Operations

📁 Source: Mathlib/Data/Set/Operations.lean

Statistics

Metric	Count
Definitions BijOn, EqOn, InjOn, InvOn, LeftInvOn, MapsTo, restrict, RightInvOn, diagonal, graphOn, image2, imageFactorization, instCompl, instSProd, kernImage, offDiag, pi, preimage, prod, range, rangeFactorization, rangeSplitting, restrictPreimage, seq, term_''_, «term_⁻¹'_»	26
Theorems out, range_coind, apply_rangeSplitting, comp_rangeSplitting, diff_eq, eq_mem_setOf, mapsTo_image, mapsTo_preimage, mem_compl_iff, mem_diagonal, mem_diagonal_iff, mem_diff, mem_diff_of_mem, mem_image, mem_image2, mem_image2_of_mem, mem_image_of_mem, mem_offDiag, mem_pi, mem_preimage, mem_prod, mem_prod_eq, mem_range, mem_range_self, mem_seq_iff, mem_setOf, mem_setOf_eq, mem_univ, mem_univ_pi, mk_mem_prod, notMem_setOf_iff, prodMk_mem_set_prod_eq, prod_eq, prod_image_left, prod_image_right, rangeFactorization_bijective, rangeFactorization_eq_iff, rangeFactorization_eq_rangeFactorization_iff, rangeFactorization_injective, rangeFactorization_surjective, restrictPreimage_coe, seq_eq_image2, setOf_mem_eq, subset_kernImage_iff	44
Total	70

Membership.mem

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
out 📖	—	Set Set.instMembership setOf	—	—	Set.mem_setOf

Set

Definitions

Name	Category	Theorems
BijOn 📖	MathDef	78 math math: Module.bijOn_reflection_of_mapsTo, Real.arcosh_bijOn, OpenPartialHomeomorph.bijOn, EisensteinSeries.gammaSet_div_gcd_to_gammaSet10_bijection, Nat.bijOn_digitsAppend', Iic_add_bij, bijOn_empty_iff_right, isAddFreimanIso_zero_iff, bijOn_empty, Real.artanh_bijOn, Matroid.Indep.exists_bijOn_of_map, SurjOn.exists_bijOn_subset, EqOn.bijOn_iff, bijOn_comm, IsMulFreimanIso.bijOn, Function.Semiconj.bijOn_range, Rat.HeightOneSpectrum.adicCompletion.padicEquiv_bijOn, Nat.bijOn_ofDigits, Iio_mul_bij, isMulFreimanIso_zero_iff, Function.bijOn_ptsOfPeriod, PartialEquiv.bijOn, isAddFreimanIso_one_iff, Function.Commute.left_bijOn_fixedPoints_comp, Equiv.bijOn_symm_image, isAddFreimanIso_two, Nat.bijOn_ofDigits', RootPairing.bijOn_reflection_root, Ico_add_bij, InjOn.bijOn_image, BijOn.insert_iff, Finite.exists_bijOn_of_encard_eq, RootPairing.bijOn_coreflection_coroot, Iic_mul_bij, isMulFreimanIso_one_iff, Equiv.bijOn, surjOn_iff_exists_bijOn_subset, Real.bijOn_sin, Real.cosh_bijOn, Real.bijOn_cos, bijOn_of_subsingleton, bijOn_comp_iff, isMulFreimanIso_two, bijOn_empty_iff_left, Equiv.bijOn_swap, Ioc_add_bij, ConjClasses.mk_bijOn, CategoryTheory.Subgroupoid.IsNormal.conjugation_bij, Function.Bijective.bijOn_univ, Ioi_add_bij, Finite.surjOn_iff_bijOn_of_mapsTo, Equiv.bijOn_symm, exists_subset_bijOn, IsAddFreimanIso.bijOn, Equiv.bijOn_image, Iio_add_bij, Nat.bijOn_digitsAppend, Finite.injOn_iff_bijOn_of_mapsTo, Equiv.swap_bijOn_exchange, Function.bijOn_periodicPts, bijOn_univ, Icc_add_bij, bijOn_singleton, bijective_iff_bijOn_univ, Ioo_add_bij, BijOn.mk, bijOn_of_subsingleton', Equiv.image_eq_iff_bijOn, Equiv.swap_bijOn_self, bijOn_id, Real.tanh_bijOn, Finpartition.exists_subset_part_bijOn, Ici_add_bij, SurjOn.bijOn_subset, Function.Commute.right_bijOn_fixedPoints_comp, InvOn.bijOn, Equiv.bijOn', Function.bijOn_fixedPoints_comp
EqOn 📖	MathDef	194 math math: exists_continuous_sum_one_of_isOpen_isCompact, completelyRegularSpace_iff_isOpen, exists_contMDiffMap_zero_one_of_isClosed, Filter.EventuallyEq.self_of_nhdsSet, eqOn_piecewise, ProbabilityTheory.eqOn_complexMGF_of_mgf', eqOn_of_isMIntegralCurveOn_Ioo, Complex.eq_const_of_exists_le, AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux, eqOn_dslope_sub_smul, Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div, OpenPartialHomeomorph.EqOnSource.eqOn, HasConstantSpeedOnWith.ratio, PeriodPair.eqOn_deriv_weierstrassPExcept_derivWeierstrassPExcept, HasProdUniformlyOn.tprod_eqOn, isMIntegralCurveOn_Ioo_eqOn_of_contMDiff, RightInvOn.eqOn, eqOn_prodMap_iff, eqOn_prod_iff, eqOn_indicator, AlgHom.le_equalizer, MDifferentiableOn.eqOn_of_isPreconnected_of_isMaxOn_norm, Continuous.exists_contDiff_approx_and_eqOn, iteratedFDerivWithin_of_isOpen, Trivialization.eqOn, exists_continuous_zero_one_of_isClosed, eqOn_union, OpenPartialHomeomorph.isLocalStructomorphWithinAt_iff, OpenPartialHomeomorph.isLocalStructomorphWithinAt_source_iff, ODE_solution_unique_of_mem_Icc, LSeries_deriv_eqOn, InjOn.cancel_left, Filter.EventuallyEq.exists_mem, hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall, PhragmenLindelof.eqOn_quadrant_II, ODE_solution_unique, PhragmenLindelof.eq_zero_on_right_half_plane_of_superexponential_decay, PhragmenLindelof.eq_zero_on_vertical_strip, PhragmenLindelof.eq_zero_on_quadrant_IV, logDeriv_eqOn_iff, ProbabilityTheory.eqOn_complexMGF_of_mgf, eqOn_abs_add_one_of_isMIntegralCurveOn_Ioo, ODE_solution_unique_of_mem_Ioo, MeasureTheory.Measure.eqOn_Ioc_of_ae_eq, eqOn_range, MDifferentiableOn.norm_eqOn_of_isPreconnected_of_isMaxOn, LinearRecurrence.sol_eq_of_eq_init, PhragmenLindelof.eqOn_quadrant_I, Manifold.IsImmersionAt.writtenInCharts, TendstoLocallyUniformlyOn.unique, extDerivWithin_extDerivWithin_eqOn, PhragmenLindelof.eqOn_quadrant_III, eqOn_univ, exists_bounded_zero_one_of_closed, ContDiffMapSupportedIn.zero_on_compl', Real.rpowIntegrand₀₁_apply_mul_eqOn_Ici, exists_continuous_one_zero_of_isCompact, AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq, ODE_solution_unique_of_mem_Icc_right, AnalyticWithinAt.exists_analyticAt, ContDiffMapSupportedIn.zero_on_compl, IsPreconnected.eqOn_const_of_mapsTo, LeftInvOn.eqOn, analyticWithinAt_iff_exists_analyticAt', PartialEquiv.EqOnSource.symm_eqOn, Function.locallyFinsuppWithin.restrict_eqOn_compl, iteratedDerivWithin_of_isOpen_eq_iterate, piecewise_eqOn, PhragmenLindelof.eq_zero_on_horizontal_strip, SmoothBumpFunction.eqOn_source, MeasureTheory.Measure.eqOn_Ico_of_ae_eq, iteratedDerivWithin_zpow, eqOn_mulIndicator', AnalyticOnNhd.eqOn_zero_of_preconnected_of_mem_closure, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, eqOn_mulIndicator, eqOn_dslope_slope, exists_continuous_zero_one_of_isCompact', Complex.eqOn_closure_of_isPreconnected_of_isMaxOn_norm, LipschitzOnWith.extend_real, Filter.eventuallyEq_principal, AnalyticOnNhd.eqOn_zero_of_preconnected_of_frequently_eq_zero, eqOn_iteratedDeriv_cotTerm, LipschitzOnWith.extend_finite_dimension, LipschitzOnWith.extend_lp_infty, cfc_eq_cfc_iff_eqOn, IsOpen.eqOn_of_fderiv_eq, iteratedDerivWithin_one_div, Complex.eqOn_closedBall_of_isMaxOn_norm, exists_bounded_mem_Icc_of_closed_of_le, LinearMap.eqOn_span_iff, Function.disjoint_mulSupport_iff, AntitoneOn.exists_antitone_extension, unique_unit_speed, IsLocalDiffeomorphAt.localInverse_eqOn_right, IsOpen.measure_eq_biSup_integral_continuous, eqOn_of_leftInvOn_of_rightInvOn, AnalyticOnNhd.eqOn_or_eventually_ne_of_preconnected, Complex.norm_eqOn_of_isPreconnected_of_isMaxOn, Function.locallyFinsuppWithin.restrict_eqOn, cfcₙ_eq_cfcₙ_iff_eqOn, HasSumUniformlyOn.tsum_eq, Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn, eqOn_of_cfc_eq_cfc, CompletelyRegularSpace.completely_regular, Function.mulSupport_disjoint_iff, PhragmenLindelof.eqOn_right_half_plane_of_superexponential_decay, ContDiffMapSupportedIn.iteratedFDeriv_zero_on_compl, exists_continuous_zero_one_of_isCompact, graphOn_inj, LipschitzOnWith.extend_pi, AlgHom.eqOn_adjoin_iff, deriv_eqOn, harmonic_is_realOfHolomorphic, MeromorphicAt.eqOn_compl_singleton_toMeromorphicNFAt, Function.support_disjoint_iff, Continuous.exists_contMDiff_approx_and_eqOn, piecewise_eqOn_compl, toMeromorphicNFOn_eq_self_on_compl, OpenPartialHomeomorph.EqOnSource.symm_eqOn_target, Complex.norm_sub_one_sq_eqOn_sphere, HasSumUniformlyOn.tsum_eqOn, HasFPowerSeriesOnBall.unique, MeasureTheory.Measure.eqOn_of_ae_eq, PhragmenLindelof.eq_zero_on_quadrant_II, eqOn_indicator', eqOn_comm, SurjOn.cancel_right, exists_smooth_zero_one_of_isClosed, MeasureTheory.Measure.eqOn_open_of_ae_eq, OpenPartialHomeomorph.isLocalStructomorphWithinAt_iff', HasFDerivWithinAt.unique_on, OpenPartialHomeomorph.subtypeRestr_symm_eqOn, PhragmenLindelof.eqOn_quadrant_IV, Complex.eqOn_of_isPreconnected_of_isMaxOn_norm, exists_continuousMap_one_of_isCompact_subset_isOpen, HasProdUniformlyOn.tprod_eq, PhragmenLindelof.eqOn_vertical_strip, eqOn_refl, HasFPowerSeriesWithinOnBall.unique, AnalyticOnNhd.eqOn_of_preconnected_of_mem_closure, iteratedDerivWithin_of_isOpen, Manifold.IsImmersionAtOfComplement.writtenInCharts, OpenPartialHomeomorph.subtypeRestr_symm_eqOn_of_le, isMIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless, exists_continuous_one_zero_of_isCompact_of_isGδ, Complex.eq_const_of_exists_max, Complex.norm_eqOn_closedBall_of_isMaxOn, Pretrivialization.eqOn, eqOn_piecewise_of_isMIntegralCurveOn_Ioo, Filter.eventuallyEq_iff_exists_mem, LinearMap.le_eqLocus, MeasureTheory.Measure.eqOn_Icc_of_ae_eq, IsCompact.measure_eq_biInf_integral_hasCompactSupport, MeasureTheory.Measure.eqOn_Ioo_of_ae_eq, ApproximatesLinearOn.exists_homeomorph_extension, HasSumLocallyUniformlyOn.tsum_eqOn, unique_unit_speed_on_Icc_zero, HasProdLocallyUniformlyOn.tprod_eqOn, AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq, WeakFEPair.f_modif_aux1, ContinuousMap.isComplete_setOf_eqOn, ODE_solution_unique_of_mem_Icc_left, completelyRegularSpace_iff, restrict_eq_restrict_iff, Nat.eq_nth_of_strictMonoOn_of_mapsTo_of_surjOn, PhragmenLindelof.eqOn_horizontal_strip, PhragmenLindelof.eq_zero_on_quadrant_I, exists_tsupport_one_of_isOpen_isClosed, exists_continuous_add_one_of_isCompact_nnreal, iteratedDerivWithin_congr_right_of_isOpen, eqOn_empty, eqOn_of_cfcₙ_eq_cfcₙ, eqOn_iteratedDerivWithin_cotTerm_integerComplement, Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div', PartialEquiv.EqOnSource.eqOn, MonotoneOn.exists_monotone_extension, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, eqOn_singleton, PhragmenLindelof.eq_zero_on_quadrant_III, IsLocalDiffeomorphAt.localInverse_eqOn_left, Complex.affine_of_mapsTo_ball_of_norm_dslope_eq_div, Real.rpowIntegrand₀₁_eqOn_mul_rpowIntegrand₀₁_one, Real.eq_Gamma_of_log_convex, AnalyticOnNhd.eqOn_zero_or_eventually_ne_zero_of_preconnected, eqOn_comp_right_iff, Real.log_doublingGamma_eq, Path.eqOn_extend_segment, Function.disjoint_support_iff, CompletelyRegularSpace.completely_regular_isOpen, Real.rpowIntegrand₀₁_eqOn_pow_div, ContDiffMapSupportedInClass.map_zero_on_compl, AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero, ContinuousMap.exists_mul_le_one_eqOn_ge
InjOn 📖	MathDef	153 math math: UV.compress_injOn, Finset.pairwiseDisjoint_singleton_iff_injOn, StrictMonoOn.injOn, iSupIndep.injOn_iInf, Matroid.comap_isBasis'_iff, LinearMap.injOn_iff_surjOn, Matroid.comapOn_isBase_iff_of_surjOn, exists_image_eq_and_injOn, imageFactorization_injective_iff, MeasureTheory.exists_decomposition_of_monotoneOn_hasDerivWithinAt, LSeries_injOn, injOn_id, Function.iterate_injOn_Iio_minimalPeriod, PrimeSpectrum.basicOpen_injOn_isIdempotentElem, zpow_left_injOn₀, Finset.erase_injOn, Finset.not_injOn_of_card_image_lt, exists_image_eq_injOn_of_subset_range, RootPairing.injOn_dualMap_subtype_span_root_coroot, Finset.erase_injOn', Function.Injective.injOn, injOn_pair, exists_eq_graphOn, PartialEquiv.injOn, Finset.insert_inj_on, norm_injOn_ray_right, Finset.card_sups_iff, Matroid.comapOn_indep_iff, injOn_circleMap_of_abs_sub_le', IsPrimitiveRoot.injOn_pow, Real.arcosh_injOn, Nat.injOn_ofDigits, Finset.pairwiseDisjoint_smul_iff, pow_injOn_Iio_orderOf, CharP.natCast_injOn_Iio, powerset_insert_injOn, Finset.injOn_of_card_image_eq, Behrend.map_injOn, injOn_iff_map_eq_zero, Matroid.comap_indep_iff, SurjOn.exists_subset_injOn_image_eq, Finite.injOn_of_encard_image_eq, injOn_preimage, Matroid.comapOn_isBase_iff, injOn_iff_injective, Int.injOn_natAbs_Iic, Real.injOn_tan, Function.invFunOn_injOn_image, fst_injOn_graph, Real.logb_injOn_pos, Finset.intervalGapsWithin_injOn, iSupIndep.injOn, EqOn.injOn_iff, Int.cast_injOn_of_ringChar_ne_two, injOn_of_eq_iff_eq, SimpleGraph.Walk.IsPath.getVert_injOn, CharP.intCast_injOn_Ico, Real.injOn_arcsin, BijOn.injOn, Matroid.comap_isBasis_iff, Matroid.comap_dep_iff, Module.End.injOn_genEigenspace, Finset.injOn_of_surjOn_of_card_le, StrictAntiOn.injOn, Filter.filter_injOn_Iic_iff_injOn, Module.End.injOn_maxGenEigenspace, Finset.card_mul_iff, Function.Injective.injOn_range, injOn_singleton, injOn_insert, IsPrimitiveRoot.injOn_pow_mul, Nat.nth_injOn, Module.injOn_dualMap_subtype_span_range_range, countable_iff_exists_injOn, Cardinal.not_injective_limitation_set, Real.rpow_left_injOn, LinearMap.disjoint_ker_iff_injOn, SimpleGraph.Walk.IsPath.getVert_injOn_iff, SmoothBumpCovering.embeddingPiTangent_injOn, Cardinal.toNat_injOn, injective_piecewise_iff, injective_iff_injOn_univ, FirstOrder.Language.model_distinctConstantsTheory, Real.injOn_sin, Submodule.exists_injOn_mkQ_image_span_eq_of_span_eq_map_mkQ_of_le_jacobson_bot, Real.logb_injOn_pos_of_base_lt_one, injOn_subtype_val, BoxIntegral.Prepartition.injOn_setOf_mem_Icc_setOf_lower_eq, injOn_union, MonotoneOn.strictMonoOn_iff_injOn, Finset.card_add_iff, injOn_circleMap_of_abs_sub_le, Finset.pairwiseDisjoint_vadd_iff, OpenPartialHomeomorph.injOn, Finset.card_image_iff, injOn_pi_polarCoord_symm, SimpleGraph.Walk.IsCycle.getVert_injOn', Finset.card_infs_iff, pairwiseDisjoint_vadd_iff, AntitoneOn.strictAnti_iff_injOn, not_injOn_infinite_finite_image, Finset.card_image₂_iff, Finite.injOn_iff_bijOn_of_mapsTo, LinearMap.injOn_of_disjoint_ker, Finset.nodup_map_iff_injOn, Finite.exists_injOn_of_encard_le, AffineIndependent.injOn_affineCombination_fintypeAffineCoords, pairwiseDisjoint_image_right_iff, Cardinal.toENat_injOn, List.injOn_insertIdx_index_of_notMem, range_injOn_strictMono, LinearIndepOn_iff_linearIndepOn_image_injOn, Real.cosh_injOn, ncard_image_iff, Matroid.comap_isBase_iff, pairwiseDisjoint_smul_iff, injOn_of_injective, Int.injOn_natAbs_Ici, pairwiseDisjoint_image_left_iff, Function.insert_injOn, range_injOn_strictAnti, norm_injOn_ray_left, Real.artanh_injOn, Finset.card_diffs_iff, injOn_iff_pairwise_ne, Subsingleton.injOn, Real.injOn_cos, LieModule.injOn_genWeightSpace, Equiv.Perm.signBijAux_injOn, sup_sdiff_injOn, Nat.mod_injOn_Ico, injOn_iff_map_eq_one, Nat.factorization_inj, nsmul_injOn_Iio_addOrderOf, injOn_univ, LinearIndepOn.injOn, MapsTo.restrict_inj, isLocallyInjective_iff_nhds, injOn_iff_invFunOn_image_image_eq_self, injOn_of_ncard_image_eq, exists_injOn_iff_injective, LeftInvOn.injOn, injOn_empty, Real.log_injOn_pos, SimpleGraph.Walk.IsCycle.getVert_injOn, Finset.insert_inj_on', injOn_of_subsingleton, Module.End.injOn_iInf_maxGenEigenspace, exists_eq_graphOn_image_fst, Real.arccos_injOn, Finset.exists_subset_injOn_image_eq_of_surjOn, pairwiseDisjoint_singleton_iff_injOn, ApproximatesLinearOn.injOn
InvOn 📖	MathDef	15 math math: SurjOn.invOn_invFunOn, Module.invOn_reflection_of_mapsTo, Finset.insert_erase_invOn, Circle.invOn_arg_exp, Equiv.invOn, InvOn.one_sub_one_add_inv, invOn_id, Nat.setInvOn_digitsAppend_ofDigits, BijOn.invOn_invFunOn, OpenPartialHomeomorph.invOn, Nat.prod_primeFactors_invOn_squarefree, Function.Commute.invOn_fixedPoints_comp, insert_erase_invOn, PartialEquiv.invOn, Function.invOn_fixedPoints_comp
LeftInvOn 📖	MathDef	9 math math: leftInvOn_id, OpenPartialHomeomorph.leftInvOn, PartialEquiv.IsImage.leftInvOn_piecewise, Function.LeftInverse.leftInvOn, OpenPartialHomeomorph.IsImage.leftInvOn_piecewise, Function.leftInvOn_invFunOn_of_subset_image_image, SurjOn.leftInvOn_of_rightInvOn, InjOn.leftInvOn_invFunOn, PartialEquiv.leftInvOn
