module

📁 Source: MathlibTest/module.lean

Statistics

Metric	Count
Definitions module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module, module	42
Theorems	0
Total	42

AddEquiv

Definitions

Name	Category	Theorems
module 📖	CompOp	1 math math: LinearEquiv.isScalarTower

AddMonoidAlgebra

Definitions

Name	Category	Theorems
module 📖	CompOp	46 math math: of'_mem_span, LaurentPolynomial.comul_T, LieAlgebra.LoopAlgebra.twoCochainOfBilinear_apply_apply, comul_single, instIsTorsionFree, counit_single, moduleFinite, LieAlgebra.LoopAlgebra.toFinsupp_symm_single, decomposeAux_coe, lsingle_apply, gradeBy.isInternal, tensorEquiv.invFun_tmul, Polynomial.toFinsuppIsoLinear_symm_apply_toFinsupp, LaurentPolynomial.leval_apply, GradesBy.decompose_single, mem_grade_iff', LieAlgebra.LoopAlgebra.toFinsupp_single_tmul, single_mem_grade, Polynomial.toFinsuppIsoLinear_apply, LaurentPolynomial.comul_C, tensorEquiv_tmul, gradeBy.gradedMonoid, tensorEquiv_symm_single, mem_grade_iff, grade.decompose_single, LieAlgebra.LoopAlgebra.residuePairing_apply_apply, lhom_ext'_iff, LaurentPolynomial.counit_C_mul_T, basis_apply, grade.isInternal, scalarTensorEquiv_symm_single, instIsCocomm, instFree, LaurentPolynomial.instIsCocomm, LaurentPolynomial.counit_T, LaurentPolynomial.counit_C, LaurentPolynomial.comul_C_mul_T, mem_span_support', decomposeAux_single, decomposeAux_eq_decompose, scalarTensorEquiv_tmul, single_mem_gradeBy, grade.gradedMonoid, LaurentPolynomial.comul_C_mul_T_self, mem_gradeBy_iff, mem_span_support

AdicCompletion

Definitions

Name	Category	Theorems
module 📖	CompOp	35 math math: map_val_apply, map_ext'_iff, ofTensorProduct_surjective_of_finite, pi_apply_coe, map_injective, sumInv_apply, sum_lof, map_comp_apply, map_comp, map_id, sumInv_comp_sum, coe_ofTensorProductEquivOfFiniteNoetherian, ofTensorProductEquivOfFiniteNoetherian_symm_of, ofTensorProduct_bijective_of_finite_of_isNoetherian, ofTensorProduct_bijective_of_pi_of_fintype, map_mk, sum_of, congr_symm_apply, tensor_map_id_left_eq_map, ofTensorProduct_tmul, map_exact, ofTensorProductEquivOfFiniteNoetherian_apply, sumEquivOfFintype_apply, map_zero, component_sumInv, piEquivOfFintype_apply, ofTensorProduct_naturality, congr_apply, tensor_map_id_left_injective_of_injective, sumEquivOfFintype_symm_apply, piEquivFin_apply, map_of, map_surjective_of_mkQ_comp_surjective, map_surjective, sum_comp_sumInv

Algebra.Extension.Cotangent

Definitions

Name	Category	Theorems
module 📖	CompOp	—

CategoryTheory.Limits.IsColimit

Definitions

Name	Category	Theorems
module 📖	CompOp	1 math math: ι_smul

CategoryTheory.Quotient.Linear

Definitions

Name	Category	Theorems
module 📖	CompOp	—

ContMDiffMap

Definitions

Name	Category	Theorems
module 📖	CompOp	13 math math: LeftInvariantDerivation.lift_zero, LeftInvariantDerivation.coe_derivation, LeftInvariantDerivation.left_invariant', LeftInvariantDerivation.lift_add, LeftInvariantDerivation.lift_smul, LeftInvariantDerivation.evalAt_coe, LeftInvariantDerivation.left_invariant'', LeftInvariantDerivation.toDerivation_injective, LeftInvariantDerivation.commutator_coe_derivation, LeftInvariantDerivation.toFun_eq_coe, LeftInvariantDerivation.instLinearMapClassContMDiffMapModelWithCornersSelfSomeENatTop, Derivation.evalAt_apply, coeFnLinearMap_apply

ContinuousLinearMap

Definitions

Name	Category	Theorems
module 📖	CompOp	642 math math: LinearMap.det_toContinuousLinearMap, LinearMap.IsSymmetric.clm_adjoint_eq, fderiv_iteratedFDeriv, LinearMap.mkContinuous₂_norm_le', derivWithin_of_bilinear, Bundle.ContMDiffRiemannianMetric.isVonNBounded, LinearMap.mkContinuous₂_apply, isPositive_iff_eq_sum_rankOne, Bundle.ContMDiffRiemannianMetric.contMDiff, continuousOn_stereoToFun, InnerProductSpace.isPositive_rankOne_self, ProbabilityTheory.isGaussian_iff_gaussian_charFun, mulLeftRight_isBoundedBilinear, MDifferentiableOn.clm_postcomp, analyticWithinAt_bilinear, HasCompactSupport.convolution_integrand_bound_left, compSL_apply, continuous₂, InnerProductSpace.toLinearIsometry_toDual, SchwartzMap.integral_sesq_fourier_fourier, Real.hasFDerivAt_fourierChar_neg_bilinear_left, MeasureTheory.convolution_eq_right', hasFDerivAt_iff_hasGradientAt, InnerProductSpace.rankOne_one_left_eq_innerSL, map_add_add, ContinuousLinearEquiv.toCompactConvergenceCLM_symm_apply, fderivWithin_fderivWithin_eq_of_mem_nhdsWithin, toLinearMap₁₂_injective, MDifferentiableAt.cle_arrowCongr, ProbabilityTheory.covarianceBilinDual_apply', hasFTaylorSeriesUpToOn_top_iff_right, ContinuousMultilinearMap.analyticAt_uncurry_of_linear, SchwartzMap.pairing_apply_apply, InnerProductSpace.isIdempotentElem_rankOne_self, StrongDual.polarSubmodule_eq_setOf, ProbabilityTheory.covarianceBilinDual_apply, Real.fderiv_fourierChar_neg_bilinear_right_apply, Continuous.convolution_integrand_fst, ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries, iteratedFDerivWithin_succ_eq_comp_right, innerSL_apply_apply, SchwartzMap.integral_bilin_fourierInv_eq, HasFTaylorSeriesUpTo.hasFDerivAt, ContinuousAlternatingMap.hasStrictFDerivAt_compContinuousLinearMap, innerSL_apply_comp_of_isSymmetric, Unitization.splitMul_apply, ContMDiffOn.clm_postcomp, toBilinForm_apply, ProbabilityTheory.covarianceBilin_apply_eq_cov, ProbabilityTheory.covarianceBilin_self, LinearMap.adjoint_eq_toCLM_adjoint, ProbabilityTheory.uncenteredCovarianceBilin_apply, LinearMap.isSelfAdjoint_toContinuousLinearMap_iff, opNorm_lsmul_apply_le, OrthonormalBasis.orthogonalProjection_eq_sum_rankOne, SchwartzMap.bilinLeftCLM_apply, InnerProductSpace.toDual_symm_apply, fpowerSeriesBilinear_apply_zero, fderiv_continuousAlternatingMap_apply_const, coprodEquiv_apply, Bundle.RiemannianMetric.isVonNBounded, ContinuousMultilinearMap.norm_compContinuousLinearMapLRight_le, MeasureTheory.charFun_toDual_symm_eq_charFunDual, ContinuousLinearEquiv.toCompactConvergenceCLM_apply, coe_restrictScalarsₗ, Differentiable.fderiv_norm_rpow, integrable_of_bilin_of_bdd_left, isometry_mul_flip, HasFPowerSeriesWithinAt.hasStrictFDerivWithinAt, adjoint_id, LinearMap.coe_toContinuousLinearMap', HasFDerivWithinAt.linear_multilinear_comp, iteratedFDerivWithin_succ_eq_comp_left, isStarNormal_iff_norm_eq_adjoint, CStarModule.innerSL_apply, hasDerivWithinAt_of_bilinear, Bundle.ContinuousLinearMap.vectorBundle, MeasureTheory.ConvolutionExistsAt.integrable, opNorm_mul_flip_apply, bilinearRestrictScalars_eq_restrictScalarsL_comp_restrictScalars, InnerProductSpace.inner_left_rankOne_apply, HasFDerivAt.norm_sq, ContMDiffAt.clm_postcomp, ContMDiffWithinAt.clm_postcomp, Real.integrable_prod_sub, eq_adjoint_iff, PiTensorProduct.toDualContinuousMultilinearMap_apply_apply, HasFDerivAt.continuousMultilinear_apply_const, InnerProductSpace.rankOne_apply, NormedSpace.Dual.toWeakDual_continuous, mkOfIsCompactOperator_mem_compactOperator, MeasureTheory.convolution_integrand_bound_right_of_le_of_subset, ContinuousMultilinearMap.norm_smulRightL_le, precompL_apply, IsSelfAdjoint.adjoint_eq, MDifferentiable.clm_postcomp, CStarMatrix.toCLM_injective, isPositive_self_comp_adjoint, LinearMap.adjoint_toContinuousLinearMap, toSesqForm_apply_coe, SchwartzMap.integral_bilin_fourierIntegral_eq, LinearIsometryEquiv.adjoint_eq_symm, toLinearMap₁₂_apply, VectorFourier.fourierPowSMulRight_eq_comp, finiteDimensional, hasFDerivWithinAt_of_bilinear, opNorm_mul, MDifferentiable.cle_arrowCongr, flipMultilinear_apply_apply, compactOperator_topologicalClosure, SchwartzMap.integral_sesq_fourier_eq, adjointAux_norm, ProbabilityTheory.covarianceBilin_apply_basisFun_self, hasStrictFDerivAt_of_bilinear, CStarMatrix.inner_toCLM_conjTranspose_left, isBoundedBilinearMap_comp, ContinuousAlternatingMap.ofSubsingletonLIE_symm_apply, Real.fourier_iteratedFDeriv, InnerProductSpace.toDualMap_apply_apply, WeakDual.toStrongDual_inj, mulLeftRight_apply, Bundle.ContinuousRiemannianMetric.symm, ContinuousMultilinearMap.curryRight_norm, analyticOn_bilinear, MDifferentiableAt.clm_postcomp, coe_innerSL_apply, RegularNormedAlgebra.isometry_mul', CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM, StrongDual.toWeakDual_inj, hasFDerivAt_ringInverse, WeakDual.coe_toStrongDual, VectorFourier.fourierIntegral_iteratedFDeriv, InnerProductSpace.nnnorm_rankOne, ContinuousLinearEquiv.arrowCongr_apply, toUniformConvergenceCLM_apply, flip_smul, InnerProductSpace.isSymmetricProjection_rankOne_self, HasFTaylorSeriesUpToOn.eventually_hasFDerivAt, ProbabilityTheory.uncenteredCovarianceBilinDual_of_not_memLp, Bundle.ContinuousRiemannianMetric.continuous, MeasureTheory.ConvolutionExistsAt.integrable_swap, hasGradientWithinAt_iff_hasFDerivWithinAt, apply_norm_sq_eq_inner_adjoint_left, VectorFourier.norm_iteratedFDeriv_fourierPowSMulRight, HasFPowerSeriesOnBall.fderiv_eq, instLocallyConvexSpace, NormedSpace.inclusionInDoubleDual_norm_eq, fderiv_of_bilinear, LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf, HasFTaylorSeriesUpToOn.hasFDerivWithinAt, Submodule.adjoint_subtypeL, LinearMap.toLinearMap_toContPerfPair, opNorm_mul_apply, norm_smulRightL, MeasureTheory.convolution_def, Pretrivialization.continuousLinearMapCoordChange_apply, InnerProductSpace.trace_rankOne, MDifferentiableWithinAt.cle_arrowCongr, MDifferentiableWithinAt.clm_precomp, ContMDiffVectorBundle.continuousLinearMap, ContinuousAlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinCLM_comp_apply, isBoundedBilinearMap_compMultilinear, hasFDerivAt_norm_rpow, compL_apply, lsmul_flip_inj, HasFPowerSeriesWithinAt.fderivWithin_eq, inCoordinates_apply_eq₂, IsPositive.conj_adjoint, flip_apply, ProbabilityTheory.covarianceBilin_zero, contDiffOn_stereoToFun, fderiv_inverse, ContMDiffOn.clm_precomp, bilinearRestrictScalars_apply_apply, InnerProductSpace.adjoint_rankOne, InnerProductSpace.rankOne_eq_zero, Real.fourierIntegral_convergent_iff', Bundle.ContMDiffRiemannianMetric.symm, InnerProductSpace.inner_right_rankOne_apply, HasFDerivWithinAt.hasGradientWithinAt, SchwartzMap.pairing_apply, innerSL_inj, fderivWithin_inv', StrongDual.toLp_of_not_memLp, fderivWithin_of_bilinear, PiTensorProduct.liftIsometry_tprodL, NormedSpace.inclusionInDoubleDual_norm_le, WeakDual.isCompact_closedBall, InnerProductSpace.rankOne_eq_rankOne_iff_comm, InnerProductSpace.toDual_apply, precompR_apply, ContinuousAlternatingMap.norm_alternatizeUncurryFinCLM_le, MeasureTheory.AEStronglyMeasurable.convolution_integrand_swap_snd', ProbabilityTheory.IsGaussian.charFun_eq', ProbabilityTheory.covarianceBilinDual_eq_covariance, SchwartzMap.integral_bilinear_deriv_right_eq_neg_left, ProbabilityTheory.covarianceBilinDual_zero, InnerProductSpace.comp_rankOne, OrthonormalBasis.starProjection_eq_sum_rankOne, InnerProductSpace.toDualMap_apply, adjoint_inner_left, hasStrictDerivAt_of_bilinear, ContinuousLinearEquiv.arrowCongr_symm, isBoundedBilinearMap, apply_norm_eq_sqrt_inner_adjoint_right, ContinuousMultilinearMap.ofSubsingletonₗᵢ_symm_apply, fderivWithin_continuousMultilinear_apply_const, Differentiable.fderiv_two, ContinuousAlternatingMap.ofSubsingletonLIE_apply, continuousMultilinearCurryRightEquiv_symm_apply, coe_smulRightₗ, lsmul_apply, Bundle.ContMDiffRiemannianMetric.pos, bilinearComp_zero_right, IsContMDiffRiemannianBundle.exists_contMDiff, InnerProductSpace.rank_rankOne, ContMDiffOn.cle_arrowCongr, InnerProductSpace.toDual_apply_apply, adjointAux_apply, inner_map_map_iff_adjoint_comp_self, ProperCone.innerDual_singleton, InnerProductSpace.rankOne_def, coe_flipₗᵢ', stereoToFun_apply, adjointAux_inner_left, SchwartzMap.convolution_apply, fderivWithin_fderivWithin_eq_of_mem_nhds, ProbabilityTheory.covarianceBilin_comm, IsStarNormal.ker_adjoint_eq_ker, SchwartzMap.integral_bilin_fourier_eq, ClosedSubmodule.orthogonal_eq_inter, hasFTaylorSeriesUpToOn_succ_iff_right, coe_flipMultilinearEquiv, NonUnitalAlgHom.coe_Lmul, ContinuousMultilinearMap.norm_map_init_le, SchwartzMap.smulRightCLM_apply_apply, MeasureTheory.condExp_stronglyMeasurable_simpleFunc_bilin, coe_restrict_scalarsL', fderivWithin_fderivWithin_eq_of_eventuallyEq, StrongDual.polarSubmodule_eq_polar, PiTensorProduct.liftEquiv_symm_apply, norm_iteratedFDeriv_le_of_bilinear, ContinuousMultilinearMap.cpolyomialOn_uncurry_of_linear, aestronglyMeasurable_comp₂, StrongDual.dualPairing_apply, HasFPowerSeriesWithinOnBall.hasSum_derivSeries_of_hasFDerivWithinAt, IsStarNormal.adjoint_apply_eq_zero_iff, LinearMap.range_toContinuousLinearMap, norm_precompR_le, isBoundedLinearMap_comp_right, hasFPowerSeriesOnBall_bilinear, WeakDual.isSeqCompact_closedBall, PiTensorProduct.liftEquiv_apply, fderivWithin_iteratedFDerivWithin, deriv_of_bilinear, Orientation.areaForm'_apply, innerSLFlip_apply_apply, nnnorm_holder_apply_apply_le, bilinear_hasTemperateGrowth, IsBoundedBilinearMap.isBoundedLinearMap_deriv, IsContinuousRiemannianBundle.exists_continuous, ContinuousAlternatingMap.fderivCompContinuousLinearMap_eq_alternatizeUncurryFin, postcomp_apply, LinearMap.coe_toContinuousLinearMap, prodMapL_apply, toSpanSingletonLE_apply, contMDiffOn_continuousLinearMapCoordChange, InnerProductSpace.enorm_rankOne, InnerProductSpace.continuousLinearMapOfBilin_zero, ContinuousMultilinearMap.compContinuousLinearMapLRight_apply, toLinearMap_innerSL_apply, ContinuousMultilinearMap.flipLinear_apply_apply, HasGradientAt.hasFDerivAt, CStarMatrix.toCLM_apply_single, Real.fourier_fderiv, hasFTaylorSeriesUpTo_succ_nat_iff_right, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, HasFDerivWithinAt.norm_sq, StrongDual.dualPairing_separatingLeft, ProperCone.dual_singleton, uncurryBilinear_apply, MeasureTheory.integral_posConvolution, VectorFourier.integral_fourierIntegral_swap, Bundle.RiemannianMetric.symm, LinearMap.toContinuousLinearMap_eq_iff_eq_toLinearMap, Unitization.norm_eq_sup, hasFTaylorSeriesUpToOn_succ_nat_iff_right, CStarMatrix.toCLM_apply_eq_sum, coeLM_apply, toPMap_adjoint_eq_adjoint_toPMap_of_dense, LinearMap.isPositive_toContinuousLinearMap_iff, MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd', ProbabilityTheory.covarianceBilin_eq_covarianceBilinDual, Bundle.ContinuousLinearMap.vectorPrebundle.isContMDiff, iteratedFDerivWithin_two_apply', HasFPowerSeriesAt.fderiv_eq, coe_restrictScalarsIsometry, memLp_of_bilin, toLinearMap_eq_iff_eq_toContinuousLinearMap, PiTensorProduct.dualSeminorms_bounded, ProbabilityTheory.covarianceBilin_self_nonneg, IsSelfAdjoint.conj_adjoint, ContinuousMultilinearMap.curryMidEquiv_apply, HasStrictFDerivAt.continuousMultilinear_apply_const, opNorm_mulLeftRight_apply_le, analyticOnNhd_bilinear, ProbabilityTheory.covarianceBilinDual_self_eq_variance, hasStrictFDerivAt_norm_sq, ContinuousMultilinearMap.cpolynomialAt_uncurry_of_linear, map_sub₂, bilinearComp_apply, CStarMatrix.toCLM_apply, InnerProductSpace.nullSubmodule_le_ker_toDualMap_left, HasFDerivAt.continuousAlternatingMap_apply_const, PiTensorProduct.injectiveSeminorm_apply, isBoundedLinearMap_comp_left, ProbabilityTheory.covarianceBilinDual_self_nonneg, ContinuousAlternatingMap.fderivCompContinuousLinearMapCLM_apply, Matrix.l2_opNNNorm_def, adjointAux_adjointAux, ContMDiffAt.clm_precomp, ProbabilityTheory.covarianceBilin_real_self, coprodEquivL_apply_apply, HasFTaylorSeriesUpToOn.hasFDerivAt, flip_add, VectorFourier.fourierSMulRight_apply, isSelfAdjoint_iff', ContinuousMultilinearMap.uncurryRight_apply, bilinearComp_zero, HasFTaylorSeriesUpToOn.shift_of_succ, ContinuousAlternatingMap.toContinuousMultilinearMapCLM_comp_fderivCompContinuousLinearMap, WeakDual.isClosed_closedBall, VectorFourier.pow_mul_norm_iteratedFDeriv_fourierIntegral_le, Real.fourier_bilin_convolution_eq, CStarMatrix.mul_entry_mul_eq_inner_toCLM, le_opNorm₂, lpPairing_eq_integral, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, InnerProductSpace.rankOne_def', opNorm_mul_le, Pretrivialization.continuousLinearMap.isLinear, ProbabilityTheory.norm_uncenteredCovarianceBilinDual_le, SchwartzMap.integral_sesq_fourierIntegral_eq, apply_norm_eq_sqrt_inner_adjoint_left, MeasureTheory.Integrable.convolution_integrand, ProbabilityTheory.covarianceBilin_apply_pi, SchwartzMap.integral_bilinear_laplacian_right_eq_left, isometry_mul, StrongDual.toLpₗ_apply, tendsto_iff_forall_eval_tendsto_topDualPairing, MeasureTheory.dist_convolution_le', MDifferentiableOn.cle_arrowCongr, InnerProductSpace.innerSL_norm, IsLocalExtrOn.linear_dependent_of_hasStrictFDerivAt, ContinuousMultilinearMap.curryRight_apply, MDifferentiableAt.clm_precomp, ProbabilityTheory.covarianceBilin_apply_basisFun, ContinuousMultilinearMap.analyticOn_uncurry_of_linear, hasGradientAt_iff_hasFDerivAt, FormalMultilinearSeries.leftInv_coeff_one, mdifferentiableOn_continuousLinearMapCoordChange, InnerProductSpace.isStarProjection_rankOne_self, HasFPowerSeriesAt.hasFDerivAt, LinearMap.mkContinuous₂_norm_le, hasFPowerSeriesAt_bilinear, StrongDual.norm_toLpₗ_le, PiTensorProduct.mapLMultilinear_apply_apply, isBoundedBilinearMap_apply, hasStrictFDerivAt_ringInverse, mul_apply', InnerProductSpace.rankOne_comp_rankOne, StrongDual.toLpₗ_of_not_memLp, InnerProductSpace.rankOne_comp, StrongDual.mem_polarSubmodule, Real.fourierIntegral_fderiv, InnerProductSpace.toMatrix_rankOne, HasFDerivWithinAt.continuousMultilinear_apply_const, PiTensorProduct.liftIsometry_comp_mapL, InnerProductSpace.isSymmetric_rankOne_self, continuousMultilinearCurryLeftEquiv_apply, restrictScalarsIsometry_toLinearMap, WStarAlgebra.exists_predual, PiTensorProduct.liftIsometry_symm_apply, opNNNorm_mul, norm_iteratedFDerivWithin_le_of_bilinear, iteratedFDeriv_succ_eq_comp_right, fderiv_continuousLinearEquiv_comp, SchwartzMap.convolution_continuous_left, adjoint_toSpanSingleton, toBilinForm_injective, ProbabilityTheory.isPosSemidef_covarianceBilinDual, IsBoundedBilinearMap.toContinuousLinearMap_apply, HasFPowerSeriesWithinOnBall.fderivWithin_eq, SchwartzMap.pairing_continuous_left, InnerProductSpace.rankOne_one_right_eq_toSpanSingleton, HasFTaylorSeriesUpToOn.hasStrictFDerivAt, innerSLFlip_apply, InnerProductSpace.toContinuousLinearMap_toDualMap, ProbabilityTheory.covarianceBilin_real, separatingDual_iff_injective, opNorm_mulLeftRight_apply_apply_le, opNorm_lsmul_le, ProbabilityTheory.covarianceBilin_of_not_memLp, WeakDual.CharacterSpace.norm_le_norm_one, innerSL_apply_norm, InnerProductSpace.toDual_apply_eq_toDualMap_apply, Real.fourier_bilin_convolution_eq_integral, coe_restrictScalarsL, norm_compL_le, ContinuousMultilinearMap.hasStrictFDerivAt_compContinuousLinearMap, fderiv_continuousMultilinear_apply_const, ProbabilityTheory.covarianceBilinDual_comm, opNNNorm_mul_flip_apply, HasFPowerSeriesOnBall.hasFDerivAt, MeasureTheory.charFun_eq_charFunDual_toDualMap, PiTensorProduct.toDualContinuousMultilinearMap_le_projectiveSeminorm, ContMDiff.clm_postcomp, fderivWithin_continuousAlternatingMap_apply_const, ContinuousLinearEquiv.arrowCongrSL_apply, MDifferentiableWithinAt.clm_postcomp, hasFDerivAt_uncurry_of_multilinear, opNorm_mul_apply_le, orthogonal_range, ProbabilityTheory.covarianceBilin_apply, HasFDerivAt.linear_multilinear_comp, MeasureTheory.convolutionExistsAt_iff_integrable_swap, ProbabilityTheory.uncenteredCovarianceBilin_of_not_memLp, norm_bilinearRestrictScalars, PiTensorProduct.liftIsometry_apply_apply, coeLMₛₗ_apply, ContinuousMultilinearMap.analyticWithinAt_uncurry_of_linear, ProbabilityTheory.covarianceBilin_map, Unitization.nnnorm_eq_sup, bilinearComp_zero_left, Bundle.ContinuousRiemannianMetric.pos, Matrix.toLin_finTwoProd_toContinuousLinearMap, IsCoercive.continuousLinearEquivOfBilin_apply, MDifferentiableOn.clm_precomp, adjoint_comp, VectorFourier.integral_bilin_fourierIntegral_eq_flip, SchwartzMap.integral_clm_comp_deriv_right_eq_neg_left, InnerProductSpace.toLinearMap_rankOne, adjointAux_inner_right, Bundle.RiemannianMetric.pos, continuousMultilinearCurryRightEquiv_symm_apply', ContinuousAlternatingMap.fderivCompContinuousLinearMap_apply, prodL_apply, LinearMap.isStarProjection_toContinuousLinearMap_iff, iteratedFDerivWithin_two_apply, Pretrivialization.continuousOn_continuousLinearMapCoordChange, DoubleCentralizer.coe_fst, Submodule.adjoint_orthogonalProjection, hasFDerivWithinAt_iff_hasGradientWithinAt, hasFDerivAt_of_bilinear, ProbabilityTheory.uncenteredCovarianceBilin_zero, LinearMap.toContPerfPair_apply, VectorFourier.fourierPowSMulRight_apply, SchwartzMap.convolution_flip, opNorm_flip, continuousOn_integral_bilinear_of_locally_integrable_of_compact_support, IsPositive.adjoint_conj, FormalMultilinearSeries.derivSeries_eq_zero, star_eq_adjoint, ProperCone.hyperplane_separation_of_notMem, ContinuousAlternatingMap.curryLeftLI_apply, norm_compSL_le, innerSL_apply, SchwartzMap.integral_bilinear_lineDerivOp_right_eq_neg_left, InnerProductSpace.isIdempotentElem_rankOne_self_iff, fderivWithin_continuousLinearEquiv_comp, StrongDual.toLp_apply, map_smul₂, coe_flipₗᵢ, ContinuousAlternatingMap.fderivCompContinuousLinearMap_of_isEmpty, IsSelfAdjoint.adjoint_conj, Real.differentiable_fourierChar_neg_bilinear_right, OrthonormalBasis.sum_rankOne_eq_id, adjoint_inner_right, HasFiniteFPowerSeriesOnBall.hasStrictFDerivAt, HasStrictFDerivAt.continuousAlternatingMap_apply_const, toSpanSingletonLE_symm_apply, norm_smulRightL_le, coe_symm_flipMultilinearEquiv, VectorFourier.norm_fourierSMulRight_le, signedDist_apply, continuousMultilinearCurryRightEquiv_apply', flipₗᵢ'_symm, HasFiniteFPowerSeriesOnBall.hasFDerivAt, HasFDerivAt.hasGradientAt, flip_zero, isAdjointPair_inner, VectorFourier.norm_fourierPowSMulRight_le, ProbabilityTheory.norm_uncenteredCovarianceBilin_le, Real.hasFDerivAt_fourierChar_neg_bilinear_right, VectorFourier.norm_fourierSMulRight, toUniformConvergenceCLM_continuous, hasStrictFDerivAt_inv', SchwartzMap.fourier_convolution_apply, iteratedFDeriv_two_apply, StrongDual.coe_toWeakDual, WeakDual.toStrongDual_apply, MDifferentiable.clm_precomp, Real.differentiable_fourierChar_neg_bilinear_left, continuousMultilinearCurryLeftEquiv_symm_apply, ContinuousMultilinearMap.smulRightL_apply, iteratedFDeriv_succ_eq_comp_left, VectorFourier.fourierPowSMulRight_iteratedFDeriv_fourierIntegral, ContinuousLinearEquiv.arrowCongrSL_symm_apply, opNorm_mulLeftRight_le, topDualPairing_apply, ContMDiffWithinAt.clm_precomp, ContinuousMultilinearMap.ofSubsingletonₗᵢ_apply, LinearMap.coe_toContinuousLinearMap_symm, continuousMultilinearCurryFin1_symm_apply, MeasureTheory.condExp_aestronglyMeasurable_bilin_of_bound, Submodule.orthogonal_eq_inter, lsmul_flip_apply, compContinuousAlternatingMapCLM_apply_apply, prodₗ_apply, coprodEquiv_symm_apply, fderiv_norm_rpow, norm_map_iff_adjoint_comp_self, norm_adjoint_comp_self, map_add₂, ContMDiff.clm_precomp, FormalMultilinearSeries.derivSeries_apply_diag, opNorm_lsmul, ProbabilityTheory.covarianceBilinDual_of_not_memLp, ContinuousMultilinearMap.fderivCompContinuousLinearMap_apply, InnerProductSpace.continuousLinearMapOfBilin_apply, NormedSpace.double_dual_bound, MeasureTheory.convolution_eq_swap, map_zero₂, Real.zero_at_infty_vector_fourierIntegral, ProbabilityTheory.uncenteredCovarianceBilinDual_apply, precomp_apply, norm_holderL_le, ContinuousAffineMap.toConstProdContinuousLinearMap_fst, HasFiniteFPowerSeriesOnBall.fderiv_eq, MeasureTheory.condExp_stronglyMeasurable_bilin_of_bound, analyticAt_bilinear, ProbabilityTheory.IsGaussian.charFunDual_eq', HasFPowerSeriesAt.hasStrictFDerivAt, toPointwiseConvergenceCLM_apply, apply_norm_sq_eq_inner_adjoint_right, innerSL_apply_coe, bilinearRestrictScalars_eq_restrictScalars_restrictScalarsL_comp, toUniformConvergenceCLM_symm_apply, hasFDerivAt_inv', exists_continuousLinearEquiv_fderiv_symm_eq, MeasureTheory.AEStronglyMeasurable.convolution_integrand', isPositive_adjoint_comp_self, map_neg₂, Matrix.l2_opNorm_def, Bundle.RiemannianMetric.continuousAt, holderL_apply_apply, MeasureTheory.AEStronglyMeasurable.convolution_integrand, norm_innerSL_le, VectorFourier.hasFDerivAt_fourierChar_smul, ContMDiffWithinAt.cle_arrowCongr, continuousMultilinearCurryRightEquiv_apply, integrable_of_bilin_of_bdd_right, ContinuousLinearEquiv.arrowCongrSL_toLinearEquiv_apply, apply_apply', HasFPowerSeriesWithinAt.hasFDerivWithinAt, ProbabilityTheory.covarianceBilinDual_of_not_memLp', SchwartzMap.fourier_convolution, ContinuousMultilinearMap.analyticOnNhd_uncurry_of_linear, flipₗᵢ_symm, exists_continuousLinearEquiv_fderivWithin_symm_eq, HasGradientWithinAt.hasFDerivWithinAt, FormalMultilinearSeries.rightInv_coeff_one, NormedSpace.Dual.coe_toWeakDual, norm_holder_apply_apply_le, VectorFourier.fourierIntegral_fderiv, fpowerSeries_apply_one, fderiv_continuousLinearEquiv_comp', smulRightL_apply_apply, ContinuousLinearMapWOT.ContinuousLinearMap.continuous_toWOT, toSpanSingletonCLE_apply_apply, opNNNorm_mul_apply, toSpanSingletonCLE_symm_apply, VectorFourier.norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le, norm_precompL_le, Real.fderiv_fourierChar_neg_bilinear_left_apply, apply_apply, ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear_apply_apply, IsSymmSndFDerivWithinAt.eq, map_smulₛₗ₂, PiTensorProduct.mapLMultilinear_opNorm, instModuleFinite, Real.fourierIntegral_iteratedFDeriv, NormedSpace.Dual.toWeakDual_inj, ContinuousLinearEquiv.arrowCongrSL_toLinearEquiv_symm_apply, MeasureTheory.AEStronglyMeasurable.convolution_integrand_swap_snd, HasCompactSupport.convolution_integrand_bound_right, HasFPowerSeriesWithinOnBall.hasFDerivWithinAt, RCLike.re_extendTo𝕜'ₗ, InnerProductSpace.symm_toEuclideanLin_rankOne, isBoundedBilinearMap_smulRight, ContMDiffAt.cle_arrowCongr, ProbabilityTheory.uncenteredCovarianceBilinDual_zero, adjoint_innerSL_apply, coprodEquivL_symm_apply, HasCompactSupport.convolution_integrand_bound_right_of_subset, toWOT_apply, norm_smulRightL_apply, DoubleCentralizer.coe_snd, fpowerSeriesBilinear_apply_one, MeasureTheory.integral_convolution, fderiv_norm_sq_apply, NormedSpace.dual_def, fderiv_norm_sq, coe_mulₗᵢ, HasFDerivWithinAt.continuousAlternatingMap_apply_const, innerSL_apply_comp, toSesqForm_apply_norm_le, MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd, PiTensorProduct.injectiveSeminorm_def, coeFn_holder, CStarMatrix.norm_def, ContMDiff.cle_arrowCongr, coe_deriv₂, InnerProductSpace.nullSubmodule_le_ker_toDualMap_right, ContinuousAlternatingMap.alternatizeUncurryFinCLM_apply, adjoint_adjoint, Bundle.ContinuousRiemannianMetric.isVonNBounded, hasDerivAt_of_bilinear, orthogonal_ker, InnerProductSpace.norm_rankOne, PiTensorProduct.mapLMultilinear_toFun_apply, CStarMatrix.inner_toCLM_conjTranspose_right, ContinuousAffineMap.toConstProdContinuousLinearMap_snd, ContinuousMultilinearMap.curryMidEquiv_symm_apply, CStarMatrix.toCLM_apply_single_apply, ContinuousMultilinearMap.uncurryRight_norm, flipAlternating_apply_apply, LinearMap.ker_toContinuousLinearMap, IsSymmSndFDerivAt.eq, norm_iteratedFDerivWithin_le_of_bilinear_aux, fderiv_inv', continuousMultilinearCurryFin1_apply

ContinuousMap

Definitions

Name	Category	Theorems
module 📖	CompOp	49 math math: toLp_inj, span_fourier_closure_eq_top, toLp_denseRange, range_toLp, UnitAddTorus.mFourierSubalgebra_coe, BoundedContinuousFunction.toContinuousMapLinearMap_apply, LocallyConstant.toContinuousMapLinearMap_apply, cfcL_apply, toLp_norm_le, cfcL_integrable, UnitAddTorus.mFourierCoeff_toLp, ContinuousCohomology.I_obj_ρ_apply, ContinuousLinearMap.const_apply_apply, WeakDual.CharacterSpace.continuousMapEval_bijective, WeakDual.CharacterSpace.homeoEval_naturality, ContinuousCohomology.I_obj_V_isModule, linearIsometryBoundedOfCompact_symm_apply, evalCLM_apply, linearIsometryBoundedOfCompact_of_compact_toEquiv, toLp_def, coeFnLinearMap_apply, ContinuousMapZero.toContinuousMapCLM_apply, adjoin_id_eq_span_one_union, toLp_norm_eq_toLp_norm_coe, fourierCoeff_toLp, ContinuousLinearMap.compLeftContinuous_apply, ContinuousCohomology.I_map_hom, cfc_eq_cfcL_mkD, fourierSubalgebra_coe, toLp_injective, coe_toLp, nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, cfcL_integral, coeFn_toLp, linearIsometryBoundedOfCompact_toAddEquiv, toLp_comp_toContinuousMap, adjoin_id_eq_span_one_add, linearIsometryBoundedOfCompact_apply_apply, instLocallyConvexSpace, PadicInt.mahlerEquiv_apply, cfc_eq_cfcL, WeakDual.CharacterSpace.continuousMapEval_apply_apply, UnitAddTorus.span_mFourier_closure_eq_top, ker_evalStarAlgHom_inter_adjoin_id, PadicInt.mahlerEquiv_symm_apply, ker_evalStarAlgHom_eq_closure_adjoin_id, MeasureTheory.ContinuousMap.inner_toLp, linearIsometryBoundedOfCompact_toIsometryEquiv, ContinuousCohomology.const_app_hom

DFinsupp

Definitions

Name	Category	Theorems
module 📖	CompOp	86 math math: coprodMap_apply, comul_single, Module.Flat.dfinsupp, iSupIndep.linearEquiv_symm_apply, iSupIndep.dfinsupp_lsum_injective, lsum_lsingle, linearIndependent_single_iff, lsingle_apply, MultilinearMap.fromDFinsuppEquiv_apply, MultilinearMap.dfinsuppFamily_single, Pi.comul_comp_dFinsuppCoeFnLinearMap, Pi.counit_coe_dFinsupp, subtypeDomainLinearMap_apply, mapRange.linearEquiv_refl, mapRange.linearMap_comp, mapRange.linearEquiv_trans, mapRange.linearEquiv_symm, Submodule.biSup_eq_range_dfinsupp_lsum, MultilinearMap.dfinsuppFamily_apply_toFun, MultilinearMap.dfinsuppFamily_apply_support', injective_pi_lapply, range_mapRangeLinearMap, Pi.counit_comp_dFinsuppCoeFnLinearMap, sigmaCurryLEquiv_apply, instIsCocomm, MultilinearMap.dfinsuppFamily_compLinearMap_lsingle, sum_mapRange_index.linearMap, MultilinearMap.dfinsuppFamily_add, MultilinearMap.dfinsuppFamily_smul, Submodule.mem_iSup_iff_exists_dfinsupp, Submodule.iSup_eq_range_dfinsupp_lsum, iSup_range_lsingle, MultilinearMap.fromDFinsuppEquiv_single, filterLinearMap_apply, counit_comp_lsingle, MultilinearMap.freeDFinsuppEquiv_single, Module.Flat.dfinsupp_iff, comul_comp_lapply, PiTensorProduct.ofDFinsuppEquiv_tprod_apply, MultilinearMap.freeDFinsuppEquiv_def, mapRange.linearEquiv_apply, linearEquivFunOnFintype_symm_apply, MultilinearMap.freeFinsuppEquiv_def, lsum_single, MultilinearMap.dfinsuppFamilyₗ_apply, MultilinearMap.dfinsupp_ext_iff, iSupIndep.linearEquiv_apply, comul_comp_lsingle, sigmaCurryLEquiv_symm_apply, MultilinearMap.dfinsuppFamily_single_left_apply, coprodMap_apply_single, lsum_symm_apply, PiTensorProduct.ofDFinsuppEquiv_tprod_single, ker_mapRangeLinearMap, sigmaFinsuppLequivDFinsupp_symm_apply, coeFnLinearMap_apply, lmk_apply, MultilinearMap.freeDFinsuppEquiv_apply, Module.Free.dfinsupp, linearEquivFunOnFintype_apply, finsuppLequivDFinsupp_apply_apply, MultilinearMap.support_dfinsuppFamily_subset, iSupIndep_iff_dfinsupp_lsum_injective, finsuppLequivDFinsupp_symm_apply, lapply_comp_lsingle_of_ne, instIsSemisimpleModuleDFinsupp, counit_single, MultilinearMap.dfinsuppFamily_zero, Module.annihilator_dfinsupp, lapply_apply, Module.instProjectiveDFinsupp, PiTensorProduct.ofDFinsuppEquiv_symm_single_tprod, Pi.comul_coe_dFinsupp, mapRange.linearMap_id, mapRange.linearMap_apply, Module.Finite.instDFinsupp, Submodule.mem_biSup_iff_exists_dfinsupp, MultilinearMap.dfinsuppFamily_single_left, lapply_comp_lsingle_same, lsum_apply_apply, linearIndependent_single, MultilinearMap.fromDFinsuppEquiv_symm_apply, lhom_ext'_iff, lsum_comp_mapRange_toSpanSingleton, domLCongr_apply, sigmaFinsuppLequivDFinsupp_apply

DirectSum.Gmodule

Definitions

Name	Category	Theorems
module 📖	CompOp	—

DirectSum.GradeZero

Definitions

Name	Category	Theorems
module 📖	CompOp	—

Equiv

Definitions

Name	Category	Theorems
module 📖	CompOp	5 math math: coalgebraIsCocomm, tensorProductAssoc_def, toLinearEquiv_continuousLinearEquiv, tensorProductComm_def, moduleIsTorsionFree

