing_ne_zero, Finsupp.coe_pointwise_smul, Ring.uniformContinuousConstSMul, Polynomial.smeval_smul, isSMulRegular_submodule_iff_right_eq_zero_of_smul, AddMonoidAlgebra.smulCommClass_symm_self, Module.Basis.isScalarTower_finsupp, Ideal.Quotient.smul_top, DualNumber.lift_smul, MvPolynomial.rTensor_apply_X_tmul, IsStrictOrderedRing.toIsStrictOrderedModule, LinearMap.restrictScalars_self, IsDivSequence.smul, RingEquiv.moduleEndSelfOp_apply, IsTopologicalSemiring.toIsModuleTopology, Algebra.TensorProduct.isScalarTower_right, MvPolynomial.rTensor_apply_tmul, IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite, IsSMulRegular.submodule, Nat.ceilRoot_def, DualNumber.exists_mul_left_or_mul_right, Matrix.mul_mul_right, MultilinearMap.freeDFinsuppEquiv_single, Matrix.vecMul_eq_sum, Polynomial.coe_smul, MultilinearMap.freeDFinsuppEquiv_def, MvPolynomial.rTensorAlgHom_toLinearMap, NonUnitalSubalgebra.instSMulCommClass, QuotSMulTop.equivQuotTensor_naturality, Algebra.TensorProduct.basisAux_tmul, AdicCompletion.transitionMap_comp_reduceModIdeal, TwoSidedIdeal.instSMulMemClass, gc_mul_ceilDiv, Module.Basis.sum_dual_apply_smul_coord, QuadraticAlgebra.C_mul_eq_smul, Finsupp.smul_single', MonoidAlgebra.smul_of, Nat.factorization_ceilRoot, MultilinearMap.freeFinsuppEquiv_def, Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner_naturality, Finset.weightedVSub_const_smul, LieAlgebra.smulCommClass, Matrix.smul_eq_mul_diagonal, Subsemiring.center.smulCommClass_right, Matrix.diagonal_toLin', MonoidAlgebra.smul_single', QuotSMulTop.map_exact, IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite, Matrix.trace_single_mul, SkewMonoidAlgebra.sum_smul_index, Polynomial.degree_smul_of_isRightRegular_leadingCoeff, MultilinearMap.freeFinsuppEquiv_apply, LinearPMap.map_smul, Module.Basis.smul_eq_map, AddMonoidAlgebra.liftNC_smul, IsEllDivSequence.smul, Matrix.detp_smul_adjp, PolyEquivTensor.toFunAlgHom_apply_tmul_eq_smul, Matrix.scalar_comm_iff, AddMonoidAlgebra.smul_single', Matrix.single_mul_eq_updateRow_zero, PolyEquivTensor.toFunBilinear_apply_apply, Module.End.instSMulCommClass', Matrix.ofNat_mulVec, Module.End.instIsScalarTower, MultilinearMap.freeFinsuppEquiv_single, MvPolynomial.rTensor_symm_apply_single, IsSemisimpleModule.exists_end_algEquiv, QuotSMulTop.equivQuotTensor_naturality_mk, Submodule.mem_smul_top_iff, KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination, MatrixModCat.isScalarTower_toModuleCat, MultilinearMap.freeDFinsuppEquiv_apply, Polynomial.map_smul, MvPolynomial.rTensor_apply_monomial_tmul, Matrix.proj_diagonal, Submodule.coe_smul, le_floorDiv_iff_mul_le, DualNumber.commute_eps_left, Nat.factorization_floorRoot, Matrix.detp_smul_add_adjp, Nat.floorRoot_def, LieAlgebra.isScalarTower, Module.End.instSMulCommClass, Submodule.restrictScalars.isScalarTower, Matrix.isAddUnit_detp_smul_mul_adjp, Module.Finite.toModuleEnd_moduleEnd_surjective, PolyEquivTensor.toFunBilinear_apply_eq_smul, Subring.center.smulCommClass_left, MonoidAlgebra.smulCommClass_symm_self, QuotSMulTop.map_first_exact_on_four_term_exact_of_isSMulRegular_last, DualNumber.commute_eps_right, Finset.weightedVSubOfPoint_const_smul, SkewMonoidAlgebra.isScalarTower_self, IsSemisimpleModule.exists_end_algEquiv_pi_matrix_end, Matrix.Semiring.isScalarTower, Matrix.natCast_mulVec, Ideal.Quotient.smulCommClass', MonoidAlgebra.liftNC_smul, IsEllSequence.smul, Matrix.diagonal_const_mulVec, Matrix.vecMulVec_mul_vecMulVec, Matrix.toLin_finTwoProd, Polynomial.fwdDiff_iter_degree_eq_factorial, QuotSMulTop.map_comp, Matrix.cons_vecMulVec, Matrix.cons_vecMul, Matrix.vecMulVec_cons, Matrix.cons_vecMul_cons, MonoidAlgebra.isScalarTower_self, Matrix.ofNat_kronecker, NonUnitalStarSubalgebra.instIsScalarTower, Ideal.Quotient