MapsTo 📖	MathDef	194 math math: mapsTo_iff_subset_preimage, StrictAnti.mapsTo_Ioi, Polynomial.rootSet_mapsTo, SurjOn.mapsTo_invFunOn, Function.mapsTo_fixedPoints_comp, Dilation.mapsTo_eball, eventually_mapsTo_of_isOpen_of_omegaLimit_subset, ContinuousMap.hasBasis_nhds, Dilation.mapsTo_closedEBall, Absorbs.eventually_nhds_zero, Antitone.mapsTo_uIcc, Real.mapsTo_cos, Real.mapsTo_cos_Ioo, mapsTo_id, LieAlgebra.mapsTo_toEnd_genWeightSpace_add_of_mem_rootSpace, AddAction.mapsTo_vadd_orbit, ContinuousMap.isOpen_setOf_mapsTo, OnePoint.continuousAt_infty, ContinuousLinearMap.nhds_zero_eq_of_basis, mapsTo', mapsTo_univ, Dilation.mapsTo_sphere, StrictMonoOn.mapsTo_Ioc, Convex.mapsTo_lineMap, mapsTo_gaugeRescale_closure, Module.End.mem_invtSubmodule_iff_mapsTo, Absorbs.eventually_nhdsNE_zero, ContinuousMap.mem_nhds_iff, RootPairing.mapsTo_reflection_root, Module.End.mapsTo_genEigenspace_of_comm, OpenPartialHomeomorph.symm_mapsTo, SurjOn.mapsTo_compl, mapsTo_iff_exists_map_subtype, mapsTo_of_subsingleton', StrictMonoOn.mapsTo_Ico, Isometry.mapsTo_closedEBall, ContinuousMap.continuous_compactOpen, Isometry.mapsTo_eball, Filter.map_mapsTo_Iic_iff_mapsTo, IsAddFreimanHom.mapsTo, mapsTo_extChartAt, OnePoint.tendsto_nhds_infty, StrictAnti.mapsTo_Ico, LipschitzWith.mapsTo_ball, eventually_cobounded_mapsTo, mapsTo_empty_iff, Filter.Tendsto.map_mapsTo_Iic, absorbs_iff_eventually_nhdsNE_zero, Isometry.mapsTo_perpBisector, Continuous.mapsTo_connectedComponent, mapsTo_iff_image_subset, Function.Semiconj.mapsTo_range, isMulFreimanHom_one_iff, Filter.map_mapsTo_Iic_iff_tendsto, ContinuousMap.compactOpen_eq, IsMulFreimanHom.mapsTo, Equiv.Perm.perm_mapsTo_inl_iff_mapsTo_inr, mapsTo_image, Isometry.mapsTo_emetric_closedBall, ContinuousMultilinearMap.hasBasis_nhds_zero, StrictAnti.mapsTo_Ioc, Dilation.mapsTo_closedBall, mapsTo_iInter, Submodule.mapsTo_smul_closure, Function.Semiconj.mapsTo_fixedPoints, Isometry.mapsTo_emetric_ball, PartialEquiv.IsImage.symm_mapsTo, Dilation.mapsTo_emetric_closedBall, exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds, LieSubmodule.mapsTo_pow_toEnd_sub_algebraMap, mapsTo_empty, ContinuousLinearMap.eventually_nhds_zero_mapsTo, PartialEquiv.IsImage.mapsTo, MonotoneOn.mapsTo_Ici, StrictMono.mapsTo_Ioc, AddAction.orbitRel.Quotient.mapsTo_vadd_orbit, mapsTo_univ_iff, mapsTo_sUnion, isAddFreimanHom_two, StrictMonoOn.mapsTo_Ioi, Equiv.Perm.perm_symm_mapsTo_iff_mapsTo, HasFDerivWithinAt.mapsTo_tangent_cone, absorbs_iff_eventually_cobounded_mapsTo, Nat.mapsTo_digitsAppend, AntitoneOn.mapsTo_Ici, StrictMono.mapsTo_Ico, mapsTo_preimage, Topology.CWComplex.mapsTo', mapsTo_iInter₂, StrictMonoOn.mapsTo_Ioo, ContinuousAlternatingMap.hasBasis_nhds_zero, isMulFreimanHom_zero_iff, MonotoneOn.mapsTo_uIcc, OpenPartialHomeomorph.IsImage.mapsTo, BoxIntegral.Box.mapsTo_insertNth_face, AntitoneOn.mapsTo_uIcc, BoxIntegral.Box.mapsTo_insertNth_face_Icc, mapsTo_sInter, isMulFreimanHom_two, Dilation.mapsTo_ball, CategoryTheory.Functor.eventualRange_mapsTo, PartialEquiv.symm_mapsTo, mapsTo_gaugeRescale_interior, OpenPartialHomeomorph.IsImage.symm_mapsTo, Polynomial.rootSet_maps_to', mapsTo_range_iff, Monotone.mapsTo_Icc, StrictMono.mapsTo_Iio, LipschitzWith.mapsTo_emetric_ball, Filter.Tendsto.basis_left, LeftInvOn.mapsTo, Antitone.mapsTo_Iic, EqOn.mapsTo_iff, mapsTo_image_iff, ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis, ContinuousAlternatingMap.hasBasis_nhds_zero_of_basis, StrictMono.mapsTo_Ioo, IsPreconnected.mapsTo_Ioi_or_Iio, Real.mapsTo_sin, mapsTo_union, LocallyCompactPair.exists_mem_nhds_isCompact_mapsTo, Topology.RelCWComplex.mapsTo, mapsTo_tangentConeAt_pi, mapsTo_range, StrictAntiOn.mapsTo_Ico, LipschitzWith.mapsTo_eball, mapsTo_of_subsingleton, mapsTo_fst_prod, ContinuousLinearMap.nhds_zero_eq, StrictMonoOn.mapsTo_Iio, Isometry.mapsTo_sphere, Real.mapsTo_sin_Ioo, Module.End.mapsTo_maxGenEigenspace_of_comm, AntitoneOn.mapsTo_Icc, BijOn.mapsTo, StrictAntiOn.mapsTo_Ioo, mapsTo_swap_prod, ContinuousMap.tendsto_nhds_compactOpen, Monotone.mapsTo_uIcc, isAddFreimanHom_zero_iff, mapsTo_iUnion₂, Filter.HasBasis.tendsto_left_iff, LipschitzWith.mapsTo_emetric_closedBall, Antitone.mapsTo_Ici, absorbs_iff_eventually_nhds_zero, Monotone.mapsTo_Iic, mapsTo_snd_prod, MulAction.orbitRel.Quotient.mapsTo_smul_orbit, mapsTo_iUnion, OpenPartialHomeomorph.mapsTo, mapsTo_inter, Dilation.mapsTo_emetric_ball, Filter.HasBasis.nhds_continuousMapConst, Function.Semiconj.mapsTo_ptsOfPeriod, StrictAntiOn.mapsTo_Ioi, StrictAnti.mapsTo_Ioo, Continuous.mapsTo_connectedComponentIn, ContinuousMap.isClosed_setOf_mapsTo, UniformConvergenceCLM.eventually_nhds_zero_mapsTo, Function.Semiconj.mapsTo_periodicPts, StrictMono.mapsTo_Ioi, Finset.intervalGapsWithin_mapsTo, ContinuousMap.nhds_compactOpen, LipschitzWith.mapsTo_closedBall, isAddFreimanHom_one_iff, Topology.CWComplex.mapsTo, AntitoneOn.mapsTo_Iic, PartialEquiv.mapsTo, Equiv.Perm.perm_inv_mapsTo_iff_mapsTo, UniformConvergenceCLM.nhds_zero_eq, IsClosed.setOf_mapsTo, RightInvOn.mapsTo, StrictAntiOn.mapsTo_Iio, MonotoneOn.mapsTo_Icc, mapsTo_univ_iff_range_subset, StrictAntiOn.mapsTo_Ioc, EuclideanGeometry.mapsTo_inversion_affineSubspace_of_mem, Isometry.mapsTo_ball, OpenPartialHomeomorph.mapsTo_extend, Filter.Tendsto.basis_both, Antitone.mapsTo_Icc, RootPairing.mapsTo_coreflection_coroot, mapsTo_prodMap_diagonal, Nat.mapsTo_ofDigits, LipschitzWith.mapsTo_closedEBall, Monotone.mapsTo_Ici, Isometry.mapsTo_closedBall, MulAction.mapsTo_smul_orbit, ContinuousMap.isClopen_setOf_mapsTo, StrictAnti.mapsTo_Iio, MonotoneOn.mapsTo_Iic, mapsTo_singleton, UniformConvergenceCLM.nhds_zero_eq_of_basis, image_eq_iff_surjOn_mapsTo
RightInvOn 📖	MathDef	13 math math: spectrumRestricts_iff, quasispectrumRestricts_iff, QuasispectrumRestricts.rightInvOn, Function.RightInverse.rightInvOn, InjOn.rightInvOn_of_leftInvOn, rightInvOn_id, SurjOn.rightInvOn_invFunOn, OpenPartialHomeomorph.rightInvOn, LeftInvOn.rightInvOn_image, SpectrumRestricts.rightInvOn, Function.LeftInverse.rightInvOn_range, ModelWithCorners.rightInvOn, PartialEquiv.rightInvOn
diagonal 📖	CompOp	24 math math: diagonal_nonempty, nhdsSet_diagonal_eq_uniformity, diagonal_eq_univ_iff, offDiag_univ, diagonal_eq_univ, prod_sdiff_diagonal, mem_diagonal_iff, isCompact_diagonal, nhdsSet_diagonal_le_uniformity, isClosed_diagonal, t2_iff_isClosed_diagonal, MeasurableEq.measurableSet_diagonal, preimage_coe_coe_diagonal, range_diag, t0Space_iff_ker_uniformity, nhdsSet_diagonal, disjoint_diagonal_offDiag, Filter.compl_diagonal_mem_prod, mem_diagonal, diagonal_subset_iff, diag_image, prod_subset_compl_diagonal_iff_disjoint, isLindelof_diagonal, mapsTo_prodMap_diagonal
graphOn 📖	CompOp	29 math math: smul_graphOn_univ, graphOn_nonempty, graphOn_insert, exists_eq_graphOn, graphOn_univ_injective, graphOn_prod_graphOn, vadd_graphOn_univ, fst_injOn_graph, ncard_graphOn, graphOn_union, exists_equiv_range_eq_graphOn_univ, vadd_graphOn, exists_eq_mgraphOn_univ, graphOn_inj, graphOn_comp, exists_range_eq_graphOn_univ, mem_graphOn, natCard_graphOn, graphOn_univ_inj, graphOn_singleton, Nonempty.graphOn, graphOn_univ_eq_range, graphOn_prod_prodMap, graphOn_empty, image_snd_graphOn, graphOn_eq_empty, image_fst_graphOn, smul_graphOn, exists_eq_graphOn_image_fst
image2 📖	CompOp	215 math math: image2_right_identity, image2_subset_iff_left, sInf_image2_eq_sInf_sSup, image2_inter_union_subset, csSup_image2_eq_csInf_csInf, isPiSystem_prod, BddAbove.image2_bddBelow, image_image2_distrib_left, image2_image_right, IsGreatest.isLeast_image2, AffineSubspace.setOf_sSameSide_eq_image2, biInter_image2, mem_upperBounds_image2_of_mem_lowerBounds, image2_subset, IsLeast.image2, DoubleCoset.doubleCoset_eq_image2, Filter.image2_mem_map₂, isLUB_image2_of_isGLB_isGLB, image2_right_comm, mem_lowerBounds_image2, isGLB_image2_of_isGLB_isLUB, Countable.image2, mem_lowerBounds_image2_of_mem_upperBounds, image2_def, image2_sInter_subset_left, IsLeast.isGreatest_image2, image2_sInter_subset_right, image2_sub, BddBelow.image2_bddAbove, image_image2_antidistrib_right, BddBelow.bddAbove_image2_of_bddAbove, TopologicalSpace.IsTopologicalBasis.prod, omegaLimit_def, image2_image_left_anticomm, image2_congr', IsGreatest.isGreatest_image2_of_isLeast, sSup_image2_eq_sInf_sInf, image2_lowerBounds_lowerBounds_subset_lowerBounds_image2, image2_inter_right, image2_congr, image2_left_comm, image_image2_antidistrib_left, image2_iUnion₂_right, image2_iInter_subset_left, sSup_image2, csSup_image2_eq_csSup_csSup, sInf_image2_eq_sInf_sInf, TopologicalSpace.IsTopologicalBasis.inf_induced, isGLB_image2_of_isLUB_isGLB, image_prod, image2_insert_right, mem_image2, BddBelow.bddBelow_image2_of_bddAbove, Finite.Set.finite_image2, image2_subset_iff_right, sUnion_image2, image2_range, image2_right, image2_inter_union_subset_union, Filter.mem_map₂_iff, image2_upperBounds_upperBounds_subset, image2_distrib_subset_left, toFinset_image2, ContinuousMap.compactOpen_eq, AffineSubspace.setOf_wOppSide_eq_image2, image_image2_right_anticomm, LipschitzOnWith.isBounded_image2, image2_left, IsCountablySpanning.prod, BddAbove.bddBelow_image2_of_bddAbove, infinite_image2, image2_distrib_subset_right, image2_vadd, Metric.ediam_eq_sSup, LipschitzOnWith.ediam_image2_le, image2_curry, generateFrom_prod, mem_upperBounds_image2_of_mem_upperBounds_of_mem_lowerBounds, image2_singleton_left, image2_iUnion_right, Cardinal.mk_image2_le, image2_left_identity, image2_div, image2_subset_left, Submodule.map₂_eq_span_image2, image2_lowerBounds_upperBounds_subset_lowerBounds_image2, Filter.le_map₂_iff, Infinite.image2_left, sInf_image2, BddAbove.bddAbove_image2_of_bddBelow, image2_assoc, Language.mul_def, BddAbove.image2, image2_image_left, image2_iInter₂_subset_left, sInter_image2, TopologicalSpace.IsTopologicalBasis.inf, mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_lowerBounds, mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_upperBounds, image2_empty_left, image_image2, Subsingleton.image2, PointedCone.tmul_subset_maxTensorProduct, sSup_image2_eq_sSup_sSup, Finite.image2, small_image2, image2_inter_subset_left, ContinuousLinearMap.isVonNBounded_iff, image2_eq_empty_iff, ContinuousLinearMap.isVonNBounded_image2_apply, iSup_image2, Filter.HasBasis.map₂, PointedCone.tmul_subset_minTensorProduct, image_subset_image2_right, image2_insert_left, omegaLimit_eq_biInter_inter, Nonempty.image2, IsLeast.isGreatest_image2_of_isGreatest, IsLeast.isLeast_image2_of_isGreatest, UniformConvergenceCLM.isVonNBounded_image2_apply, image2_upperBounds_lowerBounds_subset_lowerBounds_image2, image2_mul, image2_inter_subset_right, Submodule.map₂_span_span, csInf_image2_eq_csSup_csInf, image2_subset_iff, image2_inter_left, image2_smul, image2_eq_seq, subset_image2_diff_left, image2_comm, iUnion_image_right, sInf_image2_eq_sSup_sSup, image2_iInter_subset_right, generateFrom_prod_eq, exists_mem_image2, image2_mk_eq_prod, iUnion_image_left, image2_upperBounds_upperBounds_subset_upperBounds_image2, Finset.coe_diffs, image_image2_distrib, sInf_image2_eq_sSup_sInf, IsPiSystem.prod, image2_sUnion_left, image_subset_image2_left, generateFrom_eq_prod, image2_subset_right, image2_upperBounds_lowerBounds_subset_upperBounds_image2, seq_eq_image2, IsGreatest.image2, image2_sInter_right_subset, isGLB_image2_of_isGLB_isGLB, image2_iUnion_left, iInf_image2, image2_image_left_comm, image2_empty_right, eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset, Submodule.image2_subset_map₂, eventually_closure_subset_of_isOpen_of_omegaLimit_subset, BddBelow.image2, image_uncurry_prod, sSup_image2_eq_sSup_sInf, image2_lowerBounds_lowerBounds_subset, prod_generateFrom_generateFrom_eq, image2_vsub, image2_swap, isGLB_image2_of_isLUB_isLUB, subset_image2_diff_right, image2_image2_image2_comm, mem_image2_of_mem, image2_iInter₂_subset_right, image2_lowerBounds_upperBounds_subset_upperBounds_image2, EMetric.diam_eq_sSup, sSup_image2_eq_sInf_sSup, csInf_image2_eq_csSup_csSup, mem_upperBounds_image2_of_mem_upperBounds_of_mem_upperBounds, csSup_image2_eq_csSup_csInf, image_sup_prod, image2_union_left, csInf_image2_eq_csInf_csInf, image2_eq_iUnion, image_image2_antidistrib, AffineSubspace.setOf_sOppSide_eq_image2, csInf_image2_eq_csInf_csSup, isLUB_image2_of_isLUB_isGLB, mem_upperBounds_image2, biUnion_image2, image2_singleton_right, AffineSubspace.setOf_wSameSide_eq_image2, Infinite.image2_right, isLUB_image2_of_isGLB_isLUB, omegaLimit_subset_closure_fw_image, finite_image2, image2_union_inter_subset_union, IsGreatest.isLeast_image2_of_isLeast, image_image2_right_comm, csSup_image2_eq_csInf_csSup, image2_nonempty_iff, omegaLimit_eq_iInter, UniformConvergenceCLM.isVonNBounded_iff, image_image2_distrib_right, Finite.toFinset_image2, Finset.coe_image₂, omegaLimit_eq_iInter_inter, image2_add, image2_singleton, ContinuousMap.compactOpen_eq_generateFrom, image2_sInter_left_subset, image2_sUnion_right, image2_union_inter_subset, isLUB_image2_of_isLUB_isLUB, mem_image2_iff, image2_union_right, image_inf_prod, image2_iUnion₂_left
imageFactorization 📖	CompOp	5 math math: imageFactorization_injective_iff, surjective_onto_image, InjOn.imageFactorization_injective, imageFactorization_surjective, imageFactorization_eq
instCompl 📖	CompOp	964 math math: PredOrder.nhdsGT_eq_nhdsNE, absorbent_iff_eventually_nhdsNE_zero, mem_coLindelof', Metric.hasBasis_cobounded_compl_closedBall, tendsto_zero_iff_meromorphicOrderAt_pos, ModelWithCorners.compl_boundary, Countable.measure_restrict_compl, OnePoint.isOpen_def, MeasureTheory.mem_generateSetAlgebra_elim, indicator_self_add_compl, FirstOrder.Language.Theory.CompleteType.compl_setOf_mem, completelyRegularSpace_iff_isOpen, SetTheory.PGame.grundyValue_eq_sInf_moveLeft, MeasureTheory.setIntegral_compl₀, AlgebraicGeometry.Scheme.zeroLocus_iInf, ENNReal.iUnion_Iio_coe_nat, compl_iInter₂, compl_range_inr, tendsto_norm_sub_self_nhdsNE, isConnected_compl_of_one_lt_codim, Cardinal.compl_nonempty_of_mk_lt_mk, compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE, Ordinal.sInf_compl_lt_lift_ord_succ, Real.contDiffOn_log, Topology.IsClosedEmbedding.continuousOn_comap_finiteMeasure, even_ncard_compl_iff, eqOn_piecewise, Subalgebra.spectrum_sUnion_connectedComponentIn, ContDiff.dense_compl_range_of_finrank_lt_finrank, nhds_ne_subtype_neBot_iff, disjoint_compl_left_iff_subset, MeasureTheory.unifTight_iff_real, Convex.nhdsWithin_diff_eq_nhdsNE, Filter.compl_mem_cocardinal_of_card_lt, compl_subset_compl_of_subset, SSet.horn₃₁.desc.multicofork_π_two, Topology.IsUpper.isOpen_iff_generate_Iic_compl, Summable.sum_add_tsum_compl, frontier_compl, OpenPartialHomeomorph.eventually_ne_nhdsWithin, Topology.IsConstructible.compl, MeasureTheory.restrict_compl_singleton, punctured_nhds_smul, TopologicalSpace.Closeds.coe_compl, nhds_ne_subtype_eq_bot_iff, Metric.hasAntitoneBasis_cobounded_compl_ball, ProbabilityTheory.Kernel.borelMarkovFromReal_apply', SubMulAction.ofStabilizer_carrier, compl_union, isCompact_setOf_finiteMeasure_le_of_isCompact, MeasureTheory.probReal_compl_eq_one_sub₀, tendsto_cobounded_iff_meromorphicOrderAt_neg, Metric.mem_cocompact_iff_closedBall_compl_subset, mem_codiscreteWithin_iff_forall_mem_nhdsNE, Filter.inv_nhdsWithin_ne_zero, MeasureTheory.measure_compl_sigmaFiniteSet, HasFPowerSeriesAt.locally_ne_zero, Function.Periodic.tendsto_nhds_zero, compl_def, isClopen_compl_iff, tendsto_zpow_nhdsNE_zero_cobounded, BaireMeasurableSet.compl, preimage_boolIndicator, nhdsWithin_uIoo_right_le_nhdsNE, frontier_union_subset, isConnected_compl_singleton_of_one_lt_rank, iUnion_eq_compl_iInter_compl, mulIndicator_self_mul_compl_apply, Filter.mem_cofinite, PrimeSpectrum.isRetrocompact_zeroLocus_compl_of_fg, Function.Periodic.cuspFunction_zero_eq_limUnder_nhds_ne, compl_frontier_eq_union_interior, MeasureTheory.Measure.toSphere_apply_aux, eqOn_dslope_sub_smul, compl_iInter, Finite.multipliable_compl_iff, compl_sInter, Fin.range_succ, ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere, OnePoint.continuousAt_infty, MeasureTheory.IsSetAlgebra.compl_mem, Polynomial.isCoveringMapOn_eval, tangentConeAt_nonempty_of_properSpace, MeromorphicOn.toMeromorphicNFOn_eq_self_on_nhdsNE, TopologicalSpace.Opens.CompleteCopy.dist_eq, IsUpperSet.compl, ContinuousMap.notMem_idealOfSet, compl_image, MvPolynomial.image_comap_C_basicOpen, tendsto_norm_nhdsNE_one, instNeBotNhdsWithinComplSetSingletonOfNontrivial, toFinset_compl, subset_compl_singleton_iff, SimpleGraph.edgeFinset_top, Real.differentiableOn_log, compl_inv, Bornology.IsCobounded.closedBall_compl_subset, image_union_image_compl_eq_range, nhds_basis_closeds, MeasureTheory.measure_compl, MeasureTheory.measure_compl_sigmaFiniteSetWRT, Real.differentiableOn_arcsin, MeasureTheory.restrict_compl_sigmaFiniteSet, PeriodPair.eqOn_deriv_weierstrassPExcept_derivWeierstrassPExcept, PrimitiveSpectrum.isTopologicalBasis_relativeLower, MeasureTheory.SignedMeasure.exists_compl_positive_negative, Function.update_eventuallyEq_nhdsNE, Filter.compl_mem_kernMap, inter_eq_compl_compl_union_compl, OnePoint.isOpen_iff_of_mem, Order.Ideal.isPrime_iff, IsSemilinearSet.compl, Order.PFilter.IsPrime.compl_ideal, OnePoint.isOpen_iff_of_mem', ProbabilityTheory.Kernel.withDensity_rnDeriv_eq_zero_iff_measure_eq_zero, vadd_set_compl, MeasureTheory.average_mem_openSegment_compl_self, MeasureTheory.measurableSet_generateFrom_singleton_iff, SSet.horn₃₂.desc.multicofork_π_one, MeasureTheory.eLpNorm_top_piecewise, Real.punctured_nhds_module_neBot, AlgebraicIndependent.iff_adjoin_image_compl, MeasureTheory.measure_compl_le_add_of_le_add, Finset.isGδ_compl, IsClosed.isOpen_compl, MeasureTheory.prob_compl_eq_one_iff₀, Filter.disjoint_principal_left, Definable.compl, MeromorphicAt.eq_nhdsNE_toMeromorphicNFAt, subset_union_compl_iff_inter_subset, exists_nhds_ne_inf_principal_neBot, Iio_top, MeasureTheory.aedisjoint_compl_right, Summable.summable_compl_iff, MeasureTheory.Measure.restrict_add_restrict_compl, Filter.inv_nhdsNE_zero, stereographic'_source, Prop.compl_singleton, OnePoint.compl_infty, MeasureTheory.IsHahnDecomposition.compl, comap_norm_nhdsGT_zero, subset_closure_inter_union_closure_compl_left, HasDerivAt.eventually_ne, MulAction.smul_zpow_movedBy_eq_of_commute, SimpleGraph.neighborSet_subset_between_union, continuousWithinAt_compl_self, odd_ncard_compl_iff, MeasureTheory.NullMeasurableSet.compl_toMeasurable_compl_ae_eq, biUnion_associatedPrimes_eq_compl_regular, iInter_eq_compl_iUnion_compl, tendsto_norm_inv_nhdsNE_zero_atTop, dslope_eventuallyEq_slope_nhdsNE, PrimeSpectrum.isCompact_isOpen_iff, Absorbs.eventually_nhdsNE_zero, mulIndicator_mul_compl_eq_piecewise, compl_mem_codiscrete_iff, Filter.le_cofinite_iff_compl_singleton_mem, Topology.IsUpper.isTopologicalBasis_insert_univ_subbasis, SurjOn.mapsTo_compl, Disjoint.subset_compl_right, MeasureTheory.measure_univ_le_add_compl, Metric.ball_infDist_compl_subset, AnalyticAt.frequently_zero_iff_eventually_zero, AlgebraicGeometry.Scheme.zeroLocus_biInf, orbit_fixingAddSubgroup_compl_subset, Filter.diff_mem_inf_principal_compl, Real.continuousOn_log, AddAction.movedBy_mem_fixedBy_of_addCommute, PFun.core_eq, contDiffWithinAt_compl_self, MeasureTheory.setIntegral_compl, meromorphicOrderAt_ne_top_iff, IsLindelof.compl_mem_coLindelof, Ordinal.card_sInf_range_compl_le_lift, kernImage_preimage_eq_iff, contMDiffOn_inv₀, Finite.compl_mem_hyperfilter, meromorphicOrderAt_eq_top_iff, preimage_invFun_of_mem, compl_cIoo, FirstOrder.Field.ACF_zero_realize_iff_finite_ACF_prime_not_realize, nhdsNE_le_cofinite, Homeomorph.map_punctured_nhds_eq, OnePoint.nhdsNE_infty_neBot, disjointed_eq_inter_compl, subset_compl_comm, MeasureTheory.probReal_add_probReal_compl, Finset.insert_inj_on, mem_codiscrete', subset_compl_iff_disjoint_right, MeasureTheory.compl_cylinder, AlgebraicGeometry.Scheme.zeroLocus_singleton, MeasureTheory.Measure.sum_add_sum_compl, SimpleGraph.edgeSet_top, Filter.HasBasis.disjoint_iff_right, Affine.Simplex.altitudeFoot_mem_affineSpan_image_compl, ite_inter_closure_compl_eq_of_inter_frontier_eq, PeriodPair.compl_lattice