Finsupp

Definitions

Name	Category	Theorems
module 📖	CompOp	872 math math: TensorProduct.finsuppRight_apply, snd_sumFinsuppLEquivProdFinsupp, LinearMap.exists_finsupp_nat_of_fin_fun_injective, fst_sumFinsuppLEquivProdFinsupp, groupHomology.π_comp_H2Iso_hom_assoc, Module.Free.finsupp, Rep.coe_linearization_obj_ρ, groupHomology.mapCycles₂_comp_assoc, Orthonormal.inner_left_finsupp, mapRange.linearEquiv_refl, Module.Basis.end_repr_apply, Algebra.Presentation.differentials.comm₁₂_single, mem_supported_support, MvPolynomial.rTensor_apply_tmul_apply, groupHomology.d₁₀_single_one, restrictDom_apply, groupHomology.boundaries₂_le_cycles₂, QuadraticMap.sum_polar_sub_repr_sq, MvPolynomial.scalarRTensor_apply_monomial_tmul, Module.basisUnique_repr_eq_zero_iff, finsuppTensorFinsupp_apply, Representation.ofMulActionSelfAsModuleEquiv_symm_apply, Rep.diagonalSuccIsoFree_inv_hom_single, PolynomialModule.smul_def, IsBaseChange.finsuppPow, lmapDomain_id, NumberField.Units.fun_eq_repr, TensorProduct.equivFinsuppOfBasisLeft_symm_apply, IsBaseChange.basis_repr_comp_apply, Module.DualBases.basis_repr_symm_apply, Module.Basis.toDual_linearCombination_left, groupHomology.chainsMap_f_3_comp_chainsIso₃_apply, LinearMap.prodOfFinsuppNat_injective, sumFinsuppLEquivProdFinsupp_apply, IsAdjoinRootMonic.coeff_apply, Module.Relations.Solution.surjective_π_iff_span_eq_top, supportedEquivFinsupp_symm_single, groupHomology.d₃₂_single, Rep.coindToInd_of_support_subset_orbit, Representation.finsupp_apply, LinearMap.toMatrix_apply', mapRange.linearEquiv_toAddEquiv, groupHomology.mapCycles₁_comp_apply, groupHomology.cyclesIso₀_inv_comp_iCycles_apply, Rep.leftRegularHom_hom, Matrix.repr_toLin, KaehlerDifferential.mvPolynomialBasis_repr_D_X, linearCombination_id_surjective, TensorProduct.sum_tmul_basis_right_injective, groupHomology.eq_d₃₂_comp_inv, rank_finsupp, LinearIndependent.linearCombination_repr, exteriorPower.basis_repr_ne, Algebra.Presentation.differentials.comm₂₃, rank_finsupp_self, Representation.ofCoinvariantsTprodLeftRegular_mk_tmul_single, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_left, Module.Basis.mk_repr, Module.Flat.finsupp, Module.Relations.Solution.injective_fromQuotient_iff_ker_π_eq_span, linearEquivFunOnFinite_symm_coe, Module.Basis.repr_isUnitSMul, Span.repr_def, groupHomology.mem_cycles₂_iff, Module.FaithfullyFlat.finsupp, groupHomology.cyclesMap_comp_isoCycles₂_hom, linearIndependent_iff_injective_finsuppLinearCombination, MvPolynomial.scalarRTensor_apply_X_tmul_apply, groupHomology.comp_d₂₁_eq, groupHomology.mapCycles₁_comp_assoc, Module.Basis.prod_repr_inl, groupHomology.toCycles_comp_isoCycles₂_hom_assoc, Module.Relations.Solution.surjective_fromQuotient_iff_surjective_π, linearCombination_zero_apply, mem_supported, Module.Basis.repr_self_apply, mem_span_iff_linearCombination, IsLocalFrameOn.coeff_apply_of_mem, setBasisOfLinearIndependentOfCardEqFinrank_repr_apply, Algebra.Presentation.differentials.hom₁_single, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_right, LinearMap.sum_repr_mul_repr_mulₛₗ, Representation.ofMulAction_apply, groupHomology.d₃₂_single_one_thd, Ideal.finsuppTotal_apply_eq_of_fintype, groupHomology.isoCycles₁_inv_comp_iCycles_apply, KaehlerDifferential.quotKerTotalEquiv_symm_comp_D, strongRankCondition_iff_forall_not_injective, IntermediateField.LinearDisjoint.algebraMap_basisOfBasisRight_repr_apply, finsuppTensorFinsupp'_symm_single_eq_tmul_single_one, Rep.finsuppToCoinvariantsTensorFree_single, Module.Relations.Solution.IsPresentation.linearEquiv_symm_var, groupHomology.chains₁ToCoinvariantsKer_surjective, Rep.coinvariantsTensorFreeLEquiv_symm_apply, linearIndependent_iff_ker, Rep.standardComplex.d_eq, Module.presentationFinsupp_G, Pi.counit_comp_finsuppLcoeFun, curryLinearEquiv_symm_apply_apply, llift_apply, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_single, linearCombination_smul, groupHomology.cycles₁_eq_top_of_isTrivial, Ideal.finsuppTotal_apply, groupHomology.d₃₂_comp_d₂₁_assoc, TensorProduct.equivFinsuppOfBasisLeft_apply_tmul_apply, lmapDomain_linearCombination, range_mapRange_linearMap, Module.Basis.repr_algebraMap, Algebra.Generators.cotangentSpaceBasis_repr_one_tmul, iInf_ker_lapply_le_bot, Rep.coindFunctorIso_hom_app_hom_hom_apply_hom_hom_apply, basis_repr, Matrix.toBilin_apply, IsLocalRing.basisQuotient_repr, Module.Relations.surjective_toQuotient, Module.End.ringHomEndFinsupp_apply_apply, Representation.ofMulAction_single, groupHomology.single_one_snd_sub_single_one_fst_mem_boundaries₂, linearCombination_one_tmul, groupHomology.d₁₀ArrowIso_hom_left, Module.Basis.map_repr, groupHomology.cycles₁IsoOfIsTrivial_hom_apply, Representation.coinvariantsFinsuppLEquiv_apply, LinearMap.toMatrixAlgEquiv_apply, Algebra.TensorProduct.basis_repr_tmul, lmapDomain_comp, groupHomology.d₂₁_single_inv_mul_ρ_add_single, NumberField.mixedEmbedding.stdBasis_apply_isComplex_snd, supported_inter, LinearEquiv.finsuppUnique_symm_apply, span_le_supported_biUnion_support, groupHomology.d₁₀_comp_coinvariantsMk_apply, LinearMap.BilinForm.apply_eq_dotProduct_toMatrix_mulVec, KaehlerDifferential.kerTotal_eq, span_image_eq_map_linearCombination, LinearIndependent.repr_eq_single, IsAdjoinRootMonic.basis_repr, Representation.leftRegular_norm_eq_zero_iff, supported_eq_span_single, TensorProduct.finsuppLeft_smul', groupHomology.chainsMap_f_3_comp_chainsIso₃, groupHomology.mapCycles₁_id_comp_assoc, Algebra.TensorProduct.basisAux_map_smul, Pi.basis_repr_single, Module.Relations.Solution.fromQuotient_toQuotient, groupHomology.eq_d₂₁_comp_inv, TensorProduct.equivFinsuppOfBasisRight_apply_tmul, Module.Basis.ofEquivFun_repr_apply, supported_univ, Module.Basis.sumQuot_repr_left, groupHomology.mapCycles₁_comp, Representation.ofMulAction_self_smul_eq_mul, Module.Basis.reindexRange_repr', Module.Basis.reindexRange_repr, sigmaFinsuppLEquivPiFinsupp_symm_apply, TensorProduct.equivFinsuppOfBasisLeft_apply_tmul, PiTensorProduct.ofFinsuppEquiv'_tprod_single, TensorProduct.equivFinsuppOfBasisLeft_symm, Module.Presentation.finsupp_R, Submodule.mulLeftMap_apply, TensorProduct.finsuppScalarRight_apply, Module.Relations.Solution.ofQuotient_var, groupHomology.mapCycles₁_comp_i, TensorProduct.finsuppScalarRight_apply_tmul, Polynomial.derivativeFinsupp_apply_toFun, LieAlgebra.LoopAlgebra.toFinsupp_symm_single, Module.Basis.repr_smul, finsuppTensorFinsupp'_single_tmul_single, mapDomain.coe_linearEquiv, AddMonoidAlgebra.grade_eq_lsingle_range, NumberField.canonicalEmbedding.integralBasis_repr_apply, lcoeFun_comp_lsingle, lcoeFun_apply, basisOfLinearIndependentOfCardEqFinrank_repr_apply, PiTensorProduct.ofFinsuppEquiv_tprod_single, Module.Presentation.finsupp_G, Module.Relations.range_map, Module.Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul, groupHomology.single_one_fst_sub_single_one_snd_mem_boundaries₂, Representation.coinvariantsTprodLeftRegularLEquiv_apply, finsuppTensorFinsuppLid_symm_single_smul, ker_mapRange, groupHomology.mapCycles₂_id_comp, MvPolynomial.scalarRTensor_apply_tmul, supported_comap_lmapDomain, KaehlerDifferential.kerTotal_mkQ_single_algebraMap_one, finsuppTensorFinsupp'_symm_single_mul, LinearMap.polyCharpolyAux_map_eval, ZSpan.repr_ceil_apply, domLCongr_trans, groupHomology.d₂₁_single_inv_self_ρ_sub_self_inv, Module.Basis.addSubgroupOfClosure_repr_apply, sumFinsuppLEquivProdFinsupp_symm_apply, Module.Basis.repr_range, basis_toMatrix_basisFun_mul, instFinitePresentationFinsupp, Submodule.LinearDisjoint.linearIndependent_right_of_flat, TensorProduct.finsuppLeft_symm_apply_single, linearEquivFunOnFinite_symm_apply, finsuppTensorFinsupp'_apply_apply, Rep.coinvariantsTensorFreeLEquiv_apply, PiTensorProduct.ofFinsuppEquiv_symm_single_tprod, leftInverse_lcomapDomain_mapDomain, Module.Basis.reindexRange_repr_self, groupHomology.toCycles_comp_isoCycles₁_hom_apply, linearCombination_linear_comp, groupHomology.mapCycles₂_comp_i, Module.Basis.repr_eq_iff, mem_submodule_iff, Module.Basis.sumQuot_repr_inl, groupHomology.boundariesOfIsBoundary₁_coe, KaehlerDifferential.kerTotal_mkQ_single_add, IsBaseChange.of_basis, Algebra.Generators.cotangentSpaceBasis_repr_tmul, comul_comp_lapply, Algebra.TensorProduct.equivFinsuppOfBasis_apply, sumFinsuppLEquivProdFinsupp_symm_inl, Module.Basis.repr_linearCombination, groupHomology.eq_d₃₂_comp_inv_apply, Algebra.TensorProduct.equivFinsuppOfBasis_symm_apply, MvPolynomial.rTensor_apply_X_tmul, PiLp.basisFun_repr, Algebra.Generators.H1Cotangent.δAux_toAlgHom, linearCombination_single_index, groupHomology.single_one_fst_sub_single_one_fst_mem_boundaries₂, lift_apply, LinearIndependent.linearCombinationEquiv_apply_coe, Module.Relations.Solution.range_π, TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply, groupHomology.mapCycles₁_id_comp_apply, LinearIndependent.linearCombination_comp_repr, Polynomial.derivativeFinsupp_apply_apply, Algebra.SubmersivePresentation.sectionCotangent_eq_iff, Module.Presentation.finsupp_var, Algebra.Presentation.differentials.surjective_hom₁, MvPolynomial.rTensor_apply_tmul, codisjoint_supported_supported, supported_union, Module.Basis.linearCombination_dualBasis, groupHomology.π_comp_H2Iso_hom, FiniteDimensional.basisSingleton_repr_apply, Rep.indResAdjunction_counit_app_hom_hom, Module.Basis.equivFunL_symm_apply_repr, PowerBasis.repr_gen_pow_isIntegral, groupHomology.pOpcycles_comp_opcyclesIso_hom_apply, Rep.coindToInd_apply, groupHomology.mapCycles₁_comp_i_apply, Polynomial.derivativeFinsupp_derivative, Subalgebra.LinearDisjoint.algebraMap_basisOfBasisRight_repr_apply, lsum_single, KaehlerDifferential.ker_map, groupHomology.mapCycles₂_comp, LinearMap.CompatibleSMul.finsupp_dom, LinearMap.BilinForm.sum_repr_mul_repr_mul, linearEquivFunOnFinite_single, supported_iInter, Module.Basis.mulOpposite_repr_op, Module.Basis.smulTower'_repr_mk, instIsLocalizedModuleFinsuppLinearMap, Matrix.GeneralLinearGroup.toLin'_apply, counit_comp_lsingle, TensorProduct.finsuppScalarRight_smul, groupHomology.coe_mapCycles₂, finsuppTensorFinsupp_single, groupHomology.comp_d₁₀_eq, groupHomology.H1π_comp_map_apply, Module.Basis.repr_reindex, Module.subsingletonEquiv_symm_apply, IsLocalizedModule.map_linearCombination, rank_finsupp_self', Module.Basis.algebraMapCoeffs_repr_apply_toFun, LinearMap.toMatrix_mulVec_repr, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul, Matrix.toLin_apply_eq_zero_iff, linearDepOn_iff, groupHomology.toCycles_comp_isoCycles₁_hom_assoc, Module.Basis.algebraMapCoeffs_repr_apply_apply, Matrix.toLinearMapₛₗ₂_apply, lcomapDomain_apply, lhom_ext'_iff, LinearMap.BilinForm.dualBasis_repr_apply, IsIsotypicOfType.linearEquiv_finsupp, linearIndepOn_iff_disjoint, mapRange.linearMap_apply, Module.Basis.sumQuot_repr_inl_of_mem, mapRange.linearEquiv_symm, Module.End.ringEquivEndFinsupp_apply_apply_apply, Submodule.set_smul_eq_map, lcongr_symm, TensorProduct.finsuppScalarRight_apply_tmul_apply, Module.Basis.repr_symm_single, Representation.finsuppToCoinvariants_single_mk, groupHomology.H2π_comp_map_assoc, apply_eq_dotProduct_toMatrix₂_mulVec, TensorProduct.finsuppLeft_apply, MvPolynomial.irreducible_sumSMulX, finsuppTensorFinsuppRid_symm_single_smul, finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply, Module.Basis.restrictScalars_repr_apply, Orthonormal.inner_finsupp_eq_sum_right, Representation.finsupp_single, span_eq_range_linearCombination, Module.Basis.equivFunL_apply, isCompl_range_lmapDomain_span, Algebra.Generators.cotangentRestrict_bijective_of_isCompl, MvPolynomial.combinatorial_nullstellensatz_exists_linearCombination, Module.Relations.map_single, Module.presentationFinsupp_R, MvPolynomial.rTensorAlgHom_toLinearMap, groupHomology.d₁₀ArrowIso_inv_right, Rep.finsuppTensorRight_hom_hom, MvPolynomial.coeff_sumSMulX, linearCombinationOn_range, Algebra.TensorProduct.basisAux_tmul, Module.Basis.singleton_repr, KaehlerDifferential.derivationQuotKerTotal_apply, linearIndependent_single_of_ne_zero, LinearMap.polyCharpolyAux_map_eq_charpoly, LinearMap.polyCharpolyAux_eval_eq_toMatrix_charpoly_coeff, LinearMap.toMatrix_transpose_apply, Pi.basis_repr, groupHomology.d₁₀_comp_coinvariantsMk, groupHomology.d₂₁_comp_d₁₀_apply, groupHomology.mapCycles₂_comp_apply, Module.Relations.Quotient.linearMap_ext_iff, LinearIndependent.span_repr_eq, Rep.coindFunctorIso_inv_app_hom_hom_apply_coe, Algebra.Presentation.differentials.comm₂₃', InnerProductSpace.gramSchmidt_triangular, QuadraticAlgebra.basis_repr_apply, Module.Basis.toDual_eq_repr, Module.Basis.sum_repr_mul_repr, linearCombination_apply, Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent, NumberField.integralBasis_repr_apply, groupHomology.H2π_eq_iff, Submodule.mulRightMap_eq_mulMap_comp, Module.Basis.linearMap_repr_apply, groupHomology.H1AddEquivOfIsTrivial_single, MultilinearMap.freeFinsuppEquiv_def, groupHomology.range_d₁₀_eq_coinvariantsKer, groupHomology.isoCycles₂_hom_comp_i_apply, linearCombination_embDomain, Representation.ofMulActionSelfAsModuleEquiv_apply, KaehlerDifferential.kerTotal_map, groupHomology.eq_d₂₁_comp_inv_assoc, Rep.ofMulActionSubsingletonIsoTrivial_inv_hom, PiTensorProduct.ofFinsuppEquiv_apply, TensorProduct.finsuppScalarRight_symm_apply_single, groupHomology.cyclesMap_comp_isoCycles₂_hom_assoc, Module.Basis.mem_span_image, LinearMap.CompatibleSMul.finsupp_cod, NumberField.mixedEmbedding.stdBasis_apply_isComplex_fst, Module.Basis.localizationLocalization_repr_algebraMap, groupHomology.inhomogeneousChains.d_single, Module.Basis.dualBasis_apply, apply_eq_star_dotProduct_toMatrix₂_mulVec, Complex.coe_basisOneI_repr, linearCombination_single, Module.Presentation.CokernelData.π_lift, Module.Basis.reindexFinsetRange_repr_self, Orthonormal.inner_right_finsupp, LinearMap.toMatrix_transpose_apply', Module.Basis.repr_symm_single_one, Rep.diagonalOneIsoLeftRegular_inv_hom, coe_basis, mapRange.linearEquiv_toLinearMap, LinearMap.toMatrix_smulRight, Module.Relations.solutionFinsupp_var, counit_single, NumberField.mixedEmbedding.stdBasis_repr_eq_matrixToStdBasis_mul, RootPairing.Base.toCoweightBasis_repr_coroot, AdjoinRoot.powerBasisAux'_repr_symm_apply, groupHomology.mapCycles₂_id_comp_assoc, groupHomology.π_comp_H1Iso_hom_apply, restrictDom_comp_subtype, lsum_apply, Basis.multilinearMap_apply_apply, Rep.standardComplex.d_of, Module.Basis.coe_finTwoProd_repr, linearCombination_unique, groupHomology.toCycles_comp_isoCycles₂_hom, coe_lsum, groupHomology.cyclesMap_comp_isoCycles₂_hom_apply, linearCombination_mapDomain, Span.finsupp_linearCombination_repr, groupHomology.mapCycles₁_id_comp, lapply_comp_lsingle_same, instIsCocomm, Module.Presentation.finsupp_relation, IsAdjoinRootMonic.coeff_apply_lt, DirectSum.IsInternal.collectedBasis_repr_of_mem_ne, Module.Basis.toMatrix_apply, KaehlerDifferential.mvPolynomialBasis_repr_apply, LinearMap.exists_finsupp_nat_of_prod_injective, sigmaFinsuppLEquivPiFinsupp_apply, groupHomology.eq_d₁₀_comp_inv, Rep.diagonalSuccIsoTensorTrivial_hom_hom_single, finsuppTensorFinsuppRid_single_tmul_single, LinearIndependent.linearCombinationEquiv_symm_apply, Module.DualBases.basis_repr_apply, LieAlgebra.LoopAlgebra.toFinsupp_single_tmul, Algebra.toMatrix_lmul', groupHomology.isoShortComplexH1_inv, Module.Relations.Solution.isPresentation_iff, Polynomial.derivativeFinsupp_map, groupHomology.eq_d₁₀_comp_inv_assoc, disjoint_supported_supported, llift_symm_apply, TensorProduct.finsuppScalarLeft_apply_tmul_apply, groupHomology.chainsMap_f_1_comp_chainsIso₁_assoc, groupHomology.isoCycles₁_hom_comp_i_apply, Module.Basis.symmetricAlgebra_repr_apply, supportedEquivFinsupp_symm_apply_coe_support_val, ZLattice.exists_forall_abs_repr_le_norm, groupHomology.lsingle_comp_chainsMap_f, groupHomology.coinvariantsMk_comp_opcyclesIso₀_inv_apply, Matrix.toLin_apply, MultilinearMap.freeFinsuppEquiv_apply, linearCombination_range, groupHomology.d₃₂_single_one_fst, Submodule.mulRightMap_apply, finsuppLEquivDirectSum_single, groupHomology.d₂₁_comp_d₁₀, Orthonormal.inner_finsupp_eq_sum_left, comul_single, groupHomology.d₂₁_single_self_inv_ρ_sub_inv_self, LinearIndependent.finsuppLinearCombination_injective, Representation.ker_leftRegular_norm_eq, lapply_apply, groupHomology.chainsMap_f_3_comp_chainsIso₃_assoc, groupHomology.single_ρ_self_add_single_inv_mem_boundaries₁, groupHomology.H1ToTensorOfIsTrivial_H1π_single, mapRange.linearMap_comp, Rep.ofMulActionSubsingletonIsoTrivial_hom_hom, Module.length_finsupp, linearCombination_restrict, Algebra.Presentation.differentials.comm₁₂, Module.Basis.sumQuot_repr_inr_of_mem, coe_basisSingleOne, coe_lmapDomain, Rep.linearizationTrivialIso_inv_hom, lapply_comp_lsingle_of_ne, groupHomology.cyclesOfIsCycle₁_coe, Module.Basis.repr_injective, LinearIndependent.repr_range, Module.Basis.repr_apply_eq, groupHomology.inhomogeneousChains.ext_iff, linearCombination_comp, LinearMap.polyCharpolyAux_map_aeval, Pi.counit_coe_finsupp, groupHomology.d₂₁_apply_mem_cycles₁, NumberField.inverse_basisMatrix_mulVec_eq_repr, range_linearCombination, supported_empty, Module.Relations.ker_toQuotient, groupHomology.cyclesMap_comp_isoCycles₁_hom_apply, bilinearCombination_apply, groupHomology.toCycles_comp_isoCycles₂_hom_apply, apply_linearCombination, Module.Relations.Solution.IsPresentation.surjective_π, Module.finite_finsupp_iff, KaehlerDifferential.kerTotal_mkQ_single_mul, linearIndependent_single_iff, MultilinearMap.freeFinsuppEquiv_single, groupHomology.eq_d₃₂_comp_inv_assoc, Module.Basis.end_repr_symm_apply, InnerProductSpace.toMatrix_rankOne, MvPolynomial.rTensor_symm_apply_single, Representation.free_single_single, lcongr_symm_single, Module.Basis.apply_eq_iff, Module.Flat.iff_forall_exists_factorization, sigmaFinsuppLequivDFinsupp_symm_apply, Module.Relations.Solution.π_comp_map, domLCongr_symm, Rep.finsuppTensorRight_inv_hom, setBasisOfTopLeSpanOfCardEqFinrank_repr_apply, LinearMap.polyCharpolyAux_coeff_eval, disjoint_lsingle_lsingle, linearCombination_comp_addSingleEquiv, Matrix.toLinAlgEquiv_apply, Module.Basis.coe_ofRepr, Module.equiv_free_prod_directSum, codisjoint_supported_supported_iff, MonomialOrder.div_set, TensorProduct.equivFinsuppOfBasisRight_apply, Submodule.mulLeftMap_eq_mulMap_comp, groupHomology.mapCycles₂_hom, groupHomology.isoCycles₂_inv_comp_iCycles_apply, groupHomology.isoCycles₂_inv_comp_iCycles_assoc, KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination, mapRange.linearMap_id, lcongr_single, Module.Flat.exists_factorization_of_apply_eq_zero_of_free, Matrix.toLinearMap₂_apply, linearEquivFunOnFinite_apply, ZLattice.abs_repr_lt_of_norm_lt, groupHomology.cyclesMk₂_eq, groupHomology.chainsMap_f_1_comp_chainsIso₁, Rep.coindVEquiv_apply_hom, LieAlgebra.LoopAlgebra.residuePairing_apply_apply, KaehlerDifferential.kerTotal_mkQ_single_smul, MvPolynomial.rTensor_apply_monomial_tmul, Algebra.Generators.repr_CotangentSpaceMap, Module.Basis.dualBasis_repr, groupHomology.H1π_eq_zero_iff, groupHomology.isoCycles₁_inv_comp_iCycles_assoc, sumFinsuppLEquivProdFinsupp_symm_inr, groupHomology.π_comp_H1Iso_hom_assoc, Pi.comul_coe_finsupp, groupHomology.chainsMap_f_2_comp_chainsIso₂, groupHomology.d₂₁_single_one_fst, supported_mono, groupHomology.H2π_comp_map, NumberField.mixedEmbedding.stdBasis_apply_isReal, Module.Basis.linearCombination_repr, union_support_maximal_linearIndependent_eq_range_basis, Module.FinitePresentation.out, Module.Basis.linearCombination_coord, Representation.FiniteCyclicGroup.coinvariantsKer_leftRegular_eq_ker, Module.Basis.tensorAlgebra_repr_apply, Module.projective_def', groupHomology.H1π_comp_map_assoc, Representation.coinvariantsTprodLeftRegularLEquiv_symm_apply, Module.Basis.equivFun_apply, TensorProduct.finsuppScalarLeft_apply_tmul, groupHomology.instEpiModuleCatH1π, Module.Basis.mulOpposite_repr_eq, Module.Basis.coe_repr_symm, Module.Basis.smulTower'_repr, Module.Relations.toQuotient_map_apply, Rep.finsuppTensorLeft_inv_hom, LinearMap.snd_prodOfFinsuppNat, LinearEquiv.finsuppUnique_apply, linearCombination_fin_zero, Module.Relations.Solution.π_relation, exteriorPower.basis_repr_apply, linearEquivFunOnFinite_symm_single, Module.Basis.toMatrix_update, PiLp.basis_toMatrix_basisFun_mul, groupHomology.single_one_snd_sub_single_one_snd_mem_boundaries₂, linearCombination_equivMapDomain, mapRange.linearEquiv_apply, groupHomology.instEpiModuleCatH2π, Module.Basis.baseChange_repr_tmul, ZSpan.repr_fract_apply, finsuppLEquivDirectSum_apply, Rep.leftRegularTensorTrivialIsoFree_inv_hom, Algebra.TensorProduct.basis_repr_symm_apply, LinearMap.polyCharpoly_map_eq_charpoly, Submodule.mulLeftMap_apply_single, single_mem_span_single, KaehlerDifferential.quotKerTotalEquiv_symm_apply, finsuppTensorFinsuppLid_apply_apply, LinearIndependent.repr_ker, groupHomology.H1π_comp_map, groupHomology.chainsMap_f_hom, groupHomology.d₃₂_apply_mem_cycles₂, mapRange.linearMap_toAddMonoidHom, QuaternionAlgebra.coe_basisOneIJK_repr, Ideal.range_finsuppTotal, groupHomology.boundariesOfIsBoundary₂_coe, groupHomology.cyclesMk₁_eq, finsuppLequivDFinsupp_apply_apply, groupHomology.mapCycles₂_comp_i_assoc, domLCongr_apply, Rep.linearization_δ_hom, curryLinearEquiv_apply, QuadraticMap.apply_linearCombination', groupHomology.mapCycles₂_id_comp_apply, finsuppTensorFinsupp_symm_single, linearCombination_linearCombination, Module.Relations.Solution.IsPresentation.desc_comp_π, lift_symm_apply, groupHomology.H2π_comp_map_apply, NumberField.mixedEmbedding.latticeBasis_repr_apply, Module.Basis.ext_elem_iff, Rep.FiniteCyclicGroup.homResolutionIso_hom_f_hom_apply, KaehlerDifferential.mvPolynomialBasis_repr_D, Rep.FiniteCyclicGroup.homResolutionIso_inv_f_hom_apply_hom_hom_apply, finsuppTensorFinsuppRid_apply_apply, Module.Basis.repr_sum_self, RootPairing.Base.toWeightBasis_repr_root, Rep.finsuppTensorLeft_hom_hom, linearCombination_option, Polynomial.derivativeFinsupp_one, Algebra.leftMulMatrix_mulVec_repr, linearCombination_eq_fintype_linearCombination, KaehlerDifferential.ker_map_of_surjective, basisSingleOne_repr, supportedEquivFinsupp_symm_apply_coe, groupHomology.inhomogeneousChains.d_comp_d, Module.Basis.smulTower_repr_mk, Module.Basis.repr_support_subset_of_mem_span, Algebra.Generators.cotangentRestrict_mk, supportedEquivFinsupp_apply_support_val, finsuppLequivDFinsupp_symm_apply, lsum_comp_lsingle, Algebra.Generators.cotangentRestrict_bijective_of_basis_kaehlerDifferential, TensorProduct.finsuppRight_symm_apply_single, Module.Basis.linearMap_repr_symm_apply, KaehlerDifferential.mvPolynomialBasis_repr_comp_D, finsuppTensorFinsuppLid_single_tmul_single, Rep.indMap_hom, groupHomology.isoCycles₁_hom_comp_i_assoc, isArtinian_finsupp, span_range_eq_top_iff_surjective_finsuppLinearCombination, groupHomology.d₁₀_eq_zero_of_isTrivial, Representation.leftRegular_norm_apply, supported_iUnion, Representation.coinvariantsToFinsupp_mk_single, Module.Relations.Solution.fromQuotient_comp_toQuotient, Module.Basis.SmithNormalForm.repr_comp_embedding_eq_smul, Representation.ind_mk, Module.Basis.toDual_linearCombination_right, TensorProduct.finsuppLeft_apply_tmul, PiTensorProduct.ofFinsuppEquiv'_apply_apply, groupHomology.d₂₁_single_one_snd, LinearMap.toMatrix_toSpanSingleton, linearDepOn_iffₛ, Representation.coinvariantsFinsuppLEquiv_symm_apply, Module.finrank_finsupp_self, Polynomial.support_derivativeFinsupp_subset_range, TensorProduct.equivFinsuppOfBasisRight_symm, groupHomology.d₃₂_comp_d₂₁, groupHomology.d₃₂_single_one_snd, exteriorPower.basis_repr, groupHomology.π_comp_H2Iso_hom_apply, Module.Relations.Solution.span_relation_le_ker_π, IsBaseChange.basis_repr_comp, Rep.diagonalOneIsoLeftRegular_hom_hom, mapRange.linearEquiv_trans, OrthonormalBasis.coe_toBasis_repr_apply, mapDomain.toLinearMap_linearEquiv, Rep.coinvariantsTensorIndHom_mk_tmul_indVMk, ZLattice.normBound_spec, QuadraticMap.apply_linearCombination, groupHomology.mapCycles₁_hom, groupHomology.isoCycles₁_inv_comp_iCycles, AdjoinRoot.powerBasisAux'_repr_apply_to_fun, IsAdjoinRootMonic.coeff_apply_coe, Module.Basis.mapCoeffs_repr, Module.Basis.traceDual_repr_apply, groupHomology.cyclesMap_comp_isoCycles₁_hom_assoc, linearIndependent_single, groupHomology.single_mem_cycles₁_of_mem_invariants, groupHomology.isoShortComplexH2_inv, groupHomology.coe_mapCycles₁, groupHomology.toCycles_comp_isoCycles₁_hom, Module.Relations.toQuotient_relation, Module.Flat.exists_factorization_of_comp_eq_zero_of_free, moduleIsTorsionFree, Module.Basis.ofZLatticeComap_repr_apply, Module.projective_def, groupHomology.chainsMap_f_2_comp_chainsIso₂_assoc, Module.End.ringEquivEndFinsupp_apply_apply_support, groupHomology.mapCycles₂_comp_i_apply, Module.annihilator_finsupp, LinearMap.toMvPolynomial_eval_eq_apply, KaehlerDifferential.mvPolynomialBasis_repr_symm_single, groupHomology.boundariesToCycles₂_apply, Module.Basis.toMatrix_transpose_apply, Module.finite_finsupp_self_iff, groupHomology.cyclesOfIsCycle₂_coe, rank_finsupp', Rep.freeLift_hom, groupHomology.isoCycles₂_hom_comp_i, groupHomology.π_comp_H1Iso_hom, groupHomology.isoCycles₂_inv_comp_iCycles, Module.Basis.ofZLatticeBasis_repr_apply, groupHomology.d₁₀_single_inv, groupHomology.mkH1OfIsTrivial_apply, span_single_image, IsIsotypic.linearEquiv_finsupp, LinearMap.toMatrixAlgEquiv_apply', MonomialOrder.div, linearCombination_surjective, IntermediateField.LinearDisjoint.basisOfBasisLeft_repr_apply, MvPolynomial.scalarRTensor_symm_apply_single, lsingle_range_le_ker_lapply, groupHomology.d₁₀ArrowIso_hom_right, TensorProduct.equivFinsuppOfBasisLeft_apply, Module.Basis.prod_repr_inr, range_lmapDomain, Rep.leftRegularTensorTrivialIsoFree_hom_hom_single_tmul_single, groupHomology.single_one_mem_boundaries₁, linearCombination_comp_lmapDomain, mem_span_image_iff_linearCombination, MvPolynomial.irreducible_sumSMulXSMulY, BilinForm.apply_eq_dotProduct_toMatrix_mulVec, Module.Relations.toQuotient_map, groupHomology.d₂₁_single, mapDomain.linearEquiv_symm, Module.Basis.constr_def, groupHomology.inhomogeneousChains.d_eq, comul_comp_lsingle, groupHomology.cycles₁IsoOfIsTrivial_inv_apply, groupHomology.eq_d₂₁_comp_inv_apply, Module.Basis.toDual_apply_left, QuadraticMap.sum_repr_sq_add_sum_repr_mul_polar, MvPolynomial.rTensor_apply, groupHomology.d₁₀_comp_coinvariantsMk_assoc, Module.DualBases.lc_def, groupHomology.isoCycles₁_hom_comp_i, Module.Relations.Solution.IsPresentation.exact, NumberField.canonicalEmbedding_eq_basisMatrix_mulVec, Module.Relations.solutionFinsupp_isPresentation, groupHomology.chainsMap_f_0_comp_chainsIso₀_apply, Module.Basis.reindexFinsetRange_repr, Module.Basis.continuous_coe_repr, TensorProduct.finsuppScalarLeft_apply, Module.Basis.smulTower_repr, Module.Basis.toDual_apply_right, groupHomology.single_inv_ρ_self_add_single_mem_boundaries₁, linearCombination_comapDomain, ZSpan.repr_floor_apply, Module.Basis.repr_symm_apply, apply_linearCombination_id, Module.Basis.tensorProduct_repr_tmul_apply, Representation.free_asModule_free, groupHomology.lsingle_comp_chainsMap_f_assoc, Rep.linearizationTrivialIso_hom_hom, Basis.piTensorProduct_repr_tprod_apply, Module.Relations.Solution.π_comp_map_apply, groupHomology.single_mem_cycles₂_iff, TensorProduct.finsuppScalarLeft_symm_apply_single, Module.Basis.repr_smul', lmap_finsuppLEquivDirectSum_eq, ker_lsingle, Module.Projective.out, groupHomology.boundaries₁_le_cycles₁, Module.Basis.repr_unitsSMul, Representation.ind_apply, Module.Basis.constr_apply, Module.Finite.finsupp, Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_inv_f_hom_apply, linearCombination_eq_fintype_linearCombination_apply, Subalgebra.LinearDisjoint.basisOfBasisLeft_repr_apply, disjoint_supported_supported_iff, mem_finsuppAffineCoords_iff_linearCombination, Submodule.LinearDisjoint.linearIndependent_left_of_flat, Module.Basis.repr_eq_iff', linearDepOn_iff', Module.End.ringEquivEndFinsupp_symm_apply_apply, supportedEquivFinsupp_symm_apply_coe_apply, iSupIndep_range_lsingle, Rep.coinvariantsTensorIndInv_mk_tmul_indMk, Module.Basis.norm_repr_le_norm, MvPolynomial.scalarRTensor_apply_tmul_apply, Module.Basis.mem_span_repr_support, Pi.basisFun_repr, Algebra.leftMulMatrix_eq_repr_mul, Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op, finsuppProdLEquiv_symm_apply_apply, Module.Basis.sum_repr, lmapDomain_disjoint_ker, instFreeCarrierX₂ModuleCatProjectiveShortComplex, supportedEquivFinsupp_apply_apply, ZSpan.mem_fundamentalDomain, groupHomology.d₁₀ArrowIso_inv_left, groupHomology.H1AddEquivOfIsTrivial_symm_tmul, Module.Basis.toMatrix_mulVec_repr, groupHomology.single_mem_cycles₁_iff, groupHomology.d₂₁_single_ρ_add_single_inv_mul, single_mem_supported, TensorProduct.sum_tmul_basis_left_injective, finsetBasisOfLinearIndependentOfCardEqFinrank_repr_apply, Module.End.ringHomEndFinsupp_surjective, TensorProduct.equivFinsuppOfBasisRight_symm_apply, Module.Basis.algebraMapCoeffs_repr, Module.Basis.coord_apply, groupHomology.eq_d₁₀_comp_inv_apply, Module.Basis.mem_span_iff_repr_mem, Module.Basis.SmithNormalForm.repr_eq_zero_of_notMem_range, Module.Relations.Solution.IsPresentation.ker_π, KaehlerDifferential.quotKerTotalEquiv_apply, KaehlerDifferential.kerTotal_map', Rep.leftRegularTensorTrivialIsoFree_hom_hom, domLCongr_refl, groupHomology.d₂₁_comp_d₁₀_assoc, LinearMap.toMatrixAlgEquiv_transpose_apply, Module.Basis.repr_self, TensorProduct.finsuppRight_apply_tmul_apply, linearCombination_zero, Rep.linearization_ε_hom, lcomapDomain_eq_linearProjOfIsCompl, groupHomology.chainsMap_f_1_comp_chainsIso₁_apply, TensorProduct.finsuppRight_apply_tmul, Module.subsingletonEquiv_apply, Polynomial.degreeLT.basis_repr, Module.Basis.repr_reindex_apply, groupHomology.mapCycles₁_quotientGroupMk'_epi, groupHomology.mapCycles₁_comp_i_assoc, LinearMap.toMatrix_apply, linearDepOn_iff'ₛ, NumberField.house.basis_repr_norm_le_const_mul_house, Rep.coinvariantsTensorFreeToFinsupp_mk_tmul_single, linearIndependent_single_one, LinearMap.map_finsupp_linearCombination, Rep.FiniteCyclicGroup.leftRegular.range_applyAsHom_sub_eq_ker_linearCombination, range_restrictDom, LinearMap.toMatrixAlgEquiv_transpose_apply', Module.Relations.Solution.π_single, lmapDomain_apply, linearIndepOn_iff_linearCombinationOnₛ, Module.presentationFinsupp_var, finiteDimensional_finsupp, Rep.linearization_map_hom, domLCongr_single, groupHomology.mem_cycles₁_iff, instIsSemisimpleModuleFinsupp, TensorProduct.coe_finsuppScalarRight', LinearMap.polyCharpoly_coeff_eval, lsum_comp_mapRange_toSpanSingleton, Module.Basis.repr_unop_eq_mulOpposite_repr, groupHomology.boundariesToCycles₁_apply, groupHomology.single_mem_cycles₂_iff_inv, groupHomology.d₁₀_single, linearIndepOn_iff_linearCombinationOn, lsingle_apply, Representation.IndV.hom_ext_iff, Module.Relations.Solution.IsPresentation.π_desc_apply, Pi.comul_comp_finsuppLcoeFun, exteriorPower.basis_repr_self, TensorProduct.finsuppLeft'_apply, Polynomial.derivativeFinsupp_C, TensorProduct.finsuppRight_tmul_single, Submodule.mulRightMap_apply_single, Module.finrank_finsupp, Rep.indResHomEquiv_symm_apply_hom, groupHomology.isoCycles₂_hom_comp_i_assoc, groupHomology.comp_d₃₂_eq, Polynomial.derivativeFinsupp_X, finsuppLEquivDirectSum_symm_lof, Module.Flat.exists_factorization_of_isFinitelyPresented, groupHomology.H2π_eq_zero_iff, curryLinearEquiv_symm_apply, KaehlerDifferential.linearCombination_surjective, lsubtypeDomain_apply, HahnSeries.coeff_ofFinsuppLinearMap, Module.Basis.sumQuot_repr_inr, linearCombination_onFinset, lsum_symm_apply, Module.Basis.coord_repr_symm, Module.Relations.Solution.linearCombination_var_relation, lcongr_apply_apply, Rep.leftRegularTensorTrivialIsoFree_inv_hom_single_single, ZLattice.abs_repr_le, DirectSum.IsInternal.collectedBasis_repr_of_mem, Module.Basis.coe_sumCoords, iSup_lsingle_range, LinearMap.sum_repr_mul_repr_mul, mem_supported', lmapDomain_supported, Module.Basis.ofIsLocalizedModule_repr_apply, Rep.linearization_μ_hom, Module.Relations.Solution.ofQuotient_π, Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply, groupHomology.H1π_eq_iff, LinearMap.fst_prodOfFinsuppNat, groupHomology.d₃₂_comp_d₂₁_apply, TensorProduct.finsuppLeft_apply_tmul_apply, groupHomology.chainsMap_f, groupHomology.chainsMap_f_2_comp_chainsIso₂_apply, LinearMap.polyCharpolyAux_map_eq_toMatrix_charpoly, Representation.ofMulAction_def, groupHomology.cyclesMap_comp_isoCycles₁_hom, sigmaFinsuppLequivDFinsupp_apply, KaehlerDifferential.kerTotal_mkQ_single_algebraMap, MvPolynomial.rTensorAlgHom_apply_eq

Function.Injective

Definitions

Name	Category	Theorems
module 📖	CompOp	—

Function.Surjective

Definitions

Name	Category	Theorems
module 📖	CompOp	—

IsLocalizedModule

Definitions

Name	Category	Theorems
module 📖	CompOp	1 math math: isScalarTower_module

LieSubmodule.Quotient

Definitions

Name	Category	Theorems
module 📖	CompOp	12 math math: mk_eq_zero, mk'_ker, toEnd_comp_mk', range_mk', coe_lowerCentralSeries_ideal_quot_eq, map_mk'_eq_bot_le, surjective_mk', lieModuleHom_ext_iff, LieSubalgebra.normalizer_eq_self_iff, lieQuotientLieModule, isNoetherian, mk'_apply