.smulCommClass, NonUnitalSubalgebra.instIsScalarTowerSubtypeMem, QuotSMulTop.map_apply_mk, LaurentPolynomial.smul_eq_C_mul, fromModuleCatToModuleCatLinearEquiv_apply, ceilDiv_le_iff_le_mul, fromModuleCatToModuleCatLinearEquiv_symm_apply_coe, Nat.ceilDiv_eq_add_pred_div, Polynomial.mirror_smul, DualNumber.eps_mul_eps, MvPolynomial.rTensor_apply, TwoSidedIdeal.coe_smul, Submodule.mem_set_smul, Polynomial.eval₂_smul, Matrix.intCast_mulVec, Polynomial.isScalarTower_right, SymmetricAlgebra.IsSymmetricAlgebra.mvPolynomial, DualNumber.snd_mul, QuotSMulTop.map_id, Matrix.vecMul_cons, Polynomial.smul_X_eq_monomial, Ideal.quotTorsionOfEquivSpanSingleton_apply_mk, Matrix.mulVec_cons, Matrix.scalar_commute_iff, Module.Presentation.tautologicalRelations_relation, Matrix.smul_eq_diagonal_mul, NonUnitalStarSubalgebra.instSMulCommClass, Module.Basis.coord_unitsSMul, Ideal.Quotient.isScalarTower_right, Matrix.mul_adjp_add_detp, LaurentPolynomial.instIsScalarTowerPolynomial, Antivary.sum_mul_lt_sum_mul_comp_perm_iff, Matrix.diagonal_comp_single, Module.Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord, AddMonoidAlgebra.smulCommClass_self, polyEquivTensor_symm_apply_tmul_eq_smul, IsOrderedRing.toIsOrderedModule, Submodule.isPrimary_iff_zero_divisor_quotient_imp_nilpotent_smul, MonovaryOn.sum_mul_comp_perm_lt_sum_mul_iff, Submodule.restrictScalars_self, SkewMonoidAlgebra.liftNC_smul, Matrix.smul_mat_empty, Valuation.HasExtension.instIsScalarTowerInteger, Subsemiring.center.smulCommClass_left, MvPolynomial.rTensorAlgHom_apply_eq

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
add_smul 📖	mathematical	—	SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction Distrib.toAdd NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup	—	Module.add_smul
neg_one_smul 📖	mathematical	—	SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Ring.toSemiring Module.toDistribMulAction AddCommGroup.toAddCommMonoid NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Ring.toAddCommGroup AddMonoidWithOne.toOne AddGroupWithOne.toAddMonoidWithOne Ring.toAddGroupWithOne	—	neg_smul one_smul
neg_smul 📖	mathematical	—	SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Ring.toSemiring Module.toDistribMulAction AddCommGroup.toAddCommMonoid NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Ring.toAddCommGroup	—	eq_neg_of_add_eq_zero_left add_smul neg_add_cancel zero_smul
neg_smul_neg 📖	mathematical	—	SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Ring.toSemiring Module.toDistribMulAction AddCommGroup.toAddCommMonoid NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid Ring.toAddCommGroup	—	neg_smul smul_neg neg_neg
smul_add_one_sub_smul 📖	mathematical	—	AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Ring.toSemiring Module.toDistribMulAction SubNegMonoid.toSub AddGroup.toSubNegMonoid AddGroupWithOne.toAddGroup Ring.toAddGroupWithOne AddMonoidWithOne.toOne AddGroupWithOne.toAddMonoidWithOne	—	add_smul add_sub_cancel one_smul
sub_smul 📖	mathematical	—	SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Ring.toSemiring Module.toDistribMulAction AddCommGroup.toAddCommMonoid SubNegMonoid.toSub AddGroupWithOne.toAddGroup Ring.toAddGroupWithOne	—	sub_eq_add_neg add_smul neg_smul
two_smul 📖	mathematical	—	SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddCommMonoid.toAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction instOfNatAtLeastTwo AddMonoidWithOne.toNatCast AddCommMonoidWithOne.toAddMonoidWithOne NonAssocSemiring.toAddCommMonoidWithOne Semiring.toNonAssocSemiring Nat.instAtLeastTwoHAddOfNat AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup	—	Nat.instAtLeastTwoHAddOfNat one_add_one_eq_two add_smul one_smul

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