_diff_singleton_mem_nhds, offDiag_univ, Filter.hasBasis_cofinite, MeasureTheory.Measure.mem_cofinite, ENNReal.range_coe, Module.punctured_nhds_neBot, OpenPartialHomeomorph.IsImage.compl, tendsto_cobounded_of_meromorphicOrderAt_neg, CofiniteTopology.mem_nhds_iff, SSet.horn₃₁.desc.multicofork_pt, MeasurableSet.compl_iff, Filter.mem_coclosedCompact_iff, Metric.ball_infDist_subset_compl, Filter.inv_cobounded₀, isPathConnected_compl_singleton_of_one_lt_rank, coborder_eq_compl_frontier_iff, IsCompact.compl_mem_cocompact, hasDerivAtFilter_iff_tendsto_slope, starConvex_compl_Ici, MeasureTheory.NullMeasurableSet.compl, extChartAt_target_union_compl_range_mem_nhds_of_mem, PeriodPair.analyticOnNhd_derivWeierstrassPExcept, Ordinal.card_sInf_range_compl_le, Complex.differentiableOn_update_limUnder_of_bddAbove, OnePoint.tendsto_nhds_infty, MeasurableSpace.DynkinSystem.generateHas_compl, HasSubset.Subset.disjoint_compl_right, Filter.compl_mem_coprodᵢ, range_stereographic_symm, ENNReal.iUnion_Iic_coe_nat, PreErgodic.measure_self_or_compl_eq_zero, Nimber.add_def, mem_closure_isSwap, stabilizer_compl, Filter.map_add_left_nhdsNE, Filter.mem_cocardinal, Real.differentiableOn_rpow_const, SimpleGraph.neighborSet_union_compl_neighborSet_eq, Topology.IsLower.isOpen_iff_generate_Ici_compl, Meromorphic.MeromorphicOn.countable_compl_analyticAt, EReal.tendsto_toReal_atTop, range_diff_image, nonempty_compl_of_nontrivial, compl_singleton_mem_nhds_iff, EReal.tendsto_toReal_atBot, tendsto_norm_div_self_nhdsNE, Filter.mem_comk, image_compl_eq, hasBasis_coLindelof, SubMulAction.ofFixingSubgroup_carrier, OnePoint.compl_range_coe, Convex.radon_partition, mem_codiscrete, MeasureTheory.prob_compl_eq_one_sub₀, Function.Periodic.boundedAtFilter_cuspFunction, Filter.inf_principal_eq_bot, lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl, MeasureTheory.prob_compl_eq_one_iff, Dense.interior_compl, Topology.IsUpper.isTopologicalSpace_basis, PrimeSpectrum.mem_compl_zeroLocus_iff_notMem, exists_isClopen_of_totally_separated, Polynomial.exists_image_comap_of_monic, SimpleGraph.neighborFinset_subset_between_union, multipliable_subtype_and_compl, indicator_compl_add_self_apply, absorbs_iff_eventually_nhdsNE_zero, Function.Embedding.optionEmbeddingEquiv_apply_snd_coe, Cardinal.mk_compl_finset_of_infinite, indicator_compl', tendsto_riemannZeta_sub_one_div, SimpleGraph.neighborSet_compl, ZetaAsymptotics.tendsto_Gamma_term_aux, ncard_compl, mem_fixingAddSubgroup_compl_iff_movedBy_subset, nhdsWithin_compl_singleton_le, ContDiffOn.dense_compl_image_of_dimH_lt_finrank, MeromorphicAt.tendsto_nhds_meromorphicTrailingCoeffAt, finite_compl_fixedBy_swap, union_inter_compl_right_subset, Metric.subset_compl_thickening_compl_thickening_self, isClosed_and_discrete_iff, diff_eq_compl_inter, MeasureTheory.norm_integral_sub_setIntegral_le, IsSelfAdjoint.isConnected_spectrum_compl, MeasureTheory.setLIntegral_compl, AlgebraicGeometry.ExistsHomHomCompEqCompAux.exists_index, MeasureTheory.measure_compl_le_add_iff, isCoatom_iff, HasLineDerivAt.tendsto_slope_zero, AlgebraicGeometry.Scheme.zeroLocus_def, Filter.map_add_right_nhdsNE, MeasureTheory.prob_compl_lt_one_sub_of_lt_prob, Metric.closedBall_compl_subset_of_mem_cocompact, Filter.inv_nhdsNE, Topology.reorderRestrictProd_restrict_compl, compl_inter, ContDiffMapSupportedIn.zero_on_compl', kernImage_compl, Order.Ideal.IsPrime.compl_filter, MeasureTheory.aedisjoint_compl_left, SSet.horn₃₁.desc.multicofork_π_two_assoc, compl_neg, Complex.comap_exp_nhdsNE, mem_closure_ne_iff_frequently_within, Ultrafilter.mem_or_compl_mem, SimpleGraph.neighborFinset_subset_between_union_compl, SSet.horn₃₁.desc.multicofork_π_zero_assoc, Bool.compl_singleton, AlgebraicGeometry.Polynomial.imageOfDf_eq_comap_C_compl_zeroLocus, MeasurableSpace.DynkinSystem.has_compl, SetTheory.PGame.grundyValue_eq_sInf_moveRight, SimpleGraph.fromEdgeSet_not_isDiag, Order.Ideal.PrimePair.compl_I_eq_F, ContinuousMap.ideal_isMaximal_iff, Complex.isOpen_compl_range_intCast, ContDiffMapSupportedIn.zero_on_compl, HurwitzZeta.completedHurwitzZetaEven_residue_one, IsClopen.compl, codiscreteWithin_iff_locallyEmptyComplementWithin, compl_ordConnectedSection_ordSeparatingSet_mem_nhds, Affine.Simplex.vectorSpan_isOrtho_altitude_direction, MeasureTheory.compl_mem_measurableCylinders, Topology.isConstructible_compl, PeriodPair.differentiableOn_derivWeierstrassP, tendsto_nhds_iff_meromorphicOrderAt_nonneg, Meromorphic.countable_compl_analyticAt, subset_image_compl, Filter.hasBasis_coclosedCompact, Function.locallyFinsuppWithin.restrict_eqOn_compl, Countable.isPathConnected_compl_of_one_lt_rank, isOpen_compl_singleton, MeasureTheory.limsup_measure_compl_le_of_le_liminf_measure, Topology.reorderRestrictProd_mem_preimage_image_restrict, comap_norm_nhdsGT_zero', preimage_boolIndicator_eq_union, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, Complex.tendsto_norm_tan_atTop, SimpleGraph.Connected.exists_preconnected_induce_compl_singleton_of_finite, diff_eq, Bornology.isBounded_def, Equiv.Perm.IsSwap.finite_compl_fixedBy, Complex.tendsto_self_mul_Gamma_nhds_zero, isClosed_isNowhereDense_iff_compl, TopologicalSpace.CompactOpens.coe_compl, ite_compl, compl_subset_comm, Real.differentiableOn_mul_log, MeasureTheory.JordanDecomposition.exists_compl_positive_negative, Concept.compl_intent, NormedField.nhdsNE_neBot, compl_image_set_of, Complex.map_exp_comap_re_atBot, compl_mul_closure_one_eq, HasDerivAt.tendsto_nhdsNE, piecewise_ae_eq_restrict_compl, ENNReal.essSup_piecewise, ProjectiveSpectrum.isOpen_iff, Bornology.isCobounded_ofBounded_iff, tendsto_abs_nhdsNE_zero, mem_interior_iff_not_clusterPt_compl, IsLowerSet.compl, Equiv.image_compl, compl_cIcc, deriv.lhopital_zero_nhds, MeromorphicAt.iff_eventuallyEq_zpow_smul_analyticAt, Filter.mem_comap_iff_compl, Real.continuousOn_logb, Equiv.Set.sumCompl_symm_apply_of_mem, Function.update_eventuallyEq, meromorphicOrderAt_eq_int_iff, FirstOrder.Language.DefinableSet.coe_compl, Algebra.unramifiedLocus_eq_compl_support, Equiv.Set.sumCompl_symm_apply, MeasureTheory.measureReal_add_measureReal_compl₀, MeromorphicAt.eventuallyEq_zero_nhdsNE_of_eventuallyEq_zero_codiscreteWithin, eqOn_mulIndicator', tendsto_zero_of_meromorphicOrderAt_pos, Metric.hasBasis_cobounded_compl_ball, Topology.reorderRestrictProd_of_mem, MeasureTheory.measure_compl_sigmaFiniteSet_eq_zero_iff_sigmaFinite, kernImage_empty, MeasureTheory.restrict_compl_sigmaFiniteSet_eq_zero_or_top, UpperSet.coe_compl, coborder_eq_union_frontier_compl, MeasureTheory.eLpNormEssSup_piecewise, mem_codiscrete_accPt, continuousOn_inv₀, AnalyticOnNhd.preimage_zero_mem_codiscrete, accPt_iff_frequently_nhdsNE, eqOn_dslope_slope, Homeomorph.image_compl, MeasureTheory.integral_add_compl, union_compl_self, AlgebraicGeometry.Scheme.zeroLocus_map, EReal.range_coe, MeasureTheory.lintegral_add_compl, schnirelmannDensity_eq_one_iff, PartialEquiv.IsImage.compl, indicator_const_preimage, Metric.hasAntitoneBasis_cobounded_compl_closedBall, DirSupClosed.compl, mem_coLindelof, compl_range_some, Ne.nhdsWithin_compl_singleton, PrimeSpectrum.isOpen_iff, mulIndicator_const_preimage, coborder_eq_union_closure_compl, compl_ne_eq_singleton, SimpleGraph.Connected.induce_compl_singleton_of_degree_eq_one, compl_Ioi, MeasureTheory.measure_add_measure_compl, Cardinal.mk_sum_compl, image_subset_closure_compl_image_compl_of_isOpen, Absorbent.eventually_nhdsNE_zero, ncard_add_ncard_compl, MeromorphicAt.eventually_continuousAt, HurwitzZeta.hurwitzZeta_residue_one, LinearIndependent.notMem_span, compl_iUnion₂, Sym2.diagSet_compl_eq_fromRel_ne, MeasureTheory.mem_map_restrict_ae_iff, EReal.continuousOn_toReal, IsGδ.compl_singleton, union_inter_compl_left_subset, piecewise_compl, Ioi_bot, indicator_self_add_compl_apply, compl_singleton_mem_nhds, Finset.coe_compl, MeasureTheory.Measure.inf_apply, stableUnderGeneralization_compl_iff, Filter.principal_coprod_principal, exists_nhds_ne_neBot, homeomorphUnitSphereProd_symm_apply_coe, MeasurableSpace.DynkinSystem.has_compl_iff, compl_univ, nonunits.subset_compl_ball, tendsto_ne_zero_of_meromorphicOrderAt_eq_zero, range_extend_subset, nonempty_inter_compl_of_ncard_add_ncard_lt, ModelWithCorners.compl_interior, mem_fixingSubgroup_compl_iff_movedBy_subset, compl_prod_eq_union, MeasureTheory.prob_compl_eq_zero_iff, MeasureTheory.tendstoUniformlyOn_of_ae_tendsto', compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE, MeasurableSet.compl, Topology.continuous_reorderRestrictProd, MeasureTheory.Measure.compl_support_eq_sUnion, Countable.isConnected_compl_of_one_lt_rank, nhdsGT_le_nhdsNE, PolishSpace.IsClopenable.compl, HurwitzZeta.completedCosZeta_residue_zero, tendsto_ne_zero_iff_meromorphicOrderAt_eq_zero, spectrum.subset_singleton_zero_compl, Affine.Simplex.instNonemptyElemComplSetSingletonOfNontrivial, HasFDerivAt.eventually_notMem, MeasureTheory.measure_compl₀, disjoint_compl_right_iff_subset, Filter.tendsto_inv₀_nhdsWithin_ne_zero, HasFDerivWithinAt.tendsto_nhdsWithin_nhdsNE, PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen, Countable.dense_compl, Cardinal.mk_compl_eq_mk_compl_infinite, hasDerivAt_iff_tendsto_slope_zero, IsCompact.compl_mem_coclosedCompact_of_isClosed, compl_Iic, LocallyFinite.iInter_compl_mem_nhds, IsClosed.compl_mem_nhds, MeasureTheory.Measure.measure_compl_support, Nat.smoothNumbers_compl, compl_iUnion, MeromorphicAt.eventually_analyticAt, compl_subset_compl, Subalgebra.spectrum_isBounded_connectedComponentIn, isCoatom_singleton_compl, SimpleGraph.degree_le_between_add, MeasureTheory.IsHahnDecomposition.ge_on_compl, IsOpen.measure_eq_biSup_integral_continuous, Topology.image_snd_preimageImageRestrict, dense_compl_singleton_iff_not_open, MvPolynomial.aeval_toPolynomialAdjoinImageCompl_eq_zero, hasBasis_coclosedLindelof, aeSeq.measure_compl_aeSeqSet_eq_zero, Real.tendsto_logb_nhdsNE_zero_of_base_lt_one, compl_singleton_mem_nhdsSet_iff, Filter.mem_kernMap_iff_compl, dirSupInacc_compl, Filter.mem_ofCardinalUnion, Real.differentiableOn_negMulLog, Order.PFilter.isPrime_iff, IsOpen.isClosed_compl, homeomorphSphereProd_symm_apply_coe, MeasureTheory.OuterMeasure.isCaratheodory_compl, compl_star, smul_set_compl, compl_beattySeq', DFA.compl_def, ProbabilityTheory.cond_add_cond_compl_eq, AffineIndependent.exists_affineCombination_eq_smul_eq_of_fintype, subset_compl_iff_disjoint_left, image_compl_subset, Metric.thickening_compl_thickening_self_subset_compl, range_diff_image_subset, NormedField.tendsto_norm_inv_nhdsNE_zero_atTop, MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd, Sym2.card_diagSet_compl, Summable.tsum_subtype_add_tsum_subtype_compl, Filter.Tendsto.countable_compl_preimage_ker, restrict_ite_compl, AddAction.vadd_zsmul_movedBy_eq_of_addCommute, ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space, compl_empty_iff, PrimitiveSpectrum.preimage_upperClosure_compl_finset, isCoveringMapOn_npow, homeomorphUnitSphereProd_apply_fst_coe, MeasureTheory.unifTight_iff_ennreal, Filter.neg_nhdsNE, mem_compl_singleton_iff, restrict_dite_compl, Complex.isCoveringMapOn_exp, Nimber.not_bddAbove_compl_of_small, HasSubset.Subset.disjoint_compl_left, contDiffOn_inv, MeasureTheory.integral_add_compl₀, eventually_nhdsNE_eventually_nhds_iff, Real.tendsto_abs_tan_of_cos_eq_zero, MeasureTheory.prob_compl_le_one_sub_of_le_prob, PeriodPair.isOpen_compl_lattice_diff, MeromorphicOn.countable_compl_analyticAt_inter, indicator_one_preimage, Sym2.fromRel_subset_compl_diagSet, Bornology.isBounded_compl_iff, mem_coclosedLindelof, MeasurableSet.coe_compl, univ_pi_piecewise, Filter.hasBasis_cocardinal, exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow_of_adjoin_eq_top, compl_range_inl, Filter.mem_cocountable, Iio_union_Ioi, Tactic.ComputeAsymptotics.tendsto_nhdsNE_of_tendsto_atTop, ContDiffMapSupportedIn.iteratedFDeriv_zero_on_compl, continuousAt_iff_punctured_nhds, SSet.horn_eq_iSup, HasDerivAt.lhopital_zero_nhds, MeromorphicAt.frequently_eq_iff_eventuallyEq, ProbabilityTheory.uniformOn_add_compl_eq, ContinuousMap.setOfIdeal_eq_compl_singleton, Rat.dense_compl_compact, MeasurableSpace.measurableSet_compl, Bornology.IsBounded.compl, Function.compl_mulSupport, Ultrafilter.compl_notMem_iff, HasDerivAt.tendsto_slope, MeasureTheory.Measure.compl_mem_cofinite, Equiv.Perm.isPretransitive_of_isCycle_mem, MeromorphicAt.eventually_eq_zero_or_eventually_ne_zero, mulIndicator_compl', compl_subset_iff_union, indicator_ae_eq_restrict_compl, restrict_extend_compl_range, Complex.tendsto_norm_tan_of_cos_eq_zero, TopologicalSpace.Clopens.coe_compl, Filter.comap_eq_bot_iff_compl_range, injective_piecewise_iff, Function.Periodic.invQParam_tendsto, isLowerSet_compl, HasProd.hasProd_iff_compl, mem_codiscreteWithin, Real.differentiableOn_arccos, MeromorphicAt.eqOn_compl_singleton_toMeromorphicNFAt, isDiscrete_iff_nhdsNE, BijOn.compl, SSet.horn₃₂.desc.multicofork_π_zero_assoc, piecewise_eqOn_compl, nhdsLT_le_nhdsNE, indicator_add_compl_eq_piecewise, Metric.AreSeparated.subset_compl_right, ConjClasses.mk_bijOn, toMeromorphicNFOn_eq_self_on_compl, Sym2.diagSet_compl_eq_setOf_not_isDiag, MeasureTheory.tendstoUniformlyOn_of_ae_tendsto_of_measurable_edist', notMem_compl_iff, MeasureTheory.isTightMeasureSet_iff_exists_isCompact_measure_compl_le, Polynomial.image_comap_C_basicOpen, OnePoint.compl_image_coe, inter_subset, Disjoint.subset_compl_left, indicator_const_preimage_eq_union, MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl, ProbabilityTheory.condIndep_biSup_compl, OnePoint.nhdsNE_infty_eq, Bornology.IsCobounded.compl, Real.contDiffOn_arccos, Filter.mem_ofCountableUnion, MeromorphicOn.eventually_analyticAt_or_mem_compl, PrimitiveSpectrum.isOpen_iff, stereographic_source, PrimeSpectrum.isRetrocompact_zeroLocus_compl, Filter.compl_mem_coclosedCompact, Finite.isGδ_compl, sUnion_eq_compl_sInter_compl, nhdsWithin_uIoo_left_le_nhdsNE, indicator_compl, frontier_eq_inter_compl_interior, hasLineDerivAt_iff_tendsto_slope_zero, HasSum.hasSum_iff_compl, univ_pi_update, ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl, MeasureTheory.Measure.subset_compl_support_of_isOpen, SubAddAction.compl_def, isOpen_setOf_eventually_nhdsWithin, IsClosed.nhdsSet_le_sup, compl_sUnion, MeromorphicNFAt.eventuallyEq_nhdsNE_iff_eventuallyEq_nhds, IsClosed.nhdsSet_le_sup', IsLindeof.compl_mem_coclosedLindelof_of_isClosed, finite_compl_fixedBy_closure_iff, Subsingleton.isGδ_compl, StableUnderSpecialization.compl, ProbabilityTheory.Kernel.withDensity_rnDeriv_eq_zero_iff_apply_eq_zero, ENNReal.sum_add_tsum_compl, stableUnderSpecialization_compl_iff, Function.Periodic.qParam_tendsto, Finite.compl_mem_cofinite, PeriodPair.analyticOnNhd_derivWeierstrassP, Fin.range_succAbove, Complex.Gammaℝ_residue_zero, image_val_compl, Ordinal.lift_card_sInf_compl_le, Filter.hnot_def, SSet.horn.multicoequalizerDiagram, Multipliable.prod_mul_tprod_compl, OnePoint.hasBasis_nhds_infty, punctured_nhds_eq_nhdsWithin_sup_nhdsWithin, summable_subtype_and_compl, IsOpen.coborder_eq, Filter.HasBasis.disjoint_iff_left, MeasureTheory.measureReal_compl₀, eqOn_indicator', Function.Periodic.eventually_differentiableAt_cuspFunction_nhds_ne_zero, isUpperSet_compl, HasDerivAt.eventually_notMem, Function.Embedding.optionEmbeddingEquiv_symm_apply, Algebra.SubmersivePresentation.basisKaehler_apply, HasDerivWithinAt.tendsto_nhdsWithin_nhdsNE, mem_closure_isSwap', SSet.horn₃₁.desc.multicofork_π_three_assoc, discreteTopology_subtype_iff, inter_union_compl, MeasureTheory.NullMeasurableSet.compl_iff, TopologicalSpace.Opens.coe_compl, regularSpace_generateFrom, closure_compl, Filter.notMem_iff_inf_principal_compl, SSet.horn₃₁.desc.multicofork_π_zero, AnalyticAt.eventually_eq_or_eventually_ne, measurable_compl, ite_inter_compl_self, NormedField.tendsto_norm_zpow_nhdsNE_zero_atTop, MeasureTheory.laverage_mem_openSegment_compl_self, SubMulAction.compl_def, AnalyticAt.eventually_eq_zero_or_eventually_ne_zero, StableUnderGeneralization.compl, Ordinal.nonempty_compl_range, eventuallyConst_nhdsNE_iff_meromorphicOrderAt_sub_eq_top, Nimber.mul_def, MvPolynomial.irreducible_toPolynomialAdjoinImageCompl, SimpleGraph.isIndepSet_compl_iff_isVertexCover, BoundedContinuousFunction.dist_extend_extend, Filter.countable_compl_ker, biUnion_associatedPrimes_eq_compl_nonZeroDivisors, isOpen_singleton_iff_punctured_nhds, ProbabilityTheory.Kernel.borelMarkovFromReal_apply, frontier_eq_closure_inter_closure, dense_compl_singleton, subset_inter_union_compl_left, Topology.IsScott.isOpen_iff_Iic_compl_or_univ, Finite.infinite_compl, MeasureTheory.IsTightMeasureSet_iff_exists_isCompact_measure_compl_le, inter_compl_self, compl_empty, Metric.isCobounded_iff_closedBall_compl_subset, compl_eq_univ_diff, MeasureTheory.Measure.isOpen_compl_support, SimpleGraph.edgeSet_subset_setOf_not_isDiag, Filter.map_mul_right_nhdsNE, MeasureTheory.UnifTight.exists_measurableSet_indicator, SSet.horn₃₂.desc.multicofork_π_three, MeasureTheory.Measure.MutuallySingular.measure_compl_nullSet, LowerSet.coe_compl, MeasureTheory.measureReal_compl, punctured_nhds_smul₀, closure_sUnion_irreducibleComponents_diff_singleton, biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ, MeasureTheory.Measure.support_eq_sInter, Bornology.isCobounded_compl_iff, interior_union_inter_interior_compl_left_subset, pi_inter_compl, Filter.hasBasis_cocompact, compl_Iio, Function.Injective.compl_image_eq, compl_range_subset_kernImage, SubAddAction.ofFixingAddSubgroup_carrier, PrimeSpectrum.basicOpen_eq_zeroLocus_compl, diff_mem_nhdsWithin_compl, ProbabilityTheory.indep_biSup_compl, subset_closure_inter_union_closure_compl_right, compl_beattySeq, compl_add_closure_zero_eq_iff, Finite.summable_compl_iff, dirSupClosed_compl, Filter.compl_mem_coprod, compl_singleton_eq, Equiv.Set.sumCompl_apply_inr, Complex.compl_Iic_zero, MeasureTheory.exists_setLIntegral_compl_lt, MeasureTheory.le_measure_compl_liminf_of_limsup_measure_le, range_piecewise, PrimeSpectrum.range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk, ProbabilityTheory.Kernel.singularPart_compl_mutuallySingularSetSlice, Filter.compl_mem_hyperfilter_of_finite, HasSum.hasSum_compl_iff, Tactic.ComputeAsymptotics.tendsto_nhdsNE_of_tendsto_atTop_nhds_of_eq, Filter.compl_mem_comap, Ordinal.sInf_compl_lt_ord_succ, OnePoint.nhdsNE_coe_neBot, Filter.EventuallyEq.compl, HasDerivAt.tendsto_slope_zero, MeasureTheory.mem_ae_iff, homeomorphSphereProd_apply_snd_coe, ProjectiveSpectrum.mem_compl_zeroLocus_iff_notMem, interior_union_inter_interior_compl_right_subset, NNReal.continuousOn_rpow_const_compl_zero, PFun.compl_dom_subset_core, interior_eq_compl_closure_compl, MeasureTheory.Measure.nullMeasurableSet_compl_support, isPathConnected_compl_of_one_lt_codim, preimage_boolIndicator_false, compl_union_self, PeriodPair.differentiableOn_weierstrassPExcept, Ultrafilter.compl_mem_iff_notMem, compl_Ici, insert_image_compl_eq_range, PeriodPair.differentiableOn_weierstrassP, mulIndicator_compl_mul_self_apply, closure_eq_compl_interior_compl, Real.tendsto_abs_tan_atTop, tendsto_norm_nhdsNE_zero, perfectSpace_iff_forall_not_isolated, MeasureTheory.Measure.MutuallySingular.restrict_compl_nullSet, SimpleGraph.degree_le_between_add_compl, SSet.horn₃₂.desc.multicofork_pt, kernImage_eq_compl, MeasureTheory.tendsto_measure_compl_closedBall_of_isTightMeasureSet, exists_convex_convex_compl_subset, exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow, interior_compl, MeasureTheory.measure_add_measure_compl₀, ProbabilityTheory.uniformOn_compl, Filter.EventuallyLE.compl, homeomorphUnitSphereProd_apply_snd_coe, DirichletCharacter.LFunctionTrivChar_residue_one, MeasureTheory.restrict_compl_sigmaFiniteSetWRT, AlgebraicIndependent.iff_adjoin_image, IsCompact.exists_mem_frontier_infDist_compl_eq_dist, pure_sup_nhdsNE, Multipliable.tprod_subtype_mul_tprod_subtype_compl, MeasureTheory.VectorMeasure.restrict_add_restrict_compl, Metric.closedBall_infDist_compl_subset_closure, Subtype.preimage_coe_compl', Equiv.Set.sumCompl_apply_inl, completedRiemannZeta_residue_one, Sym2.irreflexive_iff_fromRel_subset_diagSet_compl, indicator_compl_add_self, Multipliable.multipliable_compl_iff, homeomorphSphereProd_apply_fst_coe, MeromorphicAt.eventuallyEq_nhdsNE_of_eventuallyEq_codiscreteWithin, compl_inter_self, Complex.range_exp, MeasureTheory.prob_compl_eq_one_sub, Filter.compl_diagonal_mem_prod, Topology.IsLower.isTopologicalBasis_insert_univ_subbasis, starConvex_compl_Iic, MeasureTheory.Measure.exists_null_set_measure_lt_of_disjoint, HallMarriageTheorem.hall_cond_of_compl, MapsTo.surjOn_compl, Cardinal.mk_compl_eq_mk_compl_finite_lift, PrimeSpectrum.exists_image_comap_of_finite_of_free, Function.compl_support, Filter.comap_neBot_iff_compl_range, encard_add_encard_compl, mulIndicator_const_preimage_eq_union, preimage_subtype_coe_eq_compl, insert_mem_nhds_iff, HasFDerivAt.tendsto_nhdsNE, SubAddAction.ofStabilizer_carrier, preimage_compl, upperHemicontinuous_iff_isClosed_compl_preimage_Iic_compl, union_eq_compl_compl_inter_compl, ContinuousMap.mem_idealOfSet_compl_singleton, ZMod.LFunction_residue_one, MeasurableSpace.compl_mem_generateMeasurableRec, Filter.map_mul_left_nhdsNE, MeasureTheory.ae_eq_univ, closure_compl_singleton, subset_inter_union_compl_right, Filter.compl_mem_comk, riemannZeta_residue_one, mulIndicator_compl, dense_compl_of_dimH_lt_finrank, Topology.IsLower.isTopologicalSpace_basis, Filter.le_iff_forall_inf_principal_compl, compl_mul_closure_one_eq_iff, Function.insert_injOn, PeriodPair.analyticOnNhd_weierstrassPExcept, totallySeparatedSpace_iff_exists_isClopen, isCoveringMapOn_zpow, HurwitzZeta.completedHurwitzZetaEven_residue_zero, continuousOn_zpow₀, disjoint_nhdsWithin_of_mem_discrete, Ordinal.not_bddAbove_compl_of_small, AnalyticOnNhd.preimage_zero_mem_codiscreteWithin, discreteTopology_iff_nhds_ne, ContinuousMap.idealOf_compl_singleton_isMaximal, SimpleGraph.Connected.exists_connected_induce_compl_singleton_of_finite_nontrivial, Filter.tendsto_inv₀_nhdsNE_zero, ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE, DirSupInacc.compl, interior_eq_empty_iff_dense_compl, MeasureTheory.prob_add_prob_compl, diff_subset_compl, Fintype.card_compl_set, Finite.compl_mem_cocardinal, MeromorphicOn.countable_compl_analyticAt, SimpleGraph.IsIndepSet.nonempty_mem_compl_mem_edge, Real.tendsto_log_nhdsNE_zero, Algebra.smoothLocus_eq_compl_support_inter, OnePoint.isOpen_compl_image_coe, AmpleSet.of_one_lt_codim, OnePoint.le_nhds_infty, hasDerivAt_iff_tendsto_slope, Cardinal.mk_compl_eq_mk_compl_finite, Function.support_eq_preimage, AnalyticAt.map_nhdsNE, Countable.isGδ_compl, PFun.core_res, FirstOrder.Language.Theory.CompleteType.typesWith_not, nhdsLT_sup_nhdsGT, AlgebraicGeometry.Scheme.zeroLocus_univ, AnalyticAt.frequently_eq_iff_eventually_eq, SimpleGraph.edgeSet_subset_compl_diagSet, MeasureTheory.isTightMeasureSet_iff_tendsto_measure_compl_closedBall, Filter.mem_inf_principal', EReal.nhdsWithin_top, SimpleGraph.isVertexCover_compl, Function.mulSupport_eq_preimage, accPt_iff_clusterPt, Cardinal.mk_compl_of_infinite, SSet.horn₃₂.desc.multicofork_π_zero, ProbabilityTheory.Kernel.indep_biSup_compl, MvPolynomial.isAlgebraic_of_mem_vars_of_forall_totalDegree_le, SSet.horn₃₂.desc.multicofork_π_three_assoc, Filter.mem_iff_inf_principal_compl, compl_Ioc, compl_diff, SSet.horn₃₂.desc.multicofork_π_one_assoc, punctured_nhds_vadd, mem_coclosed_Lindelof', PeriodPair.analyticOnNhd_weierstrassP, Submodule.connectedComponentIn_eq_self_of_one_lt_codim, SuccOrder.nhdsLT_eq_nhdsNE, inter_compl_nonempty_iff, Topology.reorderRestrictProd_of_compl, MeasureTheory.lintegral_piecewise, sInter_eq_compl_sUnion_compl, MeasureTheory.OuterMeasure.isCaratheodory_compl_iff, Affine.Simplex.affineSpan_pair_eq_altitude_iff, tendsto_nhds_of_meromorphicOrderAt_nonneg, Affine.Simplex.direction_altitude, nonempty_compl, prod_subset_compl_diagonal_iff_disjoint, Filter.mem_cocompact', WeakFEPair.Λ_residue_zero, SimpleGraph.neighborSet_completeEquipartiteGraph, Subtype.preimage_coe_compl, AlgebraicIndependent.iff_transcendental_adjoin_image, Real.contDiffOn_arcsin, Equiv.Set.sumCompl_symm_apply_compl, Concept.compl_extent, PrimeSpectrum.isCompact_isOpen_iff_ideal, MeasureTheory.Measure.restrict_compl_add_restrict, MeasureTheory.measureReal_add_measureReal_compl, MeromorphicAt.frequently_zero_iff_eventuallyEq_zero, LocallyConstant.ofIsClopen_fiber_one, Affine.Simplex.range_faceOpposite_points, CofiniteTopology.isOpen_iff, CofiniteTopology.isOpen_iff', MeasureTheory.prob_compl_eq_zero_iff₀, image_compl_eq_range_diff_image, isOpen_compl_iff, SSet.horn₃₁.desc.multicofork_π_three, meromorphicOrderAt_ne_top_iff_eventually_ne_zero, EReal.nhdsWithin_bot, Cardinal.mk_compl_eq_mk_compl_finite_same, WeakFEPair.Λ_residue_k, inl_compl_union_inr_compl, Finite.cofinite_inf_principal_compl, Order.Ideal.PrimePair.compl_F_eq_I, compl_add_closure_zero_eq, continuousAt_update_same, Finite.compl_mem_codiscrete, Real.tendsto_logb_nhdsNE_zero, MeasureTheory.exists_isCompact_closure_measure_compl_lt, AlgebraicGeometry.RingedSpace.zeroLocus_singleton, compl_univ_iff, Topology.restrict_compl_reorderRestrictProd, mem_compl_iff, Filter.isCompl_principal, range_extend, Filter.mem_cocompact, mem_compl, Nat.coe_maximalIdeal, MeasureTheory.SignedMeasure.toJordanDecomposition_spec, mulIndicator_self_mul_compl, compl_setOf, Equiv.Set.sumCompl_symm_apply_of_notMem, MeasureTheory.probReal_compl_eq_one_sub, AddCircle.openPartialHomeomorphCoe_target, mem_map_indicator_ae_iff_of_zero_notMem, MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite, orbit_fixingSubgroup_compl_subset, TopologicalSpace.NonemptyCompacts.range_toCompacts, tendsto_mabs_nhdsNE_one, Filter.tendsto_inv₀_cobounded', nhdsNE_sup_pure, HasFDerivAt.eventually_ne, Algebra.etaleLocus_eq_compl_support, diff_compl, MeasureTheory.VectorMeasure.of_compl, image_compl_preimage, CompletelyRegularSpace.completely_regular_isOpen, Affine.Simplex.altitude_def, MeasureTheory.MemLp.exists_eLpNorm_indicator_compl_lt, MeasureTheory.compl_mem_ae_iff, ContDiffMapSupportedInClass.map_zero_on_compl, Nat.factoredNumbers_compl, mulIndicator_compl_mul_self, PerfectSpace.not_isolated, Complex.tendsto_exp_comap_re_atBot_nhdsNE, Filter.disjoint_principal_right, MulAction.movedBy_mem_fixedBy_of_commute, Matroid.setOf_dep_eq, MeasureTheory.stoppedValue_piecewise_const', Asymptotics.isBigO_one_nhds_ne_iff, Finite.toFinset_compl, HurwitzZeta.hurwitzZetaEven_residue_one, restrict_piecewise_compl, Function.Embedding.optionEmbeddingEquiv_apply_fst, Filter.compl_notMem, SimpleGraph.card_edgeFinset_induce_compl_singleton, PeriodPair.differentiableOn_derivWeierstrassPExcept, isClosed_compl_iff, exists_mem_frontier_infDist_compl_eq_dist, HasProd.hasProd_compl_iff
instSProd 📖	CompOp	594 math math: Trivialization.image_preimage_eq_prod_univ, AddSubgroup.coe_prod, Real.continuousOn_rpowIntegrand₀₁_uncurry, tsum_setProd_singleton_right, Complex.norm_circleTransformBoundingFunction_le, Sublattice.coe_prod, Finite.toFinset_prod, ProbabilityTheory.lintegral_condCDF, image_prodMk_subset_prod_left, prod_mem_nhds, ProbabilityTheory.condDistrib_apply_of_ne_zero, prod_diff_prod, ModelWithCorners.range_prod, isOpen_prod_iff, isPiSystem_prod, sym2_eq_mk_image, mem_prod_eq, prod_subset_prod_iff, IsLowerSet.prod, prod_nhds_le_of_disjoint_cocompact, Pretrivialization.preimage_symm_proj_inter, EqOn.prodMap, Subgroup.closure_prod, egauge_prod_mk, IsOpen.trivializationDiscrete_target, IocProdIoc_preimage, encard_prod, mem_nhds_prod_iff, AntitoneOn.set_prod, Ideal.span_prod, TopologicalSpace.NonemptyCompacts.coe_prod, ContDiffBumpBase.smooth, sInter_union_sInter, Pretrivialization.target_eq, snd_image_prod_subset, exists_nhds_square, MeasureTheory.Measure.toSphere_apply_aux, sUnion_inter_sUnion, EMetric.ball_prod_same, MeasureTheory.measurableSet_predictable_Ioi_prod, offDiag_subset_prod, MeasureTheory.ProbabilityMeasure.prod_prod, Equiv.prod_assoc_image, tsum_setProd_singleton_left, zero_prod_zero, ProbabilityTheory.setIntegral_condCDF, ball_entourageProd, sSup_inf_sSup, extremePoints_prod, SetRel.prod_comp_prod_of_inter_nonempty, Complex.reProdIm_subset_iff, DirectedOn.isCofinalFor_fst_image_prod_snd_image, HasFPowerSeriesWithinAt.hasStrictFDerivWithinAt, UniqueMDiffOn.prod, sphere_prod, ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat_rat, MeasureTheory.setIntegral_prod_swap, sInter_prod, Filter.mem_prod_self_iff, ProbabilityTheory.IsRatCondKernelCDF.setIntegral, eqOn_prodMap_iff, ProbabilityTheory.setLIntegral_condKernel_univ_right, ProbabilityTheory.IsCondKernelCDF.integral, Prod.segment_subset, prod_sub_preimage_iff, sSup_prod, Trivialization.pullback_target, HasFPowerSeriesWithinOnBall.isBigO_image_sub_image_sub_deriv_principal, DirectedOn.prod, iUnion_unpair_prod, Infinite.prod_left, prod_mono_right, cauchy_iff, Submodule.prod_coe, iUnion_prod_of_monotone, ProbabilityTheory.Kernel.measurable_densityProcess_countableFiltration_aux, Trivialization.isImage_preimage_prod, NumberField.mixedEmbedding.polarSpaceCoord_target, SetRel.prod_comp_prod, MeasureTheory.setIntegral_prod_mul, ProbabilityTheory.Kernel.compProd_restrict_left, Iic_prod_eq, Subsemiring.coe_prod, ProbabilityTheory.integral_stieltjesOfMeasurableRat, MeasureTheory.Measure.nullMeasurableSet_prod, mem_nhds_prod_iff', tprod_setProd_singleton_left, Trivialization.Prod.continuous_inv_fun, AddSubmonoid.closure_prod, StarConvex.prod, TopologicalSpace.IsTopologicalBasis.prod, ProbabilityTheory.Kernel.parallelComp_apply_prod, IsCompact.nhdsSetWithin_prod_eq, biUnion_prod', Filter.eventually_nhdsSet_prod_iff, AddSubsemigroup.coe_prod, MeasureTheory.Measure.prod_prod_le, IsCompact.prod, biSup_inf_biSup, infinite_prod, uIcc_prod_uIcc, Filter.HasBasis.prod_self, prod_add_prod_comm, IsPWO.prod, Filter.prod_mem_curry, Fin.preimage_apply_01_prod, nhdsWithin_eq_comap_uniformity_of_mem, ProbabilityTheory.Kernel.setIntegral_densityProcess_of_mem, prod_preimage_eq, sub_image_prod, Filter.map_const_principal_coprod_map_id_principal, prod_mem_nhds_iff, frontier_prod_eq, prodMap_image_prod, ContinuousWithinAt.prodMap, Finset.card_sups_iff, image_mul_prod, lowerClosure_prod, centralizer_prod, Subsemigroup.coe_prod, Monotone.set_prod, Cardinal.mk_setProd, prod_subset_iff, Filter.prod_mem_prod_iff, Trivialization.target_eq, OpenSubgroup.coe_prod, univ_prod, polarCoord_target, Filter.HasBasis.prod_top, diag_preimage_prod, image_prod, prod_subset_prod_right, continuousOn_cfc_setProd, Disjoint.set_prod_left, UpperSet.coe_prod, BijOn.prodMap, Bornology.IsBounded.prod, NonarchimedeanGroup.prod_subset, offDiag_insert, LeftInvOn.prodMap, LocallyFinite.prod_right, MeasureTheory.measurableSet_predictable_singleton_bot_prod, ProbabilityTheory.Kernel.setIntegral_densityProcess_of_le, prod_image_left, continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith', prod_eq_biUnion_right, IsLUB.prod, prod_univ, Finset.pairwiseDisjoint_smul_iff, ContDiffWithinAt.prodMap, ProbabilityTheory.setLIntegral_condKernel, prod_subset_preimage_fst, SetRel.prod_comp_prod_of_disjoint, prod_mul_prod_comm, subset_tangentConeAt_prod_right, ProbabilityTheory.IsCondKernelCDF.setIntegral, OpenPartialHomeomorph.prod_target, NumberField.mixedEmbedding.polarCoordReal_source, offDiag_eq_sep_prod, PartialEquiv.prod_source, iUnion_prod_const, MeasureTheory.Measure.condKernel_apply_of_ne_zero, subset_tangentConeAt_prod_left, HahnSeries.SummableFamily.smul_support_subset_prod, prod_self_eq_iUnion_perm, prod_insert, Filter.HasBasis.prod_principal, AddSubmonoid.closure_zero_prod, iUnion₂_prod_const, interior_prod_eq, addCenter_prod, prod_image_image_eq, prod_sdiff_diagonal, Filter.Eventually.prod_nhdsSet, sInf_prod, Submodule.span_prod_le, Filter.HasAntitoneBasis.prod, TensorProduct.LinearIndepOn.tmul_of_flat_right, prod_eq_generateFrom, IsCountablySpanning.prod, mk_preimage_prod_right_eq_if, empty_prod, MeasureTheory.Measure.ext_prod₃_iff', Submonoid.coe_prod, ProbabilityTheory.IsCondKernelCDF.setLIntegral, nhds_prod_le_of_disjoint_cocompact, Filter.HasBasis.prod_same_index_mono, nhdsWithin_eq_comap_uniformity, FiberwiseLinear.source_trans_PartialHomeomorph, Complex.continuousOn_norm_circleTransformBoundingFunction, fst_image_prod_subset, latticeClosure_prod, ContinuousOn.prodMap, LocallyFinite.prod_left, ProbabilityTheory.Kernel.meas_countablePartitionSet_le_of_fst_le, Complex.preimage_equivRealProd_prod, graphOn_prod_graphOn, Filter.prod_principal_principal, subset_fst_image_prod_snd_image, Antitone.set_prod, Filter.HasBasis.uniformEquicontinuousOn_iff_right, mk_mem_convexHull_prod, image2_curry, generateFrom_prod, frontier_prod_univ_eq, neg_prod, InvOn.prodMap, singleton_prod, continuousOn_prod_of_continuousOn_lipschitzOnWith, exists_prod_set, MeasurableSet.prod, fst_image_prod, image_div_prod, RightInvOn.prodMap, MDifferentiableOn.prodMap, Countable.prod, prodMk_mem_set_prod_eq, Filter.prod_lift'_lift', IsAddFreimanIso.prodMap, prod_eq_prod_iff, ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat, prod_eq, regionBetween_subset, IsPicardLindelof.continuousOn_uncurry, Equiv.sumPiEquivProdPi_symm_preimage_univ_pi, biUnion_prod, IsSymmetricRel.prod_subset_comm, ContDiffWithinAt.prodMap', pairwiseDisjoint_prod_left, MeasureTheory.Measure.prod_restrict, MeasureTheory.Measure.prod_prod, prod_empty, ProbabilityTheory.setLIntegral_condKernel_univ_left, mk_preimage_prod_right_eq_empty, ProbabilityTheory.Kernel.setLIntegral_compProd_univ_left, inv_prod, ThreeAPFree.prod, ProbabilityTheory.Kernel.setIntegral_density_of_measurableSet, image_swap_prod, MeasureTheory.SimpleFunc.pair_preimage, Monotone.upperBounds_image_of_directedOn_prod, Filter.frequently_curry_prod_iff, vadd_set_prod, disjoint_prod, prod_nonempty_iff, Infinite.prod_right, TopologicalSpace.Compacts.coe_prod, MeasureTheory.measurableSet_predictable_Ioc_prod, nhdsKer_pair, nhds_nhds_eq_uniformity_uniformity_prod, closedBall_prod_same, Submonoid.closure_prod_one, image_prodMk_subset_prod, prod_eq_biUnion_left, AddSubmonoid.coe_prod, prod_iUnion, Filter.principal_coprod_principal, exists_monotone_Icc_subset_open_cover_unitInterval_prod_self, univ_prod_univ, biInf_prod, Prod.openSegment_subset, ProbabilityTheory.setLIntegral_toKernel_univ, insert_prod, compl_prod_eq_union, card_singleton_prod, preimage_prod_map_prod, biSup_prod, TopologicalSpace.CompactOpens.coe_prod, singleton_prod_singleton, ProbabilityTheory.Kernel.setIntegral_densityProcess, mk_preimage_prod_left, subset_prod, card_prod_singleton, tprod_setProd_singleton_right, MeasureTheory.Measure.setLIntegral_condKernel, ContMDiffOn.prodMap, ProbabilityTheory.Kernel.integral_densityProcess, LowerSet.coe_prod, MeasureTheory.Measure.IicSnd_apply, ContDiffOn.continuousOn_fderivWithin_apply, NumberField.mixedEmbedding.fundamentalDomain_stdBasis, IsCompact.nhdsSet_prod_eq, Filter.HasBasis.prod_pprod, Ici_prod_eq, NumberField.mixedEmbedding.polarCoord_source, lowerBounds_prod, le_egauge_prod, InfClosed.prod, Pretrivialization.symm_trans_target_eq, SequentiallyComplete.setSeq_sub_aux, ProbabilityTheory.Kernel.setLIntegral_density, MeasureTheory.Measure.nullMeasurableSet_prod_of_ne_zero, ball_prod_same, Finset.card_mul_iff, Homeomorph.Set.prod_apply, prod_singleton, image_smul_prod, Metric.eball_prod_same, prod_image_right, prod_preimage_left, nhdsKer_prod, add_image_prod, center_prod, range_prodMap, small_setProd, FiberwiseLinear.target_trans_openPartialHomeomorph, SetRel.prod_subset_comm, nhdsSet_prod_le_of_disjoint_cocompact, one_prod_one, IsClosed.prod, ProbabilityTheory.Kernel.integral_density, NumberField.mixedEmbedding.polarCoord_target, AddSubgroup.closure_prod, ProbabilityTheory.Kernel.lintegral_density, mem_prod, Convex.prod, InjOn.prodMap, ContinuousMap.image_coev, prod_subset_prod_left, ProbabilityTheory.Kernel.measurable_densityProcess_aux, upperClosure_prod, Trivialization.symm_trans_target_eq, nhdsWithin_prod, mk_preimage_prod_left_fn_eq_if, Complex.continuousOn_prod_circle_transform_function, Homeomorph.Set.prod_symm_apply_coe, Metric.infEDist_prod, NumberField.mixedEmbedding.polarSpaceCoord_target', Metric.hausdorffEDist_prod_le, prod_subset_prod_iff_right, union_prod, PairwiseDisjoint.prod_left, Bornology.isBounded_prod, Nonempty.prod, prod_mono_left, IsPicardLindelof.exists_forall_mem_closedBall_eq_hasDerivWithinAt_continuousOn, MeasureTheory.Measure.compProd_apply_prod, prod_inter, mfderivWithin_prodMap, prod_eq_seq, Filter.HasBasis.prod_nhds, generateFrom_prod_eq, MeasureTheory.Measure.set_prod_ae_eq, ProbabilityTheory.setLIntegral_toKernel_prod, Filter.prod_same_eq, Finite.prod, nhdsSet_prod_le, mem_nhdsWithin_prod_iff, ProbabilityTheory.Kernel.setLIntegral_compProd, MeasureTheory.Measure.ext_prod₃_iff, Filter.HasBasis.prod_same_index_anti, uncurry_preimage_prod_pi, prod_iUnion₂, image2_mk_eq_prod, Disjoint.set_prod_right, prod_pow, HasFPowerSeriesWithinAt.isBigO_image_sub_norm_mul_norm_sub, vadd_image_prod, TensorProduct.LinearIndepOn.tmul_of_flat_left, Filter.ker_prod, MeasureTheory.Measure.iInf_rat_gt_prod_Iic, Bornology.isBounded_prod_self, ncard_prod, ProbabilityTheory.integral_condCDF, Pretrivialization.trans_source, sInter_prod_sInter, AddSubmonoid.closure_prod_zero, Trivialization.prod_target, SurjOn.prodMap, intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae, ExistsContDiffBumpBase.y_smooth, IsPiSystem.prod, FiberBundle.extChartAt_target, continuousOn_prod_of_continuousOn_lipschitzOnWith', image_vsub_prod, Algebra.adjoin_prod_le, IicProdIoc_preimage, UniformSpace.mem_comp_comp, Finset.card_add_iff, generateFrom_eq_prod, IsAddFreimanHom.prodMap, NonUnitalSubalgebra.coe_prod, intervalIntegral.fderivWithin_integral_of_tendsto_ae, FiberwiseLinear.target_trans_PartialHomeomorph, infClosure_prod, Finset.pairwiseDisjoint_vadd_iff, TopologicalSpace.exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact, finite_prod, intervalIntegral.integral_hasFDerivWithinAt, Icc_prod_eq, prod_subset_prod_iff_left, MeasureTheory.Measure.setLIntegral_condKernel_univ_left, NonarchimedeanAddGroup.prod_subset, MapsTo.prodMap, mapsTo_fst_prod, Submonoid.closure_prod, prod_subset_preimage_snd, Topology.IsQuotientMap.trivializationOfSMulDisjoint_target, biInf_sup_biInf, nhdsWithin_prod_eq, preimage_entourageProd_prod, ProbabilityTheory.Kernel.setLIntegral_compProd_univ_right, Filter.prod_mem_prod, EMetric.hausdorffEdist_prod_le, map_uniformity_set_coe, NonUnitalSubsemiring.coe_prod, SimpleGraph.neighborSet_boxProd, EMetric.infEdist_prod, biSup_prod', ProbabilityTheory.Kernel.compProd_restrict, TendstoUniformlyOn.prodMap, MeasureTheory.measurableSet_predictable_singleton_prod, modelProd_range_prod_id, Finset.coe_product, prod_self_ssubset_prod_self, MeasureTheory.FiniteMeasure.prod_prod, image_uncurry_prod, Finset.card_infs_iff, addCentralizer_prod, ProbabilityTheory.setLIntegral_condKernel_eq_measure_prod, Equiv.prod_assoc_preimage, Equiv.prod_assoc_symm_image, SequentiallyComplete.setSeq_prod_subset, pairwiseDisjoint_vadd_iff, mapsTo_swap_prod, continuousOn_cfcₙ_setProd, prod_generateFrom_generateFrom_eq, IsConnected.prod, ProbabilityTheory.IsRatCondKernelCDFAux.setIntegral_iInf_rat_gt, MeasureTheory.Measure.setLIntegral_compProd, Trivialization.symm_trans_source_eq, ContMDiffWithinAt.prodMap', ContMDiffFiberwiseLinear.locality_aux₁, biInf_prod', prod_eq_empty_iff, Subring.coe_prod, Finset.card_image₂_iff, prod_eq_iff_eq, TopologicalSpace.IsSeparable.prod, smul_set_prod, Ideal.coe_prod, mk_preimage_prod_right, uIcc_prod_eq, prod_range_range_eq, Subsingleton.prod, prod_self_subset_prod_self, IsGLB.prod, IsMulFreimanHom.prodMap, Filter.HasBasis.top_prod, NonUnitalStarSubalgebra.coe_prod, prod_mono, convexHull_prod, MeasureTheory.measureReal_prod_prod, mapsTo_snd_prod, upperBounds_prod, TopologicalSpace.PositiveCompacts.coe_prod, inter_prod, OpenPartialHomeomorph.prod_source, ProbabilityTheory.Kernel.setIntegral_density, prod_addCentralizer_subset_addCentralizer_prod, pairwiseDisjoint_image_right_iff, NumberField.mixedEmbedding.polarSpaceCoord_source, offDiag_union, FiberBundleCore.continuousOn_coordChange, IsUpperSet.prod, MonotoneOn.set_prod, ProbabilityTheory.setLIntegral_condCDF, diag_preimage_prod_self, Filter.HasBasis.principal_prod, ProbabilityTheory.lintegral_stieltjesOfMeasurableRat, snd_image_prod, Finite.Set.finite_prod, MeasureTheory.Measure.IicSnd_univ, isOpen_prod_iff', nhds_eq_uniformity_prod, MDifferentiableWithinAt.prodMap, SupClosed.prod, Equiv.preimage_piEquivPiSubtypeProd_symm_pi, Metric.closedEBall_prod_same, MeasureTheory.Measure.setLIntegral_condKernel_univ_right, continuousOn_cfc_nnreal_setProd, prod_union, IsMulFreimanIso.prodMap, PartiallyWellOrderedOn.prod, ThreeGPFree.prod, ProbabilityTheory.IsCondKernelCDF.lintegral, pairwiseDisjoint_smul_iff, Fin.preimage_apply_01_prod', prod_nhdsSet_le_of_disjoint_cocompact, iUnion_prod, EMetric.closedBall_prod_same, prod_centralizer_subset_centralizer_prod, Ici_prod_Ici, nsmul_prod, image_toPullbackDiag, UniqueDiffWithinAt.prod, pairwiseDisjoint_image_left_iff, graphOn_prod_prodMap, UniqueMDiffWithinAt.prod, image_prodMk_subset_prod_right, prod_sUnion, ContMDiffWithinAt.prodMap, Icc_prod_Icc, prod_preimage_right, Finset.card_diffs_iff, Filter.eventually_curry_prod_iff, Ideal.span_prod_le, Topology.IsQuotientMap.trivializationOfVAddDisjoint_target, prod_eq_univ, contDiffOn_fderivWithin_apply, Filter.mem_prod_iff, diag_image, ProbabilityTheory.IsRatCondKernelCDFAux.setIntegral, ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat_rat, NumberField.mixedEmbedding.polarCoordReal_target, Iic_prod_Iic, Complex.polarCoord_target, PairwiseDisjoint.prod, mk_mem_prod, ContDiffOn.prodMap, Filter.map_prodMap_const_id_principal_coprod_principal, mk_preimage_prod_right_fn_eq_if, supClosure_prod, Filter.mem_prod_same_iff, NonUnitalSubring.coe_prod, NonarchimedeanGroup.prod_self_subset, FiberwiseLinear.source_trans_openPartialHomeomorph, ProbabilityTheory.Kernel.densityProcess_def, Trivialization.continuousOn_symm, mk_preimage_sym2, continuousOn_cfcₙ_nnreal_setProd, ModelWithCorners.interior_prod, MeasureTheory.NullMeasurableSet.prod, ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat, UniformSpace.hasBasis_nhds_prod, measurableSet_prod, prod_range_univ_eq, sInter_prod_sInter_subset, UniformCauchySeqOn.prodMap, prod_subset_compl_diagonal_iff_disjoint, ProbabilityTheory.Kernel.compProd_apply_prod, SimpleGraph.neighborSet_completeEquipartiteGraph, IsLocalHomeomorph.exists_lift_nhds, MDifferentiableWithinAt.prodMap', prod_univ_range_eq, TopologicalSpace.exists_open_prod_subset_of_mem_nhds_diagonal, Filter.prod_def, indicator_prod_one, hom_trivializationAt_target, IsPreconnected.prod, Pretrivialization.symm_trans_source_eq, OpenAddSubgroup.coe_prod, mk_preimage_prod_left_eq_if, MeasureTheory.Measure.restrict_prod_eq_prod_univ, Equiv.prod_assoc_symm_preimage, LinearIndepOn.tmul_of_isDomain, Filter.HasBasis.prod_same_index, nhdsSetWithin_prod_le, frontier_univ_prod_eq, FiniteExhaustion.prod_apply, MeasureTheory.Measure.setLIntegral_condKernel_eq_measure_prod, Filter.HasBasis.prod_nhds', NonarchimedeanAddGroup.prod_self_subset, toFinset_prod, Pretrivialization.target_inter_preimage_symm_source_eq, PartialEquiv.prod_target, MeasureTheory.Measure.IsCondKernel.apply_of_ne_zero, range_pair_subset, prod_sInter, IsClopen.prod, Subgroup.coe_prod, continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith, mk_preimage_prod, image_entourageProd_prod, IsOpen.prod, ProbabilityTheory.Kernel.compProd_restrict_right, MeasureTheory.Measure.ext_prod_iff, prod_inter_prod, IsSublattice.prod, Dense.prod, mk_preimage_prod_left_eq_empty, closure_prod_eq, ProbabilityTheory.setLIntegral_toKernel_Iic, prod_iInter, TangentBundle.trivializationAt_target, preimage_swap_prod, Submonoid.closure_one_prod, Filter.HasBasis.prod, MeasureTheory.Measure.measure_prod_compl_eq_zero, ProbabilityTheory.Kernel.prod_apply_prod, UniqueDiffOn.prod, Subalgebra.coe_prod, sUnion_prod_const, sInf_sup_sInf
kernImage 📖	CompOp	17 math math: preimage_kernImage, kernImage_preimage_eq_iff, isClosedMap_iff_kernImage, Filter.kernMap_principal, isClosedMap_iff_kernImage_interior, kernImage_compl, kernImage_mono, kernImage_empty, Filter.mem_comap'', isOpenMap_iff_closure_kernImage, subset_kernImage_iff, kernImage_union_preimage, isOpenMap_iff_kernImage, compl_range_subset_kernImage, Filter.mem_kernMap, kernImage_eq_compl, kernImage_preimage_union
offDiag 📖	CompOp	30 math math: Finite.einfsep, einfsep_of_fintype, offDiag_nonempty, offDiag_subset_prod, Nontrivial.offDiag_nonempty, offDiag_singleton, offDiag_univ, Finite.infsep_of_nontrivial, Nontrivial.infsep_of_fintype, Finite.infsep, offDiag_insert, Finset.coe_offDiag, Nontrivial.infsep_eq_iInf, offDiag_mono, offDiag_eq_sep_prod, mem_offDiag, prod_sdiff_diagonal, infsep_eq_iInf, infsep_of_fintype, toFinset_offDiag, Finite.offDiag, Finite.toFinset_offDiag, toFinset_off_diag, offDiag_empty, Subsingleton.offDiag_eq_empty, einfsep_eq_iInf, offDiag_eq_empty, disjoint_diagonal_offDiag, offDiag_inter, offDiag_union
pi 📖	CompOp	296 math math: MeasureTheory.Measure.univ_pi_Ioc_ae_eq_Icc, biUnion_univ_pi, pi_eq_empty_iff', piMap_mapsTo_pi, Filter.mem_pi', vadd_set_pi_of_surjective, NumberField.mixedEmbedding.fundamentalCone.interior_paramSet, isOpen_pi_iff, Filter.HasBasis.pi_self, Submonoid.coe_pi, Complex.integral_comp_pi_polarCoord_symm, IocProdIoc_preimage, MeasureTheory.Measure.pi_Ioc_ae_eq_pi_Icc, uniqueElim_preimage, univ_pi_eq_iInter, IsPWO.pi, MeasureTheory.Measure.pi'_pi, eval_image_pi_subset, Filter.pi_inf_principal_univ_pi_eq_bot, univ_pi_piecewise_univ, MeasureTheory.Measure.ae_le_set_pi, Sublattice.coe_pi, convex_pi, EMetric.diam_pi_le_of_le, le_egauge_pi, threeAPFree_pi, PartialEquiv.pi_target, MeasureTheory.volume_pi_pi, extremePoints_pi, Subgroup.coe_pi, Equiv.piCongrLeft_preimage_univ_pi, Submonoid.closure_pi, inv_pi, singleton_pi, univ_pi_eq_empty_iff, mem_convexHull_pi, NumberField.mixedEmbedding.polarSpaceCoord_target, disjoint_pi_univ_Ioc_update_left_right, MeasureTheory.Measure.univ_pi_Ioo_ae_eq_Icc, generateFrom_eq_pi, eval_preimage, NumberField.mixedEmbedding.fundamentalCone.expMap_target, vadd_set_univ_pi, pi_diff_pi_subset, AddSubmonoid.closure_pi, eval_image_univ_pi_subset, MeasureTheory.OuterMeasure.le_pi, Finset.restrict₂_preimage, set_pi_mem_nhds, egauge_univ_pi, union_pi_inter, vadd_set_pi_of_isAddUnit, MeasureTheory.Measure.pi_pi_aux, singleton_pi', Equiv.piCongrLeft_preimage_pi, nhdsKer_singleton_pi, Real.volume_pi_Ico, OpenPartialHomeomorph.pi_target, Fin.preimage_apply_01_prod, nhdsWithin_pi_univ_eq, update_mem_pi_iff, MeasureTheory.Measure.infinitePi_pi, generateFrom_pi_eq, Filter.pi_mem_pi_iff, closedBall_pi', lintegral_comp_pi_polarCoord_symm, smul_set_pi, IsPiSystem.pi, Real.volume_pi_Ioc, supClosed_pi, pi_univ_Ioo_subset, MeasureTheory.FiniteMeasure.pi_pi, empty_pi, MeasureTheory.ProbabilityMeasure.pi_pi, small_setPi, Equiv.piCongrLeft_symm_preimage_univ_pi, Filter.ker_pi, OpenPartialHomeomorph.pi_source, Filter.hasBasis_pi_same_index, NumberField.mixedEmbedding.polarCoordReal_source, Real.volume_pi_Ioc_toReal, smul_set_pi_of_surjective, infClosed_pi, isOpen_pi_iff', pi_univ_Iic, egauge_pi, measurableSet_pi_polarCoord_target, pi_update_of_notMem, isConnected_univ_pi, range_piMap, univ_pi_nonempty_iff, Filter.pi_inf_principal_univ_pi_neBot, pi_univ_Ioc_update_right, MeasureTheory.Measure.restrict_pi_pi, MeasureTheory.Measure.pi_Ico_ae_eq_pi_Icc, Equiv.piCongrLeft_symm_preimage_pi, isClosed_set_pi, Equiv.Set.univPi_symm_apply_coe, Icc_diff_pi_univ_Ioc_subset, UniqueDiffOn.pi, piMap_image_univ_pi, mem_pi, Icc_diff_pi_univ_Ioo_subset, Equiv.sumPiEquivProdPi_symm_preimage_univ_pi, isCompact_univ_pi, MeasureTheory.Measure.pi_Ioi_ae_eq_pi_Ici, Bornology.IsBounded.pi, univ_pi_empty, pi_mono', ordConnected_pi, Metric.ediam_pi_le_of_le, MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc, MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Icc, pi_eq_empty_iff, BoxIntegral.Box.coe_mk', ball_pi, update_preimage_univ_pi, MeasureTheory.Measure.univ_pi_Ioi_ae_eq_Ici, disjoint_pi, Filter.pi_inf_principal_pi_eq_bot, update_image, pi_eq_generateFrom, mem_closure_pi, Filter.univ_pi_mem_pi, PiNat.cylinder_eq_pi, Submodule.closure_coe_iSup_map_single, NumberField.mixedEmbedding.fundamentalDomain_stdBasis, Real.volume_pi_Ioo, NumberField.mixedEmbedding.polarCoord_source, Filter.pi_inf_principal_pi_neBot, dense_pi, Real.volume_pi_Ioo_toReal, isOpen_set_pi, DirectedOn.pi, Fintype.coe_piFinset, indicator_pi_one_apply, closedBall_pi, Filter.PiInfPrincipalPi.neBot, MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Ioc, convexHull_pi, NumberField.mixedEmbedding.fundamentalCone.closure_paramSet, smul_pi, isSublattice_pi, Finset.restrict_preimage_univ, smul_set_pi_of_isUnit, NumberField.mixedEmbedding.polarCoord_target, MeasureTheory.piPremeasure_pi', eval_image_pi_of_notMem, RestrictedProduct.range_coe_principal, pi_congr, nhdsWithin_pi_neBot, univ_pi_piecewise, Subgroup.closure_pi, pi_if, nhdsKer_pi, MeasureTheory.OuterMeasure.pi_pi_le, NumberField.mixedEmbedding.polarSpaceCoord_target', insert_pi, union_pi, Bornology.isBounded_pi, countable_univ_pi, UniqueDiffOn.univ_pi, pi_generateFrom_eq, pi_univ_uIcc, UniqueDiffWithinAt.pi, IsCountablySpanning.pi, pi_univ_Ioc_subset, threeGPFree_pi, uncurry_preimage_prod_pi, Filter.hasBasis_pi_principal, pi_univ_Ioc_update_union, iUnion_pi_of_monotone, image_finset_sum_pi, univ_pi_ite, nhdsWithin_pi_eq, univ_pi_update, Pi.openSegment_subset, piMap_image_pi, vadd_univ_pi, AddSubmonoid.coe_pi, IicProdIoc_preimage, pi_univ_Ioi_subset, closure_pi_set, Real.volume_pi_Ico_toReal, pi_univ_ite, mapsTo_tangentConeAt_pi, MeasureTheory.Measure.univ_pi_Iio_ae_eq_Iic, UniqueDiffWithinAt.univ_pi, vadd_pi_subset, BoxIntegral.Box.Icc_eq_pi, smul_set_univ_pi, injOn_pi_polarCoord_symm, smul_univ_pi, ZSpan.fundamentalDomain_pi_basisFun, nhdsWithin_pi_eq_bot, Finset.restrict_preimage, AddSubgroup.coe_pi, PartialEquiv.pi_source, generateFrom_pi, preimage_pi, univ_pi_eq_singleton_iff, pi_inter_compl, subsingleton_univ_pi, update_preimage_pi, MeasureTheory.comap_eval_le_generateFrom_squareCylinders_singleton, BoxIntegral.Box.coe_eq_pi, encard_pi_eq_prod_encard, uncurry_preimage_sigma_pi, pi_univ_Ioc_update_left, ordConnected_pi', Finite.pi, pi_piecewise, Submodule.coe_pi, addCenter_pi, pi_univ_Ici, iUnion_univ_pi_of_monotone, AddSubgroup.closure_pi, NumberField.mixedEmbedding.polarSpaceCoord_source, pi_subset_pi_iff, Complex.lintegral_comp_pi_polarCoord_symm, smul_pi₀, univ_pi_update_univ, set_pi_mem_nhds_iff, MeasureTheory.piPremeasure_pi_eval, Filter.mem_pi_principal, Equiv.preimage_piEquivPiSubtypeProd_symm_pi, pi_polarCoord_symm_target_ae_eq_univ, Pi.segment_subset, subset_pi_iff, pi_mono, center_pi, isPiSystem_pi, interior_pi_set, iUnion_univ_pi, Fin.preimage_apply_01_prod', pi_def, univ_pi_subset_univ_pi_iff, isTopologicalBasis_pi, pi_eq_empty, nhdsWithin_pi_eq', Filter.pi_mem_pi, MeasureTheory.squareCylinders_eq_iUnion_image, univ_pi_singleton, starConvex_pi, elim_preimage_pi, MeasureTheory.Measure.ae_eq_set_pi, PartiallyWellOrderedOn.pi, MeasureTheory.Measure.pi_Iio_ae_eq_pi_Iic, Filter.mem_pi, pi_univ, NumberField.mixedEmbedding.polarCoordReal_target, image_fintype_sum_pi, mem_univ_pi, MeasureTheory.Measure.pi_pi, pi_iUnion_eq_iInter_pi, subset_pi_eval_image, pairwiseDisjoint_pi, pi_inter_distrib, matrix_eq_pi, Equiv.Set.univPi_apply_coe, smul_set_pi₀', pi_update_of_mem, pi_nonempty_iff, vadd_pi, Disjoint.set_pi, pi_univ_Iio_subset, instNeBotNhdsWithinUnivPi, integral_comp_pi_polarCoord_symm, Filter.le_pi_principal, MeasureTheory.Measure.infinitePi_pi_univ, MeasurableSet.univ_pi, measurableSet_pi, Filter.pi_principal, BoxIntegral.Box.mem_univ_Ioc, TopologicalSpace.IsSeparable.univ_pi, MeasureTheory.Measure.infinitePi_pi_of_countable, Filter.hasBasis_pi, smul_set_pi₀, ball_pi', egauge_pi', vadd_set_pi, le_einfsep_pi_of_le, image_fintype_prod_pi, image_finset_prod_pi, eval_preimage', pi_generateFrom_eq_finite, MeasurableSet.pi, mk_preimage_tprod, pi_univ_Icc, neg_pi, piMap_image_pi_subset, isPreconnected_univ_pi, smul_pi_subset, pi_univ_Ico_subset, disjoint_univ_pi, univ_pi_eq_empty