LinearMap

Definitions

Name	Category	Theorems
module 📖	CompOp	1838 math math: Pi.comul_eq_adjoint, det_toContinuousLinearMap, LieAlgebra.IsKilling.rootSystem_toLinearMap_apply, lTensor_ker_subtype_tensorKerEquiv_symm, isSymm_zero, RootPairing.InvariantForm.apply_reflection_reflection, CliffordAlgebra.contractRight_algebraMap_mul, restrictScalars_toMatrix, Orientation.kahler_map_complex, RootPairing.rootForm_self_smul_coroot, LieDerivation.IsKilling.ad_mem_ker_killingForm_ad_range_of_mem_orthogonal, apply_symm_toPerfPair_self, SeparatingRight.toMatrix₂', RootPairing.RootPositiveForm.algebraMap_rootLength, SemimoduleCat.Hom.hom₂_apply, charpoly_def, LinearEquiv.congrRight₂_refl, iSupIndep.dfinsupp_lsum_injective, BilinForm.dualSubmoduleParing_spec, Module.End.exp_mul_of_derivation, PiTensorProduct.lift_reindex, BilinForm.zero_right, dot_self_cross, CliffordAlgebra.contractRight_algebraMap, DFinsupp.lsum_lsingle, Matrix.toLin_kronecker, Matrix.toLin'_symm, Matrix.SeparatingLeft.toBilin', detAux_def'', RootPairing.posRootForm_posForm_apply_apply, QuadraticMap.smul_toBilin, TensorProduct.LieModule.liftLie_apply, BilinForm.dualSubmoduleToDual_apply_apply, PiTensorProduct.lift.tprod, CliffordAlgebra.contractRight_eq, RootPairing.four_nsmul_coPolarization_compl_polarization_apply_root, BilinForm.toMatrix_toBilin, Module.Basis.end_repr_apply, adjoint_adjoint, Matrix.SeparatingRight.toLinearMap₂, IsReflective.regular, BilinForm.symmCompOfNondegenerate_left_apply, PolyEquivTensor.toFunBilinear_apply_eq_sum, BilinForm.dualSubmoduleToDual_injective, Rep.MonoidalClosed.linearHomEquiv_symm_hom, instIsTorsionFree, Subspace.dualAnnihilator_dualAnnihilator_eq_map, LinearEquiv.flip_apply, PiTensorProduct.liftAlgHom_apply, BilinForm.toMatrix_symm, SimpleGraph.lapMatrix_toLinearMap₂', RootPairing.coroot_eq_polarizationEquiv_apply_root, Nondegenerate.congr, dotProductEquiv_symm_apply, addMonoidHomLequivNat_symm_apply, isNilRegular_iff_natTrailingDegree_charpoly_eq_nilRank, Matrix.separatingRight_toLinearMap₂'_iff, Complex.kahler, CliffordAlgebra.changeForm_comp_changeForm, LieAlgebra.IsKilling.restrict_killingForm_eq_sum, Complex.areaForm, RootPairing.RootPositiveForm.zero_lt_posForm_apply_root, CStarModule.innerₛₗ_apply, PiTensorProduct.dualDistribEquivOfBasis_symm_apply, QuadraticMap.canLift, Polynomial.toMatrix_sylvesterMap', Matrix.toLinearMapRight'_mul, ContinuousLinearMap.toLinearMap₁₂_injective, Subspace.instModuleDualFiniteDimensional, isPositive_adjoint_comp_self, BilinForm.IsAlt.neg_eq, BilinMap.polar_toQuadraticMap, isAdjointPair_iff_comp_eq_compl₂, IsPerfPair.bijective_left, Submodule.map_dualCoannihilator_le, Matrix.separatingRight_toLinearMap₂_iff, LinearEquiv.coe_toLinearMap_flip, CommRing.Pic.mk_dual, Matrix.spectrum_toEuclideanLin, Matrix.iSup_eigenspace_toLin'_diagonal_eq_top, addMonoidHomLequivInt_apply, Module.Basis.toDual_linearCombination_left, Module.Basis.constr_comp, LinearEquiv.congrRight_symm, Module.Basis.constr_symm_apply, mapMatrixLinear_apply, BilinForm.toMatrixAux_eq, LieModule.trace_toEnd_eq_zero_of_mem_lcs, Matrix.toLinearMap₂'_comp, PiTensorProduct.lift_reindex_symm, RootPairing.root_coroot_two, Representation.finsupp_apply, toMatrix_apply', llcomp_apply, toMatrix₂_mul, BilinForm.smul_left_of_tower, Orientation.kahler_comp_rightAngleRotation', LieAlgebra.LoopAlgebra.twoCochainOfBilinear_apply_apply, Module.Basis.dualBasis_coord_toDualEquiv_apply, Matrix.repr_toLin, LieModule.traceForm_eq_sum_genWeightSpaceOf, LocalizedModule.restrictScalars_map_eq, polar_eq_iInter, tensorKer_tmul, TensorProduct.sum_tmul_basis_right_injective, Subspace.dualLift_of_subtype, RootPairing.EmbeddedG2.shortAddLongRoot_shortRoot, compRight_apply, Matrix.liftLinear_single, Orientation.inner_mul_inner_add_areaForm_mul_areaForm, Subspace.dualRestrict_comp_dualLift, Orientation.areaForm_le, BilinForm.congr_apply, LinearEquiv.conj_trans, LieModule.traceForm_eq_sum_finrank_nsmul', mul_apply_apply, BilinForm.toMatrix_compRight, RootPairing.rootSpan_dualAnnihilator_le_ker_rootForm, adjoint_innerₛₗ_apply, BilinMap.toQuadraticMap_add, Submodule.dualQuotEquivDualAnnihilator_apply, ContinuousLinearMap.toBilinForm_apply, LinearEquiv.conj_apply_apply, adjoint_eq_toCLM_adjoint, BilinForm.IsNonneg.nonneg, LieAlgebra.IsKilling.instIsReducedSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, bilinearIteratedFDerivTwo_eq_iteratedFDeriv, IsPosSemidef.add, Matrix.toLinearMap₂_toMatrix₂, toMatrix_one, IsReflective.smul_coroot, QuotSMulTop.equivTensorQuot_naturality, LieAlgebra.IsKilling.rootSystem_coroot_apply, compl₂_apply, IsReflective.apply_self_mul_coroot_apply, Module.Flat.ker_lTensor_eq, isSelfAdjoint_toContinuousLinearMap_iff, cross_cross, Matrix.separatingRight_toBilin_iff, Matrix.spectrum_toLpLin, RootPairing.InvariantForm.pairing_mul_eq_pairing_mul_swap, Submodule.quotDualCoannihilatorToDual_apply, Module.Dual.eval_comp_comp_evalEquiv_eq, CategoryTheory.Linear.comp_apply, BilinForm.toMatrix'_symm, LinearEquiv.map_mem_invtSubmodule_conj_iff, QuadraticMap.polarBilin_prod, mem_isPairSelfAdjointSubmodule, Submodule.dualPairing_apply, NonUnitalAlgHom.coe_lmul_eq_mul, Matrix.toLin_mul_apply, tensorEqLocusEquiv_apply, toMatrix_distrib_mul_action_toLinearMap, domRestrict'_apply, Module.dualMap_dualMap_eq_iff, ModuleCat.Iso.homCongr_eq_arrowCongr, AlgEquiv.linearEquivConj_mulLeft, PiToModule.fromEnd_apply, TensorPower.gMul_eq_coe_linearMap, BilinForm.IsRefl.groupSMul, BilinForm.toMatrix'_apply, Module.Basis.dual_rank_eq, BilinMap.toQuadraticMapAddMonoidHom_apply, BilinForm.SeparatingRight.toMatrix, nondegenerate_toLinearMap₂'_of_det_ne_zero', LinearEquiv.congrRight₂_apply, cross_self, CliffordAlgebra.contractRight_mul_ι, Matrix.Nondegenerate.toLinearMap₂, rTensor_injective_iff_lcomp_surjective, TensorProduct.gradedMul_assoc, contractLeft_assoc_coevaluation, Module.Basis.flag_le_ker_dual, LieModule.lowerCentralSeries_one_inf_center_le_ker_traceForm, Matrix.intrinsicStar_toLin', coe_toContinuousLinearMap', BilinMap.toQuadraticMap_zero, SemimoduleCat.Iso.homCongr_eq_arrowCongr, LinearEquiv.dualMap_trans, Module.Basis.coe_dualBasis, QuadraticMap.toBilinHom_apply, ExteriorAlgebra.liftAlternating_ι_mul, tensorKer_coe, Real.vector_fourierIntegral_eq_integral_exp_smul, RootPairing.coPolarization_apply_eq_zero_iff, Fintype.linearIndependent_iff', QuadraticMap.associated_eq_self_apply, QuadraticMap.toMatrix'_comp, trace_mul_cycle, SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix, ker_localizedMap_eq_localized'_ker, exteriorPower.alternatingMapLinearEquiv_ιMulti, star_dotProduct_toMatrix₂_mulVec, RootPairing.root_coroot_eq_pairing, IsBaseChange.linearMapLeftRightHom_comp_apply, TensorProduct.AlgebraTensorModule.lTensor_comp_cancelBaseChange, Subspace.dualAnnihilator_dualAnnihilator_eq, TensorProduct.AlgebraTensorModule.lTensor_id, zero_prodMap_dualTensorHom, sum_repr_mul_repr_mulₛₗ, TensorProduct.AlgebraTensorModule.rTensor_one, innerₛₗ_apply_apply, ringLmapEquivSelf_symm_apply, RootPairing.self_comp_coPolarization_eq_corootForm, QuadraticMap.associated_rightInverse, Matrix.toBilin_symm, spectrum_toMatrix', LieModule.trace_toEnd_genWeightSpace, Real.smul_map_diagonal_volume_pi, Algebra.traceMatrix_apply, RootPairing.span_coroot'_eq_top, toMatrix_rotation, PiTensorProduct.toDualContinuousMultilinearMap_apply_apply, BilinForm.sum_right, separatingRight_congr_iff, Module.eval_apply_injective, AlgHom.mulLeftRightMatrix.inv_comp, Submodule.dualAnnihilator_map_dualMap_le, LieAlgebra.IsKilling.ker_traceForm_eq_bot_of_isCartanSubalgebra, le_comap_range_lTensor, Submodule.dualCopairing_eq, polar_singleton, LieModule.shiftedGenWeightSpace.toEnd_eq, TensorProduct.liftAux_tmul, RootPairing.RootPositiveForm.zero_lt_posForm_iff, QuotSMulTop.equivTensorQuot_naturality_mk, RootPairing.toLinearMap_apply_CoPolarization, BilinForm.ext_iff_of_isSymm, dot_cross_self, LieModule.Cohomology.twoCochain_alt, Module.endTensorEndAlgHom_apply, Matrix.toLpLin_mul_same, mem_skewAdjointSubmodule, Submodule.quotDualCoannihilatorToDual_nondegenerate, RootPairing.Hom.weight_coweight_transpose, Module.DualBases.coe_dualBasis, CliffordAlgebra.changeForm_self_apply, LieModule.toLinearMap_maxTrivLinearMapEquivLieModuleHom, Submodule.mem_dualAnnihilator, PiTensorProduct.mul_assoc, toMatrix_baseChange, RootPairing.toPerfPair_comp_root, Submodule.dualAnnihilator_eq_top_iff, TensorProduct.adjoint_map, IsContPerfPair.bijective_left, LinearEquiv.conj_comp, TensorProduct.dualDistrib_dualDistribInvOfBasis_right_inverse, CliffordAlgebra.contractRight_comm, trace_eq_matrix_trace_of_finset, adjoint_toContinuousLinearMap, Matrix.SeparatingLeft.toLinearMap₂', rTensor_comp_flip_mk, TensorProduct.dualDistrib_apply_comm, BilinForm.IsAlt.eq_of_add_add_eq_zero, Submodule.coe_dualCoannihilator_span, Finsupp.llift_apply, ExteriorAlgebra.liftAlternating_comp, LinearEquiv.conj_refl, CharacterModule.curry_apply_apply, map_sub₂, finrank_algHom, Module.Basis.toDualEquiv_apply, CliffordAlgebra.changeFormAux_changeFormAux, Fintype.bilinearCombination_apply_single, BilinForm.nondegenerate_toMatrix_iff, toMatrixₛₗ₂'_symm, Ideal.range_mul', dotProductEquiv_apply_apply, rank_diagonal, ker_localizedMap_eq_localized₀_ker, QuotSMulTop.map_comp_mkQ, RootPairing.injOn_dualMap_subtype_span_root_coroot, RootPairing.disjoint_corootSpan_ker_corootForm, TensorProduct.flip_mk_surjective, separatingRight_toMatrix₂'_iff, Matrix.toEuclideanLin_apply, Submodule.biSup_eq_range_dfinsupp_lsum, CliffordAlgebra.changeForm.add_proof, ContinuousLinearMap.toLinearMap₁₂_apply, Algebra.toMatrix_lmul_eq, LieModule.traceForm_eq_zero_of_isNilpotent, isAdjointPair_toLinearMap₂, LinearEquiv.congrLeft_symm_apply, LieModule.Cohomology.d₂₃_apply, BilinForm.sub_left, exteriorPower.alternatingMapLinearEquiv_symm_map, LieIdeal.coe_killingCompl_top, Rep.homEquiv_apply_hom, ModN.basis_apply_eq_mkQ, Matrix.liftLinear_singleLinearMap, LieModule.Cohomology.d₁₂_apply_apply, FiniteDimensional.mem_span_of_iInf_ker_le_ker, Matrix.liftLinear_comp_singleLinearMap, ofIsComplProd_apply, mul_apply', toMatrix_symm, Matrix.toBilin_apply, toMatrix₂'_compl₁₂, RootPairing.flip_comp_polarization_eq_rootForm, prodEquiv_apply, Module.Basis.toLin_toMatrix, TensorProduct.AlgebraTensorModule.restrictScalars_lTensor, flip_injective_iff₁, Matrix.toLinearEquivRight'OfInv_symm_apply, Pi.counit_eq_adjoint, BilinForm.coeFnAddMonoidHom_apply, LocalizedModule.map_surjective, IsLocalizedModule.map_linearMap_of_isLocalization, Matrix.toLin'_mul_apply, Orientation.areaForm_to_volumeForm, Matrix.toLinearMapₛₗ₂'_single, LieAlgebra.bracket_ofTwoCocycle, Matrix.nondegenerate_toBilin'_iff, TensorProduct.AlgebraTensorModule.restrictScalars_curry, trace_eq_contract_of_basis', BilinForm.congr_comp, Algebra.traceForm_apply, CliffordAlgebra.changeFormEquiv_symm, BilinForm.apply_dualBasis_left, QuadraticMap.add_toBilin, Basis.linearEquiv_dual_iff_finiteDimensional, Orientation.areaForm_neg_orientation, toMatrix'_toLin', MultilinearMap.curryRight_apply, Module.mapEvalEquiv_apply, CliffordAlgebra.changeForm.zero_proof, RootPairing.ker_copolarization_eq_ker_corootForm, TensorProduct.toMatrix_assoc, Subspace.finrank_dualCoannihilator_eq, applyₗ'_apply_apply, Matrix.toLinearMap₂'_apply', RootPairing.exists_ge_zero_eq_rootForm, PiToModule.fromEnd_injective, coe_lTensorHom, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable, toMatrix_id, LinearIsometryEquiv.adjoint_toLinearMap_eq_symm, BilinForm.apply_eq_dotProduct_toMatrix_mulVec, Orientation.kahler_eq_zero_iff, eq_adjoint_iff_basis_left, BilinForm.isNonneg_def, ModuleCat.ExtendRestrictScalarsAdj.Counit.map_hom_apply, RootPairing.RootPositiveForm.posForm_apply_root_root_le_zero_iff, ProperCone.mem_dual, Subspace.finiteDimensional_quot_dualCoannihilator_iff, QuadraticMap.associated_prod, SemimoduleCat.ofHom₂_hom_apply_hom, toMatrix_transpose, toMatrix₂_toLinearMapₛₗ₂, IsLocalizedModule.map_surjective_iff_localizedModuleMap_surjective, LieIdeal.mem_killingCompl, Orientation.areaForm_swap, ContinuousLinearMap.holderₗ_apply_apply, isSymm_dualProd, trace_smulRight, IsRefl.ker_eq_bot_iff_ker_flip_eq_bot, Matrix.SpecialLinearGroup.toLin'_symm_to_linearMap, Orientation.kahler_swap, Matrix.toLinearMap₂'Aux_single, Matrix.vecMulBilin_apply, mk₂'ₛₗ_apply, cross_anticomm, dualMap_surjective_of_injective, Rep.ihom_ev_app_hom, IsBaseChange.endHom_toMatrix, Ideal.range_mul, ModuleCat.homLinearEquiv_symm_apply, cross_cross_eq_smul_sub_smul, Real.map_matrix_volume_pi_eq_smul_volume_pi, Submodule.dualAnnihilator_anti, BilinForm.sum_left, GradedTensorProduct.mul_def, QuadraticForm.polarBilin_tmul, LieAlgebra.hasCentralRadical_and_of_isIrreducible_of_isFaithful, LocalizedModule.map_injective, QuadraticMap.Ring.polarBilin_pi, RootPairing.toLinearMap_apply_apply_Polarization, BilinForm.Nondegenerate.congr, Rep.MonoidalClosed.linearHomEquivComm_hom, SemimoduleCat.homLinearEquiv_apply, TensorProduct.equivFinsuppOfBasisLeft_symm, IsProj.eq_conj_prodMap, flip_surjective_iff₁, tensorEqLocus_coe, ExteriorAlgebra.liftAlternatingEquiv_symm_apply, TensorProduct.curry_injective, Matrix.toLinearMap₂'_apply, lTensorHomEquivHomLTensor_apply, BilinForm.separatingLeft_toMatrix'_iff, extendScalarsOfIsLocalizationEquiv_apply, rank_lt_rank_dual', QuotSMulTop.map_surjective, Module.dual_finite, trace_mul_comm, trace_eq_contract, Orientation.kahler_comp_rightAngleRotation, toKerLocalized_isLocalizedModule, CliffordAlgebra.changeFormAux_apply_apply, LinearEquiv.arrowCongr_symm_apply, lcomp_injective_of_surjective, trace_eq_sum_inner, BilinForm.not_nondegenerate_zero, Module.FinitePresentation.linearEquivMap_symm_apply, restrictScalarsₗ_apply, Subspace.dualPairing_nondegenerate, Matrix.isNilpotent_toLin'_iff, trace_one, homTensorHomEquiv_apply, QuadraticMap.associated_apply, IsModuleTopology.continuous_bilinear_of_pi_fintype, TensorProduct.mk_apply, Matrix.toLinearEquiv'_symm_apply, BilinForm.IsSymm.eq, BilinForm.mul_toMatrix'_mul, Matrix.toLin_symm, SeparatingDual.dualMap_surjective_iff, InnerProductGeometry.norm_toLp_symm_crossProduct, dualTensorHom_apply, BilinForm.mul_toMatrix_mul, isNoetherian_linearMap_pi, Module.Invertible.rTensorEquiv_apply_apply, localized'_range_eq_range_localizedMap, lTensorHomEquivHomLTensor_toLinearMap, trace_comp_eq_mul_of_commute_of_isNilpotent, Module.Basis.linearMap_apply, Submodule.dualCoannihilator_iSup_eq, Module.comap_eval_surjective, Matrix.toLinearMap₂_compl₁₂, dualMap_id, ExteriorAlgebra.liftAlternating_ιMulti, linearIndependent_algHom_toLinearMap', CliffordAlgebra.changeForm_ι_mul, LieModule.traceForm_lieSubalgebra_mk_right, lsum_apply, Matrix.toLinearMapₛₗ₂'_symm, localized'_ker_eq_ker_localizedMap, mk₂_apply, polyCharpoly_baseChange, CliffordAlgebra.changeForm_self, isAlt_iff_eq_neg_flip, RootPairing.RootPositiveForm.exists_eq, RootPairing.corootSpan_map_flip_toPerfPair, toLinearMap_toContPerfPair, TensorProduct.AlgebraTensorModule.rTensor_comp, coevaluation_apply_one, MatrixEquivTensor.toFunBilinear_apply, isSymmetric_adjoint_mul_self, Submodule.dualRestrict_ker_eq_dualAnnihilator, IsBaseChange.transvection, toBilin'Aux_toMatrixAux, LieAlgebra.IsKilling.invtSubmoduleToLieIdeal_top, IsLocalRing.map_tensorProduct_mk_eq_top, Module.Dual.eval_apply, toMatrix₂_compl₂, DFinsupp.sum_mapRange_index.linearMap, InnerProductSpace.trace_rankOne, rTensorHomEquivHomRTensor_apply, trace_eq_contract_apply, ExteriorAlgebra.liftAlternatingEquiv_apply, dualTensorHomEquivOfBasis_symm_cancel_right, Matrix.toLpLin_symm_pow, IsLocalization.tensorProduct_isLocalizedModule, LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton', RootPairing.rootForm_root_self, IsBaseChange.det_endHom, polyCharpolyAux_map_eval, Matrix.toLinearMapₛₗ₂'_aux_eq, IsLocalizedModule.mapExtendScalars_apply_apply, CommRing.Pic.inv_eq_dual, BilinForm.isRefl_zero, IsPerfectCompl.isCompl_right, KaehlerDifferential.linearMapEquivDerivation_apply_apply, BilinForm.tensorDistribEquiv_toLinearMap, Module.preReflection_preReflection, Matrix.linfty_opNNNorm_toMatrix, Module.Basis.end_apply_apply, Module.Basis.coe_constrL, QuadraticMap.toQuadraticMap_associated, Module.Invertible.bijective_curry, Algebra.traceForm_toMatrix, trace_eq_zero_of_mapsTo_ne, cross_anticomm', Matrix.toLin_scalar, Module.rank_linearMap, adjoint_rTensor, toBilin'Aux_toMatrixAux, adjoint_inner_right, BilinForm.dotProduct_toMatrix_mulVec, BilinForm.tensorDistrib_tmul, TensorProduct.AlgebraTensorModule.mk_apply, trace_transpose', IntrinsicStar.starLinearEquiv_eq_arrowCongr, IsTensorProduct.equiv_symm_apply, Projectivization.cross_mk_of_cross_ne_zero, Matrix.toLpLin_one, RootPairing.InvariantForm.exists_apply_eq_or, Module.Dual.transpose_apply, Module.Invertible.instDual, isPosSemidef_iff_posSemidef_toMatrix, RootPairing.zero_le_rootForm, TensorProduct.AlgebraTensorModule.rTensor_tmul, Algebra.PreSubmersivePresentation.aevalDifferential_toMatrix'_eq_mapMatrix_jacobiMatrix, IsLocalizedModule.map_comp', QuadraticMap.toBilin_apply, Submodule.comap_dualAnnihilator, linearIndependent_algHom_toLinearMap, KaehlerDifferential.linearMapEquivDerivation_symm_apply, BilinForm.smul_right_of_tower, BilinForm.tensorDistribEquiv_apply, linearIndependent_toLinearMap, Matrix.piLp_ofLp_toEuclideanLin, BilinForm.toMatrix_apply, LieAlgebra.killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero, RootPairing.reflectionPerm_coroot, TensorProduct.map₂_apply_tmul, LieSubmodule.traceForm_eq_zero_of_isTrivial, PowerBasis.constr_pow_algebraMap, dualProd_apply_apply, lieEquivMatrix'_apply, TensorProduct.dualDistribEquivOfBasis_symm_apply, BilinForm.toMatrix_toBilin, Module.dual_free, isSymm_def, Matrix.toPerfectPairing_apply_apply, SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux, BilinForm.IsRefl.smul, BilinForm.IsNonneg.add, LieDerivation.exp_map_apply, compr₂_apply, minpoly_toMatrix', BilinForm.IsPosSemidef.smul, trace_prodMap, Submodule.iSup_dualAnnihilator_le_iInf, liftBaseChangeEquiv_symm_apply, LieAlgebra.IsKilling.instIsCrystallographicSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, QuadraticForm.dualProd_apply, Fintype.bilinearCombination_apply, toMatrix₂_comp, lsmul_flip_apply, Matrix.SeparatingRight.toLinearMap₂', Module.Basis.toDual_ker, killingForm_eq_zero_of_mem_zeroRoot_mem_posFitting, Module.Basis.dualBasis_apply_self, TensorProduct.lift.tmul', PointedCone.mem_dual, Submodule.mem_iSup_iff_exists_dfinsupp, Submodule.map₂_map_left, prodEquiv_symm_apply, BilinForm.add_left, tensorProductEnd_apply, lift_rank_lt_rank_dual', Matrix.toLpLin_pow, trace_eq_contract', RootPairing.toLinearMap_apply_PolarizationIn, Module.finrank_linearMap, LinearEquiv.symm_flip, InnerProductSpace.AlgebraOfCoalgebra.mul_def, BilinForm.separatingRight_toMatrix'_iff, Matrix.isUnit_toLin'_iff, Module.piEquiv_apply_apply, RootPairing.EmbeddedG2.long_eq_three_mul_short, BilinForm.smul_left, range_dualMap_eq_dualAnnihilator_ker, Submodule.dualCoannihilator_map_linearEquiv_flip, RootPairing.rootSpan_map_toPerfPair, innerₗ_apply, RootPairing.GeckConstruction.trace_toEnd_eq_zero, Matrix.toLin'_mul, Submodule.iSup_eq_range_dfinsupp_lsum, isNilpotent_toMatrix_iff, BilinForm.lieInvariant_iff, LieModule.Cohomology.twoCochain_skew, BilinForm.dotProduct_toMatrix_mulVec, Module.End.rTensorAlgHom_apply_apply, det_eq_det_toMatrix_of_finset, LieModule.traceForm_comm, LieSubmodule.trace_eq_trace_restrict_of_le_idealizer, toMatrix_reindexRange, ExteriorAlgebra.liftAlternating_apply_ιMulti, Matrix.ofLp_toLpLin, Module.range_piEquiv, Matrix.toMatrix₂Aux_toLinearMap₂'Aux, Matrix.toBilin'_symm, Module.Basis.dualBasis_equivFun, PiTensorProduct.piTensorHomMap_tprod_tprod, RootPairing.corootSpan_dualAnnihilator_map_eq, toMatrix_smulBasis_left, VertexOperator.ncoeff_apply, Module.Dual.transpose_comp, BilinForm.isSymm_iff_flip, RootPairing.corootForm_apply_apply, flip_bijective_iff₁, flip_bijective_iff₂, BilinForm.mem_dualSubmodule, range_dualMap_dual_eq_span_singleton, det_toLin', Matrix.kroneckerMapBilinear_apply_apply, SeparatingRight.congr, Ideal.pi_tensorProductMk_quotient_surjective, Matrix.rank_eq_finrank_range_toLin, Matrix.toLinearMapRight'_mul_apply, BilinForm.toMatrix_comp, CliffordAlgebra.EvenHom.contract, Module.Basis.linearCombination_dualBasis, PiTensorProduct.norm_eval_le_injectiveSeminorm, transpose_dualTensorHom, Matrix.toLin_mul, Rep.indResAdjunction_counit_app_hom_hom, FGModuleCat.FGModuleCatDual_obj, Matrix.toLin_pow, LieSubmodule.mem_baseChange_iff, QuadraticMap.associated_tmul, Module.Free.linearMap, Matrix.toLin_self, Module.Finite.linearMap, Submodule.dualCoannihilator_bot, LieDerivation.exp_apply, Finsupp.lsum_single, BilinForm.neg_left, BilinForm.sum_repr_mul_repr_mul, Orientation.kahler_neg_orientation, QuadraticForm.dualProdIsometry_invFun, separatingLeft_toMatrix₂'_iff, ModuleCat.ExtendRestrictScalarsAdj.HomEquiv.fromExtendScalars_hom_apply, CliffordAlgebra.contractLeft_contractLeft, TensorProduct.algebraMap_gradedMul, Submodule.dualCoannihilator_top, hasEigenvector_toLin'_diagonal, lcomp_apply, separatingRight_iff_flip_ker_eq_bot, Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin, Module.Dual.instIsReflecive, BilinForm.inf_orthogonal_self_le_ker_restrict, BilinForm.IsAlt.self_eq_zero, LieModule.Cohomology.d₁₂_apply_coe_apply_apply, Matrix.toLinearMapₛₗ₂'_toMatrix', continuous_of_isContPerfPair, toMatrix'_algebraMap, isNoetherian_linearMap, PiTensorProduct.liftEquiv_symm_apply, PiToModule.fromMatrix_apply_single_one, LocalizedModule.coe_map_eq, Representation.linHom_apply, QuadraticMap.associated_toQuadraticMap, dualMap_injective_of_surjective, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, dualMap_injective_iff, LinearEquiv.instIsPerfPair, Module.Basis.eval_range, BilinearForm.toMatrixAux_eq, CliffordAlgebra.contractLeft_ι_mul, QuadraticMap.two_nsmul_associated, RootPairing.rootForm_reflection_reflection_apply, dualPairing_nondegenerate, RootPairing.posRootForm_posForm_pos_of_ne_zero, Matrix.SeparatingLeft.toBilin, Matrix.charpoly_toLin, FGModuleCat.FGModuleCatCoevaluation_apply_one, toMatrixOrthonormal_symm_apply, LinearEquiv.congrLeft_apply, IsLocalizedModule.map_linearCombination, BilinMap.baseChange_tmul, Module.Flat.eqLocus_lTensor_eq, Subspace.dualRestrict_leftInverse, StrongDual.dualPairing_apply, BilinForm.IsPosSemidef.add, LinearEquiv.piRing_symm_apply, CliffordAlgebra.even.lift_ι, toMatrix_mulVec_repr, BilinForm.zero_left, toMatrix_innerₛₗ_apply, QuadraticMap.associated_left_inverse', LieModule.Cohomology.mem_twoCochain_iff, TensorProduct.lift.tmul, Matrix.toLin_apply_eq_zero_iff, TensorProduct.lTensorHomToHomLTensor_apply, IsLocalizedModule.mapEquiv_apply, Matrix.nondegenerate_toBilin_iff, range_toContinuousLinearMap, BilinForm.isSymm_neg, Matrix.toLin'_reindex, BilinForm.toDual_def, CliffordAlgebra.contractRight_mul_algebraMap, Matrix.PosSemidef.toLinearMap₂'_zero_iff, BilinForm.toMatrix'_compLeft, Matrix.toLinearMapₛₗ₂_apply, BilinForm.dualBasis_repr_apply, PiTensorProduct.liftEquiv_apply, lsmul_injective, Module.IsReflexive.bijective_dual_eval', AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_apply, ExteriorAlgebra.liftAlternating_one, Subspace.flip_quotDualCoannihilatorToDual_bijective, isSelfAdjoint_iff', Subspace.dual_finrank_eq, Submodule.apply_mem_map₂, Orientation.areaForm'_apply, Module.Basis.end_apply, Submodule.set_smul_eq_map, Matrix.Nondegenerate.toBilin, BilinForm.nondegenerate_toBilin'_iff_det_ne_zero, toMatrix'_apply, mem_span_iff_continuous_of_finite, Matrix.toLpLin_toLp, apply_eq_dotProduct_toMatrix₂_mulVec, coe_toContinuousLinearMap, IsLocalizedModule.map_injective, BilinForm.separatingRight_toMatrix_iff, RootPairing.prod_rootForm_smul_coroot_mem_range_domRestrict, flip_apply, BilinForm.SeparatingLeft.toMatrix', TensorProduct.mapBilinear_apply, LinearEquiv.conj_apply, finiteDimensional', Module.Basis.toDual_range, PiTensorProduct.dualDistribInvOfBasis_apply, Submodule.map₂_eq_span_image2, Rep.ihom_obj_ρ_apply, Matrix.ker_toLin_eq_bot, TensorProduct.AlgebraTensorModule.dualDistrib_apply, ContinuousLinearMap.toLinearMap_innerSL_apply, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, Submodule.dualAnnihilator_sup_eq, LieModule.Cohomology.d₁₂_apply_apply_ofTrivial, GradedTensorProduct.auxEquiv_mul, Fintype.linearIndependent_iff'ₛ, IsSymmetric.adjoint_eq, Module.FinitePresentation.linearEquivMap_apply, BilinMap.isSymm_iff_eq_flip, llcomp_apply', Submodule.dualCoannihilator_sup_eq, BilinForm.zero_apply, LieAlgebra.IsKilling.instIsRootSystemSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystem, QuotSMulTop.equivQuotTensor_naturality, smulRightₗ_apply, IsNonneg.smul, IsLocalizedModule.mapEquiv_symm_apply, toMatrix'_symm, Module.symm_dualMap_evalEquiv, RootPairing.algebraMap_rootFormIn, InnerProductGeometry.norm_ofLp_crossProduct, TensorProduct.AlgebraTensorModule.uncurry_apply, polyCharpolyAux_map_eq_charpoly, toContinuousLinearMap_eq_iff_eq_toLinearMap, Submodule.baseChange_eq_span, linearMap_toMatrix_mul_basis_toMatrix, Submodule.dualCopairing_apply, Module.dualPairing_apply, polyCharpolyAux_eval_eq_toMatrix_charpoly_coeff, TensorProduct.AlgebraTensorModule.coe_lTensor, BilinForm.isNonneg_zero, RootPairing.toPerfPair_flip_comp_coroot, Coalgebra.rTensor_counit_comp_comul, toMatrix_transpose_apply, BilinForm.IsSymm.sub, toMatrix_dualTensorHom, LinearEquiv.arrowCongr_apply, BilinForm.nondegenerate_iff_ker_eq_bot, ContinuousLinearMap.coeLM_apply, exteriorPower.alternatingMapLinearEquiv_apply_ιMulti, RootPairing.isCompl_rootSpan_ker_rootForm, CliffordAlgebra.changeForm_ι_mul_ι, TensorProduct.AlgebraTensorModule.homTensorHomMap_apply, RootPairing.reflectionPerm_root, isPositive_toContinuousLinearMap_iff, Matrix.range_toLin', prodMapLinear_apply, Submodule.map₂_span_singleton_eq_map_flip, RootPairing.rootForm_self_eq_zero_iff, PiTensorProduct.piTensorHomMap_tprod_eq_map, dualMap_apply', mul_toMatrix', Matrix.liftLinear_apply, TensorProduct.LieModule.coe_liftLie_eq_lift_coe, PiTensorProduct.lift_comp_reindex, ext_iff₂, PiTensorProduct.dualDistribEquivOfBasis_apply_apply, dualMap_bijective_iff, TensorPower.multilinearMapToDual_apply_tprod, HVertexOperator.of_coeff_coeff, PiTensorProduct.map₂_tprod_tprod, star_eq_adjoint, instLieModule, toMatrix_smulBasis_right, Matrix.toLpLin_symm_id, Module.Basis.toDual_eq_repr, Matrix.range_toLin_eq_top, LieModule.toLinearMap_maxTrivLinearMapEquivLieModuleHom_symm, Module.Basis.baseChange_end, Matrix.toLin_finTwoProd_apply, FDRep.char_linHom, HVertexOperator.coeff_inj_iff, BilinForm.dualBasis_eq_iff, Matrix.isPositive_toEuclideanLin_iff, addMonoidEndRingEquivInt_apply, ContinuousLinearMap.toLinearMap_eq_iff_eq_toContinuousLinearMap, Matrix.toBilin'Aux_single, LieModule.lie_traceForm_eq_zero, IsLocalizedModule.map_mk', exteriorPower.pairingDual_ιMulti_ιMulti, mul_toMatrix₂, PointwiseConvergenceCLM.coeLMₛₗ_apply, LieAlgebra.IsKilling.corootForm_rootSystem_eq_killing, QuadraticMap.discr_comp, Module.Basis.linearMap_repr_apply, exteriorPower.alternatingMapLinearEquiv_comp_ιMulti, lsmul_eq_DistribMulAction_toLinearMap, Module.eval_apply_eq_zero_iff, IsModuleTopology.continuous_bilinear_of_finite_left, lsum_single, Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner_naturality, LieModule.trace_comp_toEnd_genWeightSpace_eq, nondegenerate_congr_iff, TensorProduct.gradedMul_one, CliffordAlgebra.contractLeft_comm, Orientation.areaForm_comp_linearIsometryEquiv, FractionalIdeal.mem_dual, toMatrix'_mul, IsBaseChange.end, adjoint_toSpanSingleton, lcomp_apply', mul_basis_toMatrix, Module.Basis.toDual_toDual, Module.dualProdDualEquivDual_apply_apply, RootPairing.zero_le_posForm, BilinForm.sub_apply, IsLocalizedModule.rTensor, LinearEquiv.conj_symm_conj, killingForm_apply_apply, trace_id, toMatrix_algebraMap, spectrum_toMatrix, PowerBasis.constr_pow_gen, BilinForm.congr_refl, SeparatingRight.toMatrix₂, Module.Finite.of_isComplemented_domain, Matrix.isUnit_toLin_iff, RootPairing.InvariantForm.apply_weylGroup_smul, Subspace.quotAnnihilatorEquiv_apply, Projectivization.mk_eq_mk_iff_crossProduct_eq_zero, Matrix.diagonal_toLin', re_inner_adjoint_mul_self_nonneg, PointwiseConvergenceCLM.coeLM_apply, Module.isTorsionBySet_iff_subseteq_ker_lsmul, TensorProduct.AlgebraTensorModule.curry_apply, isPositive_self_comp_adjoint, Algebra.TensorProduct.mul_one, Matrix.charpoly_toLin', BilinForm.toMatrix_symm, BilinMap.toQuadraticMap_smul, Matrix.toLinearMap₂'_toMatrix', rank_dual_eq_card_dual_of_aleph0_le_rank', Matrix.toLinearMapₛₗ₂'_apply, IsLocalizedModule.map_apply, Algebra.TensorProduct.mk_one_injective_of_isScalarTower, Matrix.toLin'_pow, LieModule.Cohomology.mem_twoCocycle_iff, Ideal.ker_tensorProductMk_quotient, rank_dual_eq_card_dual_of_aleph0_le_rank, BilinForm.SeparatingLeft.toMatrix, TensorProduct.AlgebraTensorModule.mapBilinear_apply, BilinMap.toQuadraticMapLinearMap_apply, Matrix.l2_opNNNorm_def, RootPairing.rootForm_apply_apply, LieModule.traceForm_eq_zero_of_isTrivial, Module.Basis.dualBasis_apply, apply_toPerfPair_flip, apply_eq_star_dotProduct_toMatrix₂_mulVec, toMatrix_mul, toMatrix₂_compl₁₂, BilinForm.IsSymm.smul, Matrix.minpoly_toLin', RootPairing.iInf_ker_root'_eq, lift_rank_lt_rank_dual, eq_adjoint_iff, GradedTensorProduct.mulHom_apply, BilinForm.toMatrix_compRight, BilinForm.nondegenerate_toMatrix'_iff, Polynomial.wronskianBilin_apply, toMatrix₂_toLinearMap₂, detAux_def', toMatrix_transpose_apply', toMatrix'_intrinsicStar, BilinForm.congr_congr, finiteDimensional, isPosSemidef_zero, Orientation.nonneg_inner_and_areaForm_eq_zero_iff_sameRay, cross_apply, LinearEquiv.lieConj_apply, IsBaseChange.toDual_comp_apply, IsBaseChange.linearMapLeftRightHom_toMatrix, BilinForm.isAlt_zero, BilinForm.toMatrixAux_apply, PiTensorProduct.mapMultilinear_apply, Matrix.SpecialLinearGroup.toLin_equiv.symm_toLinearMap_eq, LinearEquiv.dualMap_symm, PiTensorProduct.dualDistrib_apply, DFinsupp.lsum_single, Matrix.toLinOfInv_symm_apply, toMatrix_smulRight, toMatrix₂Aux_eq, lsmul_apply, TensorProduct.AlgebraTensorModule.restrictScalars_rTensor, vecMulVecBilin_apply_apply, isAdjointPair_toLinearMap₂', Orientation.areaForm_comp_rightAngleRotation, Module.Dual.baseChange_apply_tmul, Matrix.toLin'_apply', QuotSMulTop.map_exact, isPairSelfAdjoint_equiv, toMatrix₂_symm, Module.End.commute_exp_left_of_commute, Configuration.ofField.crossProduct_eq_zero_of_dotProduct_eq_zero, congr_fun₂, Matrix.ker_toLin'_eq_bot_iff, vecCons₂_apply, Finsupp.lsum_apply, LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg, SeparatingLeft.toMatrix₂, toMatrix'_toLinearMap₂', TensorProduct.lift_comp_map, Derivation.llcomp_apply, BilinForm.toLin'Flip_apply, BilinForm.toMatrix_mul_basis_toMatrix, exteriorPower.pairingDual_apply_apply_eq_one, QuadraticForm.dualProdIsometry_toFun, BilinForm.mul_toMatrix', map_mul_iff, Finsupp.coe_lsum, Matrix.toLinearMap₂_symm, Matrix.coe_toEuclideanCLM_eq_toEuclideanLin, toMatrixRight'_id, leibniz_cross, Orientation.inner_rightAngleRotation_left, HVertexOperator.coeff_inj, minpoly_toMatrix, det_dualMap, PiToModule.fromEnd_apply_single_one, CharacterModule.homEquiv_symm_apply_apply_apply, BilinForm.IsRefl.neg, RootPairing.coroot_root_two, PiTensorProduct.piTensorHomMapFun₂_smul, adjoint_lTensor, LieModule.Cohomology.add_apply_apply, Matrix.nondegenerate_toLinearMap₂'_iff_nondegenerate_toLinearMap₂, multilinearCurryLeftEquiv_apply, CliffordAlgebra.evenToNeg_ι, toMatrixOrthonormal_apply, BilinMap.toQuadraticMap_list_sum, LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg_aux, lTensor_comp_mk, Module.Basis.toDual_injective, LieModule.traceForm_eq_zero_if_mem_lcs_of_mem_ucs, RootPairing.iInf_ker_coroot'_eq, Subspace.finrank_add_finrank_dualCoannihilator_eq, Module.dualProdDualEquivDual_apply, Orientation.kahler_apply_self, Module.Invertible.rTensorInv_leftInverse, RootPairing.Hom.weight_coweight_transpose_apply, BilinForm.IsSymm.add, QuadraticForm.toDualProd_apply, tendsto_iff_forall_eval_tendsto_topDualPairing, IsNonneg.nonneg, Orientation.inner_mul_areaForm_sub', AlternatingMap.curryLeftLinearMap_apply, PiTensorProduct.piTensorHomMap₂_tprod_tprod_tprod, Algebra.toMatrix_lmul', BilinForm.isAlt_neg, Submodule.map₂_span_span, BilinForm.ext_iff, Finsupp.llift_symm_apply, IsLocalizedModule.map_id, Orientation.kahler_rotation_left', Representation.linHom.invariantsEquivRepHom_symm_apply_coe, AffineMap.toConstProdLinearMap_apply, Matrix.ker_diagonal_toLin', RootPairing.RootPositiveForm.two_mul_apply_root_root, polyCharpolyAux_baseChange, coe_rTensorHom, range_dualMap_eq_dualAnnihilator_ker_of_surjective, Coalgebra.lift_lsmul_comp_counit_comp_comul, FGModuleCat.Iso.conj_eq_conj, Orientation.abs_areaForm_le, LieModule.traceForm_eq_sum_finrank_nsmul, Matrix.toBilin'_apply, LieAlgebra.conj_ad_apply, flip_injective_iff₂, RootPairing.prod_rootFormIn_smul_coroot_mem_range_PolarizationIn, toMatrix_prodMap, Matrix.toLin_apply, Matrix.toLin'_one, RootPairing.reflection_dualMap_eq_coreflection, Orientation.normSq_kahler, AlgEquiv.linearEquivConj_mulLeftRight, Module.Basis.constr_apply_fintype, LinearEquiv.symm_conj_apply, TensorProduct.LieModule.lift_apply, MultilinearMap.ofSubsingletonₗ_apply, DFinsupp.lsum_symm_apply, TensorProduct.isBaseChange, Module.isTorsionBy_iff_mem_ker_lsmul, Module.injOn_dualMap_subtype_span_range_range, Submodule.map₂_le, PiTensorProduct.mul_def, RootPairing.rootForm_self_sum_of_squares, map_zero₂, Matrix.toLin_conjTranspose, BilinForm.isRefl_neg, LinearEquiv.conj_conj_symm, LieAlgebra.IsKilling.span_weight_isNonZero_eq_top, QuadraticMap.canLift', TensorProduct.AlgebraTensorModule.lTensor_comp, homTensorHomEquiv_toLinearMap, Subspace.dualLift_of_mem, RootPairing.algebraMap_posRootForm_posForm, Submodule.dualAnnihilator_gc, AffineMap.toConstProdLinearMap_symm_apply, RootPairing.span_root'_eq_top, BilinForm.nondegenerate_congr_iff, exteriorPower.alternatingMapLinearEquiv_comp, Matrix.trace_toLin'_eq, Matrix.toLin'OfInv_symm_apply, Orientation.rotation_eq_matrix_toLin, BilinForm.flip_apply, RootPairing.orthogonal_corootSpan_eq, Module.eval_ker, isNilpotent_trace_of_isNilpotent, RootPairing.ker_rootForm_eq_dualAnnihilator, BilinForm.compRight_apply, Submodule.sup_dualAnnihilator_le_inf, Module.FinitePresentation.linearEquivMapExtendScalars_apply, IsBaseChange.endHom_apply, isSkewAdjoint_iff_neg_self_adjoint, polyCharpoly_eq_of_basis, Module.Invertible.rTensorInv_injective, TensorProduct.AlgebraTensorModule.lTensor_mul, Algebra.TensorProduct.one_mul, Ideal.constr_basisSpanSingleton, mem_polar_singleton, LinearEquiv.isUnit_det, IsSymm.eq, CliffordAlgebra.contractLeft_ι, polyCharpolyAux_map_aeval, Subspace.dualAnnihilator_iInf_eq, Algebra.traceMatrix_of_basis, LieModule.traceForm_lieSubalgebra_mk_left, LieModule.Cohomology.smul_apply_apply, CliffordAlgebra.changeForm.neg_proof, Matrix.spectrum_toLin, Submodule.quotDualCoannihilatorToDual_injective, Module.Basis.eval_injective, PolyEquivTensor.toFunBilinear_apply_apply, BilinForm.IsSymm.neg, toMatrix'_comp, Matrix.toLinearEquiv'_apply, isUnit_toMatrix'_iff, BilinForm.add_right, Matrix.toBilin_toMatrix, IsBaseChange.toDual_apply, PolyEquivTensor.toFunLinear_tmul_apply, Finsupp.bilinearCombination_apply, Matrix.nondegenerate_toLinearMap₂_iff, BilinForm.comp_congr, Matrix.toBilin'_comp, Matrix.isHermitian_iff_isSymmetric, LieModule.Cohomology.instLinearMapClassSubtypeLinearMapIdMemSubmoduleTwoCochain, Subspace.dualRestrict_surjective, RootPairing.InvariantForm.apply_eq_or_aux, IsBaseChange.endHom_one, toMatrix_directSum_collectedBasis_eq_blockDiagonal', WeakBilin.coeFn_continuous, Submodule.map₂_span_singleton_eq_map, lift_lsmul_mul_eq_lsmul_lift_lsmul, Orientation.kahler_rightAngleRotation_left, separatingLeft_congr_iff, Module.Basis.end_repr_symm_apply, Module.evalEquiv_apply, Orientation.kahler_comp_linearIsometryEquiv, IsReflective.dvd_two_mul, CliffordAlgebra.contractRight_contractRight, InnerProductSpace.toMatrix_rankOne, LieAlgebra.InvariantForm.mem_orthogonal, BilinMap.toQuadraticMap_multiset_sum, IsPerfPair.bijective_right, mul_toMatrix₂_mul, Orientation.abs_areaForm_of_orthogonal, exteriorPower.alternatingMapToDual_apply_ιMulti, BilinForm.tensorDistribEquiv_tmul, innerₗ_apply_apply, trace_tensorProduct', polar_eq_biInter_preimage, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, eq_adjoint_iff_basis, IsLocalizedModule.map_bijective_iff_localizedModuleMap_bijective, PiTensorProduct.one_mul, QuotSMulTop.equivQuotTensor_naturality_mk, Module.dual_rank_eq, toMatrix'_id, polyCharpolyAux_coeff_eval, Orientation.kahler_rotation_left, Matrix.toLinearMapₛₗ₂_symm, contractLeft_apply, SemimoduleCat.Iso.conj_eq_conj, BilinMap.tensorDistrib_tmul, Projectivization.cross_mk, BilinForm.mul_toMatrix_mul, trace_transpose, IsBaseChange.map_id_lsmul_eq_lsmul_algebraMap, IsBaseChange.endHom_comp_apply, BilinForm.toMatrix_mul_basis_toMatrix, nondegenerate_toMatrix₂'_iff, Orientation.inner_sq_add_areaForm_sq, BilinForm.toMatrix'_comp, ker_dualMap_eq_dualCoannihilator_range, CharacterModule.dual_rTensor_conj_homEquiv, mulLinearMap_apply_apply, CliffordAlgebra.changeForm_contractLeft, IsLocalization.map_eq_toLinearMap_mapₐ, triple_product_eq_det, LinearEquiv.dualMap_apply, QuadraticMap.coe_associatedHom, LieAlgebra.Extension.d₁₂_oneCochainOfTwoSplitting, mem_selfAdjointSubmodule, IsNonneg.add, finrank_range_dualMap_eq_finrank_range, LocalizedModule.map_exact, BilinForm.IsAlt.neg, bilinearIteratedFDerivWithinTwo_eq_iteratedFDeriv, Module.bijective_dual_eval, Matrix.toLinearMapRight'_apply, BilinForm.toMatrix_mul, Module.FaithfullyFlat.tensorProduct_mk_injective, Matrix.ofLp_toEuclideanLin_apply, Matrix.toLinearMap₂_apply, CliffordAlgebra.forall_mul_self_eq_iff, RootPairing.coroot_root_eq_pairing, compl₁₂_apply, rTensorHomEquivHomRTensor_toLinearMap, mem_span_of_iInf_ker_le_ker, LieAlgebra.LoopAlgebra.residuePairing_apply_apply, MultilinearMap.uncurryRight_apply, nondegenerate_toLinearMap₂'_iff_det_ne_zero, PiTensorProduct.piTensorHomMapFun₂_add, Matrix.mulVecBilin_apply, Module.Basis.dualBasis_repr, PiTensorProduct.mul_one, CategoryTheory.Abelian.Ext.bilinearCompOfLinear_apply_apply, lieEquivMatrix'_symm_apply, Matrix.proj_diagonal, Module.Dual.baseChange_baseChange, separatingDual_iff_injective, TensorProduct.dualDistribInvOfBasis_apply, trace_eq_sum_trace_restrict, map_smul₂, disjoint_ker_of_nondegenerate_restrict, dualTensorHom_prodMap_zero, IsContPerfPair.bijective_right, isUnit_toMatrix_iff, Subspace.dualEquivDual_def, LocalizedModule.map_id, TensorProduct.AlgebraTensorModule.rTensor_mul, MultilinearMap.curryMidLinearEquiv_apply, BilinForm.toMatrix_compLeft, trace_eq_matrix_trace, RootPairing.polarizationEquiv_symm_apply_coroot, BilinForm.toMatrix_apply, Matrix.linfty_opNorm_toMatrix, Matrix.toBilin'_apply', Module.Basis.linearCombination_coord, BilinForm.IsSymm.polarization, Matrix.toLinearMap₂'_single, ExteriorAlgebra.liftAlternating_comp_ιMulti, separatingRight_toLinearMap₂'_of_det_ne_zero', Algebra.TensorProduct.mul_assoc, adjoint_inner_left, TensorProduct.dualDistrib_dualDistribInvOfBasis_left_inverse, Module.Basis.toDual_apply, BilinForm.mul_toMatrix, Subspace.dualLift_rightInverse, coprodEquiv_apply, QuadraticMap.associated_linMulLin, Rep.ihom_map_hom, dualTensorHomEquivOfBasis_toLinearMap, QuadraticForm.associated_isSymm, IsBaseChange.dual, Module.apply_evalEquiv_symm_apply, Submodule.coe_dualAnnihilator_span, Orientation.kahler_mul, trace_conj', Matrix.SpecialLinearGroup.toLin'_to_linearMap, mul_toMatrix₂'_mul, toMatrix₂'_mul, IsTensorProduct.lift_eq, BilinForm.flip_flip, Algebra.toMatrix_lsmul, toMatrix_innerSL_apply, Ideal.subtype_isoBaseOfIsPrincipal_eq_mul, Matrix.separatingLeft_toLinearMap₂'_iff, QuadraticMap.exists_companion', triple_product_permutation, separatingLeft_toLinearMap₂'_of_det_ne_zero', PiTensorProduct.dualDistrib_dualDistribInvOfBasis_right_inverse, Module.End.baseChangeHom_apply_apply, PiTensorProduct.mul_tprod_tprod, LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpace, BilinForm.toMatrix_mul, Subspace.dualCopairing_nondegenerate, Module.finite_dual_iff, toMatrix_comp, Module.dual_projective, Module.Basis.coe_toDual_self, RootPairing.RootPositiveForm.exists_pos_eq, Matrix.separatingLeft_toBilin'_iff, Submodule.dualAnnihilator_top, Matrix.trace_toLin_eq, toMatrix₂_mul_basis_toMatrix, TensorProduct.AlgebraTensorModule.lift_apply, LieAlgebra.IsKilling.rootSystem_pairing_apply, Module.Basis.toMatrix_eq_toMatrix_constr, lcompₛₗ_apply, polyCharpoly_map_eq_charpoly, Ideal.pi_mkQ_rTensor, BilinMap.toQuadraticMap_sum, Orientation.kahler_map, CliffordAlgebra.contractRight_one, PolyEquivTensor.toFunBilinear_apply_eq_smul, isOrthoᵢ_def, BilinForm.isSymm_def, PiTensorProduct.lift_tprod, Matrix.toLinOfInv_apply, TensorProduct.lcurry_apply, TensorProduct.curry_apply, dualTensorHomEquivOfBasis_symm_cancel_left, IsPerfPair.id, Module.Invertible.rTensorEquiv_symm_apply_apply, Module.Flat.iff_lift_lsmul_comp_subtype_injective, PiTensorProduct.liftIsometry_apply_apply, ContinuousLinearMap.coeLMₛₗ_apply, CliffordAlgebra.foldr'Aux_apply_apply, QuotSMulTop.map_first_exact_on_four_term_exact_of_isSMulRegular_last, LinearEquiv.conj_id, Matrix.toBilin'Aux_eq, TensorProduct.lift.equiv_symm_apply, addMonoidHomLequivNat_apply, LieAlgebra.IsKilling.cartanEquivDual_apply_apply, IntrinsicStar.isSelfAdjoint_iff_toMatrix', Matrix.toLin_finTwoProd_toContinuousLinearMap, IsBaseChange.linearMapLeftRight, IsLocalization.mapExtendScalars_eq_toLinearMap_mapₐ, map_add₂, Submodule.mem_traceDual, domRestrict₂_apply, RootPairing.flip_toFun_apply, Orientation.areaForm_map, Matrix.toPerfectPairing, PiTensorProduct.norm_eval_le_projectiveSeminorm, IsProj.trace, map_neg₂, LinearEquiv.congrRight₂_trans, Subspace.map_le_dualAnnihilator_dualAnnihilator, BilinForm.isSymm_zero, TensorProduct.AlgebraTensorModule.rTensor_id, Matrix.kroneckerTMulBilinear_apply, BilinForm.orthogonal_top_eq_ker, InnerProductSpace.toLinearMap_rankOne, RootPairing.rootSpan_dualAnnihilator_map_eq_iInf_ker_root', Submodule.image2_subset_map₂, Algebra.norm_eq_zero_iff', Module.Basis.linearMap_apply_apply, addMonoidHomLequivInt_symm_apply, trace_tensorProduct, comp_dualTensorHom, QuadraticForm.dualProdProdIsometry_toFun, Matrix.toLinearEquivRight'OfInv_apply, hasEigenvalue_toLin'_diagonal_iff, IsLocalizedModule.map_injective_iff_localizedModuleMap_injective, LinearEquiv.smul_id_of_finrank_eq_one_apply, BilinForm.coe_injective, toMatrix'_toLinearMapₛₗ₂', BilinForm.smul_right, isStarProjection_toContinuousLinearMap_iff, Rep.MonoidalClosed.linearHomEquiv_hom, Matrix.toLpLin_mul, Matrix.toLin'_submatrix, TensorProduct.gradedMul_algebraMap, Matrix.separatingLeft_toLinearMap₂_iff, ModuleCat.localizedModuleMap_hom_apply, Orientation.inner_rightAngleRotationAux₁_right, ringLmapEquivSelf_apply, Matrix.toLin_toMatrix, LieModule.traceForm_genWeightSpace_eq, PiTensorProduct.lift_comp_map, TensorProduct.homTensorHomMap_apply, tensorEqLocus_tmul, LieModule.traceForm_apply_apply, cross_dot_cross, toMatrix₂_apply, ModuleCat.Hom.hom₂_apply, RootPairing.rootFormIn_self_smul_coroot, IsSymmetric.trace_eq_sum_eigenvalues, adjoint_comp, RootPairing.corootForm_self_smul_root, PiToModule.fromMatrix_apply, RootPairing.exists_form_eq_form_and_form_ne_zero, toContPerfPair_apply, LieModule.trace_toEnd_genWeightSpaceChain_eq_zero, Matrix.toEuclideanLin_apply_piLp_toLp, dualMap_apply, LieModule.Cohomology.twoCochain_val_apply, dualMap_surjective_iff, multilinearCurryLeftEquiv_symm_apply, MultilinearMap.piLinearMap_apply_apply_apply, TensorProduct.toLinearMap_symm_lid, QuadraticMap.associated_sq, dualMap_def, iSupIndep_iff_dfinsupp_lsum_injective, RootPairing.toPerfPair_conj_reflection, QuadraticMap.associated_comp, TensorProduct.tmul_of_gradedMul_of_tmul, IsLocalizedModule.map_exact, QuadraticMap.associated_isOrtho, CliffordAlgebra.contractLeft_mul_algebraMap, map_dualTensorHom, Submodule.range_dualMap_mkQ_eq, Subspace.dualLift_injective, MultilinearMap.curryMidLinearEquiv_symm_apply, Polynomial.toMatrix_sylvesterMap, smulRightₗ_apply_apply, Finsupp.lsum_comp_lsingle, IsRefl.eq_iff, TensorProduct.AlgebraTensorModule.lTensor_one, Module.finrank_linearMap_self, Matrix.separatingRight_toBilin'_iff, vecEmpty₂_apply, Orientation.areaForm_map_complex, RootPairing.InvariantForm.apply_root_root_zero_iff, LieModule.coe_maxTrivLinearMapEquivLieModuleHom_symm, Matrix.toLin_finTwoProd, BilinForm.apply_toDual_symm_apply, Matrix.SpecialLinearGroup.toLin_equiv.toLinearMap_eq, CliffordAlgebra.even.lift.aux_apply, Module.Basis.linearMap_repr_symm_apply, CliffordAlgebra.even.lift.aux_ι, BilinForm.neg_apply, RootPairing.rootForm_pos_of_ne_zero, Module.Basis.constr_basis, LinearEquiv.charpoly_conj, tensorProduct_apply, BilinForm.isPosSemidef_zero, exteriorPower.pairingDual_apply_apply_eq_one_zero, ModuleCat.homLinearEquiv_apply, Rep.coinvariantsTensorMk_apply, Orientation.inner_rightAngleRotation_right, WeakBilin.eval_continuous, QuotSMulTop.map_comp, LieModule.traceForm_apply_lie_apply, Projectivization.cross_mk_of_ne, not_separatingLeft_zero, CharacterModule.uncurry_apply, Rep.homEquiv_symm_apply_hom, QuadraticMap.Ring.associated_pi, nondegenerate_toMatrix₂_iff, Matrix.trace_kroneckerMapBilinear, AlgEquiv.linearEquivConj_mulRight, ModuleCat.Iso.conj_eq_conj, innerₛₗ_apply, Matrix.toLin_one, CliffordAlgebra.contractLeft_algebraMap_mul, BilinForm.compLeft_apply, contractRight_apply, trace_eq_contract_of_basis, range_dualMap_le_dualAnnihilator_ker, Matrix.toLin_transpose, Complex.toMatrix_conjAe, Submodule.finite_dualAnnihilator_iff, Representation.leftRegular_norm_apply, BilinForm.mul_toMatrix, RootPairing.RootPositiveForm.algebraMap_apply_eq_form_iff, TensorProduct.mk_surjective, Matrix.toLpLin_symm_comp, Module.Basis.toDual_linearCombination_right, Matrix.toLinearMapₛₗ₂_apply_basis, VertexOperator.ncoeff_of_coeff, RootPairing.restrictScalars_toLinearMap_apply_apply, ExteriorAlgebra.liftAlternating_algebraMap, Orientation.kahler_apply_apply, Orientation.inner_rightAngleRotationAux₁_left, toMatrix_toSpanSingleton, RootPairing.toPerfPair_flip_conj_coreflection, VertexOperator.ncoeff_eq_zero_of_lt_order, toMatrix₂_symm', Module.Invertible.bijective, Orientation.areaForm_apply_self, Nondegenerate.toMatrix₂', TensorProduct.equivFinsuppOfBasisRight_symm, HVertexOperator.compHahnSeries_coeff, Subspace.dualAnnihilator_le_dualAnnihilator_iff, RootPairing.InvariantForm.apply_eq_or, Matrix.Nondegenerate.toBilin', FDRep.Iso.conj_ρ, IsAlt.eq_of_add_add_eq_zero, VectorFourier.fourierIntegral_comp_add_right, BilinForm.congr_trans, toMatrix_singleton, isNonneg_def, BilinForm.toMatrix'_mul, dotProductBilin_apply_apply, trace_restrict_eq_of_forall_mem, exteriorPower.alternatingMapLinearEquiv_symm_apply, Rep.coinvariantsTensorIndHom_mk_tmul_indVMk, Rep.ihom_coev_app_hom, toMatrix₂'_compl₂, restrictScalars₁₂_injective, map_sum₂, VectorFourier.fourierIntegral_probChar, IsSymm.add, Submodule.le_dualAnnihilator_iff_le_dualCoannihilator, LieAlgebra.IsKilling.traceForm_coroot, TensorProduct.AlgebraTensorModule.curry_injective, Orientation.kahler_rotation_right, Submodule.span_eq_top_of_ne_zero, BilinForm.comp_apply, FDRep.char_dual, Module.Basis.traceDual_repr_apply, PiTensorProduct.lift.unique', Matrix.nondegenerate_toLinearMap₂'_iff, separatingRight_toMatrix₂_iff, FGModuleCat.FGModuleCatEvaluation_apply', LieModule.traceForm_eq_sum_finrank_nsmul_mul, Coalgebra.lTensor_counit_comp_comul, CliffordAlgebra.changeForm.associated_neg_proof, BilinForm.neg_right, RootPairing.RootPositiveForm.zero_lt_apply_root_root_iff, CliffordAlgebra.ofEven_ι, BilinForm.IsNonneg.smul, QuotSMulTop.map_apply_mk, Submodule.dualAnnihilator_bot, Representation.linHom.mem_invariants_iff_comm, BilinForm.linMulLin_apply, RootPairing.corootSpan_dualAnnihilator_le_ker_rootForm, mk₂'_apply, IsLocalizedModule.map_comp, Module.Dual.eval_naturality, BilinForm.comp_symmCompOfNondegenerate_apply, HVertexOperator.coeff_comp, ModuleCat.ofHom₂_hom_apply_hom, LieAlgebra.IsKilling.traceForm_eq_zero_of_mem_ker_of_mem_span_coroot, Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule_1, trace_conj, isSMulRegular_on_quot_iff_lsmul_comap_eq, IsLocalization.map_linearMap_eq_toLinearMap_mapₐ, BilinForm.congr_fun, jacobi_cross, separatingLeft_toMatrix₂_iff, Module.map_eval_injective, HVertexOperator.coeff_apply_apply, basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix, Module.Basis.toDualFlip_apply, RootPairing.EmbeddedG2.twoShortAddLongRoot_shortRoot, mem_span_iff_bound, TensorProduct.dualDistrib_apply, BilinForm.toLin_restrict_ker_eq_inf_orthogonal, BilinForm.IsAlt.sub, BilinForm.toMatrix'_toBilin', neg_cross, LocalizedModule.map_mk, PowerBasis.liftEquiv'_symm_apply_apply, PiTensorProduct.lift_comp_reindex_symm, QuadraticMap.separatingLeft_of_anisotropic, PowerBasis.constr_pow_mul, RootPairing.four_smul_rootForm_sq_eq_coxeterWeight_smul, det_toMatrix', Matrix.separatingLeft_toBilin_iff, eq_adjoint_iff_basis_right, LinearEquiv.arrowCongr_trans, LieAlgebra.IsKilling.ker_restrict_eq_bot_of_isCartanSubalgebra, Nondegenerate.toMatrix₂, Matrix.toLin'OfInv_apply, IsModuleTopology.continuous_bilinear_of_finite_right, Rep.ihom_obj_V_isModule, TensorProduct.tensorQuotEquivQuotSMul_comp_mk, BilinForm.isOrtho_def, innerₛₗ_apply_coe, BilinMap.toQuadraticMap_apply, RootPairing.rootSpan_dualAnnihilator_map_eq, BilinForm.Nondegenerate.toMatrix', isSymm_iff_basis, Module.Basis.constr_self, toMatrix_basis_equiv, restrictScalars₁₂_apply_apply, Module.evalEquiv_toLinearMap, MultilinearMap.compLinearMapMultilinear_apply, LieModule.traceForm_apply_eq_zero_of_mem_lcs_of_mem_center, topDualPairing_apply, trace_prodMap', toSeminormFamily_apply, TensorProduct.AlgebraTensorModule.lift_tmul, dualMap_comp_dualMap, separatingLeft_iff_ker_eq_bot, Matrix.SeparatingRight.toBilin', lflip_symm, ltoFun_apply, Orientation.areaForm_rightAngleRotation_right, BilinForm.apply_eq_dotProduct_toMatrix_mulVec, compBilinForm_apply_apply, lid_comp_rTensor, coe_toContinuousLinearMap_symm, Matrix.range_diagonal, BilinForm.Nondegenerate.ker_eq_bot, IsSymmetric.re_trace_eq_sum_eigenvalues, LieModule.Cohomology.mem_twoCocycle_iff_of_trivial, extendScalarsOfIsLocalizationEquiv_symm_apply, RootPairing.Base.cartanMatrixIn_mul_diagonal_eq, Module.Basis.constr_def, isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule, cardinalMk_algHom, trace_comp_comm, addMonoidEndRingEquivInt_symm_apply, trace_eq_sum_trace_restrict', BilinForm.toMatrix_compLeft, IsPositive.adjoint_eq, Matrix.maxGenEigenspace_toLin_diagonal_eq_eigenspace, Algebra.coe_lmul_eq_mul, Module.Basis.toDual_apply_left, nilRank_le_natTrailingDegree_charpoly, toPerfPair_apply, toMatrix₂'_comp, Matrix.toLinearMap₂_apply_basis, LinearPMap.mem_adjoint_domain_iff, Rep.MonoidalClosed.linearHomEquivComm_symm_hom, TensorProduct.Algebra.moduleAux_apply, toMatrix_pow, RootPairing.disjoint_rootSpan_ker_rootForm, Module.FinitePresentation.isLocalizedModule_mapExtendScalars, BilinMap.polarBilin_toQuadraticMap, IsBaseChange.linearMapLeftRightHom_apply, Algebra.trace_apply, coprodEquiv_symm_apply, BilinForm.toMatrix_comp, LinearEquiv.map_mem_invtSubmodule_iff, Matrix.SpecialLinearGroup.toLin'_symm_apply, coe_innerₛₗ_apply, trace_comp_cycle, Algebra.FormallyUnramified.comp_sec, Module.Basis.toDual_apply_right, trace_comp_cycle', Submodule.dualAnnihilator_map_linearEquiv_flip_symm, toMatrix'_one, Representation.dual_apply, Submodule.baseChange_span, BilinForm.toLin_restrict_range_dualCoannihilator_eq_orthogonal, Module.surjective_piEquiv_apply_iff, Submodule.dualAnnihilator_iSup_eq, Module.Dual.eq_of_preReflection_mapsTo', Submodule.dualAnnihilator_eq_bot_iff, TensorProduct.quotTensorEquivQuotSMul_comp_mk, Orientation.kahler_rightAngleRotation_right, PointedCone.mem_maxTensorProduct, Matrix.iSup_eigenspace_toLin_diagonal_eq_top, LinearEquiv.dualMap_refl, QuadraticMap.exists_companion, BilinForm.IsAlt.add, cross_cross_eq_smul_sub_smul', rank_lt_rank_dual, TensorProduct.rTensorHomToHomRTensor_apply, Module.Dual.congr_apply_apply, LieAlgebra.IsKilling.span_weight_eq_top, tensorKerEquiv_apply, basis_toMatrix_mul_linearMap_toMatrix, IsAlt.self_eq_zero, toMatrixₛₗ₂'_apply, Representation.dualTensorHom_comm, exact_lcomp_of_exact_of_surjective, transvection.baseChange, polar_mem_iff, Submodule.mem_biSup_iff_exists_dfinsupp, IsContPerfPair.continuous_uncurry, Matrix.toLinearMapRight'_one, isSMulRegular_iff_ker_lsmul_eq_bot, Matrix.maxGenEigenspace_toLin'_diagonal_eq_eigenspace, LieAlgebra.LoopAlgebra.twoCocycleOfBilinear_coe, IsLocalizedModule.map_surjective, QuadraticForm.associated_tmul, ModN.instModuleFinite, RootPairing.toLinearMap_apply_apply_mem_range_algebraMap, Orientation.norm_kahler, Submodule.dualRestrict_apply, TensorProduct.one_gradedMul, IsLocalizedModule.map_iso_commute, SymmetricAlgebra.IsSymmetricAlgebra.mvPolynomial, LieModule.traceForm_apply_lie_apply', Module.Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix, flip_surjective_iff₂, RootPairing.RootPositiveForm.algebraMap_posForm, lsum_piSingle, IsBaseChange.linearMapRight, Submodule.dualAnnihilator_eq_bot_iff', Module.Basis.constr_range, ker_lsmul, PiTensorProduct.mul_comm, Matrix.toLpLin_apply, Matrix.toLin'_apply, CliffordAlgebra.contractRight_ι, QuotSMulTop.map_id, LinearEquiv.arrowCongr_comp, isSMulRegular_on_quot_iff_lsmul_comap_le, Submodule.map_dualCoannihilator_linearEquiv_flip, Module.Basis.constr_apply, DFinsupp.lsum_apply_apply, BilinForm.SeparatingRight.toMatrix', TensorProduct.toLinearMap_symm_rid, Module.Dual.instLieModule, toMatrix_adjoint, Submodule.dualQuotEquivDualAnnihilator_symm_apply_mk, TensorProduct.AlgebraTensorModule.coe_rTensor, diag_toMatrix_directSum_collectedBasis_eq_zero_of_mapsTo_ne, Subspace.map_dualCoannihilator, LieAlgebra.Extension.twoCocycleOf_coe_coe, separatingLeft_toLinearMap₂'_iff_det_ne_zero, CharacterModule.homEquiv_apply_apply, map_smulₛₗ₂, Rep.ihom_obj_ρ, BilinForm.Nondegenerate.toMatrix, RootPairing.isCompl_corootSpan_ker_corootForm, dualCoannihilator_range_eq_ker_flip, im_inner_adjoint_mul_self_eq_zero, QuadraticMap.associated_flip, BilinForm.congr_symm, compr₂ₛₗ_apply, Matrix.l2_opNorm_def, signedDist_apply_linear, Matrix.toEuclideanLin_toLp, BilinForm.baseChange_tmul, toLinearMap_toPerfPair, LieAlgebra.IsKilling.ker_killingForm_eq_bot, RootPairing.corootSpan_dualAnnihilator_map_eq_iInf_ker_coroot', BilinForm.isSymm_iff_basis, PowerBasis.constr_pow_aeval, RootPairing.invtSubmodule_reflection_of_invtSubmodule_coreflection, CliffordAlgebra.EvenHom.contract_mid, det_toMatrix, Algebra.traceForm_toMatrix_powerBasis, Matrix.minpoly_toLin, basis_toMatrix_mul, Rep.coinvariantsTensorIndInv_mk_tmul_indMk, SemimoduleCat.homLinearEquiv_symm_apply, Matrix.toLin'_toMatrix', AlternatingMap.alternatizeUncurryFinLM_apply, det_toLin, posSemidef_toMatrix_iff, Representation.linHom.invariantsEquivRepHom_apply_hom, instIsLocalizedModuleLinearMapIdLocalizationLocalizedModuleMapOfFinitePresentation, trace_mul_cycle', BilinForm.nondegenerate_toBilin'_of_det_ne_zero', BilinForm.sum_apply, RootPairing.InvariantForm.apply_eq_or_of_apply_ne, QuadraticMap.associated_isSymm, FGModuleCat.Iso.conj_hom_eq_conj, RootPairing.InvariantForm.two_mul_apply_root_root, Module.Basis.toDual_eq_equivFun, SeparatingLeft.toMatrix₂', isNonneg_zero, TensorProduct.toMatrix_comm, BilinMap.toQuadraticMap_neg, dualTensorHomEquivOfBasis_apply, polar_mem, toMatrix₂'_apply, isAdjointPair_inner, ModuleCat.monoidalClosed_pre_app, IsPositive.adjoint_conj, LieModule.Cohomology.d₂₃_comp_d₁₂, Subspace.quotDualCoannihilatorToDual_bijective, ker_tensorProductMk, orthogonal_span_singleton_eq_to_lin_ker, ker_dualMap_eq_dualAnnihilator_range, PiTensorProduct.dualDistrib_dualDistribInvOfBasis_left_inverse, Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂, LinearEquiv.trans_dualMap_symm_flip, BilinForm.separatingLeft_toMatrix_iff, trace_comp_comm', TensorProduct.sum_tmul_basis_left_injective, PiTensorProduct.lift.unique, QuadraticForm.associated_baseChange, Real.volume_preserving_transvectionStruct, Matrix.toBilin'_toMatrix', trace_baseChange, CliffordAlgebra.contractLeftAux_apply_apply, Matrix.toLinearMapₛₗ₂_toMatrix₂, BilinForm.IsAlt.smul, CliffordAlgebra.contractLeft_one, le_comap_range_rTensor, toMatrix'_mulVec, RootPairing.EmbeddedG2.threeShortAddLongRoot_longRoot, trace_eq_sum_trace_restrict_of_eq_biSup, RootPairing.ker_corootForm_eq_dualAnnihilator, Subspace.dualAnnihilator_inf_eq, VertexOperator.coeff_eq_ncoeff, lsum_symm_apply, IsBaseChange.toDualBaseChange_tmul, IsLocalizedModule.map_LocalizedModules, Matrix.SeparatingLeft.toLinearMap₂, hasEigenvector_toLin_diagonal, separatingRight_toLinearMap₂'_iff_det_ne_zero, Module.rank_linearMap_self, TensorProduct.quotTensorEquivQuotSMul_symm_comp_mkQ, domRestrict₁₂_apply, dotProduct_toMatrix₂_mulVec, LieModule.range_traceForm_le_span_weight, TensorProduct.gradedComm_gradedMul, Matrix.diagonal_comp_single, FDRep.dualTensorIsoLinHom_hom_hom, IsPerfectCompl.isCompl_left, Submodule.map_dualAnnihilator_linearEquiv_flip_symm, SpecialLinearGroup.coe_dualMap, IsBaseChange.endHom_comp, traceAux_def, mem_span_iff_continuous, Algebra.leftMulMatrix_apply, InnerProductSpace.symm_toEuclideanLin_rankOne, Subspace.isCompl_dualAnnihilator, BilinForm.toMatrix'_compRight, VertexOperator.coeff_eq_zero_of_lt_order, Module.Basis.eval_ker, ExteriorAlgebra.liftAlternating_ι, TensorProduct.AlgebraTensorModule.lcurry_apply, Matrix.SeparatingRight.toBilin, LieModule.coe_maxTrivLinearMapEquivLieModuleHom, CliffordAlgebra.contractLeftAux_contractLeftAux, Subspace.comap_dualAnnihilator_dualAnnihilator, toMatrix_apply, RootPairing.ker_polarization_eq_ker_rootForm, Module.erange_coe, TensorProduct.toMatrix_map, Matrix.rank_vecMulVec, continuous_uncurry_of_isContPerfPair, Submodule.mem_traceDual_iff_isIntegral, LieAlgebra.IsKilling.rootSystem_root_apply, RootPairing.polarization_apply_eq_zero_iff, Submodule.le_dualCoannihilator_dualAnnihilator, RootPairing.EmbeddedG2.threeShortAddTwoLongRoot_longRoot, Orientation.areaForm_rightAngleRotation_left, BilinForm.restrict_apply, toMatrix_id_eq_basis_toMatrix, contractLeft_assoc_coevaluation', TensorProduct.dualDistribEquivOfBasis_apply_apply, baseChangeHom_apply, toMatrixRight'_comp, isOrtho_def, Module.dualProdDualEquivDual_symm_apply, trace_map, Module.Basis.constr_eq, Module.FinitePresentation.isLocalizedModule_map, HVertexOperator.coeff_of_coeff, lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm, applyₗ_apply_apply, dualProd.toQuadraticForm, IsAlt.neg, Subspace.finrank_add_finrank_dualAnnihilator_eq, Matrix.toBilin_comp, CliffordAlgebra.changeForm_changeForm, polyCharpoly_coeff_eval, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj, LieModuleHom.map_lie₂, BilinForm.apply_smul_sub_smul_sub_eq, SimpleGraph.mem_ker_toLin'_lapMatrix_of_connectedComponent, lsum_comp_mapRange_toSpanSingleton, Orientation.inner_mul_inner_add_areaForm_mul_areaForm', RootPairing.orthogonal_rootSpan_eq, MultilinearMap.ofSubsingletonₗ_symm_apply, det_toMatrix_eq_det_toMatrix, Algebra.TensorProduct.mul_apply, Matrix.SpecialLinearGroup.toLin'_apply, Module.Finite.of_isComplemented_codomain, Submodule.piQuotientLift_mk, QuadraticMap.polarBilin_apply_apply, toKerIsLocalized_apply_coe, ModuleCat.ihom_ev_app, adjoint_id, BilinForm.add_apply, Submodule.flip_quotDualCoannihilatorToDual_injective, lflip_apply, Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule, TensorProduct.lift.equiv_apply, IsBaseChange.linearMapLeftRightHom_comp, Rep.indResHomEquiv_symm_apply_hom, Subspace.dualPairing_eq, QuadraticForm.dualProdProdIsometry_invFun, SeparatingLeft.congr, Matrix.Nondegenerate.toLinearMap₂', AlgHom.mulLeftRightMatrix.comp_inv, Orientation.inner_mul_areaForm_sub, IsLocalizedModule.map_lTensor, Matrix.separatingRight_toLinearMap₂'_iff_separatingRight_toLinearMap₂, Module.Basis.coord_toDualEquiv_symm_apply, Finsupp.lsum_symm_apply, Module.Basis.baseChange_linearMap, BilinForm.sub_right, BilinMap.toQuadraticMap_sub, IsRefl.ker_flip, toMatrix₂Aux_apply, Module.mapEvalEquiv_symm_apply, Subspace.dualEquivDual_apply, BilinForm.iIsOrtho_def, TensorProduct.uncurry_apply, BilinForm.apply_dualBasis_right, TensorProduct.tensorQuotEquivQuotSMul_symm_comp_mkQ, Module.Dual.congr_symm_apply_apply, PointedCone.dual_singleton, TensorPower.pairingDual_tprod_tprod, sum_repr_mul_repr_mul, IsPositive.conj_adjoint, IsPerfPair.dualEval, ker_toContinuousLinearMap, TensorProduct.AlgebraTensorModule.lTensor_tmul, charpoly_toMatrix, LieAlgebra.IsKilling.instIsIrreducibleSubtypeWeightMemLieSubalgebraFinsetRootDualRootSystemOfIsSimple, toLinearMap₂'Aux_toMatrix₂Aux, isSymm_iff_isHermitian_toMatrix, trace_lie, lsmul_eq_distribSMultoLinearMap, Matrix.toBilin'_single, Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply, Module.FinitePresentation.linearEquivMapExtendScalars_symm_apply, separatingLeft_dualProd, toMatrix_toLin, range_localizedMap_eq_localized₀_range, Module.Basis.traceDual_eq_iff, IsTensorProduct.map_eq, DirectSum.gMulLHom_apply_apply, polyCharpolyAux_map_eq_toMatrix_charpoly, Matrix.toLinearEquiv_apply, LinearEquiv.piRing_apply, hasEigenvalue_toLin_diagonal_iff, PiTensorProduct.lift_symm, CliffordAlgebra.contractLeft_algebraMap, Matrix.spectrum_toLin', dualAnnihilator_ker_eq_range_flip, LieAlgebra.killingForm_of_equiv_apply

Mathlib.Linter.linter.deprecated

Definitions

Name	Category	Theorems
module 📖	CompOp	—

Matrix

Definitions

Name	Category	Theorems
module 📖	CompOp	550 math math: LinearMap.restrictScalars_toMatrix, LinearMap.SeparatingRight.toMatrix₂', LinearMap.charpoly_def, toLin_kronecker, toLin'_symm, SeparatingLeft.toBilin', LinearMap.detAux_def'', kroneckerTMulStarAlgEquiv_symm_apply, LinearMap.BilinForm.toMatrix_toBilin, Module.Basis.end_repr_apply, SeparatingRight.toLinearMap₂, BilinForm.toMatrix_symm, SimpleGraph.lapMatrix_toLinearMap₂', reindexLinearEquiv_trans, separatingRight_toLinearMap₂'_iff, Polynomial.toMatrix_sylvesterMap', IsHermitian.det_abs, toLinearMapRight'_mul, LinearMap.mapMatrix_zero, separatingRight_toLinearMap₂_iff, spectrum_toEuclideanLin, iSup_eigenspace_toLin'_diagonal_eq_top, LinearMap.mapMatrixLinear_apply, LinearMap.BilinForm.toMatrixAux_eq, toLinearMap₂'_comp, kroneckerTMulAlgEquiv_symm_single_tmul, LinearMap.toMatrix_apply', mem_pairSelfAdjointMatricesSubmodule, LinearMap.toMatrix₂_mul, LinearMap.entryLinearMap_comp_mapMatrix, repr_toLin, LinearEquiv.mapMatrix_apply, liftLinear_single, LinearMap.BilinForm.toMatrix_compRight, transposeLinearEquiv_symm, toLinearMap₂_toMatrix₂, LinearMap.toMatrix_one, separatingRight_toBilin_iff, spectrum_toLpLin, LinearMap.BilinForm.toMatrix'_symm, toLin_mul_apply, toMatrix_distrib_mul_action_toLinearMap, LinearMap.BilinForm.toMatrix'_apply, PosSemidef.sqrt_eq_zero_iff, LinearMap.BilinForm.SeparatingRight.toMatrix, LinearMap.nondegenerate_toLinearMap₂'_of_det_ne_zero', Nondegenerate.toLinearMap₂, intrinsicStar_toLin', QuadraticMap.toMatrix'_comp, mulRightLinearMap_one, SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix, star_dotProduct_toMatrix₂_mulVec, toBilin_symm, LinearMap.spectrum_toMatrix', Real.smul_map_diagonal_volume_pi, toMatrix_rotation, AlgHom.mulLeftRightMatrix.inv_comp, diagLinearMap_apply, diagonalLinearMap_apply, toLpLin_mul_same, LinearMap.toMatrix_baseChange, LinearMap.trace_eq_matrix_trace_of_finset, SeparatingLeft.toLinearMap₂', LinearMap.BilinForm.nondegenerate_toMatrix_iff, LinearMap.toMatrixₛₗ₂'_symm, LinearMap.rank_diagonal, rank_matrix_module, LinearMap.separatingRight_toMatrix₂'_iff, toEuclideanLin_apply, Algebra.toMatrix_lmul_eq, isAdjointPair_toLinearMap₂, liftLinear_singleLinearMap, LinearEquiv.mapMatrix_toLinearMap, liftLinear_comp_singleLinearMap, LinearMap.toMatrix_symm, toBilin_apply, LinearMap.toMatrix₂'_compl₁₂, LieAlgebra.SpecialLinear.singleSubSingle_add_singleSubSingle, Module.Basis.toLin_toMatrix, toLinearEquivRight'OfInv_symm_apply, toLin'_mul_apply, toLinearMapₛₗ₂'_single, nondegenerate_toBilin'_iff, LinearMap.toMatrix'_toLin', TensorProduct.toMatrix_assoc, toLinearMap₂'_apply', SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable, LinearMap.toMatrix_id, LinearMap.BilinForm.apply_eq_dotProduct_toMatrix_mulVec, MatrixEquivTensor.invFun_add, mul_reindexLinearEquiv_one, toAlgEquiv_kroneckerStarAlgEquiv, LinearMap.toMatrix_transpose, LinearMap.toMatrix₂_toLinearMapₛₗ₂, det_kroneckerMapBilinear, SpecialLinearGroup.toLin'_symm_to_linearMap, mem_skewAdjointMatricesSubmodule', vecMulBilin_apply, PosSemidef.det_sqrt, IsBaseChange.endHom_toMatrix, Real.map_matrix_volume_pi_eq_smul_volume_pi, toAlgEquiv_kroneckerTMulStarAlgEquiv, toLinearMap₂'_apply, LinearMap.BilinForm.separatingLeft_toMatrix'_iff, ModularGroup.lcRow0Extend_symm_apply, Module.Basis.matrix_apply, conjTransposeLinearEquiv_apply, isNilpotent_toLin'_iff, toLinearEquiv'_symm_apply, LinearMap.BilinForm.mul_toMatrix'_mul, toLin_symm, BilinForm.mul_toMatrix_mul, Module.Basis.linearMap_apply, toLinearMap₂_compl₁₂, mem_pairSelfAdjointMatricesSubmodule', PosSemidef.posSemidef_sqrt, toLinearMapₛₗ₂'_symm, MatrixEquivTensor.toFunBilinear_apply, LinearMap.toBilin'Aux_toMatrixAux, piLinearEquiv_apply, LinearMap.toMatrix₂_compl₂, mulRightLinearMap_eq_mulRight, toLpLin_symm_pow, LinearEquiv.mapMatrix_symm, toLinearMapₛₗ₂'_aux_eq, MatrixEquivTensor.invFun_algebraMap, linfty_opNNNorm_toMatrix, Algebra.traceForm_toMatrix, toLin_scalar, toBilin'Aux_toMatrixAux, PosSemidef.sqrt_eq_one_iff, BilinForm.dotProduct_toMatrix_mulVec, PosSemidef.sqrt_sq, toLpLin_one, LinearMap.isPosSemidef_iff_posSemidef_toMatrix, Algebra.PreSubmersivePresentation.aevalDifferential_toMatrix'_eq_mapMatrix_jacobiMatrix, singleLinearMap_apply, piLp_ofLp_toEuclideanLin, LinearMap.BilinForm.toMatrix_apply, lieEquivMatrix'_apply, BilinForm.toMatrix_toBilin, SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux, LinearMap.minpoly_toMatrix', PosSemidef.eq_sqrt_iff_sq_eq, LinearMap.toMatrix₂_comp, SeparatingRight.toLinearMap₂', toLpLin_pow, kroneckerAlgEquiv_apply, LinearMap.BilinForm.separatingRight_toMatrix'_iff, isUnit_toLin'_iff, toLin'_mul, ModularGroup.lcRow0Extend_apply, LinearMap.isNilpotent_toMatrix_iff, LinearMap.BilinForm.dotProduct_toMatrix_mulVec, LinearMap.det_eq_det_toMatrix_of_finset, LinearMap.toMatrix_reindexRange, ofLp_toLpLin, toMatrix₂Aux_toLinearMap₂'Aux, toBilin'_symm, LinearMap.toMatrix_smulBasis_left, LinearMap.det_toLin', kroneckerMapBilinear_apply_apply, rank_eq_finrank_range_toLin, toLinearMapRight'_mul_apply, BilinForm.toMatrix_comp, reindexLinearEquiv_one, toLin_mul, toLin_pow, toLin_self, kroneckerTMulLinearEquiv_one, mulRightLinearMap_zero_eq_zero, LinearMap.separatingLeft_toMatrix₂'_iff, hasEigenvector_toLin'_diagonal, nondegenerate_toBilin'_iff_nondegenerate_toBilin, reindexLinearEquiv_comp, toLinearMapₛₗ₂'_toMatrix', LinearMap.toMatrix'_algebraMap, PiToModule.fromMatrix_apply_single_one, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, BilinearForm.toMatrixAux_eq, RootPairing.GeckConstruction.span_range_h'_eq_top, SeparatingLeft.toBilin, charpoly_toLin, LinearMap.toMatrixOrthonormal_symm_apply, LinearMap.toMatrix_mulVec_repr, LinearMap.toMatrix_innerₛₗ_apply, conjTransposeLinearEquiv_symm, toLin_apply_eq_zero_iff, nondegenerate_toBilin_iff, toLin'_reindex, PosSemidef.toLinearMap₂'_zero_iff, LinearMap.BilinForm.toMatrix'_compLeft, toLinearMapₛₗ₂_apply, instNonnegSpectrumClass, Module.Basis.end_apply, CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_symm_apply, commute_mulLeftLinearMap_mulRightLinearMap, Nondegenerate.toBilin, LinearMap.BilinForm.nondegenerate_toBilin'_iff_det_ne_zero, LinearMap.toMatrix'_apply, toLpLin_toLp, apply_eq_dotProduct_toMatrix₂_mulVec, kroneckerLinearEquiv_symm_kronecker, LinearMap.BilinForm.separatingRight_toMatrix_iff, LinearMap.BilinForm.SeparatingLeft.toMatrix', LinearEquiv.mapMatrix_refl, ker_toLin_eq_bot, IntrinsicStar.isSelfAdjoint_toLin'_iff, pow_mulLeftLinearMap, LinearMap.toMatrix'_symm, LinearMap.mapMatrix_smul, mulRightLinearMap_eq_zero_iff, linearMap_toMatrix_mul_basis_toMatrix, kroneckerTMulAlgEquiv_symm_apply, LinearMap.polyCharpolyAux_eval_eq_toMatrix_charpoly_coeff, LinearMap.toMatrix_transpose_apply, CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_apply, kroneckerTMulStarAlgEquiv_apply, toMatrix_dualTensorHom, range_toLin', RootPairing.GeckConstruction.span_range_h_le_range_diagonal, LinearMap.mul_toMatrix', liftLinear_apply, pow_mulRightLinearMap, LinearMap.toMatrix_smulBasis_right, toLpLin_symm_id, range_toLin_eq_top, toLin_finTwoProd_apply, isPositive_toEuclideanLin_iff, CStarMatrix.ofMatrix_eq_ofMatrixL, LinearMap.mul_toMatrix₂, mem_skewAdjointMatricesLieSubalgebra, kroneckerTMulLinearEquiv_mul, QuadraticMap.discr_comp, Module.Basis.linearMap_repr_apply, kroneckerTMulLinearEquiv_symm_kroneckerTMul, traceLinearMap_apply, dualNumberEquiv_symm_apply, LinearMap.toMatrix'_mul, mul_basis_toMatrix, proj_comp_diagLinearMap, LinearMap.toMatrix_algebraMap, LinearMap.spectrum_toMatrix, LinearMap.SeparatingRight.toMatrix₂, isUnit_toLin_iff, uniqueLinearEquiv_apply, diagonal_toLin', charpoly_toLin', LinearMap.BilinForm.toMatrix_symm, toLinearMap₂'_toMatrix', toLinearMapₛₗ₂'_apply, toLin'_pow, LinearMap.BilinForm.SeparatingLeft.toMatrix, l2_opNNNorm_def, apply_eq_star_dotProduct_toMatrix₂_mulVec, LinearMap.toMatrix_mul, LinearMap.toMatrix₂_compl₁₂, minpoly_toLin', BilinForm.toMatrix_compRight, LinearMap.BilinForm.nondegenerate_toMatrix'_iff, LinearMap.toMatrix₂_toLinearMap₂, LinearMap.detAux_def', LinearMap.toMatrix_transpose_apply', LinearMap.toMatrix'_intrinsicStar, compLinearEquiv_symm_apply, IsBaseChange.linearMapLeftRightHom_toMatrix, LinearMap.BilinForm.toMatrixAux_apply, SpecialLinearGroup.toLin_equiv.symm_toLinearMap_eq, Module.finrank_matrix, toLinOfInv_symm_apply, LinearMap.toMatrix_smulRight, kroneckerTMulLinearEquiv_tmul, LinearMap.toMatrix₂Aux_eq, LinearMap.mapMatrix_comp, vecMulVecBilin_apply_apply, isAdjointPair_toLinearMap₂', toLin'_apply', LinearMap.toMatrix₂_symm, ker_toLin'_eq_bot_iff, PosSemidef.isUnit_sqrt_iff, LinearMap.SeparatingLeft.toMatrix₂, LinearMap.toMatrix'_toLinearMap₂', BilinForm.toMatrix_mul_basis_toMatrix, LinearMap.BilinForm.mul_toMatrix', toLinearMap₂_symm, coe_toEuclideanCLM_eq_toEuclideanLin, LinearMap.toMatrixRight'_id, LinearMap.minpoly_toMatrix, matrixEquivTensor_apply_single, nondegenerate_toLinearMap₂'_iff_nondegenerate_toLinearMap₂, LinearEquiv.mapMatrix_trans, LinearMap.toMatrixOrthonormal_apply, MatrixEquivTensor.invFun_smul, Algebra.toMatrix_lmul', mem_skewAdjointMatricesSubmodule, ker_diagonal_toLin', matrixEquivTensor_apply_symm, toBilin'_apply, LinearMap.toMatrix_prodMap, toLin_apply, toLin'_one, MatrixEquivTensor.left_inv, doublyStochastic_eq_convexHull_permMatrix, reindexLinearEquiv_mul, toLin_conjTranspose, mulRightLinearMap_apply, RootPairing.GeckConstruction.diagonal_elim_mem_span_h_iff, rank_matrix_module', PosSemidef.sq_sqrt, trace_toLin'_eq, toLin'OfInv_symm_apply, Orientation.rotation_eq_matrix_toLin, LinearEquiv.isUnit_det, mem_selfAdjointMatricesSubmodule, Algebra.traceMatrix_of_basis, spectrum_toLin, LinearMap.toMatrix'_comp, toLinearEquiv'_apply, LinearMap.isUnit_toMatrix'_iff, toBilin_toMatrix, nondegenerate_toLinearMap₂_iff, reindexLinearEquiv_apply, toBilin'_comp, isHermitian_iff_isSymmetric, PosSemidef.inv_sqrt, LinearMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal', Module.Free.matrix, Module.Basis.end_repr_symm_apply, InnerProductSpace.toMatrix_rankOne, rank_matrix', ext_linearMap_iff, LinearMap.mul_toMatrix₂_mul, toEuclideanLin_conjTranspose_eq_adjoint, stdBasis_eq_single, LinearMap.toMatrix'_id, toLinearMapₛₗ₂_symm, LinearMap.BilinForm.mul_toMatrix_mul, LinearMap.BilinForm.toMatrix_mul_basis_toMatrix, LinearMap.nondegenerate_toMatrix₂'_iff, LinearMap.BilinForm.toMatrix'_comp, LinearMap.mapMatrix_apply, mulLinearMap_apply_apply, toLinearMapRight'_apply, LinearMap.BilinForm.toMatrix_mul, rank_matrix, LieAlgebra.SpecialLinear.val_single, ofLp_toEuclideanLin_apply, toLinearMap₂_apply, mulLeftLinearMap_eq_mulLeft, LinearMap.nondegenerate_toLinearMap₂'_iff_det_ne_zero, mulVecBilin_apply, lieEquivMatrix'_symm_apply, proj_diagonal, LieAlgebra.SpecialLinear.singleSubSingle_sub_singleSubSingle, LinearMap.isUnit_toMatrix_iff, LinearMap.BilinForm.toMatrix_compLeft, LinearMap.trace_eq_matrix_trace, BilinForm.toMatrix_apply, linfty_opNorm_toMatrix, toBilin'_apply', toLinearMap₂'_single, LinearMap.separatingRight_toLinearMap₂'_of_det_ne_zero', mulLeftLinearMap_one, LinearMap.BilinForm.mul_toMatrix, reindexLinearEquiv_symm, SpecialLinearGroup.toLin'_to_linearMap, LinearMap.mul_toMatrix₂'_mul, LinearMap.toMatrix₂'_mul, Algebra.toMatrix_lsmul, toMatrix_innerSL_apply, separatingLeft_toLinearMap₂'_iff, LinearMap.separatingLeft_toLinearMap₂'_of_det_ne_zero', BilinForm.toMatrix_mul, LinearMap.toMatrix_comp, ModularGroup.tendsto_lcRow0, kroneckerMapBilinear_mul_mul, separatingLeft_toBilin'_iff, kroneckerAlgEquiv_symm_apply, trace_toLin_eq, LinearMap.toMatrix₂_mul_basis_toMatrix, Module.Basis.toMatrix_eq_toMatrix_constr, det_reindexLinearEquiv_self, toLinOfInv_apply, mulRightLinearMap_mul, toBilin'Aux_eq, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', toLin_finTwoProd_toContinuousLinearMap, finiteDimensional, LinearMap.mapMatrix_id, kroneckerTMulBilinear_apply, uniqueLinearEquiv_symm_apply, toLinearEquivRight'OfInv_apply, hasEigenvalue_toLin'_diagonal_iff, LinearMap.toMatrix'_toLinearMapₛₗ₂', toLpLin_mul, toLin'_submatrix, separatingLeft_toLinearMap₂_iff, toLin_toMatrix, LinearMap.toMatrix₂_apply, PiToModule.fromMatrix_apply, toEuclideanLin_apply_piLp_toLp, Polynomial.toMatrix_sylvesterMap, dualNumberEquiv_apply, separatingRight_toBilin'_iff, toLin_finTwoProd, SpecialLinearGroup.toLin_equiv.toLinearMap_eq, Module.Basis.linearMap_repr_symm_apply, kroneckerTMulAlgEquiv_apply, LinearMap.nondegenerate_toMatrix₂_iff, trace_kroneckerMapBilinear, LinearMap.mapMatrix_add, toLin_one, toLin_transpose, Complex.toMatrix_conjAe, BilinForm.mul_toMatrix, MatrixEquivTensor.right_inv, toLpLin_symm_comp, toLinearMapₛₗ₂_apply_basis, LinearMap.toMatrix_toSpanSingleton, LinearMap.toMatrix₂_symm', mulLeftLinearMap_zero_eq_zero, LinearMap.Nondegenerate.toMatrix₂', MatrixEquivTensor.invFun_zero, Nondegenerate.toBilin', LinearMap.toMatrix_singleton, LinearMap.BilinForm.toMatrix'_mul, LinearMap.toMatrix₂'_compl₂, entryLinearMap_apply, toLinearEquiv_kroneckerAlgEquiv, nondegenerate_toLinearMap₂'_iff, LinearMap.separatingRight_toMatrix₂_iff, reindexLinearEquiv_refl_refl, LinearMap.separatingLeft_toMatrix₂_iff, Module.Finite.matrix, basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix, LinearMap.BilinForm.toMatrix'_toBilin', LinearMap.det_toMatrix', separatingLeft_toBilin_iff, LinearMap.Nondegenerate.toMatrix₂, toLin'OfInv_apply, mulLeftLinearMap_eq_zero_iff, LinearMap.BilinForm.Nondegenerate.toMatrix', LinearMap.toMatrix_basis_equiv, reindexLinearEquiv_comp_apply, SeparatingRight.toBilin', PosSemidef.sqrt_mul_self, BilinForm.apply_eq_dotProduct_toMatrix_mulVec, range_diagonal, piLinearEquiv_symm_apply, RootPairing.Base.cartanMatrixIn_mul_diagonal_eq, mulLeftLinearMap_apply, BilinForm.toMatrix_compLeft, maxGenEigenspace_toLin_diagonal_eq_eigenspace, LinearMap.toMatrix₂'_comp, mulLeftLinearMap_mul, toLinearMap₂_apply_basis, coe_ofLinearEquiv_symm, LinearMap.toMatrix_pow, mem_selfAdjointMatricesSubmodule', LinearMap.BilinForm.toMatrix_comp, SpecialLinearGroup.toLin'_symm_apply, ModularGroup.lcRow0_apply, LinearMap.toMatrix'_one, LieAlgebra.SpecialLinear.val_singleSubSingle, iSup_eigenspace_toLin_diagonal_eq_top, basis_toMatrix_mul_linearMap_toMatrix, LinearMap.toMatrixₛₗ₂'_apply, toLinearMapRight'_one, maxGenEigenspace_toLin'_diagonal_eq_eigenspace, Module.Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix, LinearEquiv.entryLinearMap_comp_mapMatrix, toLpLin_apply, entryLinearMap_toAddHom, toLin'_apply, LinearMap.BilinForm.SeparatingRight.toMatrix', LinearMap.toMatrix_adjoint, compLinearEquiv_apply, LinearMap.diag_toMatrix_directSum_collectedBasis_eq_zero_of_mapsTo_ne, LinearMap.separatingLeft_toLinearMap₂'_iff_det_ne_zero, LinearMap.BilinForm.Nondegenerate.toMatrix, l2_opNorm_def, matrixEquivTensor_apply, toEuclideanLin_toLp, MatrixEquivTensor.toFunAlgHom_apply, entryLinearMap_toAddMonoidHom, LinearMap.det_toMatrix, Algebra.traceForm_toMatrix_powerBasis, minpoly_toLin, basis_toMatrix_mul, toLin'_toMatrix', Submodule.coe_matrix, LinearMap.det_toLin, LinearMap.posSemidef_toMatrix_iff, coe_ofLinearEquiv, LinearMap.mapMatrix_neg, LinearMap.BilinForm.nondegenerate_toBilin'_of_det_ne_zero', LinearMap.SeparatingLeft.toMatrix₂', TensorProduct.toMatrix_comm, LinearMap.toMatrix₂'_apply, kroneckerStarAlgEquiv_apply, separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂, LinearMap.BilinForm.separatingLeft_toMatrix_iff, Real.volume_preserving_transvectionStruct, toBilin'_toMatrix', toLinearMapₛₗ₂_toMatrix₂, LinearMap.toMatrix'_mulVec, SeparatingLeft.toLinearMap₂, hasEigenvector_toLin_diagonal, LinearMap.separatingRight_toLinearMap₂'_iff_det_ne_zero, dotProduct_toMatrix₂_mulVec, entryLinearMap_eq_comp, diagonal_comp_single, PosDef.posDef_sqrt, LinearMap.traceAux_def, Algebra.leftMulMatrix_apply, InnerProductSpace.symm_toEuclideanLin_rankOne, LinearMap.BilinForm.toMatrix'_compRight, rank_matrix'', LieAlgebra.SpecialLinear.singleSubSingle_sub_singleSubSingle', SeparatingRight.toBilin, transposeLinearEquiv_apply, LinearMap.toMatrix_apply, TensorProduct.toMatrix_map, rank_vecMulVec, LinearMap.toMatrix_id_eq_basis_toMatrix, mulLinearMap_eq_mul, LinearMap.toMatrixRight'_comp, kroneckerStarAlgEquiv_symm_apply, LinearMap.mapMatrix_sub, toBilin_comp, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj, SimpleGraph.mem_ker_toLin'_lapMatrix_of_connectedComponent, ModularGroup.smul_eq_lcRow0_add, LinearMap.det_toMatrix_eq_det_toMatrix, SpecialLinearGroup.toLin'_apply, PosSemidef.sqrt_eq_iff_eq_sq, Nondegenerate.toLinearMap₂', AlgHom.mulLeftRightMatrix.comp_inv, separatingRight_toLinearMap₂'_iff_separatingRight_toLinearMap₂, LinearMap.toMatrix₂Aux_apply, LinearMap.charpoly_toMatrix, LinearMap.toLinearMap₂'Aux_toMatrix₂Aux, LinearMap.isSymm_iff_isHermitian_toMatrix, toBilin'_single, LinearMap.toMatrix_toLin, LinearMap.polyCharpolyAux_map_eq_toMatrix_charpoly, toLinearEquiv_apply, hasEigenvalue_toLin_diagonal_iff, spectrum_toLin', kroneckerLinearEquiv_tmul

MeasureTheory.Lp.simpleFunc

Definitions

Name	Category	Theorems
module 📖	CompOp	27 math math: MeasureTheory.L1.SimpleFunc.norm_Integral_le_one, MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left, MeasureTheory.L1.SimpleFunc.setToL1SCLM_const, MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM, MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left', smul_toSimpleFunc, MeasureTheory.L1.setToL1_eq_setToL1SCLM, MeasureTheory.L1.SimpleFunc.setToL1S_smul_real, MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left, MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le', MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left, MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left', MeasureTheory.L1.setToL1_apply_coeToLp, MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left', MeasureTheory.L1.SimpleFunc.integral_smul, MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le, MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_measure, toLp_smul, MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left', MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left, MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left, MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono, MeasureTheory.L1.setToL1'_eq_setToL1SCLM, MeasureTheory.L1.SimpleFunc.setToL1SCLM_nonneg, MeasureTheory.L1.setToL1'_apply_coeToLp, MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left', MeasureTheory.L1.SimpleFunc.setToL1S_smul

Module.DirectLimit

Definitions

Name	Category	Theorems
module 📖	CompOp	31 math math: lift_injective, map_apply_of, lift_of, TensorProduct.directLimitRight_tmul_of, quotMk_of, congr_symm_apply_of, map_comp, of_f, ModuleCat.directLimitIsColimit_desc, Submodule.FG.lTensor.directLimit_apply', linearEquiv_of, Submodule.FG.rTensor.directLimit_apply, TensorProduct.directLimitLeft_rTensor_of, lift_comp_of, congr_apply_of, exists_of, map_id, linearEquiv_symm_mk, lift_of', Submodule.FG.lTensor.directLimit_apply, Module.fgSystem.equiv_comp_of, ModuleCat.directLimitCocone_ι_app, TensorProduct.directLimitLeft_tmul_of, TensorProduct.directLimitLeft_symm_of_tmul, TensorProduct.fromDirectLimit_of_tmul, ModuleCat.directLimitCocone_pt_isModule, TensorProduct.directLimitRight_symm_of_tmul, Submodule.FG.rTensor.directLimit_apply', hom_ext_iff, exists_of₂, TensorProduct.toDirectLimit_tmul_of

Module.IsTorsionBy

Definitions

Name	Category	Theorems
module 📖	CompOp	—

Module.IsTorsionBySet

Definitions

Name	Category	Theorems
module 📖	CompOp	2 math math: isScalarTower, isSemisimpleModule_iff

MonoidAlgebra

Definitions

Name	Category	Theorems
module 📖	CompOp	16 math math: instIsCocomm, tensorEquiv.invFun_tmul, of_mem_span_of_iff, scalarTensorEquiv_tmul, instFree, basis_apply, mem_span_support, lsingle_apply, tensorEquiv_symm_single, moduleFinite, tensorEquiv_tmul, lhom_ext'_iff, scalarTensorEquiv_symm_single, comul_single, instIsTorsionFree, counit_single

MvPolynomial

Definitions

Name	Category	Theorems
module 📖	CompOp	305 math math: pUnitAlgEquiv_symm_monomial, dvd_monomial_one_iff_exists, MonomialOrder.sPolynomial_leadingTerm_mul', Algebra.PreSubmersivePresentation.jacobiMatrix_naive, totalDegree_monomial, DirectSum.coeLinearMap_eq_dfinsuppSum, weightedHomogeneousComponent_of_isWeightedHomogeneous_same, Algebra.PreSubmersivePresentation.cotangentComplexAux_apply, rTensor_apply_tmul_apply, degrees_monomial, rTensorAlgEquiv_apply, one_def, Algebra.Generators.H1Cotangent.δAux_mul, scalarRTensor_apply_monomial_tmul, homogeneousComponent_eq_zero', universalFactorizationMap_comp_map, MonomialOrder.degree_monomial, IsWeightedHomogeneous.pderiv, X_pow_eq_monomial, divMonomial_add_modMonomial, eval₂Hom_monomial, homogeneousComponent_mem, support_monomial, monomial_finsupp_sum_index, weightedHomogeneousSubmodule_mul, mul_def, scalarRTensor_apply_X_tmul_apply, smul_monomial, Algebra.PreSubmersivePresentation.cotangentComplexAux_zero_iff, IsWeightedHomogeneous.weightedHomogeneousComponent_same, pderiv_one, pderiv_mul, monomial_zero, Polynomial.Bivariate.Polynomial.Bivariate.pderiv_one_equivMvPolynomial, Algebra.Generators.cotangentSpaceBasis_repr_one_tmul, pderiv_rename, monomial_eq_monomial_iff, eval₂_mul_monomial, weightedHomogeneousComponent_zero, homogeneousComponent_C_mul, instFree, weightedDecomposition.decompose'_apply, expand_monomial, monomial_mul_mem_coeffsIn, Algebra.Generators.H1Cotangent.δAux_C, monomial_one_dvd_iff_modMonomial_eq_zero, X_mul_pderiv_monomial, weightedHomogeneousComponent_eq_zero', homogeneousComponent_isHomogeneous, sum_homogeneousComponent, degreesLE_nsmul, coe_monomial, C_apply, single_eq_monomial, mul_X_mem_coeffsIn, MonomialOrder.leadingTerm_monomial, X_mul_mem_coeffsIn, IsHomogeneous.HomogeneousSubmodule.gcommSemiring, degreesLE_add, MonomialOrder.span_leadingTerm_eq_span_monomial, MonomialOrder.span_leadingTerm_eq_span_monomial₀, scalarRTensor_apply_tmul, C_mul_X_eq_monomial, bind₂_monomial, monomialOneHom_apply, pderiv_sumToIter, coeff_homogeneousComponent, monomial_left_inj, X_dvd_monomial, Algebra.PreSubmersivePresentation.jacobiMatrix_apply, C_mul_monomial, Algebra.Generators.H1Cotangent.δAux_monomial, vars_monomial, pow_idealOfVars_eq_span, Algebra.Generators.cotangentSpaceBasis_repr_tmul, bind₂_monomial_one, rTensor_apply_X_tmul, Algebra.SubmersivePresentation.linearIndependent_aeval_val_pderiv_relation, pderiv_X, restrictSupport_univ, Algebra.Generators.H1Cotangent.δAux_toAlgHom, algebraTensorAlgEquiv_symm_comp_aeval, finitePresentation_universalFactorizationMap, map_restrict_dom_evalₗ, tensorEquivSum_one_tmul_C, rTensor_apply_tmul, IsWeightedHomogeneous.weightedHomogeneousComponent_ne, coeff_monomial_mul, dvd_monomial_iff_exists, DirectSum.coeAddMonoidHom_eq_support_sum, mkDerivationₗ_monomial, IsWeightedHomogeneous.sum_weight_X_mul_pderiv, sum_weightedHomogeneousComponent, pderiv_def, modMonomial_add_divMonomial, MonomialOrder.span_leadingTerm_eq_span_monomial', decompose'_apply, monomial_one_mul_cancel_left_iff, coe_basisMonomials, monomial_mem_pow_idealOfVars_iff, irreducible_sumSMulX, finite_universalFactorizationMap, rTensorAlgHom_toLinearMap, homogeneousSubmodule_zero, coeff_sumSMulX, universalFactorizationMapPresentation_map, restrictSupport_mono, algebraTensorAlgEquiv_symm_X, mem_restrictDegree, MonomialOrder.monic_monomial, indicator_mem_restrictDegree, coeffsIn_mul, MonomialOrder.sPolynomial_def, Algebra.Generators.H1Cotangent.δAux_ofComp, restrictSupport_zero, as_sum, MonomialOrder.leadingCoeff_monomial, monomial_add_single, le_coeffsIn_pow, MonomialOrder.monic_monomial_one, monomial_eq_zero, eval_monomial, coeff_monomial_mul', monomial_mem_coeffsIn, mkDerivation_monomial, rename_monomial, restrictScalars_restrictSupportIdeal, instFiniteSubtypeMemSubmoduleRestrictDegreeOfFinite, mem_restrictSupport_iff, degreesLE_zero, universalFactorizationMapPresentation_relation, Algebra.PreSubmersivePresentation.aevalDifferential_single, isWeightedHomogeneous_monomial, monomial_zero', weightedHomogeneousComponent_directSum, homogeneousComponent_apply, divMonomial_mul_monomial, Algebra.Presentation.tensorModelOfHasCoeffsEquiv_tmul, weightedHomogeneousComponent_finsupp, linearIndependent_X, prod_X_pow_eq_monomial, constantCoeff_monomial, universalFactorizationMapPresentation_algebra_algebraMap, coeff_rTensorAlgHom_tmul, mem_weightedHomogeneousSubmodule, divMonomial_monomial, KaehlerDifferential.mvPolynomialBasis_repr_apply, mem_coeffsIn, Algebra.Presentation.tensorModelOfHasCoeffsHom_tmul, Polynomial.Bivariate.pderiv_zero_equivMvPolynomial, weightedHomogeneousComponent_of_isWeightedHomogeneous_ne, isHomogeneous_monomial, pderiv_C_mul, Module.Basis.symmetricAlgebra_repr_apply, decomposition.decompose'_apply, restrictSupport_nsmul, mem_degreesLE, weightedHomogeneousComponent_isWeightedHomogeneous, map_monomial, monomial_mem_restrictSupport, homogeneousComponent_eq_zero, MonomialOrder.sPolynomial_monomial_mul, mem_homogeneousSubmodule, homogeneousComponent_zero, ker_eval₂Hom_universalFactorizationMap, monomial_left_injective, monomial_sum_index, monic_monomial_eq, pderiv_monomial_single, instFiniteSubtypeMemSubmoduleRestrictTotalDegreeOfFinite, DirectSum.coeLinearMap_eq_finsum, support_monomial_subset, monomial_mul, rTensor_symm_apply_single, pderiv_X_of_ne, Polynomial.Bivariate.Polynomial.Bivariate.pderiv_zero_equivMvPolynomial, pderiv_monomial, Algebra.Generators.toComp_toAlgHom_monomial, monomial_mem_homogeneousSubmodule_pow_degree, finrank_eq_zero, dvd_monomial_mul_iff_exists, Algebra.Generators.H1Cotangent.δAux_X, mem_restrictDegree_iff_sup, weightedHomogeneousComponent_eq_zero_of_notMem, rTensor_apply_monomial_tmul, Algebra.Generators.repr_CotangentSpaceMap, pderiv_pow, mkDerivationₗ_C, tensorEquivSum_X_tmul_X, coeff_rTensorAlgHom_monomial_tmul, support_sum_monomial_coeff, mem_coeffsIn_iff_coeffs_subset, algebraTensorAlgEquiv_symm_map, weightedDecomposition.decompose'_eq, tensorEquivSum_C_tmul_one, coeffsIn_eq_span_monomial, Algebra.Generators.ofComp_toAlgHom_monomial_sumElim, Algebra.Presentation.algebraTensorAlgEquiv_symm_relation, coeffsIn_pow, monomial_eq, pderiv_map, esymm_eq_sum_monomial, HomogeneousSubmodule.gradedMonoid, instFiniteOfIsEmpty, weightedHomogeneousComponent_eq_zero, restrictSupport_add, totalDegree_monomial_le, pderiv_inr_universalFactorizationMap_X, weightedHomogeneousComponent_C_mul, Algebra.Presentation.tensorModelOfHasCoeffsEquiv_symm_tmul, mul_monomial_mem_coeffsIn, KaehlerDifferential.mvPolynomialBasis_repr_D, MonomialOrder.sPolynomial_monomial_mul', pderiv_eq_zero_of_notMem_vars, C_mul_X_pow_eq_monomial, weightedHomogeneousComponent_mem, Algebra.Generators.cotangentRestrict_mk, optionEquivLeft_monomial, Algebra.Generators.H1Cotangent.δ_eq_δAux, KaehlerDifferential.mvPolynomialBasis_repr_comp_D, algebraTensorAlgEquiv_tmul, homogeneousSubmodule_one_eq_span_X, divMonomial_monomial_mul, sum_monomial_eq, degrees_monomial_eq, pUnitAlgEquiv_monomial, coeff_monomial, coeff_mul_monomial, mem_ideal_span_monomial_image_iff_dvd, IsRegular.monomial, monomial_pow, eval₂_monomial, homogeneousSubmodule_one_pow, universalFactorizationMapPresentation_algebra_smul, IsHomogeneous.sum_X_mul_pderiv, aeval_sumElim_pderiv_inl, monomial_one_dvd_monomial_one, mul_monomial_modMonomial, tensorEquivSum_one_tmul_X, one_coeffsIn, mkDerivationₗ_X, universalFactorizationMap_freeMonic, Algebra.Generators.Hom.toAlgHom_monomial, universalFactorizationMapPresentation_val, universalFactorizationMapPresentation_σ', monomial_sum_one, scalarRTensor_symm_apply_single, irreducible_sumSMulXSMulY, mem_ideal_span_monomial_image, coeff_weightedHomogeneousComponent, rTensor_apply, C_mem_coeffsIn, disjoint_support_monomial, aeval_monomial, MonomialOrder.degree_monomial_le, coeff_mul_monomial', pderiv_X_self, pderiv_C, universalFactorizationMapPresentation_jacobiMatrix, MvRatFunc.rank_eq_max_lift, homogeneousSubmodule_mul, Polynomial.Bivariate.pderiv_one_equivMvPolynomial, lcoeff_apply, algebraTensorAlgEquiv_symm_monomial, restrictSupport_eq_span, rank_eq, SymmetricAlgebra.IsSymmetricAlgebra.mvPolynomial, finrank_eq_one, tensorEquivSum_X_tmul_one, scalarRTensor_apply_tmul_apply, coeffsIn_le, MonomialOrder.C_mul_leadingCoeff_monomial_degree, Algebra.Generators.comp_σ, tensorEquivSum_C_tmul_C, finsum_weightedHomogeneousComponent, linearMap_ext_iff, monomial_single_add, monomial_one_mul_cancel_right_iff, Polynomial.homogenizeLM_apply, degreeOf_monomial_eq, weightedHomogeneousComponent_apply, rank_eq_lift, IsHomogeneous.pderiv, mem_restrictTotalDegree, restrictTotalDegree_le_restrictDegree, Algebra.SubmersivePresentation.basisDeriv_apply, bind₁_monomial, induction_on_monomial, coe_coeffsIn, vars_monomial_single, evalₗ_apply, monomial_modMonomial, WeightedHomogeneousSubmodule.gradedMonoid, MonomialOrder.sPolynomial_mul_monomial, decomposition.decompose'_eq, universalFactorizationMapPresentation_jacobian, MonomialOrder.sPolynomial_leadingTerm_mul, Polynomial.UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap, Polynomial.homogenize_monomial, Polynomial.UniversalFactorizationRing.fromTensor_comp_universalFactorizationMap', monomial_dvd_monomial, monomial_mul_modMonomial, pderiv_inl_universalFactorizationMap_X, Algebra.Presentation.tensorModelOfHasCoeffsInv_aeval_val, rTensorAlgHom_apply_eq