preimage 📖	CompOp	2597 math math: center_units_subset, Isometry.preimage_ball, preimage_div_const_Ico, MeasurableSpace.comap_eq_generateFrom, PartialEquiv.restr_target, mapsTo_iff_subset_preimage, OrderDual.isBounded_preimage_toDual, ContinuousLinearEquiv.preimage_symm_preimage, MeasureTheory.exists_pos_preimage_ball, MeasureTheory.Measure.prod_apply_symm, Homeomorph.isClosed_preimage, IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, Trivialization.image_preimage_eq_prod_univ, StarConvex.preimage_add_right, preimage_add_right_zero, image_subset_preimage_of_inverse, CStarAlgebra.preimage_inr_Icc_zero_one, LinearIsometryEquiv.strictConvex_preimage, TopologicalSpace.IsOpenCover.isInducing_iff_restrictPreimage, LowerSemicontinuousOn.inter_biInter_preimage_Iic_eq_empty_iff_exists_finset, ContinuousLinearEquiv.symm_preimage_preimage, Convex.translate_preimage_right, preimage_mul_const_Ioo_of_neg, AddMonoidHom.coe_ker, SetRel.isTrans_preimage, AlgebraicGeometry.QuasiCompact.isCompact_preimage_singleton, GroupFilterBasis.conj, Filter.comap_hasBasis, preimage_congr, IsUpperSet.ofDual, SurjOn.comp_right, OnePoint.isOpen_def, MeasureTheory.vaddInvariantMeasure_iff, ProbabilityTheory.Kernel.prod_apply', isSublattice_preimage_ofDual, SubAddAction.map_ofFixingAddSubgroupUnion_bijective, Topology.lawsonClosed_of_lowerClosed, preimage_sub_const_Ioo, MeasurableEquiv.preimage_symm, preimage_add_const_Icc, preimage_const_div_Iio, Fin.preimage_natAdd_Iio_natAdd, preimage_const_add_Icc, Trivialization.pullback_source, IsometryEquiv.dimH_preimage, ProbabilityTheory.condIndepFun_iff_condExp_inter_preimage_eq_mul, DomAddAct.mk_vadd_indicatorConstLp, exists_open_dense_of_open_dense_subtype, preimage_const_mul_Ioi₀, AddAction.isTopologicallyTransitive_iff_dense_iUnion_preimage, DFA.reindex_apply_accept, Subring.coe_unop, ContinuousAffineEquiv.image_symm_eq_preimage, preimage_add_right_zero', MeasurableEmbedding.integrableOn_map_iff, MeasurableSet.preimage, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart', IndexedPartition.proj_fiber, MeasureTheory.SimpleFunc.sum_measure_preimage_singleton, TopologicalSpace.Opens.map_coe, preimage_kernImage, AddSubgroup.vadd_opposite_image_add_preimage', MeasurableSpace.measurable_invariants_dom, HasCompactMulSupport.isCompact_preimage, MeasureTheory.forall_measure_preimage_add_iff, ProbabilityTheory.condDistrib_ae_eq_condExp, Fin.preimage_val_Ici_val, sigma_univ, OpenPartialHomeomorph.extend_preimage_inter_eq, Isometry.preimage_perpBisector, AlgebraicGeometry.QuasiCompact.isCompact_preimage, Filter.comap_eq_lift', MeasureTheory.Content.is_add_left_invariant_outerMeasure, TopCat.GlueData.preimage_image_eq_image, Subring.coe_comap, preimage_inter, MulAction.isTopologicallyTransitive_iff_dense_iUnion_preimage, Trivialization.preimageHomeomorph_symm_apply, IsometryEquiv.preimage_sphere, preimage_subset_preimage_iff, UniqueMDiffOn.uniqueMDiffOn_preimage, IsLowerSet.ofDual, IsChain.preimage, dual_ordConnectedComponent, PrimeSpectrum.preimageEquivFiber_symm_apply_coe, ModelWithCorners.interior_open, Finset.preimage_add_left_singleton, OpenPartialHomeomorph.isOpen_extend_preimage, SimpleGraph.isVertexCover_preimage_iso, preimage_const_add_Ioo, UniformOnFun.gen_eq_preimage_restrict, ModelWithCorners.preimage_image, preimage_mul_const_Icc₀, preimage_subtype_val_Ioi, AlgebraicGeometry.Scheme.Hom.closePoints_subset_preimage_closedPoints, Cardinal.mk_preimage_of_injective_of_subset_range, OpenPartialHomeomorph.subtypeRestr_symm_trans_subtypeRestr, Topology.IsLocallyConstructible.preimage_of_isOpenEmbedding, Fin.preimage_addNat_uIcc_addNat, MeasureTheory.MeasurePreserving.measureReal_preimage, biUnion_preimage_singleton, Diffeomorph.contMDiffOn_comp_diffeomorph_iff, Fin.preimage_insertNth_Icc_of_mem, OpenPartialHomeomorph.restr_target, Topology.IsInducing.isLocallyClosed_iff, Fin.preimage_castSucc_uIoo_castSucc, seminormFromBounded_ker, IsLocalization.minimalPrimes_map, Manifold.IsImmersionAtOfComplement.target_subset_preimage_target, AlgebraicGeometry.Scheme.Hom.fiberHomeo_apply, Homeomorph.isPathConnected_preimage, Function.Fiber.map_eq_image, preimage_div_const_Icc, OrderIso.isGLB_preimage', omegaLimit_preimage_subset, Fin.preimage_castAdd_Ico_castAdd, Convex.smul_preimage, Equiv.symm_preimage_preimage, RestrictedProduct.image_coe_preimage_inclusion_subset, ContinuousLinearMap.frontier_preimage, MonoidHom.preimage_smul_setₛₗ, isOpen_extChartAt_preimage', Ioi_ofDual, AddSubgroup.closure_preimage_eq_top, IsClosedMap.isEvenlyCovered_of_openPartialHomeomorph, preimage_fst_singleton_eq_range, IsCompact.preimage_of_isOpen, IsLocalization.AtPrime.coe_orderIsoOfPrime_apply_coe, IsCoveringMapOn.homeomorph_comp_iff, OnePoint.nhdsWithin_coe, LSeries.term_convolution', IsOpen.iUnion_preimage_vadd, Ideal.isHomogeneous_iff_subset_iInter, Pretrivialization.preimage_symm_proj_inter, Function.Surjective.preimage_subset_preimage_iff, MDifferentiableWithinAt.differentiableWithinAt_mpullbackWithin_vectorField, image_swap_eq_preimage_swap, AntilipschitzWith.hausdorffMeasure_preimage_le, supClosed_preimage_ofDual, AnalyticAtWithin.compContinuousLinearMap, preimage_sumElim, OrdConnected.preimage_antitoneOn, Finset.preimage_mul_left_one, Int.preimage_Ioc, MeasureTheory.MeasurePreserving.integrableOn_comp_preimage, restrictPreimage_surjective, mdifferentiableOn_iff, Cardinal.mk_preimage_of_subset_range_lift, IocProdIoc_preimage, IsChain.preimage_iso_iff, PartialEquiv.trans_source, continuousOn_boolIndicator_iff_isClopen, Fin.preimage_natAdd_Ioi_natAdd, DiscreteTiling.PlacedTile.ext_iff_of_preimage, Fin.preimage_rev_Ico, mdifferentiableWithinAt_iff_of_mem_maximalAtlas, Function.Injective.preimage_surjective, isOpen_sum_iff, ModelWithCorners.symm_continuousWithinAt_comp_right_iff, StarConvex.affine_preimage, isOpen_induced_eq, MeasureTheory.preimage_spanningSetsIndex_singleton, OpenPartialHomeomorph.IsImage.preimage_eq', AnalyticOnNhd.compContinuousLinearMap, preimage_div_preimage_subset, Finset.sigma_image_fst_preimage_mk, preimage_boolIndicator, ordConnected_preimage, ConjClasses.carrier_eq_preimage_mk, OnePoint.isClosed_iff_of_notMem, Topology.IsInducing.isOpen_iff, AlgebraicGeometry.Scheme.Hom.range_fiberι, mdifferentiable_iff_target, AddSubgroup.coe_addSubgroupOf, Matroid.comap_loopyOn, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp, OpenPartialHomeomorph.image_eq_target_inter_inv_preimage, mk_preimage_sigma_eq_if, Cardinal.mk_preimage_equiv_lift, AddSubsemigroup.coe_toSubsemigroup_symm_apply, Subalgebra.comap_map_eq, Subalgebra.op_sSup, preimage_val_eq_univ_of_subset, uniqueElim_preimage, univ_pi_eq_iInter, AddCircle.coe_real_preimage_closedBall_eq_iUnion, MeasureTheory.Lp.indicatorConstLp_compMeasurePreserving, ContinuousOn.preimage_mem_nhdsSetWithin, preimage_mul_const_Iio_of_neg, Metric.Snowflaking.preimage_ofSnowflaking_closedBall, Equiv.restrictPreimageFinset_symm_apply_coe, Filter.tendsto_def, segment_translate_preimage, Function.LeftInverse.image_eq, MeasureTheory.measure_preimage_mul, cauchySeq_iff', ContinuousWithinAt.extChartAt_symm_preimage_inter_range_eventuallyEq, CategoryTheory.Limits.Types.range_pullbackFst, MeasureTheory.Measure.toSphere_apply_aux, UniqueMDiffWithinAt.preimage_PartialHomeomorph, map_extChartAt_symm_nhdsWithin, preimage_const_mul_Iio₀, DiscreteQuotient.fiber_subset_ofLE, Fin.preimage_castSucc_Iic_castSucc, MeasureTheory.mem_ae_map_iff, Subring.coe_op, DiffeologicalSpace.CorePlotsOn.isOpen_iff_preimages_plots, Topology.IsInducing.restrictPreimage, EuclideanGeometry.preimage_inversion_perpBisector_inversion, Topology.WithLawson.isOpen_preimage_ofLawson, Diffeomorph.uniqueMDiffOn_preimage, sym2_preimage, preimage_div_const_Iic, OpenPartialHomeomorph.extend_preimage_mem_nhds_of_mem_nhdsWithin, ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul, ContinuousLinearEquiv.comp_right_differentiableWithinAt_iff, IsClosedMap.comap_nhds_eq, QuotientGroup.preimage_image_mk, isOpenMap_iff_lowerHemicontinuous, Ideal.isHomogeneous_iff_forall_subset, MeasureTheory.measure_preimage_fst_singleton_eq_tsum, Metric.exists_forall_closedEBall_subset_aux₂, Function.LeftInverse.preimage_preimage, AddSubgroup.coe_comap, dual_ordConnectedSection, MeasurableEquiv.map_apply, Ioc_ofDual, tangentConeAt_eq_biInter_closure, MeasureTheory.MeasurePreserving.restrict_preimage, PFun.preimage_asSubtype, mk_preimage_sigma_fn_eq_if, Equiv.sigmaPreimageEquiv_symm_apply_snd_coe, TopologicalSpace.tendsto_nhds_generateFrom_iff, Subalgebra.op_sInf, MeasurableEquiv.image_preimage, preimage_surjective, ProbabilityTheory.Kernel.ae_null_of_compProd_null, compl_image, ProbabilityTheory.sum_meas_smul_cond_fiber, Nat.preimage_Ioi, encard_preimage_val_le_encard_left, Complex.closure_preimage_im, Pretrivialization.restrictPreimage_target, ProbabilityTheory.iIndepFun.measure_inter_preimage_eq_mul, Infinite.preimage', Flow.orbit_restrict, Fin.preimage_natAdd_uIoo_natAdd, MeasurableSpace.map_def, Module.freeLocus_localization, OpenPartialHomeomorph.transHomeomorph_target, mdifferentiableWithinAt_iff_target_inter, IsChain.preimage_embedding, preimage_smul_inv₀, lowerSemicontinuous_iff_isClosed_preimage, Behrend.sphere_subset_preimage_metric_sphere, Matroid.sigma_isBasis_iff, WithTop.preimage_coe_Iio_top, preimage_const_mul_Ioc_of_neg, Fin.preimage_rev_Ici, ModelWithCorners.image_eq, StrictConcaveOn.translate_right, preimage_const_mul_Ioo₀, isUpperSet_preimage_toDual_iff, preimage_const_sub_Iic, Finset.preimage_inr, ChartedSpace.LiftPropWithinAt.prop, isTopologicalBasis_subtype, Nat.preimage_Iio, OrderIso.preimage_Ioc, Topology.IsUpper.continuous_iff_Iic, map_extChartAt_symm_nhdsWithin', SimpleGraph.IsCompleteBetween.induce, ULift.up_sInf, Finset.coe_attachFin, TopologicalSpace.Opens.map_obj, Specialization.isOpen_toEquiv_preimage, MeasurableEquiv.preimage_image, OnePoint.isOpen_iff_of_mem, FiberPrebundle.continuous_trivChange, Fin.preimage_succ_Ioi_succ, Cardinal.mk_preimage_equiv, Dynamics.coverMincard_image_le, AddSubgroup.coe_unop, IsChain.preimage_iso, Subgroup.unop_closure, MeasureTheory.Measure.measure_preimage_neg, Filter.HasBasis.uniformity_nonemptyCompacts, OnePoint.isOpen_iff_of_mem', AddGroupFilterBasis.neg', MeasureTheory.Measure.pi_eval_preimage_null, Isometry.preimage_closedEBall, setOf_liouvilleWith_subset_aux, OrderEmbedding.preimage_uIoc, NonUnitalSubring.closure_preimage_le, MeasureTheory.inter_cylinder, MeasureTheory.Measure.ae_mem_finset_iff_map_eq_sum_dirac, AlgebraicGeometry.quasiCompact_iff, restrictPreimage_isClosedEmbedding, Fin.image_cast_fun, toDual_sInf, RelIso.preimage_eq_image_symm, TopCat.range_prod_map, ProbabilityTheory.Kernel.compProd_null, preimage_preimage, MeasureTheory.Measure.haar.index_elim, OpenPartialHomeomorph.trans_source', OrderEmbedding.preimage_Ici, OpenPartialHomeomorph.preimage_interior, HasFTaylorSeriesUpToOn.comp_continuousAffineMap, EReal.preimage_coe_Ioi, OpenAddSubgroup.coe_comap, CocompactMap.isCompact_preimage_of_isClosed, HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn, mulIndicator_preimage, IsometryEquiv.preimage_emetric_closedBall, Filter.comap_lift'_eq, Finsupp.comapDomain_smul_of_injective, OrderEmbedding.preimage_Icc, ContinuousOn.preimage_isClosed_of_isClosed, IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal, Nat.preimage_Ici, Topology.IsLocallyConstructible.iff_of_isOpenCover, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app', PartialEquiv.IsImage.iff_symm_preimage_eq, isClosedMap_iff_comap_nhds_le, inter_preimage_subset, prod_sub_preimage_iff, IsOpenMap.restrictPreimage, preimage_div, TopCat.GlueData.isOpen_iff, MeasureTheory.Conservative.frequently_measure_inter_ne_zero, ConcaveOn.comp_linearMap, Trivialization.pullback_target, TopologicalSpace.Fiber.sigmaIsoHom_inj, image_mul_right', extChartAt_preimage_inter_eq, preimage_sub, pairwise_disjoint_fiber, Topology.IsInducing.isSeparable_preimage, Equiv.piCongrLeft_preimage_univ_pi, preimage_div_const_Iio, MeasureTheory.measurable_measure_add_right, AmpleSet.preimage_iff, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, Finset.preimage_mul_left_singleton, image_preimage_inl_union_image_preimage_inr, NonemptyInterval.coe_dual, Fin.preimage_castSucc_Ico_castSucc, MeasureTheory.Measure.addHaar_preimage_smul, AddSubmonoid.coe_toSubmonoid_apply, map_extChartAt_nhdsWithin', LeftInvOn.image_inter', MvPolynomial.pow_idealOfVars, singleton_pi, Subtype.preimage_coe_nonempty, MvPolynomial.idealOfVars_eq_restrictSupportIdeal, preimage_const_sub_Icc, MeasureTheory.ae_measure_preimage_mul_right_lt_top, TopCat.coequalizer_isOpen_iff, EMetric.preimage_mul_right_closedBall, Fin.preimage_val_Ioc_val, SeparationQuotient.preimage_image_mk_closed, MeasureTheory.MeasurePreserving.measure_preimage_emb, AlgebraicGeometry.Scheme.Hom.irreducibleComponentsEquiv_symm_apply_coe, restrictPreimage_isOpenEmbedding, OrderEmbedding.preimage_Iic, Int.preimage_Ioi, Homeomorph.isPreconnected_preimage, Finset.coe_powerset, MonoidHom.fiberEquivKer_symm_apply, LinearIsometryEquiv.image_eq_preimage_symm, Finset.image_mul_left', restrictPreimage_bijective, Subtype.preimage_val_eq_preimage_val_iff, BijOn.subset_right, ProbabilityTheory.Kernel.deterministic_prod_apply', Trivialization.isImage_preimage_prod, CompactlyCoherentSpace.isOpen_iff, AddSubsemigroup.coe_toSubsemigroup_apply, HasProd.tprod_fiberwise, Metric.preimage_mul_right_closedEBall, Module.associatedPrimes.preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule, Topology.IsInducing.isCompact_preimage, MeasureTheory.Measure.comap_preimage, Cardinal.infinite_pigeonhole, RingSubgroupsBasis.rightMul, NumberField.mixedEmbedding.polarSpaceCoord_target, Metric.preimage_add_right_closedBall, Filter.HasAntitoneBasis.comap, Fin.preimage_natAdd_Icc_natAdd, IsLeast.dual, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, AddMonoidHom.fiberEquiv_symm_apply, MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_integrable, OrderDual.isBounded_preimage_ofDual, disjoint_image_left, MeasureTheory.SimpleFunc.restrict_lintegral, Complex.frontier_preimage_re, Fin.preimage_succ_Icc_succ, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app', TendstoUniformlyOn.comp, preimage_const_sub_Ioo, MeasureTheory.SimpleFunc.integral_eq_sum_filter, Setoid.ker_iff_mem_preimage, Fin.preimage_castSucc_uIoc_castSucc, IsAntichain.preimage_compl, LinearIsometry.preimage_closedBall, encard_preimage_val_le_encard_right, preimage_singleton_nonempty, preimage_const_mul_Icc, preimage_add_const_Ioc, Fin.preimage_succ_Iic_succ, CategoryTheory.GradedObject.instFiniteElemProdNatPreimageHAddFstSndSingletonSet, FirstOrder.Language.Substructure.coe_comap, ProbabilityTheory.Kernel.map_apply', eval_preimage, SimpleGraph.Subgraph.comap_verts, Cardinal.infinite_pigeonhole_card_lt, preimage_const_add_Ioi, preimage_eq_preimage', SubMulAction.map_ofFixingSubgroupUnion_def, NumberField.mixedEmbedding.negAt_preimage, OrderEmbedding.preimage_Ioc, WithTop.preimage_coe_top, MeasureTheory.Measure.map_apply, pi_diff_pi_subset, IndexedPartition.piecewise_preimage, MeasureTheory.Content.is_mul_left_invariant_outerMeasure, MDifferentiableWithinAt.differentiableWithinAt_writtenInExtChartAt, TopologicalSpace.Closeds.coe_preimage, preimage_const, Metric.cthickening_eq_preimage_infEdist, image_restrictPreimage, toDual_sSup, MeasurableEquiv.comap_apply, OrderEmbedding.inter_preimage_setOf_maximal_eq_of_subset, WithTop.preimage_coe_Ici, Topology.IsInducing.isClosed_preimage, Topology.IsLower.continuous_iff_Ici, Subsemiring.op_sInf, CategoryTheory.GradedObject.CofanMapObjFun.inj_iso_hom, SmoothBumpFunction.exists_r_pos_lt_subset_ball, MeasureTheory.OuterMeasure.map_apply, kernImage_preimage_eq_iff, OpenPartialHomeomorph.source_inter_preimage_inv_preimage, EReal.preimage_coe_Ioo, Cardinal.mk_preimage_down, Pretrivialization.domExtend_baseSet, PrimeSpectrum.coe_preimageHomeomorphFiber_apply_asIdeal, Finset.restrict₂_preimage, CompactlyCoherentSpace.isClosed_iff, IsAntichain.preimage_iso_iff, OrderEmbedding.inter_preimage_setOf_minimal_eq_of_subset, image_inter_preimage, preimage_invFun_of_mem, AddSubsemigroup.unop_sSup, MeasureTheory.cylinder_eq_cylinder_union, AddSubgroup.unop_sInf, preimage_mul_const_Ico, Metric.thickening_eq_preimage_infEDist, preimage_eq, NFA.cons_preimage_acceptsFrom, set_smul_sphere_zero, MeasureTheory.lintegral_indicator_const_comp, Algebra.QuasiFinite.isDiscrete_comap_preimage_singleton, Fin.preimage_castLE_uIoc_castLE, preimage_inl_range_inr, IsLowerSet.preimage, MeasurableEmbedding.measurableSet_preimage, ofDual_preimage_latticeClosure, ProbabilityTheory.iCondIndepFun_iff_condExp_inter_preimage_eq_mul, coborder_preimage, Fin.preimage_natAdd_uIoc_natAdd, Finite.preimage_embedding, preimage_add_const_uIcc, preimage_image_univ, preimage_coe_mem_nhds_subtype, Topology.IsQuotientMap.isOpen_preimage, MeasureTheory.addEquivAddHaarChar_smul_preimage, SeparationQuotient.tendsto_lift₂_nhdsWithin, LinearEquiv.image_symm_eq_preimage, extChartAt_preimage_mem_nhdsWithin', preimage_add_sphere, StrictConvex.is_linear_preimage, AddSubsemigroup.toSubsemigroup'_closure, ConvexCone.coe_comap, preimage_const_of_notMem, ContinuousOn.preimage_isClopen_of_isClopen, Nat.preimage_Ioc, QuotientGroup.preimage_image_mk_eq_mul, IsLocalization.AtPrime.coe_primeSpectrumOrderIso_apply_coe_asIdeal, CategoryTheory.HomOrthogonal.matrixDecomposition_id, UpperSemicontinuous.isClosed_preimage, isLowerSet_preimage_toDual_iff, Filter.HasBasis.uniformity_closeds, PrimeSpectrum.coe_preimageOrderIsoFiber_symm_apply_coe_asIdeal, RingFilterBasis.mul_left, MeasureTheory.measure_preimage_vadd_of_nullMeasurableSet, OpenSubgroup.coe_comap, MeasureTheory.measurable_measure_mul_right, AlgebraicGeometry.Scheme.Hom.isDiscrete_preimage_singleton, restrictPreimage_isEmbedding, UpperSemicontinuousOn.inter_biInter_preimage_Ici_eq_empty_iff_exists_finset, Real.volume_preimage_mul_left, Equiv.piCongrLeft_preimage_pi, range_restrictPreimage, PartialEquiv.trans_target', Matroid.sum'_indep_iff, ContinuousAffineEquiv.image_preimage, restrictPreimage_coe, continuousOn_iff, MeasureTheory.union_cylinder, ModelWithCorners.uniqueDiffOn_preimage, MeasureTheory.measure_mul_right_null, Filter.mem_inv, Nat.preimage_Icc, PFun.coe_preimage, IsOpenMap.preimage_closure_eq_closure_preimage, MulHom.mclosure_preimage_le, EMetric.preimage_mul_left_closedBall, preimage_snd_singleton_eq_range, AddSubgroup.unop_closure, MeasureTheory.Measure.nullMeasurableSet_preimage_fst, Finset.preimage_add_left_zero, AddSubgroup.op_sInf, Fin.preimage_addNat_Iic_addNat, Subtype.image_preimage_coe, OrderIso.map_sInf_eq_sInf_symm_preimage, IsAddUnit.preimage_vadd_setₛₗ, MapsTo.subset_preimage, Fin.preimage_apply_01_prod, mem_intrinsicFrontier, MonoidHom.closure_preimage_le, PartialEquiv.image_eq_target_inter_inv_preimage, UniqueMDiffOn.uniqueDiffOn_target_inter, MeasureTheory.MeasurePreserving.setIntegral_preimage_emb, prod_preimage_eq, Pretrivialization.restrictPreimage'_target, preimage_subtype_val_Iic, Topology.IsInducing.isLindelof_preimage, PartialEquiv.IsImage.iff_preimage_eq, LinearMap.span_preimage_le, infClosed_preimage_toDual, Submonoid.toAddSubmonoid_closure, MeasureTheory.Content.outerMeasure_preimage, CompleteLatticeHom.setPreimage_apply, Subtype.coe_preimage_self, MeasureTheory.AEDisjoint.preimage, WithTop.preimage_coe_Iio, Seminorm.closedBall_zero_eq_preimage_closedBall, WithTop.preimage_coe_Ioc, Nat.preimage_Ioo, Dynamics.IsDynCoverOf.preimage, Metric.preimage_smul_closedEBall, ModelWithCorners.boundary_open, IsOpenQuotientMap.dense_preimage_iff, MeasureTheory.Measure.restrict_map, Module.comap_freeLocus_le, Nat.preimage_floor_of_ne_zero, preimage_id, MeasureTheory.addMeasure_map_restrict_apply, ContinuousLinearMap.interior_preimage, Finset.image_add_right, ContinuousMap.spectrum_eq_preimage_range, ENat.measurable_iff, Metric.preimage_add_left_closedBall, sphere_pi, Fin.preimage_castAdd_Ici_castAdd, mapsTo_extChartAt, Affine.Simplex.interior_restrict, Metric.exists_forall_closedEBall_subset_aux₁, Nat.preimage_closedBall, Fin.preimage_cast_Ici, Finset.preimage_mul_right_one', image_diff_preimage, Diffeomorph.image_eq_preimage_symm, preimage_const_mul_Ico₀, EReal.preimage_coe_Ioo_top, preimage_inter_range, QuasiconvexOn.isPreconnected_preimage_subtype, preimage_extChartAt_eventuallyEq_compl_singleton, upperHemicontinuousWithinAt_iff_preimage_Iic, Subalgebra.op_adjoin, IsSublattice.of_dual, ContinuousAt.preimage_mem_nhds, IsometryEquiv.ediam_preimage, AlgebraicGeometry.PresheafedSpace.GlueData.pullback_base, Complex.interior_preimage_re, AlgebraicGeometry.Scheme.Hom.app_appIso_inv_assoc, preimage_const_mul_Ico_of_neg, mdifferentiableOn_iff_target, Codisjoint.preimage, MeasureTheory.mem_ae_of_mem_ae_map, TopologicalSpace.IsTopologicalBasis.inf_induced, MeasureTheory.Measure.measure_preimage_isAddLeftInvariant_eq_smul_of_hasCompactSupport, Fin.preimage_cast_Iic, NumberField.mixedEmbedding.fundamentalCone.subset_interior_normLeOne, univ_prod, Subsemiring.unop_closure, diag_preimage_prod, smul_preimage_set_subset, Metric.preimage_vadd_ball, DiscreteQuotient.fiber_eq, MeasurableEquiv.symm_preimage_preimage, Matroid.mapEmbedding_isBase_iff, OnePoint.coe_preimage_infty, PrimeSpectrum.preimage_comap_zeroLocus, Rat.preimage_cast_Ioi, preimage_eq_univ_iff, AddCircle.add_projection_respects_measure, Metric.Snowflaking.preimage_toSnowflaking_emetricClosedBall, MeasureTheory.dense_of_generateFrom_isSetRing, TopologicalSpace.IsOpenCover.isClosedEmbedding_iff_restrictPreimage, IsometryEquiv.preimage_emetric_ball, map_fst_nhdsWithin, StableUnderSpecialization.preimage, MeasureTheory.SimpleFunc.pair_preimage_singleton, preimage_sub_preimage_subset, Equiv.Perm.image_inv, Trivialization.restrictPreimage'_source, Subgroup.closure_closure_coe_preimage, ProbabilityTheory.setBernoulli_apply', Metric.Snowflaking.isBounded_preimage_ofSnowflaking_iff, QuotientAddGroup.preimage_image_mk, MeasureTheory.SimpleFunc.preimage_eq_empty_iff, StableUnderGeneralization.preimage, Submonoid.op_sSup, lowerSemicontinuousOn_iff_preimage_Ioi, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, preimage_mul_const_uIcc, MeasureTheory.MeasurePreserving.preimage_null, OpenPartialHomeomorph.eventually_nhdsWithin, AnalyticOn.compContinuousLinearMap, Int.preimage_Iio, Subring.comap_map_eq, Seminorm.ball_zero_eq_preimage_ball, ProbabilityTheory.Kernel.indepFun_iff_measure_inter_preimage_eq_mul, Topology.IsInducing.basis_nhds, Rat.preimage_cast_Icc, MeasurableSpace.measurableSet_invariants, Definable.preimage_comp, MeasureTheory.pdf.IsUniform.measure_preimage, AlgebraicGeometry.coe_opensRestrict_symm_apply, TopologicalSpace.IsOpenCover.isHomeomorph_iff_restrictPreimage, Topology.IsOpenEmbedding.isOpen_iff_preimage_isOpen, Diffeomorph.range_comp, Finsupp.supported_comap_lmapDomain, Subtype.preimage_ball, CategoryTheory.GradedObject.mapBifunctorRightUnitorCofan_inj, lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl, Interval.coe_dual, Submonoid.op_sInf, CategoryTheory.essImage_yonedaMon, AddMonoidHom.fiberEquivKer_apply, Pretrivialization.preimage_symm_proj_baseSet, AlgebraicGeometry.Scheme.Hom.isCompact_preimage_singleton, ContinuousAffineEquiv.preimage_symm, UniqueMDiffOn.uniqueMDiffOn_target_inter, MeasureTheory.Measure.addHaar_preimage_continuousLinearMap, Topology.IsConstructible.preimage_of_isOpenEmbedding, SeparationQuotient.continuousWithinAt_lift, Nonempty.preimage', range_inter_ssubset_iff_preimage_ssubset, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, OrderIso.preimage_Ioo, AddMonoidHom.coe_mker, measurableSet_preimage_down, SmoothBumpFunction.support_eq_inter_preimage, Iio_toDual, Equiv.piCongrLeft_symm_preimage_univ_pi, UpperHemicontinuousWithinAt.isInducing_comp, MeasureTheory.Measure.prod_apply_le, prod_univ, Equiv.toHomeomorph_refl, AddMonoidHom.mclosure_preimage_le, Filter.comap_smallSets, EuclideanGeometry.preimage_inversion_perpBisector, EMetric.preimage_mul_left_ball, Fin.preimage_val_Ioi_val, Fin.preimage_addNat_uIoc_addNat, Finset.card_preimage_eq_sum_card_image_eq, IsTransitiveRel.preimage_prodMap, preimage_const_add_Ici, PartialEquiv.transEquiv_target, ContinuousWithinAt.comp_inter, LeftInvOn.image_inter, OnePoint.isClosed_iff_of_mem, image_preimage_eq_range_inter, WithTop.preimage_coe_Ico, Fin.preimage_succ_Ioc_succ, RingSubgroupsBasis.leftMul, MeasureTheory.MeasurePreserving.setLIntegral_comp_preimage, MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group, MonoidHom.fiberEquiv_symm_apply, TopologicalSpace.IsOpenCover.isClosed_iff_coe_preimage, StrictConvexOn.translate_left, AlgebraicGeometry.Scheme.GlueData.isOpen_iff, Subalgebra.coe_op, prod_subset_preimage_fst, MeasureTheory.Measure.ae_eq_or_eq_iff_map_eq_dirac_add_dirac, StrictConvexOn.translate_right, Filter.mem_coprodᵢ_iff, Fin.preimage_val_uIcc_val, preimage_mul_const_Ioi, IsClosedMap.restrictPreimage, ContMDiffWithinAt.mpullbackWithin_vectorField_inter, bddBelow_preimage_toDual, Subsemiring.op_closure, Ici_ofDual, Subgroup.smul_opposite_image_mul_preimage, preimage_const_add_Ico, Pretrivialization.restrictPreimage_toFun, PartialEquiv.symm_image_target_inter_eq, MapsTo.preimage_mem_nhdsWithin, Equiv.eq_preimage_iff_image_eq, OpenPartialHomeomorph.target_inter_inv_preimage_preimage, MeasurableEquiv.restrict_map, Topology.WithLowerSet.isLowerSet_toLowerSet_preimage, Fin.preimage_succ_Iio_succ, ContinuousLinearEquiv.iteratedFDerivWithin_comp_right, OrdConnected.preimage_anti, MeasurableEmbedding.restrict_map, AlgebraicGeometry.Scheme.Hom.isConstructible_preimage, connectedComponents_preimage_singleton, Filter.HasBasis.uniformity_compacts, Measurable.measurableSet_preimage_iff_of_surjective, MvPolynomial.pow_idealOfVars_eq_span, preimage_mul_preimage_subset, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux, PrimeSpectrum.preimageEquivFiber_apply_asIdeal, Equiv.transPartialEquiv_source, MeasureTheory.dense_of_generateFrom_isSetSemiring, WeakDual.isCompact_closedBall, Topology.isQuotientMap_iff, ProbabilityTheory.Kernel.ae_kernel_lt_top, AlgebraicGeometry.ExistsHomHomCompEqCompAux.exists_index, Fin.preimage_castSucc_uIcc_castSucc, LocallyConstant.ofIsClopen_fiber_zero, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, ContinuousOn.isOpen_inter_preimage, Metric.preimage_vadd_closedEBall, PartialEquiv.IsImage.symm_preimage_eq, TopCat.range_pullback_map, AddGroupFilterBasis.conj', MeasureTheory.SimpleFunc.lintegral_restrict, AntilipschitzWith.isBounded_preimage, Topology.isQuotientMap_iff_isClosed, preimage_const_mul_Iic, preimage_mul_const_Iic_of_neg, preimage_injective, UpperHemicontinuousAt.isInducing_comp, Function.Semiconj.mapsTo_preimage, PFun.core_restrict, OrderIso.isGLB_preimage, MeasureTheory.Measure.prod_apply, IsUpperSet.preimage, AddSubsemigroup.op_sInf, CategoryTheory.GradedObject.mapBifunctorLeftUnitorCofan_inj, EMetric.exists_forall_closedBall_subset_aux₂, ContMDiffOn.mpullbackWithin_vectorField_inter, UniformEquiv.image_preimage, connectedComponents_preimage_image, Subtype.preimage_closedEBall, IsCoveringMapOn.isCoveringMap_restrictPreimage, IsGreatest.dual, isProperMap_iff_isCompact_preimage, continuous_iff_preimage_interior_subset_interior_preimage, PartialEquiv.trans_refl_restr, Fin.preimage_natAdd_Ioo_natAdd, PartialEquiv.trans_source', OpenPartialHomeomorph.continuousWithinAt_writtenInExtend_iff, PrimitiveSpectrum.hull_finsetInf, PartialEquiv.trans_refl_restr', Topology.reorderRestrictProd_restrict_compl, preimage_const_mul_Ioi_of_neg, MeasureTheory.SimpleFunc.integral_eq_sum, continuous_bool_rng, Function.Embedding.sumSet_preimage_inr, Dynamics.coverEntropy_restrict_subset, SeparatedNhds.preimage, AddSubgroup.toSubgroup_closure, Sublattice.coe_comap, Fin.preimage_castLE_Ioo_castLE, Fin.preimage_cast_uIoo, MeasureTheory.Measure.compProd_apply, Complex.interior_preimage_im, preimage_id', Manifold.LocalPresentationAt.source_subset_preimage_source, uIoo_toDual, Homeomorph.isCompact_preimage, mem_nhds_subtype, Fin.preimage_castLE_Ioi_castLE, MeasureTheory.setIntegral_map_equiv, LinearIsometryEquiv.preimage_ball, mk_preimage_prod_right_eq_if, injOn_preimage, Fin.preimage_succ_uIcc_succ, upperHemicontinuousOn_iff_preimage_Iic, OpenPartialHomeomorph.map_extend_symm_nhdsWithin, OrderIso.map_sSup_eq_sSup_symm_preimage, SupClosed.dual, Metric.Snowflaking.image_ofSnowflaking_eq_preimage, IsClosed.preimage_val, ContinuousWithinAt.comp_inter_of_eq, Topology.IsCoherentWith.isClosed_iff, MeasureTheory.setIntegral_map, extChartAt_mem_closure_interior, ConvexOn.translate_left, UniqueMDiffWithinAt.bundle_preimage', WithBot.preimage_coe_Ioc, Dynamics.coverEntropyEntourage_image_le, Module.support_subset_preimage_comap, IsometryEquiv.preimage_closedEBall, GaloisConnection.lowerBounds_u_image, LowerSemicontinuousOn.isCompact_inter_preimage_Iic, Filter.comap_lift'_eq2, preimage_const_mul_Ioc, OrderIso.bddBelow_preimage, Filter.mem_pi_of_mem, Subring.op_sSup, ContinuousMap.restrictPreimage_apply, ConcaveOn.comp_affineMap, UniqueAdd.addHom_preimage, measurableSet_swap_iff, ProbabilityTheory.comap_cond, Matroid.mapEmbedding_indep_iff, image_preimage_inter, MeasureTheory.SimpleFunc.map_lintegral, IsClosedMap.comap_nhdsSet_eq, Subsemiring.unop_sInf, preimage_div_const_Ioi, IsAddUnit.preimage_vadd_set, Rat.preimage_cast_Ico, MulAction.IsBlock.preimage, preimage_mul_left_one, CategoryTheory.essImage_yonedaGrp, extChartAt_preimage_mem_nhds, Submodule.comap_coe, EReal.preimage_coe_Icc, DFA.accepts_comap, Finset.image_add_right', WithBot.preimage_coe_Ioi, preimage_pow_subset, image_of_range_union_range_eq_univ, MeasureTheory.ae_measure_preimage_mul_right_lt_top_of_ne_zero, MeasureTheory.SimpleFunc.memLp_iff, sigma_preimage_eq, Fin.preimage_rev_Icc, OpenPartialHomeomorph.extend_target, Metric.preimage_add_left_closedEBall, IsMaxOn.on_preimage, indicator_zero_preimage, AddSubmonoid.toSubmonoid_closure, eq_preimage_iff_image_eq, finite_preimage_inl_and_inr, preimage_interior_subset_interior_preimage, IsOpen.dense_iUnion_preimage_vadd, Complex.preimage_equivRealProd_prod, Topology.reorderRestrictProd_mem_preimage_image_restrict, PrimeSpectrum.preimage_specComap_zeroLocus, preimage_const_add_Iic, preimage_boolIndicator_eq_union, Equiv.piCongrLeft_symm_preimage_pi, preimage_const_mul_Ici₀, graphOn_prod_graphOn, dual_ordConnected, MeasureTheory.Measure.map_eq_sum, Topology.IsClosedEmbedding.preimage_closedPoints, NNRat.preimage_cast_Iic, iUnion_image_preimage_sigma_mk_eq_self, MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm, SeparationQuotient.continuousOn_lift, AddGroup.preimage_vadd_set, AmpleSet.preimage, Matroid.cRk_map_eq, map_snd_nhdsWithin, Subsemigroup.toAddSubsemigroup_closure, ContMDiffWithinAt.mpullback_vectorField_preimage, Polynomial.derivativeFinsupp_derivative, DiscreteQuotient.isClopen_preimage, ProbabilityTheory.Kernel.id_prod_apply', Ultrafilter.mem_map, contMDiffWithinAt_iff_of_mem_source', MeasurableSpace.measurableSet_comap, Finsupp.embDomain_comapDomain, Algebra.QuasiFinite.iff_finite_comap_preimage_singleton, ProperCone.coe_comap, AddSubsemigroup.closure_closure_coe_preimage, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_aux, Convex.is_linear_preimage, Filter.HasBasis.cobounded_of_norm', Finsupp.mapDomain_comapDomain, IsCoveringMap.monodromy_map, ModuleFilterBasis.smul_left', preimage_smul_setₛₗ_of_isUnit_isUnit, Pretrivialization.source_eq, preimage_val_iInter, OpenPartialHomeomorph.symm_image_target_inter_eq, Matroid.sum_isBase_iff, Metric.preimage_smul_eball, Fin.preimage_castAdd_Ioo_castAdd, IsClosedMap.comap_nhds_le, Filter.sInter_comap_sets, TopologicalSpace.Opens.coe_overEquivalence_functor_obj, Nat.preimage_Iic, Nat.preimage_ceil_of_ne_zero, ofDual_sInf, Topology.IsQuotientMap.isClopen_preimage, MeasureTheory.ProbabilityMeasure.map_apply_of_aemeasurable, image_mul_right, ComplexShape.symmetryEquiv_apply_coe, seminormFromBounded_is_norm_iff, NNRat.preimage_cast_Ioi, ComplexShape.symmetryEquiv_symm_apply_coe, MeasureTheory.forall_measure_preimage_mul_iff, Metric.preimage_vadd_sphere, AddAction.IsTrivialBlock.preimage, encard_preimage_of_injective_subset_range, upperHemicontinuousAt_iff_preimage_Iic, ContMDiffWithinAt.comp', Metric.Snowflaking.preimage_toSnowflaking_closedBall, EMetric.preimage_add_right_ball, ContMDiffOn.continuousOn_tangentMapWithin, IsConnected.preimage_of_isClosedMap, Cardinal.infinite_pigeonhole_card, preimage_eq_iff_eq_image, contMDiff_iff_target, Homeomorph.preimage_symm, AlgebraicGeometry.Scheme.Pullback.range_map, MDifferentiableWithinAt.mpullbackWithin_vectorField_inter_of_eq, TopologicalSpace.Compacts.coe_equiv_apply_eq_preimage, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'_assoc, QuasispectrumRestricts.subset_preimage, Finsupp.comapDomain_surjective, MonoidHom.fiberEquivKer_apply, Finsupp.comapDomain_mapDomain, SSet.degenerate_le_preimage, preimage_mul_const_Iic, image_sigmaMk_preimage_sigmaMap_subset, Equiv.Perm.preimage_inv, AddSubgroup.op_closure, IsOpen.dense_iUnion_preimage_smul, Topology.WithUpperSet.isUpperSet_toUpperSet_preimage, Rat.preimage_cast_uIcc, QuasilinearOn.isPreconnected_preimage_subtype, VectorField.mlieBracketWithin_def, preimage_const_mul_Iio_of_neg, Metric.preimage_mul_right_eball, ContMDiffWithinAt.mpullback_vectorField_preimage_of_eq, Fin.preimage_val_Iic_val, Fin.preimage_castAdd_Ioi_castAdd, NumberField.mixedEmbedding.fundamentalCone.normLeOne_eq_preimage_image, vadd_preimage_set_subset, Trivialization.restrictPreimage'_baseSet, Subgroup.coe_toAddSubgroup_symm_apply, OpenPartialHomeomorph.IsImage.iff_preimage_eq', OpenPartialHomeomorph.preimage_frontier, Trivialization.preimage_subset_source, IsOpenMap.preimage_interior_eq_interior_preimage, MeasureTheory.OuterMeasure.map_ofFunction, indicator_comp_right, upperSemicontinuousOn_iff_preimage_Iio, PartialEquiv.target_subset_preimage_source, MeasureTheory.L1.SimpleFunc.norm_eq_sum_mul, prod_quotient_preimage_eq_image, WeakDual.isSeqCompact_closedBall, Filter.map_generate_le_generate_preimage_preimage, MeasureTheory.smulInvariantMeasure_iff, prod_eq, DomMulAct.mk_smul_indicatorConstLp, DenseRange.subset_closure_image_preimage_of_isOpen, Fin.preimage_val_Ico_val, isOpen_extChartAt_preimage, Fin.preimage_cast_Ioi, EuclideanGeometry.preimage_inversion_sphere_dist_center, MeasureTheory.Measure.integrable_measure_prodMk_left, MeasureTheory.SimpleFunc.measurableSet_fiber', ProbabilityTheory.Kernel.measurable_kernel_prodMk_left', Topology.WithScott.isOpen_iff_isUpperSet_and_scottHausdorff_open', MeasureTheory.SimpleFunc.lintegral_eq_of_subset', GroupFilterBasis.conj', AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_range, mdifferentiable_iff, Equiv.sumPiEquivProdPi_symm_preimage_univ_pi, MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_opNorm, Subfield.coe_comap, preimage_sub_const_Ico, MeasureTheory.Measure.measure_ae_null_of_prod_null, Fin.preimage_addNat_Ioi_addNat, Fin.preimage_castAdd_uIoo_castAdd, CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_symm_apply, Fin.preimage_addNat_Iio_addNat, Metric.cthickening_eq_preimage_infEDist, OpenPartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_right, RingFilterBasis.mul_left', Fin.preimage_natAdd_Ioc_natAdd, CategoryTheory.Subfunctor.preimage_obj, TopologicalSpace.Opens.overEquivalence_unitIso_hom_app_left, Dynamics.le_coverEntropyEntourage_image, Algebra.quasiFiniteAt_iff_isOpen_singleton_fiber, Cardinal.mk_preimage_of_subset_range, uIcc_ofDual, preimage_sub_const_Iio, Complex.frontier_preimage_im, QuotientGroup.dense_preimage_mk, OrderIso.preimage_Iio, StrictConvex.linear_preimage, measurableSet_sum_iff, AnalyticOnNhd.preimage_zero_mem_codiscrete, Submonoid.coe_op, preimage_vadd_neg, isLowerSet_preimage_ofDual_iff, SeminormFamily.basisSets_neg, Fin.image_rev_fun, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, ContinuousWithinAt.preimage_mem_nhdsWithin', image_preimage, Algebra.smoothLocus_comap_of_isLocalization, Topology.IsClosedEmbedding.setIntegral_map, extChartAt_preimage_mem_nhdsWithin, AddGroupFilterBasis.neg, IsChain.preimage_relIso, Topology.WithUpper.isOpen_preimage_ofUpper, Topology.WithLower.isOpen_def, UniqueMDiffWithinAt.preimage_openPartialHomeomorph, Fin.preimage_castAdd_Iio_castAdd, TopologicalSpace.Opens.IsBasis.of_isInducing, mk_preimage_prod_right_eq_empty, indicator_const_preimage, ConvexOn.translate_right, image_preimage_eq, Submodule.span_preimage_eq, TopologicalSpace.Fiber.sigmaIsoHom_apply, MeasureTheory.disjoint_cylinder_iff, Fin.preimage_succ_Ico_succ, IsEvenlyCovered.toTrivialization_apply, Topology.WithLawson.isOpen_def, NumberField.mixedEmbedding.fundamentalCone.closure_normLeOne_subset, Subalgebra.unop_sSup, mulTSupport_comp_subset_preimage, Seminorm.preimage_metric_closedBall, preimage_const_div_Ioi, CategoryTheory.Functor.toPreimages_map, Finset.preimage_compl, AEMeasurable.nullMeasurableSet_preimage, Ico_toDual, extChartAt_preimage_mem_nhds', preimage_comp, MeasureTheory.SimpleFunc.pair_preimage, QuotientAddGroup.sound, Algebra.QuasiFinite.finite_comap_preimage, Subsemiring.isHomogeneous_iff_subset_iInter, Profinite.exists_isClopen_of_cofiltered, Isometry.preimage_sphere, SpectrumRestricts.subset_preimage, Icc_ofDual, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app'_assoc, NNRat.preimage_cast_uIcc, mulIndicator_const_preimage, Metric.preimage_smul_sphere, OnePoint.isOpen_iff_of_notMem, NonUnitalRingHom.sclosure_preimage_le, EReal.preimage_coe_Ici, preimage_add_preimage_subset, AddSubmonoid.unop_sInf, AddGroup.preimage_vadd_setₛₗ, NNRat.preimage_cast_Icc, Ioo_ofDual, Infinite.preimage, preimage_range, IsCompact.preimage_vadd, Subsemigroup.coe_unop, OpenPartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq_aux, StarConvex.preimage_smul, CategoryTheory.HomOrthogonal.matrixDecompositionLinearEquiv_apply, MeasureTheory.SimpleFunc.FinMeasSupp.meas_preimage_singleton_ne_zero, Metric.preimage_mul_left_ball, Finset.preimage_mul_right_one, IsCoveringMapOn.restrictPreimage, OrderIso.symm_preimage_preimage, MeasureTheory.exists_ne_forall_mem_nhds_pos_measure_preimage, OpenPartialHomeomorph.map_nhdsWithin_preimage_eq, contMDiffWithinAt_iff_source_of_mem_maximalAtlas, AddSubsemigroup.op_sSup, MeasureTheory.mem_map_restrict_ae_iff, ProbabilityTheory.iIndepFun.cond, Fin.preimage_castLE_Ioc_castLE, InfClosed.preimage, Subtype.preimage_coe_inter_self, preimage_val_image_val_eq_self, Fin.preimage_rev_Ioi, Matroid.comap_closure_eq, CategoryTheory.GradedObject.CofanMapObjFun.