NormedAddGroupHom

Definitions

Name	Category	Theorems
module 📖	CompOp	—

PUnit

Definitions

Name	Category	Theorems
module 📖	CompOp	9 math math: Submodule.botEquivPUnit_apply, IsNoetherian.equivPUnitOfProdInjective_apply, Submodule.botEquivPUnit_symm_apply, rank_punit, CliffordAlgebraRing.ι_eq_zero, CliffordAlgebraRing.involute_eq_id, CliffordAlgebraRing.reverse_apply, CliffordAlgebraRing.reverse_eq_id, IsNoetherian.equivPUnitOfProdInjective_symm_apply

Pi

Definitions

Name	Category	Theorems
module 📖	CompOp	468 math math: comul_eq_adjoint, DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply, Rep.resCoindHomEquiv_apply_hom, ContinuousLinearMap.sum_comp_single, PiLp.continuousLinearEquiv_symm_apply, hasStrictFDerivAt_apply, differentiableWithinAt_finCons', iSupIndep.dfinsupp_lsum_injective, ModuleCat.biproductIsoPi_inv_comp_π, DFinsupp.lsum_lsingle, MultilinearMap.fromDFinsuppEquiv_apply, LinearMap.toAddMonoidHom_proj, hasStrictFDerivAt_list_prod_finRange', dotProductEquiv_symm_apply, LinearEquiv.piCongrRight_apply, LinearIsometryEquiv.piLpCongrRight_symm, ContinuousMultilinearMap.norm_iteratedFDerivComponent_le, Submodule.fg_pi, hasFDerivAtFilter_pi, hasFDerivAt_multiset_prod, LinearMap.ker_pi, AlternatingMap.coe_pi, QuadraticMap.posDef_pi_iff, AffineMap.pi_lineMap_apply, differentiableWithinAt_pi'', LinearMap.IsSymmetric.diagonalization_apply_self_apply, comul_comp_dFinsuppCoeFnLinearMap, counit_coe_dFinsupp, ContinuousAlternatingMap.opNNNorm_pi, LinearMap.det_pi, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, QuadraticMap.Isometry.proj_comp_single_of_same, LinearEquiv.piCongrRight_trans, differentiableOn_pi, linearIndependent_single, MultilinearMap.piFamily_add, AffineMap.proj_linear, locallyConvexSpace, Submodule.pi_univ_bot, ContinuousAlternatingMap.piLIE_apply_apply, QuadraticMap.Isometry.single_apply, AdicCompletion.incl_apply, ModuleCat.instSmallSubtypeForallCarrierObjMemSubmoduleSectionsSubmodule, Submodule.quotientPi_aux.map_smul, ContinuousMultilinearMap.piLinearEquiv_symm_apply, ContinuousAlternatingMap.coe_pi, ExteriorAlgebra.liftAlternating_ι_mul, Fintype.linearIndependent_iff', Algebra.TensorProduct.piRightHom_mul, HasFDerivWithinAt.continuousMultilinearMapCompContinuousLinearMap, LinearEquiv.piFinTwo_symm_apply, IsLocalFrameOn.coeff_apply_of_mem, Submodule.iSup_map_single_le, comul_comp_single, cfcₙ_map_pi, hasDerivWithinAt_pi, LinearMap.iSup_range_single, LinearIsometryEquiv.piLpCongrRight_apply, ContinuousLinearMap.single_apply, OrthonormalBasis.measurePreserving_repr, differentiable_finCons', LinearEquiv.piCurry_apply, counit_comp_finsuppLcoeFun, ExteriorAlgebra.liftAlternating_comp, LinearMap.iInf_ker_proj, hasStrictFDerivAt_pi', HasFDerivAt.continuousMultilinearMapCompContinuousLinearMap, mem_convexHull_pi, PiLp.coe_continuousLinearEquiv, ContinuousMultilinearMap.linearDeriv_apply, Submodule.biSup_eq_range_dfinsupp_lsum, Rep.resCoindAdjunction_unit_app_hom_hom, ContinuousAlternatingMap.piEquiv_apply, hasFDerivWithinAt_pi', fderivWithin_pi, quasispectrum_eq, Fin.consEquivL_apply, MeasureTheory.charFunDual_pi', differentiable_apply, LinearMap.iInf_ker_proj_le_iSup_range_single, Submodule.mem_pi, counit_eq_adjoint, AdicCompletion.pi_apply_coe, MultilinearMap.fromDirectSumEquiv_symm_apply, ModuleCat.HasLimit.productLimitCone_cone_π, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace, compRightL_apply, DFinsupp.injective_pi_lapply, CStarMatrix.inner_toCLM_conjTranspose_left, DFinsupp.range_mapRangeLinearMap, counit_comp_dFinsuppCoeFnLinearMap, ContinuousLinearMap.pi_apply, EuclideanSpace.volume_closedBall_fin_three, basis_repr_single, ContinuousLinearEquiv.piUnique_symm_apply, intrinsicStar_comul_commSemiring, hasStrictFDerivAt_multiset_prod, Module.Free.pi, Finsupp.sigmaFinsuppLEquivPiFinsupp_symm_apply, QuadraticMap.Ring.polarBilin_pi, ContinuousLinearMap.det_pi, PiLp.sumPiLpEquivProdLpPiLp_symm_apply_ofLp, ContinuousLinearMap.iInf_ker_proj, DirectSum.ker_lmap, ExteriorAlgebra.liftAlternatingEquiv_symm_apply, hasFDerivAt_finCons, hasFDerivWithinAt_finCons, IsLocalFrameOn.coeff_congr, ContinuousLinearEquiv.piUnique_apply, PiLp.volume_preserving_toLp, Module.rankAtStalk_pi, ContinuousLinearMap.proj_apply, AffineMap.pi_apply, ExteriorAlgebra.liftAlternating_ιMulti, LinearMap.lsum_apply, basis_apply, Matrix.piLinearEquiv_apply, DFinsupp.sum_mapRange_index.linearMap, Ideal.pi_mkQ_surjective, ExteriorAlgebra.liftAlternatingEquiv_apply, LinearMap.iSup_range_single_eq_iInf_ker_proj, differentiableOn_finCons', LinearMap.proj_apply, differentiableAt_apply, Submodule.biInf_comap_proj, ContinuousLinearMap.pi_eq_zero, ContinuousLinearMap.pi_proj, ContinuousAlternatingMap.piEquiv_symm_apply, differentiableAt_pi'', intrinsicStar_comul, LinearEquiv.piUnique_symm_apply, PiLp.hasFDerivAt_toLp, CFC.nnrpow_map_pi, deriv_pi, EuclideanSpace.volume_closedBall_fin_two, hasFDerivAt_apply, LinearMap.IsSymmetric.diagonalization_symm_apply, LinearEquiv.piUnique_apply, LinearEquiv.sumPiEquivProdPi_apply, hasFTaylorSeriesUpToOn_pi', AffineMap.proj_apply, Submodule.mem_iSup_iff_exists_dfinsupp, hasFDerivWithinAt_finCons', Submodule.iSup_eq_range_dfinsupp_lsum, ExteriorAlgebra.liftAlternating_apply_ιMulti, MultilinearMap.fromDFinsuppEquiv_single, Submodule.pi_mono, Module.jacobson_pi_le, Ideal.pi_tensorProductMk_quotient_surjective, hasFDerivAt_list_prod', hasStrictFDerivAt_finCons, LinearMap.coe_proj, Submodule.pi_empty, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed, ContinuousAlternatingMap.hasStrictFDerivAt, LinearMap.pi_apply, UniqueDiffOn.pi, LinearMap.pi_eq_zero, differentiableOn_pi'', LinearIsometryEquiv.piLpCurry_apply, hasFDerivAt_finCons', instPolynormableSpaceForall, hasStrictFDerivAt_piLp, TensorProduct.piRight_symm_apply, IsLocalizedModule.pi, hasStrictDerivAt_finCons', hasStrictDerivAt_finCons, MultilinearMap.map_add_sub_map_add_sub_linearDeriv, ContinuousMultilinearMap.piLinearEquiv_apply, ExteriorAlgebra.liftAlternating_one, LinearIsometryEquiv.piLpCongrRight_single, DirectSum.linearEquivFunOnFintype_lof, CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_symm_apply, MultilinearMap.freeDFinsuppEquiv_def, ModuleCat.piIsoPi_inv_kernel_ι_apply, ContinuousLinearMap.continuousMultilinearMapOption_apply_eq_self, hasFDerivAt_list_prod_finRange', Fintype.linearIndependent_iff'ₛ, TensorProduct.piRight_symm_single, PiLp.volume_preserving_ofLp, ContinuousMultilinearMap.hasStrictFDerivAt, ContinuousLinearMap.coe_pi, ContinuousMultilinearMap.pi_apply, AffineMap.proj_pi, hasFDerivWithinAt_piLp, Submodule.pi_top, differentiableOn_piLp, Module.jacobson_pi_eq_bot, DFinsupp.linearEquivFunOnFintype_symm_apply, CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_apply, basis_repr, ContinuousMultilinearMap.opNorm_pi, differentiableAt_finCons, QuadraticMap.Isometry.proj_comp_single_of_ne, IsBaseChange.pi, IsLocalFrameOn.eventually_eq_sum_coeff_smul, LinearMap.lsum_single, Matrix.proj_comp_diagLinearMap, Submodule.quotientPi_apply, differentiableAt_piLp, Matrix.diagonal_toLin', differentiable_pi, ContinuousAlternatingMap.pi_apply, LinearEquiv.piCongrLeft'_symm_apply, LinearEquiv.piCongrRight_symm, LinearMap.pi_ext_iff, Ideal.ker_tensorProductMk_quotient, TensorProduct.piRightHom_tmul, fderivWithin_continuousMultilinearMapCompContinuousLinearMap, hasFDerivAt_single, MultilinearMap.fromDirectSumEquiv_lof, Submodule.closure_coe_iSup_map_single, FiniteDimensional.finiteDimensional_pi', QuadraticMap.pi_apply_single, IsLocalFrameOn.coeff_apply_of_notMem, DFinsupp.lsum_single, LinearIsometryEquiv.piLpCongrRight_refl, LinearMap.pi_proj_comp, WithCStarModule.pi_inner, ContinuousAlternatingMap.toContinuousMultilinearMapCLM_comp_fderivCompContinuousLinearMap, MultilinearMap.dfinsuppFamilyₗ_apply, coe_lpPiLpₗᵢ, Submodule.pi_liftQ_eq_liftQ_pi, fderiv_update, CStarMatrix.mul_entry_mul_eq_inner_toCLM, MultilinearMap.map_add_eq_map_add_linearDeriv_add, MultilinearMap.piFamily_apply, Finsupp.sigmaFinsuppLEquivPiFinsupp_apply, orthonormalBasis.toBasis, IsLocalFrameOn.coeff_apply_zero_at, ContDiffMapSupportedIn.isUniformEmbedding_pi_structureMapCLM, ContinuousMultilinearMap.piEquiv_apply, WithCStarModule.inner_single_right, convexHull_pi, Matrix.ker_diagonal_toLin', differentiableWithinAt_pi, LinearEquiv.sumPiEquivProdPi_symm_apply, EuclideanSpace.volume_ball_fin_three, PiLp.projₗ_apply, ContinuousLinearMap.norm_single, DFinsupp.lsum_symm_apply, MultilinearMap.piFamilyₗ_apply, MultilinearMap.fromDirectSumEquiv_apply, derivWithin_pi, QuadraticMap.Isometry.proj_apply, ContinuousAlternatingMap.piLIE_symm_apply_apply, OrthonormalBasis.measurePreserving_repr_symm, hasStrictFDerivAt_finCons', MultilinearMap.piFamily_single_left_apply, LinearIsometryEquiv.piLpCurry_symm_apply, hasStrictFDerivAt_list_prod_attach', ContinuousAlternatingMap.piLinearEquiv_symm_apply, Module.finrank_pi_fintype, hasFDerivAt_update, counit_coe_finsupp, Algebra.TensorProduct.piRightHom_one, LinearMap.pi_comp, Submodule.iSup_map_single, ProbabilityTheory.iIndepFun.hasGaussianLaw, UniqueDiffOn.univ_pi, Submodule.quotientPi_symm_apply, hasDerivAtFilter_finCons', PiLp.hasStrictFDerivAt_apply, DFinsupp.ker_mapRangeLinearMap, hasDerivWithinAt_finCons', ContinuousMultilinearMap.norm_iteratedFDeriv_le', hasStrictFDerivAt_finset_prod, UniqueDiffWithinAt.pi, LinearMap.proj_pi, hasFDerivAtFilter_finCons, ContinuousLinearMap.pi_comp, pi_midpoint_apply, Submodule.quotientPi_aux.map_add, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, DFinsupp.coeFnLinearMap_apply, MultilinearMap.piFamily_single_left, ContinuousMultilinearMap.iteratedFDerivComponent_apply, LinearMap.pi_proj, EuclideanSpace.volume_preserving_symm_measurableEquiv_toLp, LinearMap.coe_single, ContinuousMultilinearMap.hasFDerivAt, DFinsupp.lmk_apply, coe_lpPiLpₗᵢ_symm, differentiable_piLp, ContinuousLinearMap.proj_pi, ContinuousMultilinearMap.piₗᵢ_symm_apply, QuadraticMap.pi_apply, Subalgebra.pi_toSubsemiring, EuclideanSpace.volume_ball_fin_two, instIsSemisimpleModuleForallOfFinite, CFC.nnrpow_eq_nnrpow_pi, SmoothBumpCovering.comp_embeddingPiTangent_mfderiv, LinearMap.pi_ext'_iff, MultilinearMap.piFamily_smul, Matrix.proj_diagonal, fderiv_continuousMultilinearMapCompContinuousLinearMap, comul_coe_finsupp, ModuleCat.HasLimit.productLimitCone_cone_pt_isModule, Module.annihilator_pi, IsLocalFrameOn.eq_iff_coeff, EuclideanSpace.volume_ball, Algebra.TensorProduct.piRight_tmul, ContinuousMultilinearMap.opNNNorm_pi, ExteriorAlgebra.liftAlternating_comp_ιMulti, LinearEquiv.piCurry_symm_apply, ContinuousMultilinearMap.coe_pi, ContinuousMultilinearMap.hasStrictFDerivAt_compContinuousLinearMap, ProbabilityTheory.iIndepFun_iff_charFunDual_pi', ModuleCat.piIsoPi_hom_ker_subtype_apply, LinearMap.intrinsicStar_single, Module.length_pi_of_fintype, AffineMap.pi_comp, Submodule.piQuotientLift_single, ContinuousLinearMap.hasFDerivAt_uncurry_of_multilinear, ModuleCat.piIsoPi_hom_ker_subtype, Subalgebra.coe_pi, Ideal.pi_mkQ_rTensor, HasStrictFDerivAt.continuousMultilinearMapCompContinuousLinearMap, PiLp.hasFDerivAt_ofLp, LinearMap.proj_comp_single, UniqueDiffWithinAt.univ_pi, ModuleCat.piIsoPi_inv_kernel_ι, differentiableAt_finCons', Submodule.quotientPi_aux.left_inv, DFinsupp.linearEquivFunOnFintype_apply, withSeminorms_pi, ContinuousLinearMap.coe_proj, hasDerivAtFilter_finCons, comul_comp_proj, instModuleIsTorsionFree, DirectSum.coeFnLinearMap_apply, MeasureTheory.charFunDual_eq_pi_iff', hasStrictDerivAt_pi, LinearEquiv.piFinTwo_apply, DirectSum.range_lmap, iSupIndep_iff_dfinsupp_lsum_injective, WithCStarModule.inner_single_left, AffineMap.pi_eq_zero, ContinuousLinearMap.norm_pi_le_of_le, hasFDerivWithinAt_pi, ModuleCat.HasLimit.lift_hom_apply, differentiableWithinAt_apply, isNoetherian_pi, ContinuousAlternatingMap.hasFDerivWithinAt, ContinuousMultilinearMap.piEquiv_symm_apply, PiLp.hasFDerivAt_apply, volume_euclideanSpace_eq_dirac, LinearMap.single_apply, ContinuousLinearMap.norm_single_le_one, QuadraticMap.Ring.associated_pi, Module.Injective.pi, hasStrictFDerivAt_list_prod, hasDerivAtFilter_pi, PiLp.proj_apply, ExteriorAlgebra.liftAlternating_algebraMap, comul_single, Submodule.le_comap_single_pi, hasDerivAt_finCons', Submodule.coe_pi, Submodule.quotientPiLift_mk, differentiable_pi'', MultilinearMap.piFamily_single, PiLp.coe_symm_continuousLinearEquiv, differentiableOn_apply, ModuleCat.biproductIsoPi_inv_comp_π_apply, differentiableAt_pi, Finset.expect_apply, ContinuousLinearMap.pi_proj_comp, Module.pi_induction, LinearMap.iSup_range_single_le_iInf_ker_proj, LinearMap.disjoint_single_single, IsLocalFrameOn.coeff_eq_of_eq, hasFDerivAtFilter_finCons', LinearMap.pi_zero, isArtinian_pi, Submodule.quotientPi_aux.right_inv, Module.Finite.pi, AdicCompletion.piEquivOfFintype_apply, MeasureTheory.charFunDual_pi, DirectSum.linearEquivFunOnFintype_symm_single, comul_coe_dFinsupp, MultilinearMap.linearDeriv_apply, QuadraticMap.nonneg_pi_iff, CFC.sqrt_map_pi, MultilinearMap.pi_ext_iff, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed_symm, ContinuousMultilinearMap.piₗᵢ_apply, Matrix.range_diagonal, hasStrictFDerivAt_list_prod', Matrix.piLinearEquiv_symm_apply, LinearMap.proj_surjective, AffineMap.pi_zero, differentiableWithinAt_piLp, PiLp.hasStrictFDerivAt_ofLp, ContinuousMultilinearMap.fderivCompContinuousLinearMap_apply, differentiable_finCons, AlternatingMap.pi_apply, ContinuousAlternatingMap.opNorm_pi, Submodule.mem_biSup_iff_exists_dfinsupp, PiLp.continuousLinearEquiv_apply, MultilinearMap.pi_apply, LinearEquiv.piCongrRight_refl, ContinuousLinearMap.coe_pi', counit_comp_single, Submodule.topologicalClosure_iSup_map_single, LinearMap.lsum_piSingle, IsLocalFrameOn.coeff_sum_eq, ContinuousAlternatingMap.hasFDerivAt, TensorProduct.piRight_apply, ContinuousLinearMap.pi_zero, LinearMap.proj_comp_single_ne, differentiableOn_finCons, LinearEquiv.piCongrLeft'_apply, Module.pi_induction', DFinsupp.lsum_apply_apply, hasDerivAt_pi, hasDerivWithinAt_finCons, QuadraticMap.Equivalent.pi, QuadraticMap.IsometryEquiv.pi_toLinearEquiv, QuadraticMap.Ring.polar_pi, fderiv_pi, MultilinearMap.piFamily_compLinearMap_lsingle, instIsCocomm, OrthonormalBasis.measurePreserving_measurableEquiv, LinearMap.ker_single, Submodule.iInf_comap_proj, AffineMap.pi_linear, hasFDerivAtFilter_pi', ModuleCat.HasLimit.productLimitCone_isLimit_lift, ContinuousAlternatingMap.piLinearEquiv_apply, Module.Finite.pi_iff, LinearMap.lsum_symm_apply, hasFDerivAt_pi', Matrix.entryLinearMap_eq_comp, ContinuousMultilinearMap.hasStrictFDerivAt_uncurry, Matrix.diagonal_comp_single, rank_pi, AdicCompletion.piEquivFin_apply, EuclideanSpace.volume_closedBall, Fin.consEquivL_symm_apply, fderiv_single, hasFDerivAt_list_prod_attach', MultilinearMap.fromDFinsuppEquiv_symm_apply, ContinuousLinearEquiv.piCongrRight_symm_apply, hasStrictFDerivAt_pi, Module.FinitePresentation.pi, counit_single, ExteriorAlgebra.liftAlternating_ι, ContinuousLinearEquiv.piCongrRight_apply, hasDerivAt_finCons, ContinuousLinearEquiv.piFinTwo_symm_apply, Fin.consLinearEquiv_symm_apply, PiLp.sumPiLpEquivProdLpPiLp_apply_ofLp, DirectSum.linearEquivFunOnFintype_symm_coe, lsum_comp_mapRange_toSpanSingleton, DirectSum.linearEquivFunOnFintype_apply, Submodule.piQuotientLift_mk, comul_comp_finsuppLcoeFun, hasFDerivAt_pi, differentiableWithinAt_finCons, hasFDerivWithinAt_apply, CStarMatrix.inner_toCLM_conjTranspose_right, ContinuousLinearEquiv.piFinTwo_apply, Fin.consLinearEquiv_apply, LinearMap.proj_comp_single_same, PiLp.hasStrictFDerivAt_toLp, hasFDerivAt_finset_prod, MultilinearMap.piFamily_zero

Pi.Function

Definitions

Name	Category	Theorems
module 📖	CompOp	1162 math math: Matrix.l2_opNorm_toEuclideanCLM, OrthonormalBasis.singleton_repr, Pi.comul_eq_adjoint, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_left_iff, Matrix.UnitaryGroup.toLin'_one, Rep.resCoindHomEquiv_symm_apply_hom, Finset.weightedVSub_sdiff, Rep.resCoindHomEquiv_apply_hom, groupCohomology.instEpiModuleCatH2π, Lagrange.interpolate_one, Affine.Simplex.affineCombination_mem_closedInterior_iff, rank_fun_infinite, Finset.weightedVSubOfPoint_map, Algebra.SubmersivePresentation.cotangentComplexAux_injective, Module.Basis.equivFun_self, Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_left_iff, PiLp.basisFun_equivFun, dot_self_cross, Finset.affineCombination_vsub, Lagrange.interpolate_eq_of_values_eq_on, Matrix.toLin'_symm, QuaternionAlgebra.coe_linearEquivTuple_symm, Matrix.SeparatingLeft.toBilin', NumberField.mixedEmbedding.normAtComplexPlaces_mixedSpaceOfRealSpace, IsIsotypic.linearEquiv_fun, LinearMap.toMatrixAlgEquiv'_mul, Polynomial.ofFn_zero, groupCohomology.toCocycles_comp_isoCocycles₁_hom, Module.Basis.map_equivFun, groupCohomology.isoCocycles₁_hom_comp_i_apply, Finset.affineCombination_affineCombinationSingleWeights, LinearMap.funLeft_id, groupCohomology.mem_cocycles₂_def, NumberField.Units.finrank_eq_rank, TannakaDuality.FiniteGroup.toRightFDRepComp_in_rightRegular, SimpleGraph.lapMatrix_toLinearMap₂', tendsto_tsum_div_pow_atTop_integral, NumberField.canonicalEmbedding.mem_span_latticeBasis, hasStrictFDerivAt_list_prod_finRange', groupCohomology.d₂₃_hom_apply, Algebra.PreSubmersivePresentation.cotangentComplexAux_apply, dotProductEquiv_symm_apply, Matrix.separatingRight_toLinearMap₂'_iff, ZSpan.volume_real_fundamentalDomain, Lagrange.eval_interpolate_not_at_node', Matrix.det_smul_inv_mulVec_eq_cramer, Matrix.UnitaryGroup.toGL_mul, Finset.weightedVSubOfPoint_apply, Finset.affineCombination_apply_eq_lineMap_sum, Matrix.toLinearMapRight'_mul, Finset.centroid_eq_affineCombination_fintype, hasFDerivAt_multiset_prod, Module.Basis.equivFun_symm_apply, OrthonormalBasis.repr_injective, Matrix.cramer_transpose_row_self, groupCohomology.d₀₁_comp_d₁₂, Matrix.spectrum_toEuclideanLin, NumberField.canonicalEmbedding.latticeBasis_apply, Matrix.iSup_eigenspace_toLin'_diagonal_eq_top, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_completeFamily_of_ne, Matrix.cstar_nnnorm_def, groupCohomology.toCocycles_comp_isoCocycles₂_hom_apply, Module.Basis.constr_comp, Module.Basis.constr_symm_apply, Rep.resCoindAdjunction_counit_app_hom_hom, IsAdjoinRootMonic.coeff_apply, Matrix.toLinearMap₂'_comp, Matrix.l2_opNorm_mulVec, Finset.weightedVSubOfPoint_insert, groupCohomology.eq_d₀₁_comp_inv, Representation.finsupp_apply, groupCohomology.H1π_comp_map_assoc, ZMod.LFunction_one_sub, groupCohomology.π_comp_H1Iso_hom_assoc, TensorProduct.sum_tmul_basis_right_injective, Ideal.map_pi, ZMod.dft_even_iff, Finset.affineCombination_sdiff_sub, ZMod.invDFT_apply, RCLike.linearIndependent_of_ne_zero_of_wInner_one_eq_zero, Matrix.liftLinear_single, groupCohomology.eq_d₁₂_comp_inv, TensorProduct.piScalarRight_symm_algebraMap, LinearMap.BilinForm.toMatrix_basisFun, Rep.indToCoindAux_self, groupCohomology.mapCocycles₂_comp_i, NumberField.mixedEmbedding.fundamentalCone.abs_det_completeBasis_equivFunL_symm, Finset.weightedVSubOfPoint_congr, Rep.diagonalHomEquiv_symm_apply, cross_cross, IsBaseChange.of_fintype_basis_eq, FunOnFinite.linearMap_comp, Matrix.spectrum_toLpLin, Finsupp.linearEquivFunOnFinite_symm_coe, BoxIntegral.unitPartition.mem_smul_span_iff, NumberField.mixedEmbedding.fundamentalCone.completeBasis_apply_of_ne, LinearMap.BilinForm.toMatrix'_symm, Matrix.cramer_row_self, groupCohomology.coe_mapCocycles₁, PiToModule.fromEnd_apply, LieModule.instIsFaithfulMatrixForall, LinearMap.BilinForm.toMatrix'_apply, groupCohomology.d₁₂_hom_apply, LinearMap.nondegenerate_toLinearMap₂'_of_det_ne_zero', ZMod.dft_mul_const, cross_self, groupCohomology.coboundariesToCocycles₁_apply, groupCohomology.cochainsMap_f_3_comp_cochainsIso₃_assoc, Affine.Simplex.wSameSide_affineSpan_faceOpposite_iff, Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_right_iff, Lagrange.eval_interpolate_not_at_node, Algebra.PreSubmersivePresentation.cotangentComplexAux_zero_iff, Matrix.intrinsicStar_toLin', Polynomial.ofFn_eq_sum_monomial, Matrix.coe_mulVecLin, Matrix.mulVecLin_apply, continuous_equivFun_basis, Affine.Simplex.ExcenterExists.affineCombination_eq_excenter_iff, Lagrange.interpolate_eq_nodalWeight_mul_nodal_div_X_sub_C, Module.le_rank_iff_exists_linearMap, Lagrange.interpolate_eq_sum_interpolate_insert_sdiff, QuadraticMap.toMatrix'_comp, Matrix.GeneralLinearGroup.coe_toLin, SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix, Finset.weightedVSub_apply_const, groupCohomology.cochainsMap_f_2_comp_cochainsIso₂, star_dotProduct_toMatrix₂_mulVec, isStablyFiniteRing_iff_injective_of_surjective, IsLocalFrameOn.coeff_apply_of_mem, Finset.weightedVSubOfPoint_indicator_subset, range_vecMulLinear, groupCohomology.comp_d₁₂_eq, groupCohomology.mem_cocycles₁_of_addMonoidHom, NumberField.mixedEmbedding.commMap_apply_of_isComplex, groupCohomology.cocycles₂.d₂₃_apply, groupCohomology.d₀₁_hom_apply, OrthonormalBasis.repr_self, LinearMap.spectrum_toMatrix', Real.smul_map_diagonal_volume_pi, Algebra.TensorProduct.piScalarRight_tmul, AffineIndependent.affineCombination_eq_iff_eq, Matrix.diagLinearMap_apply, Matrix.diagonalLinearMap_apply, strongRankCondition_iff_forall_not_injective, dot_cross_self, groupCohomology.dArrowIso₀₁_inv_right, groupCohomology.d₁₂_comp_d₂₃_apply, Matrix.toLpLin_mul_same, groupCohomology.eq_d₂₃_comp_inv_assoc, Algebra.PreSubmersivePresentation.isUnit_jacobian_iff_aevalDifferential_bijective, groupCohomology.eq_d₂₃_comp_inv_apply, QuadraticAlgebra.linearEquivTuple_apply, QuadraticForm.equivalent_signType_weighted_sum_squared, OrthonormalBasis.measurePreserving_repr, groupCohomology.eq_d₁₂_comp_inv_apply, Finset.weightedVSub_map, RootPairing.GeckConstruction.mem_ωConjLieSubmodule_iff, Matrix.linearIndependent_rows_of_invertible, Matrix.coe_vecMulLinear, CStarMatrix.toCLM_injective, Finset.affineCombination_apply_const, Matrix.SeparatingLeft.toLinearMap₂', Finsupp.llift_apply, LDL.lowerInv_eq_gramSchmidtBasis, Matrix.sum_cramer, mem_vectorSpan_iff_eq_weightedVSub, Fintype.bilinearCombination_apply_single, LinearMap.toMatrixₛₗ₂'_symm, QuadraticMap.discr_smul, dotProductEquiv_apply_apply, LinearMap.rank_diagonal, NumberField.mixedEmbedding.negAt_apply_snd, LinearMap.separatingRight_toMatrix₂'_iff, Matrix.toEuclideanLin_apply, Rep.resCoindAdjunction_unit_app_hom_hom, groupCohomology.mem_cocycles₁_def, IsIsotypic.submodule_linearEquiv_fun, LinearMap.toMatrixAlgEquiv'_id, ProbabilityTheory.IsGaussianProcess.hasGaussianLaw, Matrix.liftLinear_singleLinearMap, Euclidean.closedBall_eq_image, NumberField.mixedEmbedding.negAt_preimage, Matrix.liftLinear_comp_singleLinearMap, QuadraticForm.equivalent_one_neg_one_weighted_sum_squared, groupCohomology.cochainsMap_f_1_comp_cochainsIso₁_apply, Rep.coindFunctorIso_hom_app_hom_hom_apply_hom_hom_apply, LinearMap.toMatrix₂'_compl₁₂, groupCohomology.mapCocycles₂_comp_i_apply, groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_apply, CStarMatrix.norm_def', NumberField.mixedEmbedding.iUnion_negAt_plusPart_ae, BoxIntegral.unitPartition.integralSum_eq_tsum_div, Quaternion.linearIsometryEquivTuple_symm_apply, Matrix.toLinearEquivRight'OfInv_symm_apply, groupCohomology.cochainsMap_f_3_comp_cochainsIso₃_apply, Pi.counit_eq_adjoint, Matrix.toLin'_mul_apply, Finset.sum_smul_vsub_eq_weightedVSubOfPoint_sub, Matrix.toLinearMapₛₗ₂'_single, Matrix.nondegenerate_toBilin'_iff, Lagrange.eq_interpolate_iff, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_apply, groupCohomology.coboundaries₁_eq_bot_of_isTrivial, ContinuousLinearMapWOT.inducingFn_apply, linearIndependent_monoidHom, CStarMatrix.inner_toCLM_conjTranspose_left, LinearMap.toMatrix'_toLin', tendsto_card_div_pow_atTop_volume, Pi.linearIndependent_single_of_ne_zero, EuclideanSpace.instFactEqNatFinrankFin, differentiable_euclidean, NumberField.mixedEmbedding.stdBasis_apply_isComplex_snd, Matrix.toLinearMap₂'_apply', Affine.Simplex.excenter_eq_affineCombination, Finset.weightedVSubOfPoint_vadd, PiToModule.fromEnd_injective, groupCohomology.cocycles₂_map_one_fst, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable, NumberField.mixedEmbedding.negAt_signSet_apply_isComplex, groupCohomology.mapCocycles₂_comp_i_assoc, LinearEquiv.coe_curry, Matrix.ker_mulVecLin_conjTranspose_mul_self, groupCohomology.H1IsoOfIsTrivial_inv_apply, FirstOrder.Field.lift_genericMonicPoly, NumberField.mixedEmbedding.euclidean.stdOrthonormalBasis_map_eq, Matrix.mulVecLin_add, Matrix.linearIndependent_cols_of_isUnit, LinearMap.funLeft_comp, Pi.intrinsicStar_comul_commSemiring, hasStrictFDerivAt_multiset_prod, Module.Basis.ofEquivFun_repr_apply, NumberField.mixedEmbedding.normAtComplexPlaces_polarSpaceCoord_symm, groupCohomology.cocycles₁IsoOfIsTrivial_hom_hom_apply_apply, Lagrange.interpolate_eq_sum, Matrix.SpecialLinearGroup.toLin'_symm_to_linearMap, Matrix.toLinearMap₂'Aux_single, groupCohomology.H2π_comp_map_apply, Matrix.vecMulBilin_apply, cross_anticomm, RootPairing.Base.exists_mem_span_pairingIn_ne_zero_and_pairwise_ne, CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM, cross_cross_eq_smul_sub_smul, Real.map_matrix_volume_pi_eq_smul_volume_pi, affineCombination_mem_affineSpan_pair, ContinuousLinearEquiv.finTwoArrow_symm_apply, groupCohomology.dArrowIso₀₁_hom_right, NumberField.mixedEmbedding.fundamentalCone.logMap_expMapBasis, Finset.weightedVSubOfPoint_filter_of_ne, TensorProduct.equivFinsuppOfBasisLeft_symm, groupCohomology.toCocycles_comp_isoCocycles₂_hom, Module.Basis.coe_toOrthonormalBasis_repr, Matrix.toLinearMap₂'_apply, Affine.Simplex.affineCombination_mem_interior_face_iff_mem_Ioo, LinearMap.funLeft_surjective_of_injective, LinearMap.BilinForm.separatingLeft_toMatrix'_iff, Matrix.linfty_opNNNorm_eq_opNNNorm, ModularGroup.lcRow0Extend_symm_apply, groupCohomology.cochainsMap_f_1_comp_cochainsIso₁_assoc, Finset.affineCombination_congr, Affine.Simplex.circumcenter_eq_affineCombination_of_pointsWithCircumcenter, groupCohomology.cocyclesOfIsCocycle₁_coe, Matrix.isNilpotent_toLin'_iff, Finset.affineCombination_eq_linear_combination, Module.Basis.constrL_apply, IsModuleTopology.continuous_bilinear_of_pi_fintype, Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero, IsLocalFrameOn.coeff_congr, groupCohomology.coboundaries₂_le_cocycles₂, NumberField.Units.span_basisOfIsMaxRank, Matrix.toLinearEquiv'_symm_apply, LinearMap.BilinForm.mul_toMatrix'_mul, Finset.weightedVSub_empty, NumberField.canonicalEmbedding.integralBasis_repr_apply, InnerProductGeometry.norm_toLp_symm_crossProduct, isNoetherian_linearMap_pi, Finsupp.lcoeFun_comp_lsingle, FunOnFinite.continuous_linearMap, NumberField.Units.basisOfIsMaxRank_apply, Rep.coindVEquiv_symm_apply_coe, Finset.sum_smul_const_vsub_eq_vsub_affineCombination, Affine.Simplex.affineCombination_mem_closedInterior_face_iff_nonneg, Finsupp.lcoeFun_apply, Module.Finite.exists_nat_not_surjective, groupCohomology.H1IsoOfIsTrivial_H1π_apply_apply, LinearIsometryEquiv.piLpCongrLeft_apply, NumberField.mixedEmbedding.negAt_signSet_apply_isReal, groupCohomology.comp_d₂₃_eq, Matrix.cramer_apply, mem_affineSpan_iff_eq_affineCombination, LinearEquiv.funUnique_symm_apply, Matrix.toLinearMapₛₗ₂'_symm, Matrix.cRank_toNat_eq_finrank, OrthonormalBasis.coe_ofRepr, groupCohomology.coboundaries₂.val_eq_coe, convexHull_basis_eq_stdSimplex, LinearMap.toBilin'Aux_toMatrixAux, Finset.weightedVSub_weightedVSubVSubWeights, Matrix.linearIndependent_cols_of_det_ne_zero, Complex.orthonormalBasisOneI_repr_apply, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_completeFamily_of_eq, ZMod.dft_comp_unitMul, Finset.weightedVSubOfPoint_sdiff, groupCohomology.d₁₂_apply_mem_cocycles₂, Matrix.toLpLin_symm_pow, groupCohomology.d₀₁_apply_mem_cocycles₁, Finset.affineCombination_of_eq_one_of_eq_zero, affineCombination_mem_affineSpan, Matrix.toLinearMapₛₗ₂'_aux_eq, stdSimplex.map_coe, Rep.indToCoindAux_fst_mul_inv, Matrix.ker_mulVecLin_eq_bot_iff, dist_affineCombination_lt_of_strictConvexSpace, Matrix.linfty_opNNNorm_toMatrix, Module.Basis.coe_constrL, strongRankCondition_iff_forall_rank_lt_aleph0, Matrix.toEuclideanCLM_toLp, NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis, groupCohomology.subtype_comp_d₀₁_apply, ZSpan.volume_fundamentalDomain, basis_toMatrix_basisFun_mul, cross_anticomm', Affine.Simplex.mongePoint_eq_affineCombination_of_pointsWithCircumcenter, groupCohomology.H2π_eq_iff, toBilin'Aux_toMatrixAux, groupCohomology.comp_d₀₁_eq, Finsupp.linearEquivFunOnFinite_symm_apply, Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_left_iff, BilinForm.dotProduct_toMatrix_mulVec, Finset.eq_weightedVSubOfPoint_subset_iff_eq_weightedVSubOfPoint_subtype, QuadraticMap.proj_apply, Projectivization.cross_mk_of_cross_ne_zero, Matrix.toLpLin_one, groupCohomology.cocycles₂_map_one_snd, LinearRecurrence.solSpace_rank, TannakaDuality.FiniteGroup.sumSMulInv_apply, Algebra.PreSubmersivePresentation.aevalDifferential_toMatrix'_eq_mapMatrix_jacobiMatrix, Finset.attach_affineCombination_coe, Matrix.piLp_ofLp_toEuclideanLin, eq_affineCombination_of_mem_affineSpan_image, PowerBasis.constr_pow_algebraMap, AnalyticOnNhd.eval_linearMap, lieEquivMatrix'_apply, Matrix.toPerfectPairing_apply_apply, Matrix.mulVecLin_zero, SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux, FunOnFinite.linearMap_apply_apply, Finset.map_affineCombination, Rep.indToCoindAux_comm, OrthonormalBasis.tensorProduct_repr_tmul_apply', LinearEquiv.sumArrowLequivProdArrow_symm_apply_inr, instFinitePresentationForall, groupCohomology.dArrowIso₀₁_inv_left, LinearEquiv.funCongrLeft_apply, groupCohomology.π_comp_H1Iso_hom, NumberField.mixedEmbedding.fundamentalCone.prod_deriv_expMap_single, LinearMap.minpoly_toMatrix', Matrix.linearIndependent_rows_of_isUnit, Fintype.bilinearCombination_apply, Matrix.l2_opNNNorm_mulVec, NumberField.mixedEmbedding.span_idealLatticeBasis, NumberField.mixedEmbedding.fundamentalCone.norm_expMapBasis, Lagrange.degree_interpolate_erase_lt, Affine.Simplex.wOppSide_affineSpan_faceOpposite_iff, Matrix.SeparatingRight.toLinearMap₂', Algebra.SubmersivePresentation.linearIndependent_aeval_val_pderiv_relation, groupCohomology.cocycles₂_ρ_map_inv_sub_map_inv, PiLp.basisFun_repr, PointwiseConvergenceCLM.isInducing_inducingFn, Matrix.toLpLin_pow, LinearMap.BilinForm.separatingRight_toMatrix'_iff, Matrix.isUnit_toLin'_iff, Module.piEquiv_apply_apply, IsIsotypicOfType.linearEquiv_fun, Finset.weightedVSub_smul, RootPairing.GeckConstruction.trace_toEnd_eq_zero, Matrix.toLin'_mul, Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_right_iff, ModularGroup.lcRow0Extend_apply, Matrix.instLieModuleForall, LinearMap.BilinForm.dotProduct_toMatrix_mulVec, hasDerivAt_update, Lagrange.degree_interpolate_lt, OrthonormalBasis.equiv_apply, groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_assoc, LinearMap.vecCons_apply, Matrix.ofLp_toLpLin, Module.range_piEquiv, Matrix.toMatrix₂Aux_toLinearMap₂'Aux, Matrix.toBilin'_symm, Module.Basis.dualBasis_equivFun, Representation.coe_coindV, Module.Basis.coe_toOrthonormalBasis_repr_symm, RootPairing.EmbeddedG2.allRoots_eq_map_allCoeffs, VertexOperator.ncoeff_apply, Algebra.SubmersivePresentation.sectionCotangent_eq_iff, Matrix.mulVecLin_one, LinearMap.det_toLin', MvPolynomial.map_restrict_dom_evalₗ, groupCohomology.instEpiModuleCatH1π, LinearMap.mapMatrixModule_apply, NumberField.mixedEmbedding.fundamentalCone.completeBasis_apply_of_eq, Module.FinitePresentation.exists_fin, hasFDerivAt_list_prod', Matrix.toLinearMapRight'_mul_apply, groupCohomology.H2π_comp_map, Matrix.eRank_toNat_eq_finrank, Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one, NumberField.mixedEmbedding.mem_span_latticeBasis, Module.Basis.equivFunL_symm_apply_repr, OrthonormalBasis.repr_apply_apply, Rep.coindToInd_apply, Finsupp.lsum_single, QuadraticForm.equivalent_one_zero_neg_one_weighted_sum_squared, Finsupp.linearEquivFunOnFinite_single, LinearMap.separatingLeft_toMatrix₂'_iff, QuadraticForm.equivalent_weightedSumSquares, TannakaDuality.FiniteGroup.leftRegular_apply, ContinuousMap.coeFnLinearMap_apply, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed, Matrix.cramer_zero, hasEigenvector_toLin'_diagonal, Matrix.GeneralLinearGroup.toLin'_apply, LinearMap.toMatrixAlgEquiv'_apply, Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin, groupCohomology.isoCocycles₂_hom_comp_i, NumberField.mixedEmbedding.negAt_apply_isReal_and_notMem, ContinuousLinearMapWOT.isInducing_inducingFn, Matrix.vecMulLinear_apply, Matrix.toLinearMapₛₗ₂'_toMatrix', Affine.Simplex.affineCombination_mem_setInterior_face_iff_mem, LinearMap.toMatrix'_algebraMap, Matrix.cramer_reindex, PiToModule.fromMatrix_apply_single_one, NumberField.Units.dirichletUnitTheorem.unitLattice_span_eq_top, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply, NumberField.mixedEmbedding.fundamentalCone.expMap_basis_of_eq, NumberField.mixedEmbedding.volume_preserving_negAt, IsBaseChange.of_fintype_basis, LinearEquiv.piRing_symm_apply, LinearMap.toMatrix_innerₛₗ_apply, groupCohomology.dArrowIso₀₁_hom_left, LinearMap.funLeft_injective_of_surjective, NumberField.mixedEmbedding.negAt_apply_isComplex, Algebra.PreSubmersivePresentation.jacobian_eq_det_aevalDifferential, Matrix.toBilin_basisFun, Matrix.ker_mulVecLin_transpose_mul_self, Matrix.toLin'_reindex, groupCohomology.cocyclesMap_comp_isoCocycles₁_hom, AffineBasis.toMatrix_vecMul_coords, ContinuousLinearEquiv.finTwoArrow_apply, Matrix.PosSemidef.toLinearMap₂'_zero_iff, LinearMap.BilinForm.toMatrix'_compLeft, groupCohomology.eq_d₀₁_comp_inv_apply, Module.finrank_pi, PiLp.basisFun_apply, span_range_eq_top_iff_surjective_fintypeLinearCombination, NumberField.mixedEmbedding.normAtComplexPlaces_normAtAllPlaces, groupCohomology.cocycles₁_map_inv, Rep.freeLiftLEquiv_apply, Submodule.set_smul_eq_map, groupCohomology.mapCocycles₁_one, finrank_euclideanSpace, LinearMap.BilinForm.nondegenerate_toBilin'_iff_det_ne_zero, LinearMap.toMatrix'_apply, Matrix.toLpLin_toLp, Rep.indToCoindAux_mul_fst, Matrix.rank_eq_finrank_span_row, Lagrange.eq_interpolate, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_apply, ZMod.invDFT_def', NumberField.mixedEmbedding.mem_negAt_plusPart_of_mem, NumberField.Units.regOfFamily_of_isMaxRank, Module.Basis.equivFunL_apply, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, hasFDerivAt_list_prod_finRange', CStarMatrix.toCLM_apply_single, Finset.affineCombination_map, AffineBasis.det_smul_coords_eq_cramer_coords, ContinuousLinearMapWOT.isEmbedding_inducingFn, LinearMap.toMatrix'_symm, eq_affineCombination_of_mem_affineSpan_of_fintype, AffineBasis.coord_apply_combination_of_notMem, InnerProductGeometry.norm_ofLp_crossProduct, groupCohomology.mem_cocycles₂_iff, Matrix.range_mulVecLin, Matrix.ofLp_toEuclideanCLM, CStarMatrix.toCLM_apply_eq_sum, ProbabilityTheory.IsGaussianProcess.hasGaussianLaw_increments, groupCohomology.H2π_comp_map_assoc, NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_mem_fundamentalCone_iff, Finset.affineCombination_indicator_subset, Matrix.range_toLin', RootPairing.GeckConstruction.span_range_h_le_range_diagonal, LinearEquiv.finTwoArrow_apply, LinearMap.pi_apply_eq_sum_univ, ZMod.dft_apply, Rep.coindFunctorIso_inv_app_hom_hom_apply_coe, groupCohomology.d₀₁_ker_eq_invariants, LinearMap.mul_toMatrix', Matrix.liftLinear_apply, LinearMap.toMatrix₂_basisFun, AdicCompletion.ofTensorProduct_bijective_of_pi_of_fintype, HVertexOperator.of_coeff_coeff, Polynomial.toFn_zero, Polynomial.ofFn_zero', Matrix.mul_adjugate_apply, IsLocalFrameOn.eventually_eq_sum_coeff_smul, Matrix.toLpLin_symm_id, differentiableWithinAt_euclidean, HVertexOperator.coeff_inj_iff, Matrix.isPositive_toEuclideanLin_iff, NumberField.mixedEmbedding.fundamentalCone.hasFDerivAt_expMap, Matrix.toBilin'Aux_single, QuadraticMap.discr_comp, Affine.Simplex.affineCombination_mem_interior_iff, groupCohomology.mem_cocycles₁_iff, Matrix.sum_cramer_apply, groupCohomology.inhomogeneousCochains.d_comp_d, ZMod.invDFT_apply', Matrix.UnitaryGroup.coe_toGL, Matrix.linearIndependent_cols_iff_isUnit, Finset.weightedVSub_const_smul, LinearMap.toMatrix'_mul, Affine.Simplex.affineCombination_eq_touchpoint_iff, groupCohomology.π_comp_H0IsoOfIsTrivial_hom_apply, Configuration.ofField.instNondegenerateProjectivizationForallFinOfNatNat, RootPairing.GeckConstruction.instIsIrreducible, groupCohomology.toCocycles_comp_isoCocycles₁_hom_assoc, LinearEquiv.coe_curry_symm, groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply, PowerBasis.constr_pow_gen, Finset.sum_smul_const_vsub_eq_neg_weightedVSub, groupCohomology.isoCocycles₁_inv_comp_iCocycles, Projectivization.mk_eq_mk_iff_crossProduct_eq_zero, Matrix.diagonal_toLin', CStarMatrix.toCLM_apply, IsAdjoinRootMonic.ext_elem_iff, Matrix.charpoly_toLin', Lagrange.iterate_derivative_interpolate, deriv_update, Matrix.toLinearMap₂'_toMatrix', Matrix.toLinearMapₛₗ₂'_apply, Matrix.toLin'_pow, groupCohomology.cochainsMap_f_0_comp_cochainsIso₀_apply, Finset.affineCombination_linear, groupCohomology.cocyclesMap_comp_isoCocycles₂_hom, Finset.affineCombination_apply, LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply, NumberField.mixedEmbedding.stdBasis_apply_isComplex_fst, Matrix.toLinAlgEquiv'_apply, Matrix.l2_opNNNorm_def, groupCohomology.coboundariesOfIsCoboundary₁_coe, Matrix.minpoly_toLin', affineCombination_mem_affineSpan_image, groupCohomology.d₁₂_comp_d₂₃_assoc, LinearMap.BilinForm.nondegenerate_toMatrix'_iff, Finset.affineCombination_affineCombinationLineMapWeights, EuclideanSpace.basisFun_repr, NumberField.mixedEmbedding.span_latticeBasis, LinearMap.toMatrix'_intrinsicStar, Algebra.PreSubmersivePresentation.aevalDifferential_single, cross_apply, Finset.affineCombination_filter_of_ne, IsLocalFrameOn.coeff_apply_of_notMem, Fintype.range_linearCombination, Complex.orthonormalBasisOneI_repr_symm_apply, Pi.basisFun_det, LinearMap.toMatrixAlgEquiv'_toLinAlgEquiv', Matrix.represents_iff, groupCohomology.cochainsMap_f_2_comp_cochainsIso₂_apply, Submodule.IsLattice.finrank_of_pi, vecMulVecBilin_apply_apply, NumberField.mixedEmbedding.stdBasis_repr_eq_matrixToStdBasis_mul, RootPairing.GeckConstruction.ωConjLieSubmodule_eq_top_iff, isAdjointPair_toLinearMap₂', Matrix.toLin'_apply', Configuration.ofField.crossProduct_eq_zero_of_dotProduct_eq_zero, Matrix.Represents.congr_fun, ZMod.dft_comp_neg, Rep.coindIso_inv_hom_hom, Matrix.ker_toLin'_eq_bot_iff, LinearMap.vecCons₂_apply, Finsupp.lsum_apply, LinearMap.toMatrix'_toLinearMap₂', Finset.weightedVSubOfPoint_smul, groupCohomology.cocycles₁_map_mul_of_isTrivial, Polynomial.toFn_comp_ofFn_eq_id, groupCohomology.subtype_comp_d₀₁_assoc, LinearMap.BilinForm.mul_toMatrix', NumberField.mixedEmbedding.fundamentalCone.norm_normAtAllPlaces, Finsupp.coe_lsum, CStarMatrix.mul_entry_mul_eq_inner_toCLM, Matrix.coe_toEuclideanCLM_eq_toEuclideanLin, QuadraticForm.equivalent_sum_squares, Submodule.IsLattice.rank_of_pi, LinearMap.toMatrixRight'_id, leibniz_cross, HVertexOperator.coeff_inj, PiToModule.fromEnd_apply_single_one, Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one, Rep.indToCoindAux_mul_snd, groupCohomology.cocycles₁IsoOfIsTrivial_inv_hom_apply_coe, Matrix.nondegenerate_toLinearMap₂'_iff_nondegenerate_toLinearMap₂, Matrix.charpoly_mulVecLin, IsAdjoinRootMonic.coeff_apply_lt, LinearEquiv.finTwoArrow_symm_apply, Affine.Simplex.sSameSide_affineSpan_faceOpposite_iff, Affine.Simplex.sSameSide_affineSpan_faceOpposite_of_sign_eq, OrthonormalBasis.sum_repr_symm, groupCohomology.cocyclesOfIsMulCocycle₂_coe, IsLocalFrameOn.coeff_apply_zero_at, LinearMap.compLeft_apply, eq_affineCombination_of_mem_affineSpan, Affine.Simplex.affineCombination_mem_interior_face_iff_pos, groupCohomology.coboundariesOfIsMulCoboundary₁_coe, DirichletCharacter.IsPrimitive.fourierTransform_eq_inv_mul_gaussSum, groupCohomology.d₁₂_comp_d₂₃, Finsupp.llift_symm_apply, Matrix.ker_diagonal_toLin', AffineBasis.toMatrix_inv_vecMul_toMatrix, Matrix.toBilin'_apply, Matrix.adjugate_def, NumberField.mixedEmbedding.fundamentalCone.hasFDerivAt_expMapBasis, Matrix.PosDef.toQuadraticForm', groupCohomology.H1π_eq_zero_iff, Matrix.toLin'_one, Module.Basis.constr_apply_fintype, Matrix.cramer_submatrix_equiv, groupCohomology.cochainsMap_f, groupCohomology.coboundaries₁.val_eq_coe, Matrix.mulVecLin_submatrix, inhomogeneousCochains.d_hom_apply, ZMod.dft_dft, mem_affineSpan_iff_eq_weightedVSubOfPoint_vadd, NumberField.mixedEmbedding.negAt_symm, groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_apply, Module.finrank_fin_fun, RootPairing.GeckConstruction.instIsTriangularizableSubtypeMatrixSumMemFinsetSupportLieSubalgebraLieAlgebraCartanSubalgebra'Forall, RootPairing.GeckConstruction.diagonal_elim_mem_span_h_iff, IsAdjoinRootMonic.coeff_injective, endVecAlgEquivMatrixEnd_apply_apply, OrthonormalBasis.measurePreserving_repr_symm, groupCohomology.cocyclesMk₁_eq, instInvertibleReprₛ, EuclideanSpace.piLpCongrLeft_single, Matrix.trace_toLin'_eq, Finset.weightedVSub_filter_of_ne, NumberField.mixedEmbedding.negAt_apply_isReal_and_mem, Matrix.toLin'OfInv_symm_apply, QuadraticForm.posDef_of_toMatrix', Finset.weightedVSub_vadd, QuadraticAlgebra.linearEquivTuple_symm_apply, hasStrictFDerivAt_list_prod_attach', LinearIndependent.fintypeLinearCombination_injective, NumberField.mixedEmbedding.volume_negAt_plusPart, deriv_single, hasFDerivWithinAt_euclidean, Module.Basis.parallelepiped_eq_map, TensorProduct.piScalarRight_symm_single, Ideal.constr_basisSpanSingleton, Affine.Simplex.incenter_eq_affineCombination, FiniteDimensional.finiteDimensional_pi, Finset.weightedVSubOfPoint_apply_const, RCLike.linearIndependent_of_ne_zero_of_wInner_cWeight_eq_zero, groupCohomology.coboundariesToCocycles₂_apply, linearIndependent_iff_injective_fintypeLinearCombination, Polynomial.injective_ofFn, Module.Basis.coePiBasisFun.toMatrix_eq_transpose, LinearMap.toMatrix'_comp, Matrix.toLinearEquiv'_apply, LinearMap.isUnit_toMatrix'_iff, ZLattice.volume_image_eq_volume_div_covolume, groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_apply, Real.map_linearMap_volume_pi_eq_smul_volume_pi, Finsupp.bilinearCombination_apply, groupCohomology.cocyclesOfIsMulCocycle₁_coe, Matrix.toEuclideanLin_eq_toLin, groupCohomology.H2π_eq_zero_iff, Matrix.cramer_subsingleton_apply, Euclidean.closedBall_eq_preimage, Matrix.toBilin'_comp, Matrix.isHermitian_iff_isSymmetric, groupCohomology.mapCocycles₁_comp_i_assoc, NumberField.mixedEmbedding.iUnion_negAt_plusPart_union, hasStrictFDerivAt_finset_prod, groupCohomology.cocycles₁.val_eq_coe, Lagrange.degree_interpolate_le, PiLp.basisFun_eq_pi_basisFun, groupCohomology.H1π_comp_map_apply, Matrix.cramer_is_linear, groupCohomology.cocycles₁_map_one, MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar, InnerProductSpace.toMatrix_rankOne, Affine.Simplex.affineCombination_mem_affineSpan_faceOpposite_iff, differentiableAt_euclidean, NumberField.canonicalEmbedding.mem_rat_span_latticeBasis, groupCohomology.π_comp_H2Iso_hom_assoc, AffineBasis.coords_apply, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, Module.Finite.exists_comp_eq_id_of_projective, groupCohomology.toCocycles_comp_isoCocycles₂_hom_assoc, Finset.weightedVSubOfPoint_eq_of_weights_eq, Matrix.toLpLinAlgEquiv_apply_apply_ofLp, LinearMap.toMatrix'_id, weightedVSub_mem_vectorSpan, EuclideanGeometry.inner_weightedVSub, Projectivization.orthogonal_mk, Projectivization.cross_mk, LinearMap.nondegenerate_toMatrix₂'_iff, QuadraticMap.weightedSumSquares_apply, LinearMap.BilinForm.toMatrix'_comp, differentiableOn_euclidean, triple_product_eq_det, LinearEquiv.funCongrLeft_id, Matrix.toLinearMap₂_basisFun, NumberField.mixedEmbedding.injective_mixedSpaceOfRealSpace, LinearEquiv.funUnique_apply, LinearEquiv.sumArrowLequivProdArrow_apply_fst, Affine.Simplex.sOppSide_affineSpan_faceOpposite_of_pos_of_neg, Matrix.toLinearMapRight'_apply, Finset.weightedVSub_congr, Continuous.matrix_cramer, Matrix.ofLp_toEuclideanLin_apply, Finsupp.linearEquivFunOnFinite_apply, LinearMap.vecEmpty_apply, Rep.coindVEquiv_apply_hom, Matrix.linearIndependent_rows_of_det_ne_zero, LinearMap.nondegenerate_toLinearMap₂'_iff_det_ne_zero, Matrix.mulVecBilin_apply, groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_assoc, lieEquivMatrix'_symm_apply, Matrix.proj_diagonal, Pi.comul_coe_finsupp, Euclidean.ball_eq_preimage, NumberField.mixedEmbedding.stdBasis_apply_isReal, IsLocalFrameOn.eq_iff_coeff, ZMod.completedLFunction_one_sub_odd, OrthonormalBasis.repr_symm_single, groupCohomology.cocycles₂.val_eq_coe, Algebra.SubmersivePresentation.cotangentComplexAux_surjective, LinearMap.continuous_on_pi, Matrix.linfty_opNorm_toMatrix, groupCohomology.eq_d₂₃_comp_inv, MvPolynomial.ker_evalₗ, Matrix.toBilin'_apply', Affine.Simplex.point_eq_affineCombination_of_pointsWithCircumcenter, Matrix.toLinearMap₂'_single, FunOnFinite.linearMap_id, LinearMap.separatingRight_toLinearMap₂'_of_det_ne_zero', Polynomial.eval_eq_sum_degreeLTEquiv, IsAdjoinRootMonic.coeff_algebraMap, Module.Basis.equivFun_apply, Finset.eq_affineCombination_subset_iff_eq_affineCombination_subtype, FixedPoints.linearIndependent_smul_of_linearIndependent, Lagrange.interpolate_eq_add_interpolate_erase, Matrix.SpecialLinearGroup.toLin'_to_linearMap, groupCohomology.isoCocycles₂_inv_comp_iCocycles, LinearMap.mul_toMatrix₂'_mul, LinearMap.toMatrix₂'_mul, Finset.eq_weightedVSub_subset_iff_eq_weightedVSub_subtype, ZLattice.volume_image_eq_volume_div_covolume', Submodule.fg_iff_exists_fin_linearMap, toMatrix_innerSL_apply, Finset.weightedVSub_eq_linear_combination, Matrix.separatingLeft_toLinearMap₂'_iff, Finsupp.linearEquivFunOnFinite_symm_single, triple_product_permutation, LinearMap.separatingLeft_toLinearMap₂'_of_det_ne_zero', PiLp.basis_toMatrix_basisFun_mul, Polynomial.ofFn_degree_lt, Pi.basisFun_apply, Lagrange.interpolate_apply, affineCombination_mem_convexHull, inhomogeneousCochains.d_eq, Lagrange.interpolate_eq_iff_values_eq_on, crossProduct_ne_zero_iff_linearIndependent, Matrix.separatingLeft_toBilin'_iff, groupCohomology.cocyclesMk₂_eq, Module.Basis.toMatrix_eq_toMatrix_constr, Finset.weightedVSub_vadd_affineCombination, Matrix.mulVec_cramer, Matrix.cramerMap_is_linear, affineCombination_mem_affineSpan_of_nonempty, NumberField.mixedEmbedding.latticeBasis_apply, strongRankCondition_iff_forall_zero_lt_finrank, Affine.Simplex.centroid_eq_affineCombination, groupCohomology.toCocycles_comp_isoCocycles₁_hom_apply, MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv_aux, Finset.weightedVSub_apply, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_right_iff, Matrix.toBilin'Aux_eq, rank_fin_fun, Finset.weightedVSubOfPoint_eq_of_sum_eq_zero, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', groupCohomology.isoCocycles₁_hom_comp_i, Affine.Simplex.affineCombination_touchpointWeights, LinearMap.toMatrixAlgEquiv'_symm, ZMod.dft_const_mul, QuadraticForm.equivalent_sign_ne_zero_weighted_sum_squared, Finset.weightedVSubOfPoint_const_smul, Matrix.toPerfectPairing, det_fderivPiPolarCoordSymm, NumberField.mixedEmbedding.commMap_apply_of_isReal, groupCohomology.mapCocycles₁_comp_i_apply, LinearEquiv.sumArrowLequivProdArrow_apply_snd, Rep.coindMap_hom, InnerProductSpace.gramSchmidtOrthonormalBasis_inv_triangular', NumberField.mixedEmbedding.hasFDerivAt_polarCoordReal_symm, groupCohomology.cocycles₂_ext_iff, Matrix.toLinearEquivRight'OfInv_apply, hasEigenvalue_toLin'_diagonal_iff, ContinuousLinearMapWOT.continuous_inducingFn, LinearMap.toMatrix'_toLinearMapₛₗ₂', BilinForm.toMatrix_basisFun, Matrix.toLpLin_mul, groupCohomology.cochainsMap_f_3_comp_cochainsIso₃, Matrix.toLin'_submatrix, IsAdjoinRootMonic.coeff_one, stdSimplex.image_linearMap, hasFDerivAt_pi_polarCoord_symm, NumberField.mixedEmbedding.latticeBasis_repr_apply, Affine.Simplex.mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter, cross_dot_cross, LinearMap.ker_compLeft, mem_fintypeAffineCoords_iff_sum, PiToModule.fromMatrix_apply, Polynomial.surjective_toFn, Matrix.toEuclideanLin_apply_piLp_toLp, Lagrange.interpolate_empty, Finset.attach_affineCombination_of_injective, Algebra.SubmersivePresentation.aevalDifferentialEquiv_apply, groupCohomology.cochainsMap_f_hom, Matrix.linfty_opNorm_eq_opNorm, groupCohomology.coboundaries₁_ext_iff, span_flip_eq_top_iff_linearIndependent, Finsupp.linearCombination_eq_fintype_linearCombination, max_aleph0_card_le_rank_fun_nat, NumberField.mixedEmbedding.normAtPlace_negAt, Lagrange.eq_interpolate_of_eval_eq, Matrix.cramer_one, Algebra.TensorProduct.equivPiOfFiniteBasis_symm_apply, Finsupp.lsum_comp_lsingle, Matrix.separatingRight_toBilin'_iff, LinearMap.vecEmpty₂_apply, groupCohomology.π_comp_H2Iso_hom_apply, stdSimplex.barycenter_eq_centerMass, LinearMap.range_compLeft, Matrix.toLinAlgEquiv'_toMatrixAlgEquiv', NumberField.mixedEmbedding.finrank, LinearMap.funLeft_apply, Module.Basis.constr_basis, QuaternionAlgebra.coe_linearEquivTuple, NumberField.mixedEmbedding.fractionalIdealLatticeBasis_apply, isStablyFiniteRing_iff_isDedekindFiniteMonoid_moduleEnd, Projectivization.cross_mk_of_ne, LinearEquiv.sumArrowLequivProdArrow_symm_apply_inl, endVecAlgEquivMatrixEnd_symm_apply_apply, Representation.coind_apply, FunOnFinite.linearMap_piSingle, NumberField.mixedEmbedding.negAt_apply_norm_isReal, Affine.Simplex.affineCombination_mem_closedInterior_face_iff_mem_Icc, Pi.linearIndependent_single_one, Matrix.toLinAlgEquiv'_one, OrthonormalBasis.coe_toBasis_repr, hasStrictFDerivAt_list_prod, groupCohomology.π_comp_H1Iso_hom_apply, OrthonormalBasis.sum_repr, groupCohomology.d₀₁_comp_d₁₂_assoc, Matrix.toLpLin_symm_comp, VertexOperator.ncoeff_of_coeff, Lagrange.eval_interpolate_at_node, VertexOperator.ncoeff_eq_zero_of_lt_order, ZMod.invDFT_def, Matrix.toLin_eq_toLin', TensorProduct.equivFinsuppOfBasisRight_symm, Polynomial.ofFn_comp_toFn_eq_id_of_natDegree_lt, NumberField.mixedEmbedding.fundamentalCone.expMapBasis_apply, LinearIsometryEquiv.piLpCongrLeft_single, HVertexOperator.compHahnSeries_coeff, Matrix.Nondegenerate.toBilin', Lagrange.interpolate_poly_eq_self, LinearMap.BilinForm.toMatrix'_mul, dotProductBilin_apply_apply, Algebra.traceMatrix_of_basis_mulVec, hasDerivAt_single, OrthonormalBasis.coe_toBasis_repr_apply, affineCombination_eq_centerMass, instIsStablyFiniteRingEndForallOfFinite, LinearMap.toMatrix₂'_compl₂, Matrix.toLinAlgEquiv'_symm, AffineIndependent.injOn_affineCombination_fintypeAffineCoords, IsAdjoinRootMonic.coeff_apply_coe, ZSpan.discreteTopology_pi_basisFun, Affine.Simplex.signedInfDist_affineCombination, Matrix.nondegenerate_toLinearMap₂'_iff, Matrix.SpecialLinearGroup.toLin'_injective, TensorProduct.piScalarRight_apply, Matrix.mulVecLin_reindex, EuclideanGeometry.dist_affineCombination, LocallyConstant.coeFnₗ_apply, groupCohomology.d₀₁_comp_d₁₂_apply, Finset.sum_smul_vsub_const_eq_weightedVSub, Finset.sum_smul_vsub_const_eq_affineCombination_vsub, Matrix.cramer_smul, Matrix.UnitaryGroup.toLin'_mul, IsLocalFrameOn.coeff_eq_of_eq, AddChar.linearIndependent, HVertexOperator.coeff_comp, groupCohomology.cocyclesIso₀_hom_comp_f_apply, Function.ExtendByZero.linearMap_apply, groupCohomology.subtype_comp_d₀₁, jacobi_cross, HVertexOperator.coeff_apply_apply, Set.Finite.convexHull_eq_image, Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_right_iff, Rep.indToCoindAux_snd_mul_inv, tendsto_card_div_pow_atTop_volume', fromModuleCatToModuleCatLinearEquiv_apply, LinearMap.BilinForm.toMatrix'_toBilin', Fintype.linearCombination_apply_single, SimplexCategory.coe_toTopMap, Fintype.linearCombination_apply, neg_cross, groupCohomology.isoCocycles₁_inv_comp_iCocycles_assoc, fromModuleCatToModuleCatLinearEquiv_symm_apply_coe, groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom, PowerBasis.liftEquiv'_symm_apply_apply, PowerBasis.constr_pow_mul, NumberField.mixedEmbedding.fundamentalCone.abs_det_fderiv_expMapBasis, LinearMap.det_toMatrix', ContinuousAlternatingMap.differentiable, NumberField.mixedEmbedding.euclidean.finrank, convexHull_rangle_single_eq_stdSimplex, Matrix.mulVec_injective_iff, MeasureTheory.lpNorm_expect_le, groupCohomology.coboundaries₁_le_cocycles₁, Module.Finite.exists_fin', Module.length_pi, Matrix.toLin'OfInv_apply, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_expMap_symm, Basis.equivFun_symm_single, AffineBasis.affineCombination_coord_eq_self, NumberField.mixedEmbedding.measurableSet_negAt_plusPart, Module.finrank_fintype_fun_eq_card, rank_fun, Module.Basis.constr_self, Matrix.UnitaryGroup.toGL_one, NumberField.mixedEmbedding.euclidean.volumePreserving_toMixed_symm, groupCohomology.cochainsMap_f_2_comp_cochainsIso₂_assoc, Finset.sum_smul_vsub_eq_affineCombination_vsub, Finset.weightedVSub_subtype_eq_filter, Matrix.SeparatingRight.toBilin', IsAdjoinRootMonic.coeff_apply_le, LinearMap.ltoFun_apply, Module.Basis.opNNNorm_le, groupCohomology.isoCocycles₂_hom_comp_i_apply, Rep.diagonalHomEquiv_apply, Matrix.range_diagonal, ZMod.completedLFunction_one_sub_even, hasStrictFDerivAt_list_prod', Finset.sum_smul_const_vsub_eq_sub_weightedVSubOfPoint, Module.Basis.constr_def, rank_fun', Rep.freeLiftLEquiv_symm_apply, AddChar.coe_complexBasis, LinearMap.toMatrix₂'_comp, groupCohomology.coboundaries₂_ext_iff, Polynomial.ofFn_coeff_eq_val_of_lt, Finset.weightedVSubOfPoint_erase, IsAdjoinRootMonic.coeff_root, Matrix.coe_ofLinearEquiv_symm, AffineBasis.coord_apply_combination_of_mem, Rep.indToCoindAux_of_not_rel, groupCohomology.cocyclesOfIsCocycle₂_coe, LinearMap.toMatrix_eq_toMatrix', groupCohomology.cocyclesIso₀_inv_comp_iCocycles_apply, Matrix.SpecialLinearGroup.toLin'_symm_apply, Finset.weightedVSub_sdiff_sub, groupCohomology.H1π_comp_map, Pi.basisFun_equivFun, LinearMap.toMatrix'_one, NumberField.mixedEmbedding.normAtAllPlaces_mixedSpaceOfRealSpace, NumberField.mixedEmbedding.det_fderivPolarCoordRealSymm, Matrix.cstar_norm_def, rank_fun_eq_lift_mul, Module.Basis.opNorm_le, Module.surjective_piEquiv_apply_iff, PiLp.basisFun_map, AffineIndependent.affineCombination_eq_lineMap_iff_weight_lineMap, TannakaDuality.FiniteGroup.map_mul_toRightFDRepComp, Configuration.ofField.mem_iff, cross_cross_eq_smul_sub_smul', Quaternion.linearIsometryEquivTuple_apply, LinearMap.toMatrixₛₗ₂'_apply, Module.Basis.coe_ofEquivFun, Matrix.toLinearMapRight'_one, Matrix.maxGenEigenspace_toLin'_diagonal_eq_eigenspace, DirichletCharacter.fourierTransform_eq_gaussSum_mulShift, Matrix.cramer_transpose_apply, Finset.sum_smul_vsub_const_eq_weightedVSubOfPoint_sub, NumberField.mixedEmbedding.disjoint_span_commMap_ker, Finset.weightedVSub_indicator_subset, Module.Basis.sum_equivFun, SymmetricAlgebra.IsSymmetricAlgebra.mvPolynomial, Module.Finite.exists_fin_quot_equiv, TensorProduct.piScalarRightHom_tmul, groupCohomology.isoShortComplexH1_inv, Algebra.TensorProduct.equivPiOfFiniteBasis_apply, IsLocalFrameOn.coeff_sum_eq, Module.Basis.constr_range, Matrix.toLpLin_apply, ZMod.dft_eq_fourier, Algebra.TensorProduct.piScalarRight_tmul_apply, ZMod.dft_odd_iff, Matrix.toLin'_apply, AddChar.complexBasis_apply, Module.Basis.constr_apply, Pi.mem_span_range_single_inl_iff, Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero, Finsupp.linearCombination_eq_fintype_linearCombination_apply, Affine.Simplex.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter, LinearMap.separatingLeft_toLinearMap₂'_iff_det_ne_zero, Finset.weightedVSubOfPoint_subtype_eq_filter, weightedVSub_mem_vectorSpan_pair, strongRankCondition_iff_succ, Matrix.l2_opNorm_def, groupCohomology.cochainsMap_f_1_comp_cochainsIso₁, Matrix.toEuclideanLin_toLp, LinearIsometryEquiv.piLpCongrLeft_symm, groupCohomology.cocycles₁_ext_iff, PowerBasis.constr_pow_aeval, LinearMap.CompatibleSMul.pi, IsBaseChange.finitePow, TannakaDuality.FiniteGroup.ofRightFDRep_hom, Matrix.linearIndependent_cols_of_invertible, Matrix.toLin'_toMatrix', LinearEquiv.funCongrLeft_comp, groupCohomology.eq_d₁₂_comp_inv_assoc, Matrix.coe_ofLinearEquiv, Affine.Simplex.affineCombination_mem_setInterior_iff, ZMod.LFunction_dft, LinearMap.BilinForm.nondegenerate_toBilin'_of_det_ne_zero', Pi.basisFun_repr, Polynomial.ofFn_coeff_eq_zero_of_ge, Module.Basis.toDual_eq_equivFun, Pi.orthonormalBasis_repr, ZMod.dft_const_smul, LinearMap.toMatrix₂'_apply, Matrix.vecMul_injective_iff, NumberField.mixedEmbedding.fundamentalCone.linearIndependent_completeFamily, NumberField.mixedEmbedding.commMap_canonical_eq_mixed, Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂, RootPairing.GeckConstruction.coe_genWeightSpace_zero_eq_span_range_u, TensorProduct.sum_tmul_basis_left_injective, ZMod.dft_smul_const, NumberField.mixedEmbedding.norm_negAt, Real.volume_preserving_transvectionStruct, Matrix.toLpLinAlgEquiv_symm_apply, Matrix.toBilin'_toMatrix', groupCohomology.isoShortComplexH2_inv, groupCohomology.isoCocycles₂_hom_comp_i_assoc, Matrix.rank_eq_finrank_span_cols, LinearMap.toMatrix'_mulVec, VertexOperator.coeff_eq_ncoeff, Rep.coindIso_hom_hom_hom, Finset.centroid_def, LinearMap.separatingRight_toLinearMap₂'_iff_det_ne_zero, Module.Basis.coord_equivFun_symm, ZMod.dft_def, dotProduct_toMatrix₂_mulVec, Matrix.entryLinearMap_eq_comp, Affine.Simplex.sOppSide_affineSpan_faceOpposite_iff, Matrix.diagonal_comp_single, groupCohomology.coe_mapCocycles₂, LinearRecurrence.is_sol_iff_mem_solSpace, groupCohomology.eq_d₀₁_comp_inv_assoc, AdicCompletion.piEquivFin_apply, LinearMap.const_apply, finrank_euclideanSpace_fin, QuadraticMap.basisRepr_apply, isNoetherian_pi', Affine.Simplex.centroid_eq_affineCombination_of_pointsWithCircumcenter, hasFDerivAt_list_prod_attach', groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_assoc, groupCohomology.H1π_eq_iff, NumberField.mixedEmbedding.normAtPlace_mixedSpaceOfRealSpace, OrthonormalBasis.repr_reindex, InnerProductSpace.symm_toEuclideanLin_rankOne, LinearMap.BilinForm.toMatrix'_compRight, groupCohomology.isoCocycles₁_hom_comp_i_assoc, VertexOperator.coeff_eq_zero_of_lt_order, AnalyticOnNhd.eval_continuousLinearMap, Affine.Simplex.faceOppositeCentroid_eq_affineCombination, Module.Free.function, Finset.affineCombination_subtype_eq_filter, ContinuousLinearEquiv.coe_funUnique_symm, Polynomial.ofFn_natDegree_lt, Matrix.rank_vecMulVec, NumberField.mixedEmbedding.fundamentalCone.expMap_basis_of_ne, MvPolynomial.range_evalᵢ, TannakaDuality.FiniteGroup.rightRegular_apply, groupCohomology.coboundariesOfIsCoboundary₂_coe, NumberField.mixedEmbedding.disjoint_negAt_plusPart, Algebra.SubmersivePresentation.cotangentEquiv_apply, LinearMap.toMatrixRight'_comp, OrthonormalBasis.tensorProduct_repr_tmul_apply, convexHull_range_eq_exists_affineCombination, Module.Basis.constr_eq, QuadraticMap.toMatrix'_smul, HVertexOperator.coeff_of_coeff, Matrix.toLpLin_eq_toLin, Algebra.SubmersivePresentation.basisDeriv_apply, Lagrange.interpolate_singleton, SimpleGraph.lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj, isArtinian_pi', SimpleGraph.mem_ker_toLin'_lapMatrix_of_connectedComponent, groupCohomology.mapCocycles₁_comp_i, Matrix.SpecialLinearGroup.toLin'_apply, ContinuousLinearEquiv.coe_funUnique, Matrix.GeneralLinearGroup.toLin_apply, Matrix.mulVecLin_mul, MvPolynomial.evalₗ_apply, CStarMatrix.norm_def, Pi.comul_comp_finsuppLcoeFun, Matrix.linearIndependent_rows_iff_isUnit, groupCohomology.cocycles₁.d₁₂_apply, NumberField.mixedEmbedding.mem_rat_span_latticeBasis, Pi.basisFun_det_apply, Matrix.Nondegenerate.toLinearMap₂', groupCohomology.π_comp_H2Iso_hom, Finset.sum_smul_vsub_eq_weightedVSub_sub, Finset.weightedVSubOfPoint_sdiff_sub, Matrix.separatingRight_toLinearMap₂'_iff_separatingRight_toLinearMap₂, Finsupp.lsum_symm_apply, IsAdjoinRootMonic.coeff_root_pow, NumberField.mixedEmbedding.fundamentalCone.logMap_normAtAllPlaces, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_normLeOne, ZMod.dft_apply_zero, LinearMap.toMatrixAlgEquiv'_comp, BoxIntegral.unitPartition.tag_mem_smul_span, parallelepiped_single, NumberField.mixedEmbedding.normAtAllPlaces_normAtAllPlaces, ContMDiffMap.coeFnLinearMap_apply, Matrix.cramer_eq_adjugate_mulVec, groupCohomology.isoCocycles₂_inv_comp_iCocycles_assoc, CStarMatrix.inner_toCLM_conjTranspose_right, LieModule.toEnd_matrix, Matrix.det_smul_inv_vecMul_eq_cramer_transpose, CStarMatrix.toCLM_apply_single_apply, LinearMap.toLinearMap₂'Aux_toMatrix₂Aux, QuadraticForm.equivalent_weightedSumSquares_units_of_nondegenerate', Matrix.toBilin'_single, LinearEquiv.funCongrLeft_symm, NumberField.mixedEmbedding.mixedSpaceOfRealSpace_apply, LinearEquiv.piRing_apply, Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_left_iff, groupCohomology.d₀₁_eq_zero, Matrix.spectrum_toLin', hasStrictFDerivAt_euclidean, hasFDerivAt_finset_prod

Polynomial

Definitions

Name	Category	Theorems
module 📖	CompOp	590 math math: fderiv, PowerSeries.coeff_mul_eq_coeff_trunc_mul_trunc₂, MvPolynomial.pUnitAlgEquiv_symm_monomial, support_monomial, degreeLT.addLinearEquiv_symm_apply_inr, Lagrange.interpolate_one, taylor_eval, Lagrange.eval_iterate_derivative_eq_sum, eraseLead_monomial, PowerSeries.IsWeierstrassFactorizationAt.algEquivQuotient_symm_apply, Derivation.apply_eval_eq, Lagrange.interpolate_eq_of_values_eq_on, derivative_X_sq, smeval_monomial_mul, ofFn_zero, PolyEquivTensor.toFunBilinear_apply_eq_sum, lcoeff_comp_mapAlgHom_eq, coe_taylorAlgHom, monomial_eq_monomial_iff, coeff_derivative, degreeLT.instFreeSubtypeMemSubmodule, Sequence.basis_natDegree_strictMono, Monic.free_quotient, hasseDeriv_zero', Lagrange.eval_interpolate_not_at_node', derivWithin_aeval, derivative_intCast_mul, toMatrix_sylvesterMap', eval_minpolyDiv_self, Ideal.mem_ofPolynomial, expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul, derivative_C_mul, eval₂_minpolyDiv_self, iterate_derivative_zero, card_rootSet_le_derivative, PowerSeries.coeff_trunc, derivative_mul, PowerSeries.trunc_mul_trunc, Chebyshev.one_sub_X_sq_mul_derivative_derivative_U_eq_poly_in_U, opRingEquiv_op_monomial, natDegree_monomial_le, Sequence.span, PowerSeries.trunc_X_pow_self_mul, Lagrange.derivative_nodal, monomial_mem_lifts, taylor_mul, rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero, sum_bernoulli, C_mul_X_eq_monomial, eval₂_minpolyDiv_of_eval₂_eq_zero, hasseDeriv_one', derivative_comp_one_sub_X, derivative_pow_eq_zero, PolyEquivTensor.right_inv, PolynomialModule.aeval_equivPolynomial, degreeLT_mono, roots_monomial, aeval_iterate_derivative_of_ge, derivative_expand, derivative_sum, isCoveringMapOn_eval, derivative_pow_succ, Sequence.basis_eq_self, sum_smul_minpolyDiv_eq_X_pow, derivative_ofNat, monomial_one_one_eq_X, Splits.eval_derivative, derivative_smul, Splits.monomial, RatFunc.laurent_algebraMap, iterate_derivative_eq_factorial_smul_sum, iterate_derivative_X_add_pow, eval_minpolyDiv_of_aeval_eq_zero, derivative_X, sumIDeriv_map, Lagrange.eval_interpolate_not_at_node, mirror_monomial, dvd_iterate_derivative_pow, coeffs_monomial, KaehlerDifferential.polynomialEquiv_D, ofFn_eq_sum_monomial, Lagrange.interpolate_eq_nodalWeight_mul_nodal_div_X_sub_C, abc, finrank_quotient_span_eq_natDegree', Lagrange.interpolate_eq_sum_interpolate_insert_sdiff, Splits.eval_root_derivative, RatFunc.laurent_div, degreeLT.basis_val, derivative_monomial, aeval_sumIDeriv, sumIDeriv_C, Chebyshev.T_derivative_eq_U, coeff_taylor_natDegree, aeval_iterate_derivative_self, Chebyshev.iterate_derivative_T_eval_one_recurrence, toFinsupp_monomial, coe_monomial, rootMultiplicity_sub_one_le_derivative_rootMultiplicity, PolyEquivTensor.toFunAlgHom_apply_tmul, adjoin_monomial_eq_reesAlgebra, monomial_mul_C, X_pow_eq_monomial, Bivariate.Polynomial.Bivariate.pderiv_one_equivMvPolynomial, Matrix.derivative_det_one_add_X_smul, derivative_rootMultiplicity_of_root, taylor_injective, iterate_derivative_mul_X, sumIDeriv_eq_self_add, self_sub_monomial_natDegree_leadingCoeff, iterate_derivative_X, eval₂_monomial, Splits.eval_derivative_eq_eval_mul_sum, divByMonic_add_X_sub_C_mul_derivative_divByMonic_eq_derivative, derivative'_apply, mem_support_derivative, PowerSeries.eq_shift_mul_X_pow_add_trunc, monomial_pow, toLaurent_C_mul_T, card_roots_le_derivative, Lagrange.eq_interpolate_iff, derivative_eval, PolynomialModule.monomial_smul_single, X_pow_smul_rTensor_monomial, smul_monomial, iterate_derivative_X_sub_pow, PowerSeries.degree_trunc_lt, IsAdjoinRootMonic.basis_repr, FirstOrder.Field.lift_genericMonicPoly, Lagrange.interpolate_eq_sum, degreeLT.addLinearEquiv_castAdd, KaehlerDifferential.polynomialEquiv_comp_D, Algebra.discr_powerBasis_eq_norm, map_monomial, matPolyEquiv_coeff_apply_aux_1, degreeLT.addLinearEquiv_apply_fst, derivative_X_sub_C_sq, smeval.linearMap_apply, eval_derivative_of_splits, isAlt_wronskianBilin, divRadical_dvd_derivative, aeval_add_of_sq_eq_zero, monomial_mem_adjoin_monomial, derivativeFinsupp_apply_toFun, deriv_aeval, ofFinsupp_single, le_trailingDegree_monomial, iterate_derivative_derivative_mul_X_sq, Derivation.comp_aeval_eq, PowerSeries.trunc_trunc_mul_trunc, C_mul_X_pow_eq_monomial, PowerSeries.trunc_mul_C, taylor_zero', monomial_eq_zero_iff, card_roots_toFinset_le_card_roots_derivative_diff_roots_succ, mkDerivation_one_eq_derivative', derivative_X_add_C, hermite_succ, PolyEquivTensor.toFunLinear_mul_tmul_mul, hermite_eq_iterate, hasseDeriv_comp, iterate_derivative_natCast_mul, AdjoinRoot.mk_leftInverse, natTrailingDegree_monomial, bernsteinPolynomial.iterate_derivative_at_1_eq_zero_of_lt, finrank_quotient_span_eq_natDegree, eval_monomial_one_add_sub, map_taylor, Derivation.mapCoeffs_monomial, degreeLT.addLinearEquiv_apply_snd, modByMonicHom_apply, addHom_ext'_iff, PowerSeries.trunc_X_of, derivative_X_pow_succ, degree_monomial_le, pow_sub_one_dvd_derivative_of_pow_dvd, coeff_monomial_of_ne, derivative_X_sub_C_pow, Lagrange.degree_interpolate_erase_lt, derivWithin, Chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T, adjSylvester_comp_sylveserMap, exists_iterate_derivative_eq_factorial_smul, monomial_injective, taylor_eq_zero, Lagrange.degree_interpolate_lt, RatFunc.laurentAux_div, traceForm_dualSubmodule_adjoin, polyEquivTensor_symm_apply_tmul, degreeLT.instFiniteSubtypeMemSubmodule, IsAdjoinRootMonic.modByMonicHom_map, derivativeFinsupp_apply_apply, taylor_apply, eval_derivative_eq_eval_mul_sum_of_splits, iterate_derivative_comp_one_sub_X, PowerSeries.IsWeierstrassDivisorAt.mod'_mk_eq_mod, degree_derivative_lt, derivative_natCast_mul, fderiv_aeval, Splits.eval_derivative_div_eval_of_ne_zero, valuation_inv_monomial_eq_valuation_X_zpow, sum_monomial_index, derivativeFinsupp_derivative, sum_monomial_eq, PowerSeries.trunc_trunc_mul, degree_derivative_eq, C_mul_monomial, natDegree_hasseDeriv_le, as_sum_range, det_taylorLinearEquiv_toLinearMap, monomial_mul_X, rootSet_derivative_subset_convexHull_rootSet, mapAlgHom_monomial, Chebyshev.iterate_derivative_T_eval_zero_recurrence, iterate_derivative_X_pow_eq_natCast_mul, natDegree_taylor, degree_taylor, lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors, hasFDerivAt, WeierstrassCurve.Affine.derivative_addPolynomial_slope, degreeLT.addLinearEquiv_apply, Lagrange.eq_interpolate, separable_def', derivative_of_natDegree_zero, toFinsuppIsoLinear_symm_apply_toFinsupp, Splits.taylor, coe_polyEquivTensor'_symm, reesAlgebra.monomial_mem, lt_rootMultiplicity_iff_isRoot_iterate_derivative, PowerSeries.derivativeFun_coe, coeff_monomial, Differential.mapCoeffs_monomial, taylor_coeff_zero, mul_eq_sum_sum, coe_polyEquivTensor', PowerSeries.eq_X_pow_mul_shift_add_trunc, iterate_derivative_C, PowerSeries.trunc_trunc_of_le, taylor_X_pow, toFn_zero, ofFn_zero', derivative_C_mul_X, eval_iterate_derivative_rootMultiplicity, lcoeff_apply, iterate_derivative_derivative_mul_X, Chebyshev.add_one_mul_self_mul_T_eq_poly_in_T, iterate_derivative_eq_zero, hasseDeriv_coeff, derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors, monomial_zero_one, hasseDeriv_mul, PowerSeries.trunc_one_left, eraseLead_add_monomial_natDegree_leadingCoeff, RatFunc.laurentAux_algebraMap, AdjoinRoot.modByMonicHom_mk, iterate_derivative_X_pow_eq_smul, Chebyshev.derivative_U_eval_one, monomial_one_eq_iff, StandardEtalePresentation.toSubmersivePresentation_jacobian, degreeLT.addLinearEquiv_apply', X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic, hasseDeriv_zero, coeff_monomial_zero_mul, Lagrange.iterate_derivative_interpolate, pow_sub_dvd_iterate_derivative_pow, hasFDerivAt_aeval, natDegree_derivative_le, iterate_derivative_eq_sum, Chebyshev.iterate_derivative_U_eval_zero_recurrence, wronskianBilin_apply, PolyEquivTensor.invFun_add, PowerSeries.trunc_C, sumIDeriv_apply, sylvesterDeriv_updateRow, dvd_derivative_iff, coeff_mul_monomial_zero, resultant_taylor, Ideal.is_fg_degreeLE, degreeLT.addLinearEquiv_symm_apply_inl, degreeLT.addLinearEquiv_symm_apply_inr_basis, natDegree_hasseDeriv, AdjoinRoot.powerBasisAux'_repr_symm_apply, Bivariate.swap_monomial_monomial, eval_derivative_div_eval_of_ne_zero_of_splits, Chebyshev.one_sub_X_sq_mul_iterate_derivative_U_eval, PolynomialModule.equivPolynomial_single, toFn_comp_ofFn_eq_id, supNorm_monomial, derivative_comp, gaussNorm_monomial, taylor_taylor, expand_monomial, instIsTorsionFree, derivative_sq, Monic.finite_quotient, hasseDeriv_monomial, PowerSeries.trunc_trunc_pow, Bivariate.pderiv_zero_equivMvPolynomial, derivativeFinsupp_map, monomial_add, one_lt_rootMultiplicity_iff_isRoot_gcd, hasDerivWithinAt_aeval, coeff_monomial_same, hasStrictDerivAt_aeval, span_of_finite_le_degreeLT, bernsteinPolynomial.iterate_derivative_succ_at_0_eq_zero, iterate_derivative_eq_zero_of_degree_lt, toFinsuppIsoLinear_apply, bernsteinPolynomial.linearIndependent_aux, isRoot_iterate_derivative_of_lt_rootMultiplicity, iterate_derivative_prod_X_sub_C, opRingEquiv_symm_monomial, toAddCircle_monomial_eq_smul_fourier, PolynomialModule.equivPolynomialSelf_apply_eq, degreeLT.addLinearEquiv_symm_apply, Chebyshev.iterate_derivative_U_eval_one_recurrence, coe_basisMonomials, PowerSeries.trunc_apply, isNilpotent_monomial_iff, as_sum_range', natDegree_monomial, PolyEquivTensor.toFunAlgHom_apply_tmul_eq_smul, factorial_mul_shiftedLegendre_eq, Chebyshev.one_sub_X_sq_mul_derivative_derivative_T_eq_poly_in_T, IsAdjoinRootMonic.map_modByMonicHom, separable_def, iterate_derivative_smul, sumIDeriv_X, contentIdeal_monomial, injective_ofFn, PolyEquivTensor.toFunBilinear_apply_apply, PolyEquivTensor.toFunLinear_tmul_apply, coeff_iterate_derivative, Lagrange.degree_interpolate_le, bernoulli_def, derivative_eval₂_C, lhom_ext'_iff, PowerSeries.derivative_coe, Bivariate.Polynomial.Bivariate.pderiv_zero_equivMvPolynomial, X_pow_mul_monomial, aeval_derivative_of_splits, lsum_apply, IsDistinguishedAt.algEquivQuotient_symm_apply, derivative_pow, derivative_prod, LindemannWeierstrass.integral_exp_mul_eval, valuation_monomial_eq_valuation_X_pow, eval_sumIDeriv_of_pos, Chebyshev.iterate_derivative_U_eval_one, iterate_derivative_C_mul, PolyEquivTensor.invFun_monomial, mkDerivation_one_eq_derivative, degreeLT.addLinearEquiv_symm_apply', iterate_derivative_mul_X_pow, IsAdjoinRootMonic.modByMonicHom_root_pow, taylor_coeff_one, fderivWithin_aeval, PolyEquivTensor.toFunLinear_one_tmul_one, fderivWithin, monomial_neg, hasDerivAt, monomial_zero_right, iterate_derivative_sub, signVariations_monomial, Derivation.map_aeval, PolyEquivTensor.left_inv, monomial_mul_X_pow, aeval_derivative_mem_differentIdeal, iterate_derivative_X_pow_eq_C_mul, derivative_X_pow, sumIDeriv_derivative, X_mul_monomial, Lagrange.interpolate_eq_add_interpolate_erase, StandardEtalePair.HasMap.isUnit_derivative_f, aeval_sumIDeriv_eq_eval, IsCoprime.wronskian_eq_zero_iff, KaehlerDifferential.polynomial_D_apply, leval_apply, exists_derivative_mul_eq_and_isIntegral_coeff, ofFn_degree_lt, taylor_monomial, Lagrange.interpolate_apply, valuation_aeval_monomial_eq_valuation_pow, Lagrange.interpolate_eq_iff_values_eq_on, factorial_smul_hasseDeriv, mem_ker_modByMonic, Chebyshev.one_sub_X_sq_mul_iterate_derivative_T_eq_poly_in_T, hasDerivAt_aeval, aeval_root_derivative_of_splits, iterate_derivative_neg, polyEquivTensor_apply, PolyEquivTensor.toFunBilinear_apply_eq_smul, instFree, sum_taylor_eq, taylor_eval_sub, derivative_prod_finset, Differential.deriv_aeval_eq, hasseDeriv_C, coeff_list_sum, smeval_monomial, homogenize_monomial_of_lt, StandardEtalePair.cond, derivative_sub, hasStrictDerivAt, aroots_monomial, det_taylorLinearEquiv, PowerSeries.trunc_one, derivative_monomial_succ, resultant_deriv, PowerSeries.trunc_succ, monomial_one_right_eq_X_pow, Derivation.compAEval_eq, hasseDeriv_apply_one, surjective_toFn, Lagrange.interpolate_empty, bernsteinPolynomial.derivative_succ, pow_sub_dvd_iterate_derivative_of_pow_dvd, iterate_derivative_X_sub_pow_self, derivativeFinsupp_one, degreeLT_eq_span_X_pow, Lagrange.eq_interpolate_of_eval_eq, taylor_zero, toMatrix_sylvesterMap, PowerSeries.coeff_mul_eq_coeff_trunc_mul_trunc, MvPolynomial.optionEquivLeft_monomial, IsAdjoinRootMonic.modByMonicHom_root, derivative_neg, deriv, PowerSeries.trunc_coe_eq_self, hasseDeriv_natDegree_eq_C, hasFDerivWithinAt_aeval, derivative_C_mul_X_pow, derivative_C, derivative_bernoulli_add_one, hasseDeriv_apply, addSubmonoid_closure_setOf_eq_monomial, eval₂_derivative_of_splits, sum_modByMonic_coeff, Chebyshev.iterate_derivative_U_eval_one_eq_div, aeval_monomial, Chebyshev.iterate_derivative_T_eval_one, Lagrange.eval_interpolate_at_node, sylvesterMap_apply_coe, degreeLT.addLinearEquiv_natAdd, MvPolynomial.pUnitAlgEquiv_monomial, conductor_mul_differentIdeal, support_derivativeFinsupp_subset_range, ofFn_comp_toFn_eq_id_of_natDegree_lt, monomial_zero_left, Lagrange.interpolate_poly_eq_self, taylor_coeff, taylor_inj, derivative_zero, AdjoinRoot.powerBasisAux'_repr_apply_to_fun, succ_signVariations_X_sub_C_mul_monomial, derivative_X_add_C_sq, one_lt_rootMultiplicity_iff_isRoot_iterate_derivative, span_le_degreeLE_of_finite, as_sum_support, taylorLinearEquiv_symm, natTrailingDegree_monomial_le, PowerSeries.eval₂_trunc_eq_sum_range, degreeLT.addLinearEquiv_symm_apply_inl_basis, PowerSeries.trunc_one_X, leval_coe_eq_smeval, PolynomialModule.monomial_smul_apply, PowerSeries.IsWeierstrassDivisorAt.mk_mod'_eq_self, logMahlerMeasure_monomial, PowerSeries.trunc_trunc, rootSet_monomial, monomial_left_inj, RatFunc.taylor_mem_nonZeroDivisors, newtonMap_apply, Derivation.apply_aeval_eq', natDegree_iterate_derivative, eval_add_of_sq_eq_zero, derivative_X_add_C_pow, natDegree_monomial_eq, iterate_derivative_mul, aeval_iterate_derivative_of_lt, derivative_map, iterate_derivative_intCast_mul, taylor_mem_degreeLT, leadingCoeff_monomial, Lagrange.eval_nodal_derivative_eval_node_eq, Module.Basis.traceDual_powerBasis_eq, card_support_le_one_iff_monomial, coeff_mul_monomial, LaurentPolynomial.trunc_C_mul_T, derivative_natCast, hasseDeriv_X, derivative_bernoulli, ofFn_coeff_eq_val_of_lt, PolynomialModule.monomial_smul_lsingle, degreeLT.basisProd_natAdd, PowerSeries.WithPiTopology.tendsto_trunc_atTop, Sequence.linearIndependent, monomial_natDegree_leadingCoeff_eq_self, degreeLE_mono, bernsteinPolynomial.linearIndependent, bernsteinPolynomial.derivative_zero, iterate_derivative_map, sumIDeriv_apply_of_le, Bivariate.pderiv_one_equivMvPolynomial, support_monomial', mem_degreeLE, bernsteinPolynomial.iterate_derivative_at_0_eq_zero_of_lt, taylor_X, bernsteinPolynomial.iterate_derivative_at_0, Chebyshev.derivative_U_eval_one_eq_div, hasFDerivWithinAt, degreeLE_eq_span_X_pow, iterate_derivative_one, natDegree_derivative_lt, Chebyshev.one_sub_X_sq_mul_iterate_derivative_U_eq_poly_in_U, RatFunc.algebraMap_monomial, instFiniteDimensionalQuotientPolynomialIdealSpanSingletonSetSmithCoeffs, taylor_pow, smul_X_eq_monomial, Sequence.basis_degree_strictMono, one_lt_rootMultiplicity_iff_isRoot, monomial_sub, coeff_monomial_succ, PowerSeries.trunc_sub, degree_monomial, PowerSeries.trunc_derivativeFun, sylveserMap_comp_adjSylvester, PowerSeries.trunc_map, ker_modByMonicHom, PowerSeries.coeff_coe_trunc_of_lt, isMonicOfDegree_monomial_one, PowerSeries.trunc_zero', hasseDeriv_eq_zero_of_lt_natDegree, ofFn_coeff_eq_zero_of_ge, degreeLT.basisProd_castAdd, Matrix.derivative_det_one_add_X_smul_aux, trailingDegree_monomial, hasDerivWithinAt, IsAdjoinRootMonic.liftPolyₗ_apply, PowerSeries.trunc_derivative, bernsteinPolynomial.derivative_succ_aux, derivative_X_sub_C, card_roots_toFinset_le_derivative, iterate_derivative_sum, Chebyshev.one_sub_X_sq_mul_iterate_derivative_T_eval, RatFunc.laurentAux_ofFractionRing_mk, sumIDeriv_apply_of_lt, erase_monomial, homogenizeLM_apply, degree_derivative_le, Chebyshev.iterate_derivative_T_eval_one_eq_div, isRoot_of_isRoot_iff_dvd_derivative_mul, derivative_one, taylorLinearEquiv_apply_coe, PowerSeries.natDegree_trunc_lt, Chebyshev.derivative_T_eval_one, taylor_C, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, mem_degreeLT, derivative_apply, PowerSeries.trunc_X, degreeLT.basis_repr, polyEquivTensor_symm_apply_tmul_eq_smul, monomial_comp, ofFn_natDegree_lt, monomial_mul_monomial, mkDerivation_apply, not_finite, coeff_monomial_mul, Lagrange.interpolate_singleton, PowerSeries.trunc_C_mul, Chebyshev.add_one_mul_T_eq_poly_in_U, coeffList_monomial, derivative_C_mul_X_sq, content_monomial, derivativeFinsupp_C, taylor_one, derivativeFinsupp_X, eval_multiset_prod_X_sub_C_derivative, lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors', bernsteinPolynomial.iterate_derivative_at_1, Lagrange.nodalWeight_eq_eval_derivative_nodal, derivative_intCast, leadingCoeff_taylor, homogenize_monomial, derivative_add, monomial_add_erase, eval_monomial, aeval_sumIDeriv_of_pos, Derivation.apply_aeval_eq, PowerSeries.trunc_derivative'

PresheafOfModules.Sheafify

Definitions

Name	Category	Theorems
module 📖	CompOp	—

RestrictScalars

Definitions

Name	Category	Theorems
module 📖	CompOp	21 math math: ContinuousLinearMap.norm_extendTo𝕜, Representation.smul_ofModule_asModule, Subrepresentation.mem_asSubmodule'_iff, LinearMap.extendTo𝕜_apply, Subrepresentation.submoduleSubrepresentationOrderIso_apply, Representation.isSimpleModule_iff_irreducible_ofModule, Representation.ofModule_asModule_act, ContinuousLinearMap.extendTo𝕜_apply, ringEquiv_map_smul, lsmul_apply_apply, addEquiv_symm_map_smul_smul, isScalarTower, Representation.ofModule_asAlgebraHom_apply_apply, Representation.is_simple_module_iff_irreducible_ofModule, Subrepresentation.mem_ofSubmodule_iff, isCentralScalar, addEquiv_symm_map_algebraMap_smul, Representation.isSemisimpleModule_iff_isSemisimpleRepresentation_ofModule, smul_def, Subrepresentation.submoduleSubrepresentationOrderIso_symm_apply, addEquiv_map_smul

Submodule

Definitions

Name	Category	Theorems
module 📖	CompOp	1544 math math: DirectSum.IsInternal.ofBijective_coeLinearMap_same, TensorProduct.mapInclIsometry_apply, finrank_sup_add_finrank_inf_eq, Ideal.toCotangent_to_quotient_square, LinearMap.lTensor_ker_subtype_tensorKerEquiv_symm, DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply, KaehlerDifferential.kerCotangentToTensor_toCotangent, Rep.resCoindHomEquiv_symm_apply_hom, Rep.resCoindHomEquiv_apply_hom, groupCohomology.instEpiModuleCatH2π, Polynomial.degreeLT.addLinearEquiv_symm_apply_inr, groupHomology.π_comp_H2Iso_hom_assoc, Rep.invariantsAdjunction_homEquiv_symm_apply_hom, Affine.Simplex.altitudeFoot_restrict, TensorProduct.forall_vanishesTrivially_iff_forall_fg_rTensor_injective, LieDerivation.IsKilling.ad_mem_ker_killingForm_ad_range_of_mem_orthogonal, LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff, rTensorOne_symm_apply, iSupIndep.linearEquiv_symm_apply, topEquiv_symm_apply_coe, iSupIndep.dfinsupp_lsum_injective, FG.rTensor.directedSystem, Algebra.idealMap_eq_ofEq_comp_toLocalized₀, coe_continuous_linearProjOfClosedCompl', groupHomology.mapCycles₂_comp_assoc, Module.End.isSemisimple_restrict_iff, RootPairing.posRootForm_posForm_apply_apply, Module.Basis.mem_submodule_iff, LinearMap.BilinForm.dualSubmoduleToDual_apply_apply, OrthonormalBasis.orthogonalProjection_eq_sum, IsIsotypic.linearEquiv_fun, finrank_span_finset_eq_card, finrank_span_set_eq_card, linearProjOfIsCompl_apply_left, groupCohomology.toCocycles_comp_isoCocycles₁_hom, subtype_comp_inclusion, Algebra.Extension.Cotangent.span_eq_top_of_span_eq_ker, MvPolynomial.DirectSum.coeLinearMap_eq_dfinsuppSum, IsSemisimpleModule.exists_submodule_linearEquiv_quotient, Affine.Simplex.orthogonalProjectionSpan_map, LinearMap.BilinForm.dualSubmoduleToDual_injective, EuclideanGeometry.vsub_orthogonalProjection_mem_direction_orthogonal, ExteriorAlgebra.GradedAlgebra.liftι_eq, groupCohomology.isoCocycles₁_hom_comp_i_apply, IsSemisimpleModule.submodule, AffineMap.restrict.bijective, RootPairing.root'In_corootSpanMem_eq_pairingIn, RootPairing.rootForm_restrict_nondegenerate_of_ordered, Polynomial.degreeLT.instFreeSubtypeMemSubmodule, finrank_add_finrank_le_of_disjoint, orthogonalDecomposition_symm_apply, Ideal.finrank_eq_finrank, Algebra.PreSubmersivePresentation.cotangentComplexAux_apply, LinearMap.subtype_comp_codRestrict, exteriorPower.basis_apply, LinearMap.IsProj.subtype_comp_codRestrict, LinearMap.subtype_comp_restrict, LinearMap.IsProj.codRestrict_apply_cod, RootPairing.instIsRootSystemSubtypeMemSubmoduleSpanRangeCoeEmbeddingRootCorootRestrictScalars', Finsupp.restrictDom_apply, rank_le_card_isVisible, RootPairing.RootPositiveForm.zero_lt_posForm_apply_root, exteriorPower.linearMap_ext_iff, Ideal.toCotangent_eq, RootPairing.finrank_corootSpan_eq', mulMap_tmul, comm_trans_lTensorOne, isNoetherian_submodule_right, Polynomial.toMatrix_sylvesterMap', exists_fg_le_eq_rTensor_inclusion, LinearMap.IsRefl.domRestrict, Affine.Simplex.finiteDimensional_direction_altitude, TensorProduct.range_mapIncl, LinearMap.BilinForm.ker_restrict_eq_of_codisjoint, IsIsotypic.isotypicComponents, toLinearEquiv_orthogonalDecomposition_symm, Module.End.isNilpotent_restrict_genEigenspace_nat, Affine.Simplex.orthogonalProjection_vadd_smul_vsub_orthogonalProjection, LinearMap.injective_restrict_iff_disjoint, covBy_iff_quot_is_simple, lift_rank_map_le, MeasureTheory.norm_condExpL2_le, CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_apply, tensorEquivSpan_apply_tmul, groupCohomology.toCocycles_comp_isoCocycles₂_hom_apply, linearMap_eq_zero_iff_of_eq_span, ContinuousLinearMap.ker_codRestrict, groupCohomology.cocyclesIso₀_hom_comp_f, Rep.resCoindAdjunction_counit_app_hom_hom, Finsupp.supportedEquivFinsupp_symm_single, LinearDisjoint.not_linearIndependent_pair_of_flat_left, Rep.coindToInd_of_support_subset_orbit, groupCohomology.H1π_comp_map_assoc, re_inner_starProjection_eq_normSq, groupHomology.mapCycles₁_comp_apply, DirectSum.decomposeAlgEquiv_symm_apply, Affine.Simplex.restrict_reindex, finiteDimensional_vectorSpan_range, LinearMap.le_ker_iff_comp_subtype_eq_zero, MeasureTheory.inner_condExpL2_left_eq_right, RootPairing.restrictScalars_coe_coroot, groupCohomology.π_comp_H0Iso_hom, Affine.Simplex.faceOppositeCentroid_restrict, linearProjOfIsCompl_apply_right, groupCohomology.π_comp_H1Iso_hom_assoc, prodEquivOfIsCompl_symm_apply_snd_eq_zero, exteriorPower.coe_basis, LinearMap.tensorKer_tmul, LinearMap.IsSymmetric.diagonalization_apply_self_apply, finrank_vectorSpan_image_finset_le, EuclideanGeometry.orthogonalProjection_congr, Subspace.dualLift_of_subtype, mulMap_one_right_eq, exteriorPower.ιMulti_family_linearIndependent_field, ContinuousLinearEquiv.snd_equivOfRightInverse, span_setOf_mem_eq_top, toLinearEquiv_orthogonalDecomposition, LinearPMap.mk_apply, MeasureTheory.condExpL2_ae_eq_zero_of_ae_eq_zero, Subspace.dualRestrict_comp_dualLift, groupCohomology.mapCocycles₂_comp_i, dualQuotEquivDualAnnihilator_apply, ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm, LinearIndependent.linearCombination_repr, exteriorPower.basis_repr_ne, orthogonalProjection_mem_subspace_eq_self, LinearMap.range_domRestrict_le_range, Module.Basis.mk_repr, LinearMap.submoduleComap_apply_coe, finiteDimensional_direction_affineSpan_of_finite, Module.Flat.ker_lTensor_eq, KaehlerDifferential.cotangentComplexBaseChange_tmul, groupCohomology.H0IsoOfIsTrivial_hom, LinearMap.ofIsCompl_smul, quotDualCoannihilatorToDual_apply, injective_inclusionSpan, IsLattice.rank', bot_lt_isotypicComponent, dualPairing_apply, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_i_hom, LinearMap.tensorEqLocusEquiv_apply, groupCohomology.coe_mapCocycles₁, LinearMap.domRestrict'_apply, Affine.Simplex.restrict_map_restrict, EuclideanGeometry.orthogonalProjection_contLinear, RootPairing.range_polarization_domRestrict_le_span_coroot, RootPairing.finrank_range_polarization_eq_finrank_span_coroot, EuclideanGeometry.dist_orthogonalProjection_eq_iff_oangle_eq, groupHomology.cyclesMap_comp_isoCycles₂_hom, AffineSubspace.coe_subtypeA, finrank_vectorSpan_le_iff_not_affineIndependent, groupCohomology.coboundariesToCocycles₁_apply, groupHomology.mapCycles₁_comp_assoc, LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal, groupHomology.toCycles_comp_isoCycles₂_hom_assoc, AffineSubspace.subtype_apply, Algebra.PreSubmersivePresentation.cotangentComplexAux_zero_iff, le_linearEquiv_of_sSup_eq_top, FG.lTensor.directedSystem, Module.Basis.coe_mkFinConsOfLE, rank_map_le, subtypeₗᵢ_toContinuousLinearMap, LinearMap.tensorKer_coe, LinearMap.BilinForm.restrict_nondegenerate_orthogonal_spanSingleton, Module.End.mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo, finrank_quotient_add_finrank, isIsotypicOfType_submodule_iff, val_mulMap'_tmul, isSemisimple_torsionBy_of_irreducible, SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix, eq_isotypicComponent_iff, RootPairing.coroot'In_rootSpanMem_eq_pairingIn, exteriorPower.alternatingMapLinearEquiv_ιMulti, KaehlerDifferential.kerToTensor_apply, coe_subtypeL', Polynomial.degreeLT.basis_val, selfAdjointPart_comp_subtype_skewAdjoint, setBasisOfLinearIndependentOfCardEqFinrank_repr_apply, IsSemisimpleRing.exists_linearEquiv_ideal_of_isSimpleModule, Algebra.Presentation.differentials.hom₁_single, LinearMap.rank_eq_of_surjective, TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul, Algebra.Generators.exists_presentation_of_basis_cotangent, lift_rank_range_of_injective, Ideal.mem_toCotangent_ker, groupHomology.δ₀_apply, isArtinian_iff_submodule_quotient, Module.End.eigenspace_restrict_le_eigenspace, LinearMap.IsSymmetric.directSum_decompose_apply, linearProjOfIsCompl_comp_surjective_of_exact, dualCopairing_eq, EuclideanGeometry.reflection_vadd_smul_vsub_orthogonalProjection, Ideal.finsuppTotal_apply_eq_of_fintype, groupHomology.isoCycles₁_inv_comp_iCycles_apply, rank_span_set, RootPairing.RootPositiveForm.zero_lt_posForm_iff, eq_isotypicComponent_of_le, groupCohomology.map_H0Iso_hom_f_apply, EuclideanGeometry.inter_eq_singleton_orthogonalProjection, groupCohomology.H0IsoOfIsTrivial_inv_apply, quotDualCoannihilatorToDual_nondegenerate, rank_bot, groupHomology.chains₁ToCoinvariantsKer_surjective, LinearMap.submoduleMap_coe_apply, Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange, AffineSubspace.linear_topEquiv, toLinearPMap_range, Affine.Simplex.orthogonalProjectionSpan_eq_point, LieHom.quotKerEquivRange_toFun, isSimpleModule_iff_isAtom, FiniteDimensional.span_of_finite, MeasureTheory.MemLp.condExpL2_ae_eq_condExp, TensorProduct.map₂_eq_range_lift_comp_mapIncl, LinearMap.exists_basis_basis_of_span_eq_top_of_mem_algebraMap, isArtinian_of_le, exteriorPower.ιMulti_span_of_span, biSup_comap_subtype_eq_top, LinearPMap.adjointAux_unique, RootPairing.injOn_dualMap_subtype_span_root_coroot, LinearMap.range_domRestrict_eq_range_iff, Ideal.finsuppTotal_apply, TensorProduct.exists_finite_submodule_of_finite', biSup_eq_range_dfinsupp_lsum, Rep.resCoindAdjunction_unit_app_hom_hom, IsOrtho.orthogonalProjection_comp_subtypeL, RootPairing.restrictScalars_pairing, EuclideanGeometry.orthogonalProjection_map, LieModule.Cohomology.d₂₃_apply, LieAlgebra.InvariantForm.restrict_nondegenerate, exteriorPower.alternatingMapLinearEquiv_symm_map, span_range_inclusion_restrictScalars_eq_top, isNoetherian_submodule, selfAdjointPart_comp_subtype_selfAdjoint, exteriorPower.instFree, Affine.Simplex.circumradius_restrict, HahnEmbedding.Partial.truncLT_mem_range_extendFun, mulMap_comm, Module.Flat.out, RootPairing.finrank_rootSpan_map_polarization_eq_finrank_corootSpan, LieModule.Cohomology.d₁₂_apply_apply, Affine.Simplex.medial_restrict, Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_zero_iff, equivSubtypeMap_symm_apply, botEquivPUnit_apply, LinearMap.ofIsComplProd_apply, Rep.coindFunctorIso_hom_app_hom_hom_apply_hom_hom_apply, AffineIndependent.finrank_vectorSpan_add_one, coe_orthogonalDecomposition_symm, groupCohomology.mapCocycles₂_comp_i_apply, Projectivization.finrank_submodule, ker_subtype, groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_apply, MeasureTheory.aestronglyMeasurable_condExpL2, orthogonalProjection_bot, linearProjOfIsCompl_apply_right', EuclideanGeometry.angle_orthogonalProjection_self, EuclideanGeometry.orthogonalProjection_apply_mem, LinearMap.restrict_comp, isNoetherian_of_le, IsSimpleRing.exists_ringEquiv_matrix_end_mulOpposite, coe_isComplEquivProj_symm_apply, finrank_add_inf_finrank_orthogonal, groupHomology.cycles₁IsoOfIsTrivial_hom_apply, LieAlgebra.bracket_ofTwoCocycle, LinearMap.exact_subtype_mkQ, AffineSubspace.coe_subtypeₐᵢ, Algebra.Extension.Cotangent.val_mk, linearDisjoint_iff, GradedTensorProduct.auxEquiv_one, toLocalized₀_apply_coe, EuclideanGeometry.dist_orthogonalProjection_eq_iff_angle_eq, LinearMap.IsSymm.domRestrict, Subspace.finrank_dualCoannihilator_eq, sup_eq_range, MeasureTheory.condExpL2_indicator_ae_eq_smul, exteriorPower.ιMultiDual_apply_diag, Representation.instIsTrivialSubtypeMemSubgroupSubmoduleInvariantsCompLinearMapIdSubtypeToInvariants, LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_right, groupCohomology.cocyclesIso₀_inv_comp_iCocycles, Algebra.TensorProduct.linearEquivIncludeRange_toLinearMap, retractionOfSectionOfKerSqZero_comp_kerToTensor, groupCohomology.mapCocycles₂_comp_i_assoc, DirectSum.IsInternal.collectedBasis_coe, rank_quotient_add_rank, Ideal.toCotangent_range, RootPairing.RootPositiveForm.posForm_apply_root_root_le_zero_iff, Module.Invertible.exists_linearEquiv_ideal, Subspace.finiteDimensional_quot_dualCoannihilator_iff, groupCohomology.H1IsoOfIsTrivial_inv_apply, LinearIndependent.repr_eq_single, FirstOrder.Field.lift_genericMonicPoly, Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange_residueField, groupHomology.mapCycles₁_id_comp_assoc, rank_submodule_le_one_iff, AffineIndependent.card_le_finrank_succ, Coplanar.finrank_le_two, Algebra.Generators.exists_presentation_of_free_cotangent, goursat, coe_prodEquivOfIsCompl', Affine.Simplex.finrank_direction_altitude, HahnEmbedding.Seed.strictMono_coeff, subtypeₗᵢ_toLinearMap, groupCohomology.cocycles₁IsoOfIsTrivial_hom_hom_apply_apply, Module.Basis.sumQuot_repr_left, LinearEquiv.ofSubmodules_apply, LinearMap.lTensor_range, groupCohomology.H2π_comp_map_apply, groupHomology.mapCycles₁_comp, topEquiv_apply, Polynomial.degreeLT.addLinearEquiv_castAdd, subtype_apply, LinearEquiv.toSpanNonzeroSingleton_homothety, MeasureTheory.integral_condExpL2_eq, IsCompactOperator.restrict', finiteDimensional_vectorSpan_image_of_finite, RootPairing.instFiniteSubtypeMemSubmoduleRootSpanOfFinite, orzechProperty_iff, Affine.Simplex.ninePointCircle_restrict, LinearMap.tensorKerEquivOfSurjective_symm_tmul, Module.End.pow_restrict, groupCohomology.toCocycles_comp_isoCocycles₂_hom, LinearMap.IsProj.eq_conj_prodMap, LinearMap.tensorEqLocus_coe, LinearDisjoint.not_linearIndependent_pair_of_flat_right, length_lt, finrank_lt_finrank_of_lt, LinearMap.mem_submoduleImage_of_le, Polynomial.degreeLT.addLinearEquiv_apply_fst, mulLeftMap_apply, LinearMap.restrict_commute, MeasureTheory.setIntegral_condExpL2_indicator, ModuleCat.imageIsoRange_hom_subtype, OrthonormalBasis.orthogonalProjection_apply_eq_sum, groupHomology.mapCycles₁_comp_i, LinearMap.toKerLocalized_isLocalizedModule, submoduleOf_sup_of_le, Affine.Simplex.face_restrict, rank_range_le, DirectSum.coeLinearMap_of, ContinuousLinearMap.coe_equivRange, rank_le, coe_prodEquivOfIsCompl, groupCohomology.shortComplexH0_f, DirectSum.coeLinearMap_eq_dfinsuppSum, RootPairing.polarizationIn_Injective, finrank_vectorSpan_insert_le, LinearIsometryEquiv.coe_ofEq_apply, Subspace.dualPairing_nondegenerate, Affine.Simplex.interior_restrict, ExteriorAlgebra.GradedAlgebra.ι_sq_zero, AddMonoidAlgebra.decomposeAux_coe, LinearMap.ker_rangeRestrict, LinearMap.IsSymmetric.card_filter_eigenvalues_eq, AffineSubspace.inclusion_linear, LinearMap.compMultilinearMap_codRestrict, Module.Flat.iff_rTensor_injective, HasStrictFDerivAt.to_implicitFunctionOfComplemented, EuclideanGeometry.orthogonalProjection_apply', Affine.Simplex.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq, Rep.subtype_hom, MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le, Rep.coindVEquiv_symm_apply_coe, Ideal.map_includeRight_eq, inf_genEigenspace, finiteDimensional_sup, isIsotypic_submodule_iff, finiteDimensional_direction_map, torsionBy_torsionBy_eq_top, TensorProduct.exists_finite_submodule_left_of_setFinite, TensorProduct.quotientTensorEquiv_symm_apply_mk_tmul, groupCohomology.H1IsoOfIsTrivial_H1π_apply_apply, LinearEquiv.mem_dilatransvections_iff_rank, ModuleCat.imageIsoRange_inv_image_ι_apply, basisOfLinearIndependentOfCardEqFinrank_repr_apply, EuclideanGeometry.orthogonalProjection_orthogonalProjection_of_le, strictMono_comap_prod_map, Matrix.cRank_toNat_eq_finrank, RootPairing.PolarizationIn_eq, EuclideanGeometry.Sphere.isTangent_iff_isTangentAt_orthogonalProjection, Module.Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul, dualRestrict_ker_eq_dualAnnihilator, LinearIsometry.equivRange_apply_coe, FiniteDimensional.span_finset, Subalgebra.finrank_toSubmodule, MeasureTheory.norm_condExpL2_coe_le, isArtinian_span_of_finite, MeasureTheory.setLIntegral_nnnorm_condExpL2_indicator_le, Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_iff, IsIsotypic.isotypicComponent, LinearMap.finrank_range_of_inj, LinearMap.restrict_sub, Rep.invariantsAdjunction_unit_app, groupHomology.mapCycles₂_id_comp, coe_linearProjOfIsCompl_apply, inclusion_apply, LinearEquiv.ofLeftInverse_symm_apply, LinearMap.range_rangeRestrict, isNoetherian_iff_submodule_quotient, surjective_tensorToSpan, orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, torsion_torsion_eq_top, LinearIsometryEquiv.ofTop_toLinearEquiv, finrank_span_le_card, isLocalizedModule, range_subtypeL, botEquivPUnit_symm_apply, Module.length_top, span_range_inclusion_eq_top, GradedTensorProduct.auxEquiv_symm_one, groupCohomology.H2π_eq_iff, GradedTensorProduct.auxEquiv_comm, LinearMap.comap_leq_ker_subToSupQuotient, HilbertBasis.hasSum_orthogonalProjection, LinearMap.rTensor_injective_iff_subtype, collinear_iff_finrank_le_one, LinearMap.subtype_compAlternatingMap_codRestrict, groupCohomology.δ₁_apply, LinearIsometryEquiv.ofTop_symm_apply_coe, LinearMap.submoduleImage_apply_of_le, LinearRecurrence.solSpace_rank, LinearEquiv.ofSubmodule'_symm_apply, groupHomology.toCycles_comp_isoCycles₁_hom_apply, IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp, Affine.Simplex.altitude_restrict_eq_comap_subtype, MeasureTheory.lintegral_nnnorm_condExpL2_le, exteriorPower.finrank_eq, AffineSubspace.isometryEquivMap.toAffineMap_eq, groupHomology.mapCycles₂_comp_i, AffineSubspace.signedInfDist_eq_signedDist_of_mem, Polynomial.degreeLT.addLinearEquiv_apply_snd, groupCohomology.map_H0Iso_hom_f, LinearMap.finrank_genEigenspace_le, Module.Basis.sumQuot_repr_inl, LinearMap.IsSymmetric.diagonalization_symm_apply, ContinuousLinearMap.coe_codRestrict, LieModule.weight_vector_multiplication, Affine.Simplex.signedInfDist_apply_self, LinearMap.quotientInfEquivSupQuotient_apply_mk, SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux, finiteDimensional_vectorSpan_of_finite, groupCohomology.π_comp_H1Iso_hom, Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_zero_iff, Module.End.eigenspace_restrict_eq_bot, AffineSubspace.signedInfDist_eq_signedDist_orthogonalProjection, exteriorPower.map_apply_ιMulti_family, AffineEquiv.linear_ofEq, LinearMap.submoduleOf_span_singleton_of_mem, norm_orthogonalProjection_apply_le, instPolynormableSpaceSubtypeMemSubmodule, Algebra.Extension.Hom.sub_one_tmul, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_assoc_apply, mem_iSup_iff_exists_dfinsupp, FDRep.average_char_eq_finrank_invariants, isArtinian_range, TensorProduct.inner_mapIncl_mapIncl, RootPairing.toLinearMap_apply_PolarizationIn, exteriorPower.zeroEquiv_naturality, Collinear.finiteDimensional_direction_affineSpan, KaehlerDifferential.D_apply, finiteDimensional_direction_affineSpan_insert_set, TensorProduct.exists_finite_submodule_right_of_setFinite, HahnEmbedding.IsPartial.truncLT_mem_range, Affine.Simplex.orthogonalProjectionSpan_congr, LinearIndependent.linearCombinationEquiv_apply_coe, Polynomial.adjSylvester_comp_sylveserMap, iSup_eq_range_dfinsupp_lsum, Module.Basis.span_apply, LinearMap.ker_codRestrict, MeasureTheory.lpMeasToLpTrimLie_symm_indicator, groupHomology.mapCycles₁_id_comp_apply, groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_assoc, CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe, RootPairing.instFiniteSubtypeMemSubmoduleCorootSpanOfFinite, ContinuousLinearMap.equivRange_symm_apply, RootPairing.finrank_corootSpanIn, LinearIndependent.linearCombination_comp_repr, coe_inclusion, LinearIsometry.extend_apply, instLocallyConvexSpaceSubtypeMemSubmodule, Polynomial.degreeLT.instFiniteSubtypeMemSubmodule, finrank_strictMono, Module.End.IsSemisimple.restrict, Algebra.Extension.Cotangent.mk_eq_zero_iff, linearEquiv_det_reflection, groupCohomology.instEpiModuleCatH1π, LinearPMap.apply_comp_inclusion, LieSubmodule.toEnd_comp_subtype_mem, Coplanar.finiteDimensional_direction_affineSpan, IsSemisimpleModule.sup, Matrix.rank_eq_finrank_range_toLin, Affine.Simplex.ExcenterExists.touchpointWeights_restrict, groupCohomology.H2π_comp_map, Matrix.eRank_toNat_eq_finrank, CharacterModule.intSpanEquivQuotAddOrderOf_apply, Ideal.mapCotangent_toCotangent, groupHomology.π_comp_H2Iso_hom, stereoToFun_apply, prodEquivOfIsCompl_symm_apply_fst_eq_zero, exteriorPower.ιMulti_apply_coe, affineIndependent_iff_not_finrank_vectorSpan_le, coe_tensorSpanEquivSpan_apply_tmul, FG.lTensor.directLimit_apply', dualRestrict_def, ContinuousLinearMap.coe_projKerOfRightInverse_apply, Rep.coindToInd_apply, groupHomology.mapCycles₁_comp_i_apply, Module.Basis.SmithNormalForm.snf, Rep.FiniteCyclicGroup.groupHomologyπEven_eq_iff, groupHomology.mapCycles₂_comp, AlgHom.toKerIsLocalization_apply, LinearMap.restrict_eq_codRestrict_domRestrict, ModuleCat.kernelIsoKer_inv_kernel_ι_apply, IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite, HahnEmbedding.Partial.truncLT_mem_range_sSupFun, FG.rTensor.directLimit_apply, groupCohomology.isoCocycles₂_hom_comp_i, coe_continuous_linearProjOfClosedCompl, EuclideanGeometry.orthogonalProjection_subtype, LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict, LieModule.Cohomology.d₁₂_apply_coe_apply_apply, coe_orthogonalDecomposition, EuclideanGeometry.two_zsmul_oangle_self_orthogonalProjection, rank_submodule_le_one_iff', groupCohomology.π_comp_H0Iso_hom_apply, finiteDimensional_direction_affineSpan_insert, finrank_orthogonal_span_singleton, inclusionSpan_apply_coe, groupHomology.coe_mapCycles₂, comap_subtype_self, exteriorPower.oneEquiv_naturality, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, one_le_finrank_iff, groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply, groupHomology.H1π_comp_map_apply, Affine.Simplex.ExcenterExists.excenter_restrict, IsPrincipal.generator_submoduleImage_dvd_of_mem, norm_subtypeL, RootPairing.posRootForm_posForm_pos_of_ne_zero, Module.Flat.eqLocus_lTensor_eq, LinearMap.lift_rank_eq_of_surjective, Subspace.dualRestrict_leftInverse, TensorProduct.exists_finite_submodule_left_of_finite, DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem_ne, rank_mono, LinearMap.comap_codRestrict, ContinuousLinearMap.coe_rangeRestrict, setLike.coe_galgebra_toFun, norm_sq_eq_add_norm_sq_projection, finiteDimensional_direction_affineSpan_range, Polynomial.det_taylorLinearEquiv_toLinearMap, ContinuousLinearMap.equivRange_symm_toLinearEquiv, EuclideanGeometry.orthogonalProjection_vadd_eq_self, groupCohomology.cocyclesMap_comp_isoCocycles₁_hom, groupHomology.toCycles_comp_isoCycles₁_hom_assoc, LinearEquiv.ofInjective_apply, Module.length_submodule, comm_trans_rTensorOne, starProjection_apply, linearIndependent_span, Subspace.flip_quotDualCoannihilatorToDual_bijective, groupCohomology.cocyclesIso₀_inv_comp_iCocycles_assoc, comap_equiv_self_of_inj_of_le_apply, rank_quotient_add_rank_of_isDomain, Module.Basis.sumQuot_repr_inl_of_mem, exists_fg_le_subset_range_rTensor_subtype, DirectSum.coeAlgHom_of, TensorProduct.rTensor_injective_of_forall_vanishesTrivially, isArtinian_of_fg_of_artinian, set_smul_eq_map, groupCohomology.mapCocycles₁_one, RingHom.toKerIsLocalization_apply, EuclideanGeometry.oangle_orthogonalProjection_self, Affine.Simplex.map_altitude_restrict, Affine.Simplex.faceOpposite_restrict, groupHomology.H2π_comp_map_assoc, Polynomial.degreeLT.addLinearEquiv_apply, fst_orthogonalDecomposition_apply, LinearEquiv.mem_dilatransvections_iff_finrank, Matrix.rank_eq_finrank_span_row, RootPairing.prod_rootForm_smul_coroot_mem_range_domRestrict, finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply, LinearEquiv.ofSubmodules_symm_apply, IsSemisimpleModule.exists_quotient_linearEquiv_submodule, LinearMap.kerComplementEquivRange_apply_coe, LinearDisjoint.rank_inf_le_one_of_commute_of_flat_right, restrictScalarsEquiv_symm_apply, finiteDimensional_vectorSpan_insert_set, LinearMap.quotientInfEquivSupQuotient_injective, isArtinian_of_submodule_of_artinian, Module.Basis.restrictScalars_repr_apply, CliffordAlgebra.GradedAlgebra.ι_apply, EuclideanGeometry.dist_orthogonalProjection_eq_dist_iff_eq_of_mem, LieModule.Cohomology.d₁₂_apply_apply_ofTrivial, GradedTensorProduct.auxEquiv_mul, lTensorOne'_tmul, RootPairing.CoPolarizationIn_eq, AffineSubspace.abs_signedInfDist_eq_dist_of_mem_affineSpan_insert, Affine.Simplex.exradius_restrict, rTensorOne'_tmul_one, inner_orthogonalProjection_eq_of_mem_right, DirectSum.decompose_symm_algebraMap, LinearMap.ker_domRestrict, finiteDimensional_vectorSpan_singleton, rTensorOne'_tmul, finiteDimensional_inf_right, Finsupp.linearCombinationOn_range, Algebra.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex, TensorProduct.exists_finite_submodule_right_of_setFinite', realPart_comp_subtype_selfAdjoint, groupCohomology.π_comp_H0Iso_hom_assoc, projective_units, RootPairing.algebraMap_rootFormIn, map_subtype_embedding_eq, ModuleCat.imageIsoRange_hom_subtype_assoc, LinearDisjoint.injective, LinearMap.mem_submoduleImage, dualCopairing_apply, LinearEquiv.toSpanNonzeroSingleton_apply, Affine.Simplex.orthogonalProjection_circumcenter, Module.Flat.instSubalgebraToSubmodule, ContinuousLinearMap.coe_linearMap_equivRange, LinearMap.BilinForm.nondegenerate_restrict_iff_disjoint_ker, groupCohomology.H2π_comp_map_assoc, ker_inclusion, AffineIndependent.finrank_vectorSpan_image_finset, exteriorPower.alternatingMapLinearEquiv_apply_ιMulti, groupHomology.mapCycles₂_comp_apply, LinearIsometryEquiv.toSpanUnitSingleton_apply, comap_subtype_eq_top, AffineSubspace.subtypeA_toAffineMap, LinearIndependent.span_repr_eq, Affine.Simplex.exists_forall_dist_eq_iff_exists_excenterExists_and_eq_excenter, prodEquivOfIsCompl_symm_apply, Rep.coindFunctorIso_inv_app_hom_hom_apply_coe, injective_tensorToSpan, projective_of_isUnit, Module.Basis.restrictScalars_apply, instIsSemisimpleModuleSubtypeMemSubmoduleIsotypicComponent, HasStrictFDerivAt.to_implicitFunction, exteriorPower.ιMultiDual_apply_ιMulti, natAbs_det_equiv, Module.Basis.mem_submodule_iff', Module.End.IsFinitelySemisimple.restrict, map_subtype_span_singleton, Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq, LinearPMap.adjointDomainMkCLM_apply, Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent, exteriorPower.pairingDual_ιMulti_ιMulti, groupHomology.H2π_eq_iff, mulRightMap_eq_mulMap_comp, le_isotypicComponent, isNoetherian_submodule', exteriorPower.alternatingMapLinearEquiv_comp_ιMulti, rank_span_finset_le, groupHomology.H1AddEquivOfIsTrivial_single, MeasureTheory.integral_condExpL2_eq_of_fin_meas_real, finrank_map_le, finrank_span_eq_card, LinearIsometryEquiv.ofTop_apply, prodComm_trans_prodEquivOfIsCompl, groupHomology.isoCycles₂_hom_comp_i_apply, DirectSum.decompose_algebraMap, LinearMap.compLeftInverse_apply_of_bdd, groupCohomology.π_comp_H0IsoOfIsTrivial_hom_apply, RootPairing.zero_le_posForm, finrank_eq_zero, Module.length_bot, rTensorOne_tmul, ContinuousLinearEquiv.toSpanNonzeroSingleton_symm_apply, groupCohomology.toCocycles_comp_isoCocycles₁_hom_assoc, exists_fun_fin_finrank_span_eq, groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_assoc, LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_left, Module.Finite.of_isComplemented_domain, Polynomial.degreeLT.addLinearEquiv_apply', affineIndependent_iff_le_finrank_vectorSpan, Subspace.quotAnnihilatorEquiv_apply, skewAdjointPart_comp_subtype_selfAdjoint, groupCohomology.isoCocycles₁_inv_comp_iCocycles, unitsToPic_apply, Affine.Simplex.circumcenter_restrict, AffineMap.restrict.injective, coe_mulMap_comp_eq, Affine.Simplex.height_restrict, exteriorPower.oneEquiv_symm_apply, ModuleCat.imageIsoRange_inv_image_ι, LinearMap.finrank_range_le, isIdempotentElemEquiv_apply_coe, TensorProduct.exists_finite_submodule_left_of_finite', EuclideanGeometry.orthogonalProjection_affineSpan_singleton, LinearEquiv.submoduleMap_apply, ContinuousLinearEquiv.ofSubmodule'_apply, Module.Flat.iff_lTensor_injective, groupHomology.cyclesMap_comp_isoCycles₂_hom_assoc, LieModule.Cohomology.mem_twoCocycle_iff, groupCohomology.cocyclesMap_comp_isoCocycles₂_hom, exteriorPower.map_comp_ιMulti, AffineSubspace.subtype_injective, Module.End.isFinitelySemisimple_iff', AffineSubspace.subtype_linear, TensorProduct.quotientTensorQuotientEquiv_apply_tmul_mk_tmul_mk, orthogonalProjection_apply_eq_linearProjOfIsCompl, TensorProduct.tensorQuotientEquiv_apply_mk_tmul, TensorProduct.exists_finite_submodule_of_finite, MeasureTheory.condExpL2_indicator_of_measurable, GradedTensorProduct.mulHom_apply, EuclideanGeometry.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq, MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictDegreeOfFinite, ContinuousMultilinearMap.codRestrict_apply_coe, AffineSubspace.topEquiv_symm_apply_coe, coe_isComplEquivProj_apply, IsCompactOperator.restrict, prodEquivOfIsCompl_symm_apply_left, sndL_comp_coe_orthogonalDecomposition, RootPairing.PolarizationIn_apply, injective_subtype, linearProjOfIsCompl_isCompl_projection, quotientEquivOfIsCompl_apply_mk_coe, MeasureTheory.condExpL2_indicator_eq_toSpanSingleton_comp, DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem, LinearMap.coe_domRestrict, Module.End.isNilpotent_restrict_maxGenEigenspace_sub_algebraMap, linearProjOfIsCompl_comp_subtype, IsSemisimpleModule.sSup_simples_le, rank_eq_zero, IsLattice.finrank_of_pi, Polynomial.degreeLT.addLinearEquiv_symm_apply_inl, coe_subtypeₗᵢ, Polynomial.degreeLT.addLinearEquiv_symm_apply_inr_basis, MvPolynomial.weightedHomogeneousComponent_directSum, IsSemisimpleModule.exists_simple_submodule, groupHomology.mapCycles₂_id_comp_assoc, groupHomology.π_comp_H1Iso_hom_apply, Rep.coindIso_inv_hom_hom, LinearMap.quotKerEquivRange_symm_apply_image, Finsupp.restrictDom_comp_subtype, Module.Finite.span_singleton, MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le_real, rTensor_mkQ, map_le_isotypicComponent, Collinear.finiteDimensional_vectorSpan, groupCohomology.map_id_comp_H0Iso_hom_apply, exteriorPower.pairingDual_apply_apply_eq_one, groupCohomology.subtype_comp_d₀₁_assoc, Module.Flat.tensorProduct_mapIncl_injective_of_left, imaginaryPart_comp_subtype_selfAdjoint, iSupIndep.linearEquiv_apply, groupHomology.toCycles_comp_isoCycles₂_hom, ContinuousLinearEquiv.coord_toSpanNonzeroSingleton, HasRankNullity.rank_quotient_add_rank, groupCohomology.map_id_comp_H0Iso_hom, Module.Flat.iff_lTensor_injectiveₛ, groupHomology.cyclesMap_comp_isoCycles₂_hom_apply, CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_assoc_apply, Affine.Simplex.incenter_restrict, IsLattice.rank_of_pi, groupHomology.mapCycles₁_id_comp, IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite, norm_orthogonalProjection_apply, toENat_rank_span_set, groupCohomology.cocycles₁IsoOfIsTrivial_inv_hom_apply_coe, EuclideanGeometry.orthogonalProjection_vadd_smul_vsub_orthogonalProjection, torsionBySet_torsionBySet_eq_top, rTensorOne_tmul_one, toLinearMap_prodEquivOfIsCompl_symm, finrank_eq_rank, Subspace.finrank_add_finrank_dualCoannihilator_eq, Function.Exact.linearMap_rangeRestrict, Affine.Simplex.orthogonalProjectionSpan_faceOpposite_eq_point_rev, Module.Finite.range, TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective, LinearIndependent.linearCombinationEquiv_symm_apply, equivSubtypeMap_apply, Algebra.Extension.Hom.mapKer_apply_coe, EuclideanGeometry.dist_orthogonalProjection_eq_infNndist, isIdempotentElemEquiv_symm_apply_coe, FiniteDimensional.finiteDimensional_submodule, Algebra.Extension.contangentEquiv_tmul, det_reflection, Representation.linHom.invariantsEquivRepHom_symm_apply_coe, LinearMap.quotKerEquivRange_apply_mk, Algebra.Generators.cMulXSubOneCotangent_eq, rank_span_le, ContinuousLinearEquiv.toSpanNonzeroSingleton_coord, Ideal.toCotangent_apply, inner_orthogonalProjection_eq_of_mem_left, LieAlgebra.InvariantForm.restrict_orthogonal_nondegenerate, rank_submodule_eq_one_iff, groupHomology.isoCycles₁_hom_comp_i_apply, AffineMap.restrict.surjective, Affine.Simplex.inradius_restrict, Finsupp.supportedEquivFinsupp_symm_apply_coe_support_val, groupCohomology.cocyclesIso₀_hom_comp_f_assoc, AffineMap.restrict.coe_apply, linearMap_eq_iff_of_eq_span, CategoryTheory.ShortComplex.exact_iff_surjective_moduleCatToCycles, RootPairing.prod_rootFormIn_smul_coroot_mem_range_PolarizationIn, ModuleCat.imageIsoRange_inv_image_ι_assoc, IsSemisimpleModule.sSup_simples_eq_top, groupCohomology.H1π_eq_zero_iff, Ideal.toCotangent_eq_zero, exteriorPower.ιMulti_family_span_of_span, toLinearMap_orthogonalProjection_eq_linearProjOfIsCompl, mulRightMap_apply, LieHom.quotKerEquivRange_invFun, IsLattice.free, Module.injOn_dualMap_subtype_span_range_range, groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_apply, LieSubalgebra.coe_ofLe, Real.Convex.dimH_eq_finrank_vectorSpan, quotientEquivOfIsCompl_symm_apply, GradedTensorProduct.auxEquiv_tmul, ModuleCat.kernelIsoKer_hom_ker_subtype, exteriorPower.map_injective_field, groupHomology.H1ToTensorOfIsTrivial_H1π_single, CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_assoc_apply, EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangent, Subspace.dualLift_of_mem, LinearEquiv.coe_ofEq_apply, Polynomial.degreeLT.addLinearEquiv_symm_apply, coord_norm', RootPairing.algebraMap_posRootForm_posForm, Rep.FiniteCyclicGroup.groupHomologyπEven_eq_zero_iff, groupCohomology.cocyclesMk₁_eq, Algebra.idealMap_isLocalizedModule, DirectSum.decomposeLinearEquiv_apply, Subalgebra.finiteDimensional_toSubmodule, exteriorPower.alternatingMapLinearEquiv_comp, collinear_iff_rank_le_one, Finsupp.linearCombination_restrict, LinearEquiv.ofSubmodule'_toLinearMap, finrank_top, DirectSum.decomposeLinearEquiv_symm_comp_lof, exteriorPower.basis_coord, EuclideanGeometry.orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem, LinearIndependent.repr_range, Algebra.Extension.Cotangent.mk_eq_mk_iff_sub_mem, Rep.quotientToInvariantsFunctor_obj_V, IsSemisimpleModule.exists_end_ringEquiv, Module.isPrincipal_submodule_iff, Function.Exact.iff_linearMap_rangeRestrict, RootPairing.range_polarizationIn, RootPairing.restrictScalars_coe_root, exteriorPower.presentation_relation, mulMap_one_left_eq, LinearMap.finrank_maxGenEigenspace, groupCohomology.coboundariesToCocycles₂_apply, EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq, MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictTotalDegreeOfFinite, quotDualCoannihilatorToDual_injective, Affine.Simplex.centroid_restrict, MultilinearMap.codRestrict_apply_coe, groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_apply, groupHomology.cyclesMap_comp_isoCycles₁_hom_apply, groupHomology.toCycles_comp_isoCycles₂_hom_apply, TensorProduct.exists_finite_submodule_of_setFinite', groupCohomology.H2π_eq_zero_iff, MvPolynomial.DirectSum.coeLinearMap_eq_finsum, LinearIsometryEquiv.submoduleMap_symm_apply_coe, Subspace.dualRestrict_surjective, Coplanar.finiteDimensional_vectorSpan, groupCohomology.mapCocycles₁_comp_i_assoc, LinearMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal', ker_subtypeL, LinearMap.kerComplementEquivRange_symm_apply, groupCohomology.H1π_comp_map_apply, LinearMap.lift_rank_range_add_rank_ker, mulMap_comm_of_commute, TensorAlgebra.GradedAlgebra.ι_apply, MeasureTheory.integrable_condExpL2_of_isFiniteMeasure, HahnEmbedding.ArchimedeanStrata.isInternal_stratum', groupCohomology.cocyclesMap_cocyclesIso₀_hom_f, EuclideanGeometry.dist_orthogonalProjection_eq_infDist, exteriorPower.instFinite, EuclideanGeometry.orthogonalProjection_mem, LinearEquiv.coord_self, exteriorPower.alternatingMapToDual_apply_ιMulti, groupCohomology.π_comp_H2Iso_hom_assoc, Affine.Simplex.orthogonalProjectionSpan_restrict, isArtinian_submodule', KaehlerDifferential.D_tensorProductTo, groupCohomology.toCocycles_comp_isoCocycles₂_hom_assoc, LinearEquiv.ofEq_symm, IsSemisimpleModule.exists_end_algEquiv, mulMap_eq_mul'_comp_mapIncl, DirectSum.decomposeLinearEquiv_symm_apply, LinearEquiv.ofLeftInverse_apply, setBasisOfTopLeSpanOfCardEqFinrank_repr_apply, Algebra.Extension.Cotangent.mk_surjective, Affine.Simplex.setInterior_restrict, AffineSubspace.finiteDimensional_sup, rank_add_le_rank_add_rank, RootPairing.RootPositiveForm.isSymm_posForm, mem_smul_top_iff, DirectSum.decomposeAlgEquiv_apply, AffineIndependent.finrank_vectorSpan, IsCompl.projection_apply, LinearMap.comp_codRestrict, mulLeftMap_eq_mulMap_comp, Representation.subrepresentation_apply, groupHomology.mapCycles₂_hom, MeasureTheory.integrable_condExpL2_indicator, rank_quotient_add_rank_le, LieAlgebra.Extension.d₁₂_oneCochainOfTwoSplitting, groupHomology.isoCycles₂_inv_comp_iCycles_apply, LinearMap.finrank_range_dualMap_eq_finrank_range, Affine.Simplex.excenterWeights_restrict, groupHomology.isoCycles₂_inv_comp_iCycles_assoc, EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff, AffineEquiv.ofEq_symm, ModuleCat.kernelIsoKer_hom_ker_subtype_apply, HahnEmbedding.ArchimedeanStrata.iSupIndep_stratum', LinearMap.IsProj.eq_conj_prod_map', LinearEquiv.submoduleMap_symm_apply, LinearMap.map_codRestrict, EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq, groupHomology.cyclesMk₂_eq, LinearMap.BilinForm.finrank_orthogonal, Rep.coindVEquiv_apply_hom, DirectSum.IsInternal.card_filter_subordinateOrthonormalBasisIndex_eq, ContinuousMultilinearMap.codRestrict_toMultilinearMap, Polynomial.degreeLT.addLinearEquiv_symm_apply', LinearDisjoint.rank_le_one_of_flat_of_self, ContinuousLinearMap.projKerOfRightInverse_comp_inv, LinearEquiv.toSpanNonzeroSingleton_symm_apply_smul, Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_is_extension, instInvertibleSubtypeMemVal, AffineSubspace.linear_equivMapOfInjective, groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_assoc, EuclideanGeometry.Sphere.IsTangentAt.eq_orthogonalProjection, groupHomology.H1π_eq_zero_iff, groupHomology.isoCycles₁_inv_comp_iCycles_assoc, MeasureTheory.MemLp.condExpL2_ae_eq_condExp', LinearMap.BilinForm.IsSymm.restrict, groupHomology.π_comp_H1Iso_hom_assoc, Affine.Simplex.median_restrict, isNoetherian_of_fg_of_noetherian, groupHomology.H2π_comp_map, CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_apply, Subspace.dualEquivDual_def, affineSpan_coe_preimage_eq_top, TensorProduct.quotientTensorEquiv_apply_tmul_mk, exists_finset_span_eq_linearIndepOn, exteriorPower.zeroEquiv_ιMulti, LinearMap.surjective_comp_linearProjOfIsCompl, LinearMap.exact_subtype_ker_map, Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_iff, HahnEmbedding.Seed.truncLT_mem_range_baseEmbedding, Polynomial.eval_eq_sum_degreeLTEquiv, mulMap'_surjective, subtypeL_apply, finrank_span_eq_finrank_span, groupHomology.H1π_comp_map_assoc, Subspace.dualLift_rightInverse, Ideal.natAbs_det_equiv, EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq, finrank_map_subtype_eq, groupHomology.instEpiModuleCatH1π, ker_orthogonalProjection, Affine.Simplex.closedInterior_restrict, groupCohomology.isoCocycles₂_inv_comp_iCocycles, LinearMap.ofIsCompl_add, LinearEquiv.ofSubmodule'_apply, FG.directedSystem, EuclideanGeometry.orthogonalProjection_orthogonalProjection, exteriorPower.map_injective, Ideal.subtype_isoBaseOfIsPrincipal_eq_mul, LinearDisjoint.rank_le_one_of_commute_of_flat_of_self, groupCohomology.δ₀_apply, EuclideanGeometry.orthogonalProjection_vsub_mem_direction, exteriorPower.basis_repr_apply, LinearIndependent.repr_eq, mk_quotientEquivOfIsCompl_apply, prodEquivOfIsCompl_symm_apply_right, coe_mapIic_apply, submoduleOf_eq_top, Subspace.dualCopairing_nondegenerate, exteriorPower.presentation_R, finiteDimensional_direction_affineSpan_image_of_finite, groupHomology.instEpiModuleCatH2π, ContinuousLinearEquiv.submoduleMap_symm_apply, Affine.Simplex.orthogonalProjection_eq_circumcenter_of_dist_eq, AffineSubspace.topEquiv_apply, LinearEquiv.ofTop_symm_apply, groupCohomology.cocyclesMk₂_eq, Affine.Simplex.mongePoint_restrict, rank_span_of_finset, finrank_mono, mulLeftMap_apply_single, ContinuousLinearMap.range_eq_map_coprodSubtypeLEquivOfIsCompl, mem_isotypicComponents_iff, LinearIndependent.repr_ker, groupHomology.H1π_comp_map, Affine.Simplex.eulerPoint_restrict, finrank_add_le_finrank_add_finrank, EuclideanGeometry.vsub_orthogonalProjection_mem_direction, mulMap_map_comp_eq, Module.Flat.iff_lift_lsmul_comp_subtype_injective, AffineSubspace.coe_inclusion_apply, Ideal.range_finsuppTotal, groupCohomology.toCocycles_comp_isoCocycles₁_hom_apply, Affine.Simplex.ExcenterExists.touchpoint_restrict, Affine.Simplex.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection, Module.Finite.top, IsSemisimpleRing.exists_ringEquiv_pi_matrix_end_mulOpposite, finrank_le_one_iff_isPrincipal, groupHomology.cyclesMk₁_eq, Module.End.isNilpotent_restrict_sub_algebraMap, FG.lTensor.directLimit_apply, finrank_span_eq_finrank, LinearMap.IsProj.codRestrict_ker, Collinear.finrank_le_one, Module.fgSystem.equiv_comp_of, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply, groupHomology.mapCycles₂_comp_i_assoc, RootPairing.posRootForm_posForm_nondegenerate, Affine.Simplex.excenterWeightsUnnorm_restrict, groupCohomology.isoCocycles₁_hom_comp_i, range_inclusion, Module.Flat.instSubtypeMemSubmoduleSubmoduleAlgebra, ContinuousLinearMap.projKerOfRightInverse_apply_idem, LinearMap.domRestrict₂_apply, LinearMap.IsProj.trace, IsLocalRing.CotangentSpace.map_eq_top_iff, rank_range_of_injective, basisOfPid_bot, EuclideanGeometry.Sphere.IsTangent.isTangentAt, groupCohomology.mapCocycles₁_comp_i_apply, ContinuousLinearMap.coe_codRestrict_apply, Rep.coindMap_hom, LinearMap.restrict_smul_one, groupHomology.mapCycles₂_id_comp_apply, LinearMap.range_domRestrict, Polynomial.det_taylorLinearEquiv, IsSemisimpleModule.exists_end_algEquiv_pi_matrix_end, instIsTorsionFree, coe_subtype, mulMap_comp_lTensor, LinearEquiv.rank_map_eq, isNoetherian_span_of_finite, RootPairing.algebraMap_coroot'In_apply, ContinuousLinearMap.IsPositive.orthogonalProjection_comp, LinearMap.linearEquiv_of_ne_zero, LinearMap.IsSymm.nondegenerate_restrict_of_isCompl_ker, dimH_orthogonalProjection_le, LinearMap.range_ofIsCompl, RootPairing.rootForm_restrict_nondegenerate_of_isAnisotropic, derivationOfSectionOfKerSqZero_apply_coe, torsionBy_isTorsionBy, Module.Finite.iff_fg, groupHomology.H2π_comp_map_apply, lTensorOne_symm_apply, LinearMap.tensorEqLocus_tmul, exteriorPower.map_comp_ιMulti_family, EuclideanGeometry.reflection_apply', fg_top, RootPairing.rootFormIn_self_smul_coroot, tensorToSpan_apply_tmul, exteriorPower.map_id, LinearMap.ofIsCompl_left_apply, EuclideanGeometry.oangle_self_orthogonalProjection, isCompl_comap_subtype_of_isCompl_of_le, le_isotypicComponent_iff, Projectivization.instFiniteDimensionalSubtypeMemSubmoduleSubmodule, Module.Finite.of_fg, fg_iff_finiteDimensional, iSupIndep_iff_dfinsupp_lsum_injective, Finsupp.supportedEquivFinsupp_symm_apply_coe, Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_iff, LinearEquiv.toSpanNonzeroSingleton_one, lTensorOne'_one_tmul, LinearMap.compAlternatingMap_codRestrict, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_K, LinearMap.coe_quotientInfToSupQuotient, Subspace.dualLift_injective, Algebra.Generators.cotangentRestrict_mk, MatrixModCat.toModuleCat_map, Polynomial.toMatrix_sylvesterMap, TensorProduct.exists_finite_submodule_right_of_finite, LinearMap.subtype_compMultilinearMap_codRestrict, Finsupp.supportedEquivFinsupp_apply_support_val, AffineEquiv.coe_ofEq_apply, finiteDimensional_bot, ContinuousLinearMap.exist_extension_of_finiteDimensional_range, groupCohomology.π_comp_H2Iso_hom_apply, MeasureTheory.hausdorffMeasure_orthogonalProjection_le, CliffordAlgebra.even.lift.aux_apply, finrank_add_finrank_orthogonal, instIsSemisimpleModuleSubtypeMemSubmoduleValSetIsotypicComponents, AffineMap.restrict.linear, map_subtype_top, exteriorPower.pairingDual_apply_apply_eq_one_zero, lipschitzWith_orthogonalProjection, Affine.Simplex.abs_signedInfDist_eq_dist_of_mem_affineSpan_range, LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces, exists_fg_le_subset_range_rTensor_inclusion, span_span_coe_preimage, AffineSubspace.coe_subtype, Representation.coind_apply, groupHomology.isoCycles₁_hom_comp_i_assoc, ContinuousLinearEquiv.submoduleMap_apply, ClosedComplemented.exists_submodule_equiv_prod, Rep.invariantsFunctor_map_hom, Module.End.iInf_maxGenEigenspace_restrict_map_subtype_eq, finrank_span_singleton, Affine.Simplex.orthogonalProjectionSpan_reindex, ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection, Module.Finite.map, rank_le_one_iff_isPrincipal, subalgebra_top_finrank_eq_submodule_top_finrank, stereographic_apply, MeasureTheory.inner_condExpL2_eq_inner_fun, finite_dualAnnihilator_iff, linearProjOfIsCompl_comp_bijective_of_exact, RootPairing.RootPositiveForm.algebraMap_apply_eq_form_iff, groupCohomology.π_comp_H1Iso_hom_apply, MeasureTheory.integrableOn_condExpL2_of_measure_ne_top, LinearMap.rTensor_range, Module.Basis.SmithNormalForm.repr_comp_embedding_eq_smul, TensorProduct.exists_finite_submodule_right_of_finite', IsLocalRing.CotangentSpace.span_image_eq_top_iff, AlternatingMap.codRestrict_apply_coe, LinearEquiv.coord_apply_smul, inf_iInf_maxGenEigenspace_of_forall_mapsTo, RootPairing.restrictScalars_toLinearMap_apply_apply, LinearMap.finrank_maxGenEigenspace_zero_eq, fstL_comp_coe_orthogonalDecomposition, RootPairing.finrank_corootSpan_eq, LinearMap.ofIsCompl_right_apply, Polynomial.sylvesterMap_apply_coe, Polynomial.degreeLT.addLinearEquiv_natAdd, EuclideanGeometry.orthogonalProjection_linear, ContinuousLinearEquiv.ofSubmodule'_symm_apply, inclusion_injective, mulMap_comp_map_inclusion, orthogonalProjection_orthogonal_apply_eq_zero, AffineSubspace.signedInfDist_def, LinearMap.submoduleComap_surjective_of_surjective, IsSemisimpleModule.exists_sSupIndep_sSup_simples_eq_top, exteriorPower.basis_repr, exists_extension_norm_eq, CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_apply, finrank_le, groupHomology.π_comp_H2Iso_hom_apply, isArtinian_sup, lTensorOne_tmul, instIsSimpleModuleSubtypeMemSubmoduleValSetSetOfNonemptyLinearEquivId, Module.Flat.iff_rTensor_injectiveₛ, LinearDisjoint.val_mulMap_tmul, LinearMap.trace_restrict_eq_of_forall_mem, Affine.Simplex.orthogonalProjectionSpan_eulerPoint_mem_ninePointCircle, Module.Dual.finrank_ker_add_one_of_ne_zero, Module.Finite.bot, Algebra.Extension.Hom.sub_tmul, exteriorPower.alternatingMapLinearEquiv_symm_apply, groupHomology.mapCycles₁_hom, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_π_hom, LinearMap.exists_extend, isNoetherian_submodule_left, groupHomology.isoCycles₁_inv_comp_iCycles, Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisors, affineIndependent_iff_finrank_vectorSpan_eq, orthogonalProjectionFn_eq, RootPairing.finrank_rootSpanIn, Affine.Simplex.signedInfDist_affineCombination, Ideal.cotangentEquivIdeal_symm_apply, Module.flat_iff, Polynomial.taylorLinearEquiv_symm, groupHomology.cyclesMap_comp_isoCycles₁_hom_assoc, DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne, EuclideanGeometry.angle_self_orthogonalProjection, exists_fg_le_eq_rTensor_subtype, groupHomology.coe_mapCycles₁, groupHomology.toCycles_comp_isoCycles₁_hom, Polynomial.degreeLT.addLinearEquiv_symm_apply_inl_basis, iSup_eq_toSubmodule_range, TensorProduct.exists_finite_submodule_of_setFinite, TensorProduct.exists_finite_submodule_left_of_setFinite', RootPairing.RootPositiveForm.zero_lt_apply_root_root_iff, exteriorPower.map_apply_ιMulti, ContinuousAlternatingMap.codRestrict_apply_coe, LinearMap.finrank_eq_of_isPerfPair, LinearMap.ker_restrict, RootPairing.posRootForm_posForm_anisotropic, groupHomology.mapCycles₂_comp_i_apply, rank_quotient_add_rank_of_divisionRing, Module.End.isNilpotent_restrict_genEigenspace_top, EuclideanGeometry.orthogonalProjection_eq_self_iff, groupCohomology.cocyclesIso₀_hom_comp_f_apply, groupHomology.boundariesToCycles₂_apply, Module.Finite.span_finset, Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule_1, groupCohomology.subtype_comp_d₀₁, AddMonoidAlgebra.decomposeAux_single, EuclideanGeometry.reflection_subtype, map_range_rTensor_subtype_lid, LinearMap.det_eq_det_mul_det, isNoetherian_sup, FiniteDimensional.instSubtypeMemSubmoduleMap, ContinuousLinearEquiv.coord_norm', LinearEquiv.coe_ofTop_symm_apply, IsLattice.finite, groupHomology.isoCycles₂_hom_comp_i, LinearMap.injective_domRestrict_iff, groupHomology.π_comp_H1Iso_hom, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_H, fstEquiv_symm_apply_coe, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_assoc_apply, snd_orthogonalDecomposition_apply, groupHomology.isoCycles₂_inv_comp_iCycles, LinearMap.BilinForm.toLin_restrict_ker_eq_inf_orthogonal, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_apply, groupHomology.mkH1OfIsTrivial_apply, fromModuleCatToModuleCatLinearEquiv_apply, Module.Finite.span_of_finite, IsIsotypic.linearEquiv_finsupp, groupCohomology.isoCocycles₁_inv_comp_iCocycles_assoc, fromModuleCatToModuleCatLinearEquiv_symm_apply_coe, groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom, LinearMap.surjective_rangeRestrict, AffineSubspace.inclusion_rfl, IsSemisimpleModule.eq_bot_or_exists_simple_le, linearProjOfIsCompl_range, LinearMap.restrict_coe_apply, LieAlgebra.IsKilling.ker_restrict_eq_bot_of_isCartanSubalgebra, AlgHom.toKerIsLocalization_isLocalizedModule, FractionalIdeal.equivNum_apply, isNoetherian_of_submodule_of_noetherian, orthogonalDecomposition_apply, map_comap_subtype, span_range_subtype_eq_top_iff, spanRank_top, exteriorPower.map_comp, Module.End.isNilpotent.restrict, groupHomology.π_comp_H1Iso_inv_apply, lift_rank_range_le, LinearMap.quotientInfEquivSupQuotient_symm_apply_left, exteriorPower.ιMultiDual_apply_nondiag, LinearMap.codRestrict_apply, sSup_simples_eq_top_iff_isSemisimpleModule, AddMonoidAlgebra.decomposeAux_eq_decompose, exteriorPower.presentation_var, groupCohomology.isoCocycles₂_hom_comp_i_apply, LinearIsometryEquiv.ofEq_rfl, mulMap_op, LinearEquiv.ofEq_rfl, EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal, Module.End.genEigenspace_restrict_eq_top, isNoetherian_top_iff, LieModule.Cohomology.mem_twoCocycle_iff_of_trivial, MeasureTheory.eLpNorm_condExpL2_le, finrank_vectorSpan_range_add_one_le, CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_assoc_apply, Module.Flat.tensorProduct_mapIncl_injective_of_right, ContinuousLinearEquiv.toSpanNonzeroSingleton_apply_coe, mem_span_set_iff_exists_finsupp_le_finrank, ContinuousLinearEquiv.coord_self, LinearMap.ofIsCompl_subtype_zero_eq, Derivation.liftKaehlerDifferential_apply, LinearEquiv.finrank_map_eq, groupHomology.cycles₁IsoOfIsTrivial_inv_apply, map_subtype_range_inclusion, finrank_bot, ContinuousLinearEquiv.coord_norm, FiniteDimensional.subalgebra_toSubmodule, exteriorPower.map_surjective, KaehlerDifferential.DLinearMap_apply, LinearPMap.mem_adjoint_domain_iff, Module.Basis.coe_mkFinCons, CliffordAlgebra.GradedAlgebra.lift_ι_eq, linearProjOfIsCompl_ker, EuclideanGeometry.eq_orthogonalProjection_of_eq_subspace, LinearEquiv.lift_rank_map_eq, EuclideanGeometry.orthogonalProjection_apply, Module.Basis.extendOfIsLattice_apply, groupHomology.isoCycles₁_hom_comp_i, groupCohomology.cocyclesIso₀_inv_comp_iCocycles_apply, torsion_isTorsion, rank_span, norm_orthogonalProjection, AffineSubspace.signedInfDist_apply_self, EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection, EuclideanGeometry.orthogonalProjection_mem_orthogonal, Polynomial.degreeLT.basisProd_natAdd, Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange, exteriorPower.presentation_G, groupCohomology.H1π_comp_map, lTensorOne_one_tmul, LinearMap.finrank_maxGenEigenspace_eq, LinearMap.IsProj.codRestrict_apply, LinearMap.BilinForm.toLin_restrict_range_dualCoannihilator_eq_orthogonal, LinearMap.nondegenerate_restrict_of_disjoint_orthogonal, rank_le_of_isSMulRegular, LinearPMap.coe_vadd, Module.Dual.eq_of_preReflection_mapsTo', finrank_lt, rank_sup_add_rank_inf_eq, MeasureTheory.condExpIndSMul_ae_eq_smul, rank_top, Module.fgSystem.instDirectedSystemSubtypeSubmoduleFGMemValCoeLinearMapId, isNoetherian_range, linearProjOfIsCompl_apply_eq_zero_iff, coplanar_iff_finrank_le_two, LinearMap.tensorKerEquiv_apply, finite_sup, LinearMap.range_codRestrict, mem_biSup_iff_exists_dfinsupp, exteriorPower.ιMulti_family_linearIndependent_ofBasis, LieAlgebra.LoopAlgebra.twoCocycleOfBilinear_coe, groupCohomology.cocyclesMk₀_eq, EuclideanGeometry.two_zsmul_oangle_orthogonalProjection_self, range_subtype, NumberField.instIsLocalizedModuleIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmoduleValFractionalIdealNonZeroDivisorsRestrictScalarsSubtype, subtype_injective, dualRestrict_apply, LinearMap.BilinForm.finrank_add_finrank_orthogonal, IsSemisimpleModule.exists_linearEquiv_dfinsupp, Module.Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix, comapSubtypeEquivOfLe_apply_coe, RootPairing.RootPositiveForm.algebraMap_posForm, LinearDisjoint.rank_inf_le_one_of_commute_of_flat_left, ModuleCat.mono_as_hom'_subtype, DirectSum.isInternal_biSup_submodule_of_iSupIndep, LinearMap.ker_le_range_iff, Ideal.map_toCotangent_ker, LinearMap.ofIsCompl_zero, AffineSubspace.equivMapOfInjective_toFun, exteriorPower.ιMulti_family_span, Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero, toLocalized'_apply_coe, orthogonalProjection_comp_subtypeL_eq_zero_iff, dualQuotEquivDualAnnihilator_symm_apply_mk, LieDerivation.IsKilling.killingForm_restrict_range_ad_nondegenerate, Ideal.quotTorsionOfEquivSpanSingleton_apply_mk, Module.Basis.sumQuot_inl, LinearEquiv.ofInjective_symm_apply, LieAlgebra.Extension.twoCocycleOf_coe_coe, EuclideanGeometry.coe_orthogonalProjection_eq_iff_mem, ExteriorAlgebra.GradedAlgebra.ι_apply, Algebra.SubmersivePresentation.basisCotangent_apply, linearProjOfIsCompl_surjective, LinearMap.exists_extend_of_notMem, RootPairing.posRootForm_rootFormIn_posDef, HahnEmbedding.Partial.truncLT_eval_mem_range_extendFun, LinearMap.comp_ker_subtype, mulMap_range, isFullyInvariant_iff_le_imp_isotypicComponent_le, TensorProduct.tensorQuotientEquiv_symm_apply_tmul_mk, Affine.Simplex.map_subtype_restrict, Algebra.kerTensorProductMapIdToAlgHomEquiv_symm_apply, exteriorPower.oneEquiv_ιMulti, Affine.Simplex.restrict_map_subtype, Algebra.Extension.Cotangent.map_mk, RootPairing.CoPolarizationIn_apply, orthogonalProjection_eq_zero_iff, CliffordAlgebra.GradedAlgebra.ι_sq_scalar, Polynomial.sylveserMap_comp_adjSylvester, groupCohomology.map_H0Iso_hom_f_assoc, exteriorPower.ιMulti_span, map_equivMapOfInjective_symm_apply, Finsupp.supportedEquivFinsupp_symm_apply_coe_apply, orthogonalProjection_starProjection_of_le, Real.exists_extension_norm_eq, LinearMap.finrank_range_add_finrank_ker, exteriorPower.toTensorPower_apply_ιMulti, coe_orthogonalProjection_apply, ModuleCat.kernelIsoKer_inv_kernel_ι, IsIsotypicOfType.isotypicComponent, torsion'_torsion'_eq_top, Representation.linHom.invariantsEquivRepHom_apply_hom, Module.Flat.iff_rTensor_injective', ModuleCat.imageIsoRange_hom_subtype_apply, skewAdjointPart_comp_subtype_skewAdjoint, Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op, Polynomial.degreeLT.basisProd_castAdd, MatrixModCat.toModuleCat_obj_isModule, finrank_add_eq_of_isCompl, Finsupp.supportedEquivFinsupp_apply_apply, FG.rTensor.directLimit_apply', LinearMap.restrict_eq_domRestrict_codRestrict, finiteDimensional_of_le, groupHomology.H1AddEquivOfIsTrivial_symm_tmul, LieModule.Cohomology.d₂₃_comp_d₁₂, Subspace.quotDualCoannihilatorToDual_bijective, Ideal.toCotangent_surjective, lTensor_mkQ, norm_subtypeL_le, IsSemisimpleModule.range, ContinuousLinearEquiv.fst_equivOfRightInverse, LinearMap.quotientInfEquivSupQuotient_symm_apply_right, exteriorPower.zeroEquiv_symm_apply, finsetBasisOfLinearIndependentOfCardEqFinrank_repr_apply, Affine.Simplex.restrict_map_inclusion, basis_of_pid_aux, HahnEmbedding.Partial.apply_of_mem_stratum, finiteDimensional_direction_affineSpan_singleton, EuclideanGeometry.dist_orthogonalProjection_line_eq_iff_two_zsmul_oangle_eq, sndEquiv_apply, RingHom.toKerIsLocalization_isLocalizedModule, LinearPMap.adjointAux_inner, sndEquiv_symm_apply_coe, Algebra.idealMap_apply_coe, comapSubtypeEquivOfLe_symm_apply, groupCohomology.map_id_comp_H0Iso_hom_assoc, Module.End.genEigenspace_restrict, groupCohomology.isoCocycles₂_hom_comp_i_assoc, Matrix.rank_eq_finrank_span_cols, EuclideanGeometry.orthogonalProjection_eq_iff_mem, EuclideanGeometry.dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq, coe_subtypeL, Rep.coindIso_hom_hom_hom, orthogonalProjection_coe_eq_linearProjOfIsCompl, quotient_prod_linearEquiv, LinearMap.domRestrict₁₂_apply, Ideal.map_includeLeft_eq, MeasureTheory.lpMeasToLpTrimLie_symm_toLp, groupCohomology.coe_mapCocycles₂, orthogonalProjection_norm_le, ContinuousLinearEquiv.equivOfRightInverse_symm_apply, LinearMap.injective_rangeRestrict_iff, LinearDisjoint.rank_inf_le_one_of_flat_right, LinearMap.quotientInfEquivSupQuotient_surjective, LinearPMap.graph_map_snd_eq_range, ModuleCat.toKernelSubobject_arrow, orthogonalProjection_orthogonal, LinearMap.finrank_eigenspace_le, Module.Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord, LinearMap.domRestrict_apply, LinearMap.surjective_comp_subtype_of_isComplemented, Polynomial.taylorLinearEquiv_apply_coe, MeasureTheory.norm_condExpL2_le_one, Module.finrank_quotient_add_finrank_le, groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_assoc, Module.Flat.iff_lTensor_injective', groupCohomology.H1π_eq_iff, DirectSum.decompose_smul, MeasureTheory.condExpL2_comp_continuousLinearMap, LinearEquiv.ofTop_apply, groupCohomology.isoCocycles₁_hom_comp_i_assoc, mulMap_comp_rTensor, Algebra.Extension.Cotangent.ker_mk, torsionBySet_isTorsionBySet, orthogonalProjection_orthogonalComplement_singleton_eq_zero, Polynomial.degreeLT.basis_repr, IsSemisimpleModule.exists_end_ringEquiv_pi_matrix_end, Module.End.pos_finrank_genEigenspace_of_hasEigenvalue, LinearIsometry.submoduleMap_apply_coe, LinearDisjoint.rank_inf_le_one_of_flat_left, EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangentAt, orthogonalProjection_eq_linearProjOfIsCompl, LinearPMap.adjoint_apply_of_dense, groupHomology.mapCycles₁_quotientGroupMk'_epi, RootPairing.range_polarizationIn_le_span_coroot, groupHomology.mapCycles₁_comp_i_assoc, exists_smith_normal_form_of_le, MeasureTheory.condExpL2_const_inner, LinearPMap.toFun_eq_coe, map_subtype_le, AffineEquiv.ofEq_rfl, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_apply, LinearMap.finiteDimensional_range, coe_prodEquivOfClosedCompl, Module.Basis.restrictScalars_toMatrix, TensorProduct.exists_of_fg, Subalgebra.rank_toSubmodule, groupHomology.π_comp_H2Iso_inv_apply, LinearMap.BilinForm.restrict_apply, DirectSum.range_coeLinearMap, submoduleOf_self, EuclideanGeometry.dist_set_eq_iff_dist_orthogonalProjection_eq, Finsupp.range_restrictDom, LinearMap.BilinForm.nondegenerate_restrict_of_disjoint_orthogonal, linearIndepOn_iff_linearCombinationOnₛ, finiteDimensional_inf_left, EuclideanGeometry.Sphere.finrank_orthRadius, ContinuousLinearEquiv.ofSubmodule'_toContinuousLinearMap, LinearMap.rank_range_add_rank_ker, finrank_vectorSpan_insert_le_set, coe_equivMapOfInjective_apply, disjoint_iff_comap_eq_bot, LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm, Subspace.finrank_add_finrank_dualAnnihilator_eq, TensorProduct.toLinearMap_mapInclIsometry, Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_zero_iff, lsum_comp_mapRange_toSpanSingleton, RootPairing.algebraMap_root'In_apply, groupCohomology.mapCocycles₁_comp_i, groupHomology.boundariesToCycles₁_apply, OrthogonalFamily.projection_directSum_coeAddHom, TensorProduct.range_mapIncl_mono, Module.Finite.of_isComplemented_codomain, MatrixModCat.fromMatrixLinear_apply_coe, linearIndepOn_iff_linearCombinationOn, Module.Flat.instInvertibleSubtypeMemSubmoduleSubmoduleAlgebra, LinearMap.toKerIsLocalized_apply_coe, LinearMap.submoduleMap_surjective, hasFDerivAt_stereoInvFunAux_comp_coe, LinearMap.finrank_le_of_isSMulRegular, LinearIsometryEquiv.ofEq_symm, finiteDimensional_vectorSpan_insert, LinearPMap.inverse_domain, exteriorPower.basis_repr_self, MeasureTheory.condExpL2_indicator_nonneg, flip_quotDualCoannihilatorToDual_injective, coe_prodEquivOfClosedCompl_symm, Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule, Algebra.Extension.cotangentComplex_mk, mulRightMap_apply_single, Subspace.dualPairing_eq, groupHomology.isoCycles₂_hom_comp_i_assoc, subalgebra_top_rank_eq_submodule_top_rank, fstEquiv_apply, groupCohomology.π_comp_H2Iso_hom, FiniteDimensional.span_singleton, Ideal.to_quotient_square_comp_toCotangent, retractionOfSectionOfKerSqZero_tmul_D, DirectSum.IsInternal.subordinateOrthonormalBasisIndex_def, groupHomology.H2π_eq_zero_iff, finrank_vectorSpan_range_le, ContinuousLinearEquiv.ofSubmodules_apply, LinearMap.ofIsCompl_eq_add, span_range_inclusionSpan, LinearMap.surjective_domRestrict_iff, Subspace.dualEquivDual_apply, instIsLocalizedModuleSubtypeMemLocalized₀ToLocalized₀, Affine.Simplex.restrict_points_coe, Ideal.isoBaseOfIsPrincipal_apply, groupCohomology.isoCocycles₂_inv_comp_iCocycles_assoc, DirectSum.IsInternal.collectedBasis_repr_of_mem, Affine.Simplex.excenterExists_restrict, restrictScalarsEquiv_apply, rank_range_of_surjective, groupHomology.δ₁_apply, LinearMap.restrict_apply, EuclideanGeometry.orthogonalProjection_mem_subspace_eq_self, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply, groupHomology.H1π_eq_iff, nonempty_basis_of_pid, LinearIsometryEquiv.submoduleMap_apply_coe, comap_subtype_le_iff, isSemisimpleModule_iff_exists_linearEquiv_dfinsupp, groupHomology.cyclesMap_comp_isoCycles₁_hom, ContinuousLinearEquiv.ofSubmodules_symm_apply

Submodule.Quotient

Definitions

Name	Category	Theorems
module 📖	CompOp	309 math math: Ideal.toCotangent_to_quotient_square, TopModuleCat.hom_cokerπ, CategoryTheory.ShortComplex.pOpcycles_comp_moduleCatOpcyclesIso_hom_apply, LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff, lTensor.inverse_comp_lTensor, IsSemisimpleModule.exists_submodule_linearEquiv_quotient, Module.support_quotient, Module.Presentation.cokernelSolution_var, Module.End.IsNilpotent.mapQ, Submodule.annihilator_quotient, covBy_iff_quot_is_simple, ModuleCat.epi_as_hom''_mkQ, RingTheory.Sequence.isWeaklyRegular_cons_iff, Module.isTorsionBySet_quotient_set_smul, Ring.coe_jacobson_quotient, ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom_apply, Submodule.mapQ_apply, Submodule.dualQuotEquivDualAnnihilator_apply, Submodule.factor_mk, Module.isTorsionBySet_quotient_ideal_smul, QuotSMulTop.equivTensorQuot_naturality, Submodule.top_eq_ofList_cons_smul_iff, Submodule.quotDualCoannihilatorToDual_apply, Submodule.dualPairing_apply, lTensor.inverse_of_rightInverse_comp_lTensor, AdicCompletion.incl_apply, Ideal.range_cotangentToQuotientSquare, Module.supportDim_add_length_eq_supportDim_of_isRegular, Submodule.finrank_quotient_add_finrank, Module.exists_smul_eq_zero_and_mk_eq, TensorProduct.quotTensorEquivQuotSMul_comp_mkQ_rTensor, TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul, isArtinian_iff_submodule_quotient, Module.Grassmannian.rankAtStalk_eq, Submodule.dualCopairing_eq, RingTheory.Sequence.isRegular_cons_iff, KaehlerDifferential.quotKerTotalEquiv_symm_comp_D, QuotSMulTop.equivTensorQuot_naturality_mk, IsLocalRing.map_mkQ_eq, Submodule.quotDualCoannihilatorToDual_nondegenerate, Submodule.quotEquivOfEqBot_symm_apply, Module.Presentation.cokernel_relation, LieHom.quotKerEquivRange_toFun, Module.equiv_directSum_of_isTorsion, QuotSMulTop.map_comp_mkQ, TensorProduct.tensorQuotEquivQuotSMul_comp_mkQ_lTensor, LinearMap.range_mkQ_comp, rTensor.inverse_of_rightInverse_apply, LinearMap.exact_subtype_mkQ, Module.Presentation.cokernel_R, Submodule.rank_quotient_add_rank, Subspace.finiteDimensional_quot_dualCoannihilator_iff, Module.Presentation.cokernelSolution.isPresentation, AdicCompletion.factor_eval_eq_evalₐ, Submodule.goursat, Submodule.quotientQuotientEquivQuotientAux_mk, FiniteDimensional.finiteDimensional_quotient, lTensor.inverse_of_rightInverse_apply, Submodule.mapQ_pow, Submodule.range_mkQ, QuotSMulTop.map_surjective, Submodule.isOpenQuotientMap_mkQ, Submodule.factor_eq_factor, Subspace.dualPairing_nondegenerate, CategoryTheory.ShortComplex.pOpcycles_comp_moduleCatOpcyclesIso_hom, Hausdorffification.instIsHausdorff, Submodule.linearMap_qext_iff, TensorProduct.quotientTensorEquiv_symm_apply_mk_tmul, Ideal.Quotient.span_singleton_one, Submodule.strictMono_comap_prod_map, equiv_symm, ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom, LinearMap.surjective_range_liftQ, Ideal.pi_mkQ_surjective, KaehlerDifferential.kerTotal_mkQ_single_algebraMap_one, isNoetherian_iff_submodule_quotient, lTensor.inverse_apply, LinearMap.comap_leq_ker_subToSupQuotient, Submodule.map_mkQ_eq_top, Module.supportDim_quotSMulTop_succ_eq_of_notMem_minimalPrimes_of_mem_maximalIdeal, isArtinian_of_quotient_of_artinian, LinearMap.quotientInfEquivSupQuotient_apply_mk, KaehlerDifferential.kerTotal_mkQ_single_add, AdicCompletion.range_eval, Ideal.Quotient.smul_top, Function.Exact.exact_mapQ_iff, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_assoc_apply, Submodule.mapQ_zero, AdicCompletion.transitionMap_comp_eval, CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe, Ideal.quotientToQuotientRangePowQuotSucc_mk, ModuleCat.smulShortComplex_g, Submodule.coe_quotEquivOfEqBot_symm, Ideal.pi_tensorProductMk_quotient_surjective, CharacterModule.intSpanEquivQuotAddOrderOf_apply, Submodule.annihilator_map_mkQ_eq_colon, AdicCompletion.coe_eval, Submodule.ker_liftQ_eq_bot', TensorProduct.tensorQuotEquivQuotSMul_tmul_mk, isFiniteLength_quotient_span_singleton, Submodule.mkQ_apply, Ideal.exact_mulQuot_quotOfMul, Submodule.card_quotient_mul_card_quotient, Submodule.factor_comp, Subspace.flip_quotDualCoannihilatorToDual_bijective, rank_quotient_add_rank_of_isDomain, Submodule.quotEquivOfEqBot_apply_mk, IsSemisimpleModule.exists_quotient_linearEquiv_submodule, LinearMap.quotientInfEquivSupQuotient_injective, Ideal.Quotient.torsionBy_eq_span_singleton, ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv, QuotSMulTop.equivQuotTensor_naturality, KaehlerDifferential.derivationQuotKerTotal_apply, AdicCompletion.transitionMap_comp_reduceModIdeal, Submodule.dualCopairing_apply, Submodule.le_comap_mkQ, LinearMap.exact_smul_id_smul_top_mkQ, associatedPrimes.eq_singleton_of_isPrimary, restrictScalarsEquiv_symm_mk, Submodule.mapQ_eq_factor, LinearMap.injective_range_liftQ_of_exact, Module.Grassmannian.finite_quotient, Module.Presentation.cokernel_var, IsLocalizedModule.toLocalizedQuotient', Ideal.quotientToQuotientRangePowQuotSucc_surjective, Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner_naturality, Submodule.quotientPi_apply, rTensor.inverse_of_rightInverse_comp_rTensor, Subspace.quotAnnihilatorEquiv_apply, Ideal.ker_tensorProductMk_quotient, instIsLocalizedModuleQuotientSubmoduleLocalizedModuleLocalizationLocalizedToLocalizedQuotient, TensorProduct.quotientTensorQuotientEquiv_apply_tmul_mk_tmul_mk, TensorProduct.tensorQuotientEquiv_apply_mk_tmul, rTensor.inverse_comp_rTensor, LinearMap.liftQ₂_mk, Submodule.quotientEquivOfIsCompl_apply_mk_coe, Submodule.comapMkQOrderEmbedding_eq, QuotSMulTop.map_exact, groupHomology.π_comp_H1Iso_hom_apply, LinearMap.quotKerEquivRange_symm_apply_image, rTensor_mkQ, TensorProduct.tensorQuotEquivQuotSMul_symm_mk, RingTheory.Sequence.isWeaklyRegular_append_iff, equiv_symm_apply, Submodule.pi_liftQ_eq_liftQ_pi, HasRankNullity.rank_quotient_add_rank, Module.supportDim_quotSMulTop_succ_eq_supportDim, CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_assoc_apply, Submodule.comap_map_mkQ, Submodule.ker_liftQ_eq_bot, Module.IsTorsionBySet.quotient, Ideal.quotOfMul_surjective, LinearMap.quotKerEquivRange_apply_mk, isSimpleModule_iff_quot_maximal, Ideal.annihilator_quotient, LieHom.quotKerEquivRange_invFun, Submodule.isOpenMap_mkQ, CategoryTheory.ShortComplex.pOpcycles_comp_moduleCatOpcyclesIso_hom_assoc, Submodule.quotientEquivOfIsCompl_symm_apply, Ideal.cotangentEquivIdeal_apply, Ideal.quotientToQuotientRangePowQuotSucc_injective, TensorProduct.quotTensorEquivQuotSMul_symm_mk, Submodule.quotDualCoannihilatorToDual_injective, Submodule.quotientPi_symm_apply, Module.jacobson_quotient_of_le, Submodule.liftQ_apply, KaehlerDifferential.kerTotal_mkQ_single_mul, Submodule.isQuotientEquivQuotientPrime_iff, Module.Presentation.cokernel_G, Module.supportDim_le_supportDim_quotSMulTop_succ, QuotSMulTop.equivQuotTensor_naturality_mk, Module.exists_isPrincipal_quotient_of_finite, LinearMap.exact_map_mkQ_range, AdicCompletion.mk_smul_top_ofAlgEquiv_symm, Module.supportDim_quotSMulTop_succ_eq_supportDim_mem_jacobson, Submodule.mkQ_map_self, Module.support_quotSMulTop, Module.equiv_free_prod_directSum, AdicCompletion.factor_eval_liftRingHom, rank_quotient_add_rank_le, ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv_apply, Algebra.TensorProduct.quotIdealMapEquivTensorQuot_symm_tmul, KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination, Submodule.toLocalizedQuotient'_mk, KaehlerDifferential.kerTotal_mkQ_single_smul, Hausdorffification.lift_comp_of, CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_apply, TensorProduct.quotientTensorEquiv_apply_tmul_mk, Submodule.QuotientTorsion.instIsTorsionFree, Module.isTorsionBySet_quotient_iff, AdicCompletion.transitionMap_map_pow, isAssociatedPrime_iff_exists_injective_linearMap, rank_quotient_le, Submodule.mk_quotientEquivOfIsCompl_apply, Subspace.dualCopairing_nondegenerate, Submodule.map_liftQ, Submodule.piQuotientLift_single, Submodule.liftQSpanSingleton_apply, Ideal.pi_mkQ_rTensor, KaehlerDifferential.quotKerTotalEquiv_symm_apply, LinearMap.quotKerEquivOfSurjective_symm_apply, Submodule.mapQ_mkQ, rTensor.inverse_apply, Module.Basis.sumQuot_inr, QuotSMulTop.map_first_exact_on_four_term_exact_of_isSMulRegular_last, Module.isTorsionBy_quotient_iff, Submodule.factor_comp_apply, Submodule.quotEquivOfEq_mk, AdicCompletion.transitionMap_ideal_mk, AdicCompletion.transitionMap_map_mul, Function.Exact.linearEquivOfSurjective_apply, Module.supportDim_le_supportDim_quotSMulTop_succ_of_mem_jacobson, LinearMap.coe_quotientInfToSupQuotient, Submodule.range_dualMap_mkQ_eq, groupCohomology.π_comp_H2Iso_hom_apply, Algebra.TensorProduct.quotIdealMapEquivTensorQuot_mk, Hausdorffification.lift_of, QuotSMulTop.map_comp, Submodule.finite_dualAnnihilator_iff, Representation.quotient_apply, groupCohomology.π_comp_H1Iso_hom_apply, linearIndepOn_union_iff_quotient, Submodule.ker_mkQ, groupHomology.π_comp_H2Iso_hom_apply, Submodule.quotientPiLift_mk, Submodule.mkQ_surjective, AdicCompletion.factor_evalₐ_eq_eval, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_π_hom, Function.Exact.linearEquivOfSurjective_symm_apply, Module.length_quotient, Ideal.cotangentEquivIdeal_symm_apply, isSimpleModule_iff_isCoatom, QuotSMulTop.map_apply_mk, Module.supportDim_quotient_eq_ringKrullDim, AdicCompletion.eval_of, rank_quotient_add_rank_of_divisionRing, LinearMap.det_eq_det_mul_det, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_H, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_apply, Module.exists_surjective_quotient_of_finite, TensorProduct.tensorQuotEquivQuotSMul_comp_mk, Module.Grassmannian.projective_quotient, groupHomology.π_comp_H1Iso_inv_apply, LinearMap.quotientInfEquivSupQuotient_symm_apply_left, AdicCompletion.mk_apply_coe, IsLocalRing.map_mkQ_eq_top, RingTheory.Sequence.map_first_exact_on_four_term_right_exact_of_isSMulRegular_last, Submodule.mapQ_comp, Submodule.goursat_surjective, AdicCompletion.transitionMap_comp_eval_apply, ModuleCat.smulShortComplex_X₃_isModule, TensorProduct.quotTensorEquivQuotSMul_comp_mk, Submodule.finrank_quotient_le, Module.supportDim_quotSMulTop_succ_eq_of_notMem_minimalPrimes_of_mem_jacobson, rank_quotient_eq_of_le_torsion, Module.Finite.exists_fin_quot_equiv, Submodule.liftQ_mkQ, Submodule.dualAnnihilator_eq_bot_iff', QuotSMulTop.map_id, LinearMap.ker_le_range_iff, Submodule.dualQuotEquivDualAnnihilator_symm_apply_mk, Ideal.quotTorsionOfEquivSpanSingleton_apply_mk, Submodule.ker_liftQ, Submodule.QuotientTorsion.torsion_eq_bot, TensorProduct.tensorQuotientEquiv_symm_apply_tmul_mk, LinearMap.ker_eq_bot_range_liftQ_iff, AdicCompletion.eval_surjective, Rep.mkQ_hom, AdicCompletion.transitionMap_map_one, Module.jacobson_quotient_jacobson, Subspace.quotDualCoannihilatorToDual_bijective, LinearMap.ker_tensorProductMk, Module.supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes, lTensor_mkQ, LinearMap.quotientInfEquivSupQuotient_symm_apply_right, Module.torsion_by_prime_power_decomposition, AdicCompletion.eval_apply, IsSemisimpleModule.quotient, equiv_trans, KaehlerDifferential.quotKerTotalEquiv_apply, TensorProduct.quotTensorEquivQuotSMul_symm_comp_mkQ, quotient_prod_linearEquiv, restrictScalarsEquiv_mk, LinearMap.quotientInfEquivSupQuotient_surjective, Module.finrank_quotient_add_finrank_le, LinearMap.quotKerEquivOfSurjective_apply_mk, Submodule.comap_liftQ, groupHomology.π_comp_H2Iso_inv_apply, Submodule.isPrimary_iff_zero_divisor_quotient_imp_nilpotent_smul, AdicCompletion.of_apply, Module.isTorsionBy_quotient_element_smul, Module.IsTorsionBy.quotient, Submodule.piQuotientLift_mk, Submodule.flip_quotDualCoannihilatorToDual_injective, Subspace.dualPairing_eq, Submodule.mapQ_id, AdicCompletion.eval_comp_of, Ideal.to_quotient_square_comp_toCotangent, Submodule.factor_comp_mk, Module.Basis.sumQuot_repr_inr, TensorProduct.quotTensorEquivQuotSMul_mk_tmul, TensorProduct.tensorQuotEquivQuotSMul_symm_comp_mkQ, Submodule.quotientQuotientEquivQuotientAux_mk_mk, LinearIndepOn.quotient_iff_union, Submodule.factor_surjective, equiv_apply, Submodule.range_liftQ, Ideal.mulQuot_injective, KaehlerDifferential.kerTotal_mkQ_single_algebraMap

SubmoduleClass

Definitions

Name	Category	Theorems
module 📖	CompOp	90 math math: LieSubmodule.coe_smul, LieModule.shiftedGenWeightSpace.shift_apply, NonUnitalSubalgebraClass.coe_subtype, TensorProduct.LieModule.mapIncl_def, LieModule.traceForm_eq_sum_genWeightSpaceOf, LieModule.traceForm_eq_sum_finrank_nsmul', LieModule.shiftedGenWeightSpace.instSubtypeMemLieSubmodule, LieModule.isNilpotent_toEnd_genWeightSpace_zero, NonUnitalStarSubalgebra.unitizationStarAlgEquiv_apply_coe, LieModule.trace_toEnd_genWeightSpace, LieModule.shiftedGenWeightSpace.toEnd_eq, LieModule.toLinearMap_maxTrivLinearMapEquivLieModuleHom, LieModule.instIsTriangularizableSubtypeMemLieSubmodule_1, LieSubmodule.traceForm_eq_of_le_idealizer, LieSubmodule.incl_apply, LieSubmodule.lowerCentralSeries_eq_bot_iff_lcs_eq_bot, NonUnitalSubalgebra.unitization_injective, LieSubmodule.lowerCentralSeries_map_eq_lcs, NonUnitalStarSubalgebra.unitization_range, LieSubmodule.map_incl_top, LieSubmodule.subsingleton_of_bot, NonUnitalSubalgebra.unitization_range, LieSubmodule.comap_incl_eq_top, LieAlgebra.coe_rootSpaceWeightSpaceProduct_tmul, LieSubmodule.comap_incl_eq_bot, LieSubmodule.ucs_comap_incl, LieSubmodule.instIsTorsionFreeSubtypeMem, LieSubmodule.range_incl, LieSubmodule.map_incl_lt_iff_lt_top, LieSubmodule.trace_eq_trace_restrict_of_le_idealizer, NonUnitalStarSubalgebra.toNonUnitalSubalgebra_subtype, NonUnitalSubalgebra.toSubring_subtype, NonUnitalStarSubalgebra.range_val, NonUnitalSubalgebraClass.subtype_injective, NonUnitalStarSubalgebra.unitization_injective, NonUnitalStarSubalgebra.unitization_apply, LieSubmodule.ker_incl, LieSubmodule.instIsArtinianSubtypeMem, NonUnitalStarSubalgebraClass.coe_subtype, LieSubmodule.toEnd_restrict_eq_toEnd, NonUnitalSubalgebra.unitizationAlgEquiv_apply_coe, LieModuleHom.codRestrict_apply, LieModule.toLinearMap_maxTrivLinearMapEquivLieModuleHom_symm, LieModule.trace_comp_toEnd_genWeightSpace_eq, LieSubmodule.coe_inclusion, LieModuleHom.comp_ker_incl, NonUnitalSubalgebra.unitization_apply, LieSubmodule.map_incl_le, LieModule.maxTrivEquiv_of_equiv_symm_eq_symm, LieSubmodule.lowerCentralSeries_eq_lcs_comap, LieModule.traceForm_eq_sum_finrank_nsmul, LieSubmodule.coe_toEnd, LieModuleEquiv.ofTop_apply, LieModule.isNilpotent_toEnd_sub_algebraMap, NonUnitalStarSubalgebraClass.subtype_injective, LieSubmodule.incl_eq_val, LieModule.maxTrivEquiv_of_refl_eq_refl, NonUnitalStarSubalgebra.toSubring_subtype, LieSubmodule.inclusion_apply, LieModule.traceForm_genWeightSpace_eq, LieAlgebra.IsKilling.finrank_rootSpace_eq_one, LieModule.trace_toEnd_genWeightSpaceChain_eq_zero, LieModule.zero_lt_finrank_genWeightSpace, SMulMemClass.subtype_injective, LieSubmodule.lieIdeal_oper_eq_tensor_map_range, NonUnitalStarAlgHom.subtype_comp_codRestrict, LieModule.coe_maxTrivLinearMapEquivLieModuleHom_symm, NonUnitalSubalgebra.toNonUnitalSubsemiring_subtype, LieSubmodule.coe_toEnd_pow, LieModule.traceForm_eq_sum_finrank_nsmul_mul, LieModule.coe_maxTrivEquiv_apply, SMulMemClass.subtype_apply, LieSubmodule.instIsNoetherianSubtypeMem, LieSubmodule.inclusion_injective, NonUnitalSubalgebraClass.subtype_apply, SMulMemClass.coe_subtype, LieSubmodule.incl_coe, LieModule.coe_maxTrivHom_apply, LieSubmodule.instLieModule, LieSubmodule.injective_incl, LieSubmodule.map_comap_incl, LieModule.shiftedGenWeightSpace.shift_symm_apply, NonUnitalStarSubalgebraClass.subtype_apply, LieSubmodule.comap_incl_self, LieAlgebra.rootSpaceProduct_tmul, LieModule.posFittingComp_map_incl_sup_of_codisjoint, LieModule.coe_maxTrivLinearMapEquivLieModuleHom, NonUnitalSubalgebra.range_val, NonUnitalAlgHom.subtype_comp_codRestrict, LieModule.genWeightSpace_genWeightSpaceOf_map_incl

Subsemiring

Definitions

Name	Category	Theorems
module 📖	CompOp	—

SymmetricPower

Definitions

Name	Category	Theorems
module 📖	CompOp	4 math math: span_tprod_eq_top, range_mk, tprod_equiv, domDomCongr_tprod

TensorProduct.Algebra

Definitions

Name	Category	Theorems
module 📖	CompOp	7 math math: smul_def, Subbimodule.coe_mk, Subbimodule.coe_toSubbimoduleInt, Subbimodule.coe_toSubbimoduleNat, Subbimodule.coe_toSubmodule, Subbimodule.coe_toSubmodule', Subbimodule.coe_baseChange

TrivSqZeroExt

Definitions

Name	Category	Theorems
module 📖	CompOp	13 math math: fstCLM_apply, lift_inlAlgHom_inrHom, lift_comp_inrHom, algHom_ext'_iff, map_comp_inrHom, sndCLM_apply, inrCLM_apply, inrHom_apply, inlCLM_apply, sndHom_comp_map, sndHom_apply, DualNumber.algHom_ext'_iff, liftEquiv_symm_apply_coe

ULift

Definitions

Name	Category	Theorems
module 📖	CompOp	—

---

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