ιMapObj_iso_inv, TangentBundle.trivializationAt_source, MeasureTheory.Measure.fst_apply, spectrum.preimage_algebraMap, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, OpenPartialHomeomorph.image_source_inter_eq, preimage_add_closedBall, Homeomorph.image_symm, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, isClosedMap_iff_comap_nhdsSet_le, QuotientAddGroup.preimage_image_mk_eq_add, ContinuousAlgEquiv.image_eq_preimage_symm, update_preimage_univ_pi, TopCat.isClosed_iff_of_isColimit, preimage_liftCover, Subring.op_sInf, preimage_ite, WithBot.preimage_coe_Iio, AddSubsemigroup.unop_sInf, UniformEquiv.image_symm, MeasureTheory.NullMeasurableSet.preimage, IsCoveringMap.monodromy_refl, MeasureTheory.SimpleFunc.integrable_iff, IsGLB.dual, Int.preimage_Iic, ContDiffOn.comp_inter, Matroid.restrictSubtype_inter_indep_iff, LinearIsometry.preimage_ball, Metric.Snowflaking.ediam_preimage_ofSnowflaking, Topology.IsClosedEmbedding.isClosed_iff_preimage_isClosed, Manifold.IsImmersionAtOfComplement.source_subset_preimage_source, preimage_mul_const_Ioo₀, Subring.unop_sInf, measurable_measure_prodMk_left_finite, MeasureTheory.SMulInvariantMeasure.measure_preimage_smul, FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter, IsProperMap.isCompact_preimage, Isometry.preimage_closedBall, Isometry.preimage_setOf_dist, Metric.preimage_add_left_ball, image_preimage_eq_inter_range, StructureGroupoid.liftPropWithinAt_self_target, AddSubgroup.coe_op, biUnion_range_preimage_singleton, Fin.preimage_succ_Ioo_succ, MeasureTheory.MeasurePreserving.measure_preimage_equiv, uIcc_toDual, AddCircle.coe_real_preimage_closedBall_inter_eq, Metric.preimage_smul_closedBall, AddSubmonoid.unop_closure, Complex.closure_preimage_re, IsOpen.inter_preimage_val_iff, Finsupp.degree_preimage_add, Filter.EventuallyEq.preimage, Ioc_toDual, Metric.preimage_vadd_closedBall, EReal.preimage_coe_Ioi_bot, MonoidHom.coe_mker, UniqueMul.mulHom_preimage, preimage_prod_map_prod, preimage_iterate_eq, Function.mulSupport_comp_eq_preimage, Subsemigroup.coe_comap, PartialEquiv.source_inter_preimage_inv_preimage, Metric.preimage_mul_left_closedEBall, Matroid.sum_indep_iff, InjOn.preimage_image_inter, Quotient.preimage_mem_nhds, Equiv.image_eq_preimage, LinearEquiv.image_eq_preimage_symm, PartialEquiv.EqOnSource.source_inter_preimage_eq, Function.Surjective.restrictPreimage, MeasureTheory.measurableSet_preimage_stoppedValue_inter, mapsTo_preimage, ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range, Function.support_comp_eq_preimage, ProbabilityTheory.Kernel.iIndepFun.measure_inter_preimage_eq_mul, subset_preimage_univ, Filter.mem_coprod_iff, MDifferentiableWithinAt.mfderivWithin, MeasureTheory.Measure.restrict_map_of_aemeasurable, preimage_nsmul_subset, ProbabilityTheory.IdentDistrib.measure_preimage_eq, contMDiffWithinAt_iff, CategoryTheory.Subfunctor.map, Rat.preimage_cast_Iic, Subsemiring.comap_toSubmonoid, EReal.preimage_coe_Ioc, Filter.comap_lift_eq2, Topology.IsOpenEmbedding.restrictPreimage, NonUnitalSubalgebra.coe_comap, Subgroup.unop_sInf, mk_preimage_prod_left, SurjOn.image_preimage, ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff, Fin.preimage_rev_Ioc, OrderEmbedding.preimage_uIcc, tangentConeAt_def, injective_iff_injective_of_iUnion_eq_univ, Ordinal.accPt_subtype, Subtype.preimage_coe_self_inter, Fin.preimage_val_uIoo_val, EMetric.exists_forall_closedBall_subset_aux₁, Fin.preimage_natAdd_uIcc_natAdd, Equiv.preimage_subset, upperHemicontinuous_iff_isOpen_preimage_Iic, image_add_right, continuous_iff_isClosed, upperHemicontinuous_iff_preimage_Iic, Metric.Snowflaking.image_toSnowflaking_eq_preimage, IsPiSystem.comap, IsometryEquiv.hausdorffMeasure_preimage, ContinuousWithinAt.preimage_mem_nhdsWithin, Fin.preimage_insertNth_Icc_of_notMem, Filter.comap_pure, MeasureTheory.FiniteMeasure.prod_apply, isOpen_deltaGenerated_iff, WithBot.preimage_coe_bot, preimage_mem_irreducibleComponents, UniformCauchySeqOn.comp, IsAntichain.preimage_iso, Subtype.preimage_closedBall, Equiv.Set.image_symm_preimage, MeasureTheory.measure_preimage_snd_singleton_eq_sum, MeasureTheory.measure_lintegral_div_measure, CategoryTheory.HomOrthogonal.matrixDecomposition_comp, SSet.Subcomplex.preimage_obj, MeasureTheory.measure_preimage_smul_null, AddSubmonoid.unop_sSup, Homeomorph.preimage_interior, MeasureTheory.tendsto_measure_symmDiff_preimage_nhds_zero, StrictConvex.affine_preimage, Filter.mem_comap, NumberField.mixedEmbedding.fundamentalDomain_stdBasis, Fin.preimage_natAdd_Ico_natAdd, Pretrivialization.restrictPreimage'_baseSet, Trivialization.source_inter_preimage_target_inter, preimage_smul, preimage_one, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp, Matroid.sum_isBasis_iff, WithBot.preimage_coe_Ici, ProbabilityTheory.lintegral_toKernel_mem, Submonoid.coe_comap, MeasureTheory.Measure.haar.addIndex_defined, ConcaveOn.translate_right, Metric.Snowflaking.preimage_toSnowflaking_eball, IsRetrocompact.preimage_of_isClosedEmbedding, preimage_empty, preimage_singleton_eq_empty, OpenPartialHomeomorph.extend_preimage_mem_nhds, isAntichain_preimage_subtypeVal, preimage_val_sInter_eq_sInter, ContDiffWithinAt.comp_continuousLinearMap, PartialEquiv.symm_image_target_inter_eq', ProbabilityTheory.cond_iInter, ProbabilityTheory.Kernel.parallelComp_apply', WithBot.preimage_coe_Ico, Metric.preimage_mul_right_ball, ContinuousOn.comp_inter, preimage_add_const_Ioo, MeasureTheory.measure_map_restrict_apply, preimage_smul_inv, Topology.image_snd_preimageImageRestrict, Filter.HasBasis.comap, Topology.IsConstructible.preimage_of_isClosedEmbedding, Topology.IsGeneratedBy.isOpen_iff, preimage_boolIndicator_true, preimage_eq_empty, Manifold.IsImmersionAt.source_subset_preimage_source, preimage_mul_const_Ici, QuotientGroup.preimage_image_mk_eq_iUnion_image, preimage_mul_const_Iic₀, isClosed_induced_iff, TopologicalSpace.Opens.overEquivalence_counitIso_inv_app, preimage_eq_core, RingFilterBasis.mul_right, Filter.tendsto_finset_preimage_atTop_atTop, WeakDual.isClosed_closedBall, Fin.image_rev, preimage_inr_image_inl, Filter.ker_comap, preimage_const_div_Ici, preimage_sigmaMap_sigma, Finset.coe_eraseNone, Iic_toDual, ProbabilityTheory.hasFiniteIntegral_prodMk_left, OrderIso.preimage_Icc, Rat.preimage_cast_Ici, range_codRestrict, preimage_const_mul_Ici_of_neg, WithTop.preimage_coe_Iic, AddSubgroup.op_sSup, dual_ordSeparatingSet, MeasureTheory.NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const, DeltaGeneratedSpace.isOpen_iff, preimage_mul_const_Ici_of_neg, preimage_mul_const_Ioc_of_neg, preimage_union, Submodule.coe_torsionBySet, HasFTaylorSeriesUpToOn.compContinuousLinearMap, prod_preimage_left, Matrix.SpecialLinearGroup.range_toGL, OpenPartialHomeomorph.trans_ofSet, range_subtype_map, Pretrivialization.restrictPreimage_source, TopologicalSpace.Opens.overEquivalence_unitIso_inv_app_left, Filter.Tendsto.countable_compl_preimage_ker, preimage_setOf_eq, RingHom.sclosure_preimage_le, isChain_preimage_subtypeVal, Fin.preimage_castLE_Iic_castLE, IsOpen.preimage_val, Fin.preimage_addNat_Icc_addNat, preimage_singleton_false, PrimitiveSpectrum.preimage_upperClosure_compl_finset, mdifferentiableWithinAt_iff', Matroid.mapSetEmbedding_indep_iff, IsCoveringMapOn.isLocalHomeomorphOn, Subgroup.toAddSubgroup_closure, IsAntichain.preimage_embedding, UniqueMDiffOn.tangentBundle_proj_preimage, preimage_mul_const_Iio, Topology.IsEmbedding.restrictPreimage, preimage_vadd_setₛₗ', preimage_sub_const_Ioc, Equiv.set_forall_iff, Subring.closure_preimage_le, AddSubmonoid.closure_closure_coe_preimage, AddSubsemigroup.coe_unop, Matroid.map_closure_eq, AlgebraicGeometry.locallyQuasiFinite_iff_isDiscrete_preimage_singleton, HasCountableSeparatingOn.subtype_iff, Equiv.preimage_symm_preimage, preimage_mul_right_singleton, StructureGroupoid.LocalInvariantProp.right_invariance', Fin.preimage_castLE_Icc_castLE, preimage_add_const_Ioi, isOpen_induced, CategoryTheory.GradedObject.CofanMapObjFun.inj_iso_hom_assoc, NNRat.preimage_cast_Ici, LinearIsometry.preimage_sphere, preimage_sub_const_Ioi, Subgroup.op_sInf, Pretrivialization.domExtend_invFun, preimage_sub_const_Ici, Continuous.frontier_preimage_subset, ProbabilityTheory.IndepFun.measure_inter_preimage_eq_mul, preimage_div_const_Ioc, Subalgebra.unop_adjoin, Subsemigroup.closure_closure_coe_preimage, Finset.restrict_preimage_univ, Subgroup.coe_toAddSubgroup_apply, ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff', Metric.Snowflaking.preimage_toSnowflaking_closedEBall, OrdConnected.preimage_coe_nnreal_ennreal, Finset.preimage_neg, tsupport_comp_subset_preimage, AddEquiv.addDissociated_preimage, Topology.IsEmbedding.isSeparable_preimage, preimage_subtype_val_Ico, Metric.Snowflaking.preimage_ofSnowflaking_ball, MeasureTheory.Measure.le_map_apply, preimage_subtype_val_Ioo, StrictConvex.preimage_smul, EReal.preimage_coe_Ioo_bot, image_add_left, MeasureTheory.isClosed_setOf_preimage_ae_eq, Ideal.coe_comap, contDiffWithinAtProp_self_target, ContinuousLinearEquiv.comp_right_differentiableOn_iff, unitInterval.image_coe_preimage_symm, AddSubmonoid.coe_unop, indicator_one_preimage, PrimeSpectrum.preimage_comap_zeroLocus_aux, Topology.IsOpenEmbedding.map_nhdsWithin_preimage_eq, preimage_const_mul_Icc₀, EMetric.preimage_mul_right_ball, contMDiffWithinAt_iff', OpenPartialHomeomorph.isOpen_extend_preimage', MeasureTheory.Measure.measure_preimage_of_map_eq_self, Subsemiring.isHomogeneous_iff_forall_subset, disjoint_image_right, OrderIso.bddAbove_preimage, TopCat.pullback_fst_image_snd_preimage, preimage_add_left_singleton, Equiv.preimage_eq_iff_eq_image, AddMonoidHom.preimage_vadd_setₛₗ, Topology.isConstructible_preimage_iff_of_isOpenEmbedding, MeasureTheory.setLIntegral_map, Matroid.sum'_isBasis_iff, OrderEmbedding.preimage_Iio, sigma_preimage_left, AddSubmonoid.op_sSup, MeasurableEmbedding.comap_restrict, ProbabilityTheory.condIndepFun_iff_condIndepSet_preimage, OrderTopology.continuous_iff, ContinuousLinearEquiv.image_eq_preimage_symm, MeasureTheory.integral_simpleFunc_larger_space, Function.Fiber.mem_iff_eq_image, Topology.IsQuotientMap.isClosed_preimage, mk_preimage_prod_left_fn_eq_if, MeasureTheory.measure_preimage_smul_of_nullMeasurableSet, PartialEquiv.image_source_inter_eq', MonoidHom.mclosure_preimage_le, ContinuousLinearEquiv.contDiffWithinAt_comp_iff, uniqueMDiffWithinAt_iff, IsometryEquiv.diam_preimage, MeasureTheory.integrableOn_map_equiv, preimage_const_mul_uIcc, Submodule.coe_torsionBy, subset_kernImage_iff, StrictConvex.linearIsometry_preimage, Function.Bijective.restrictPreimage, preimage_add, preimage_mono, map_extChartAt_nhdsWithin, OpenPartialHomeomorph.image_source_inter_eq', MDifferentiableOn.mpullback_vectorField_preimage, HasFPowerSeriesWithinOnBall.compContinuousLinearMap, bddBelow_preimage_ofDual, BddBelow.dual, mem_nhds_induced, Rat.preimage_cast_Iio, preimage_vadd_setₛₗ_of_isAddUnit_isAddUnit, infClosed_preimage_ofDual, MeasureTheory.Measure.haar.addIndex_elim, preimage_image_sigmaMk_of_ne, tendsto_cofinite_cocompact_iff, Fin.preimage_castAdd_Icc_castAdd, ContinuousLinearEquiv.image_symm_eq_preimage, Equiv.image_eq_preimage_symm, Nontrivial.preimage, preimage_coe_coe_diagonal, SSet.Subcomplex.fromPreimage_app_coe, lowerHemicontinuous_iff_isClosed_preimage_Iic, Homeomorph.isConnected_preimage, Filter.HasBasis.cobounded_of_norm, Rat.preimage_cast_uIoc, Filter.generate_image_preimage_le_comap, IsEvenlyCovered.restrictPreimage, Function.Injective.restrictPreimage, EReal.preimage_coe_Ico_top, Matroid.sigma_isBase_iff, Diffeomorph.contMDiffWithinAt_comp_diffeomorph_iff, ContinuousOn.preimage_mem_nhdsSetWithin_of_mem_nhdsSet, insert_pi, compact_covered_by_add_left_translates, OpenPartialHomeomorph.trans_source, AntilipschitzWith.dimH_preimage_le, preimage_mul_right_one, preimage_const_sub_Ioi, Isometry.hausdorffMeasure_preimage, isClosed_preimage_val_coborder, WithTop.preimage_coe_Ioo, Function.Embedding.sumSet_preimage_inl, ContinuousOn.preimage_interior_subset_interior_preimage, MeromorphicNFOn.zero_set_eq_divisor_support, Topology.lawsonClosed_of_scottClosed, OpenPartialHomeomorph.trans_target', contMDiffOn_iff, IsOpenMap.interior_preimage_subset_preimage_interior, preimage_closedPoints_subset, Filter.smallSets_comap, PrimeSpectrum.coe_primesOverOrderIsoFiber_apply_asIdeal, CategoryTheory.HomOrthogonal.matrixDecomposition_apply, measurable_measure_prodMk_left, UniqueMDiffOn.uniqueDiffOn_inter_preimage, ConcaveOn.translate_left, mdifferentiableWithinAt_iff_target_inter', Subsemigroup.op_closure, MeasureTheory.FiniteMeasure.map_apply, IsUnit.preimage_smul_set, IsStrongAntichain.preimage, Euclidean.closedBall_eq_preimage, Trivialization.restrictPreimage'_target, OrderIso.isLUB_preimage, Trivialization.pullback_baseSet, IsOpen.isOpen_inter_preimage_of_fderiv_eq_zero, StarConvex.preimage_add_left, TopologicalSpace.IsTopologicalBasis.isInducing, OpenPartialHomeomorph.isOpen_inter_preimage, GroupFilterBasis.inv, Trivialization.restrictPreimage'_apply, Algebra.adjoin_adjoin_coe_preimage, OpenPartialHomeomorph.symm_trans_restr, SetRel.isRefl_preimage, contMDiff_iff, SubAddAction.val_preimage_orbit, UniqueMDiffWithinAt.bundle_preimage, Fin.preimage_cast_Iio, AffineSubspace.comap_span, induced_generateFrom_eq, AntivaryOn.comp_right, Filter.mem_vadd_filter, MeasureTheory.Measure.QuasiMeasurePreserving.preimage_ae_eq, ULift.down_sSup, Iic_ofDual, preimage_mul_const_Ici₀, preimage_add_left_zero', Matroid.restrictSubtype_indep_iff_of_subset, IsCompact.preimage_smul, preimage_zero, MeasurableEquiv.image_symm, bijective_iff_bijective_of_iUnion_eq_univ, AnalyticAt.preimage_of_nhdsNE, preimage_const_sub_Ico, HasCompactSupport.isCompact_preimage, AlgebraicGeometry.IsFinite.finite_preimage_singleton, MeasureTheory.MeasurePreserving.aeconst_preimage, uniqueMDiffWithinAt_iff_inter_range, LinearMap.polar_eq_biInter_preimage, Ioo_toDual, mem_omegaLimit_iff_frequently, Bundle.TotalSpace.range_mk, MeasurableEmbedding.map_apply, OpenPartialHomeomorph.IsImage.iff_symm_preimage_eq, piecewise_comp, unitInterval.subtype_Ici_eq_Icc, Fin.preimage_castSucc_Iio_castSucc, SeparationQuotient.preimage_mk_closure, MeasureTheory.ae_measure_preimage_add_right_lt_top_of_ne_zero, QuotientAddGroup.preimage_image_coe, Cardinal.exists_infinite_fiber, isOpen_coinduced, Matroid.comap_ground_eq, Iio_ofDual, Homeomorph.image_eq_preimage_symm, UpperHalfPlane.atImInfty_basis, ContMDiffOn.comp', preimage_val_sInter, ProbabilityTheory.Kernel.snd_apply', Affine.Simplex.setInterior_restrict, Trivialization.Prod.continuous_to_fun, image_add_left', image_mul_left', image_sumElim, Subsemiring.op_sSup, IsClosed.inter_preimage_val_iff, Fin.preimage_rev_Iio, ordConnected_dual, preimage_add_right_singleton, PartialEquiv.symm_image_eq_source_inter_preimage, indicator_const_preimage_eq_union, OpenPartialHomeomorph.symm_image_eq_source_inter_preimage, measure_eq_measure_preimage_add_measure_tsum_Ico_zpow, unitInterval.subtype_Iio_eq_Ico, tendsto_nhds, QuasiconcaveOn.isPreconnected_preimage_subtype, RingEquiv.image_eq_preimage, TopCat.GlueData.preimage_range, Diffeology.isOpen_iff_preimages_plots, uncurry_preimage_prod_pi, mdifferentiableWithinAt_iff_of_mem_source', CompactlyCoherentSpace.isOpen_iff_forall_compactSpace, preimage_range_inter, OrderIso.preimage_Ici, Metric.Snowflaking.preimage_toSnowflaking_ball, VectorField.contMDiffWithinAt_mpullbackWithin_extChartAt_symm, Finset.preimage_add_right_singleton, preimage_eq_empty_iff, Measurable.measurableSet_preimage_iff_preimage_val, mdifferentiableWithinAt_iff_source_of_mem_maximalAtlas, HasFPowerSeriesWithinAt.isBigO_sub_partialSum_pow, CategoryTheory.HomOrthogonal.matrixDecompositionAddEquiv_symm_apply, Fin.preimage_cast_uIoc, Function.Injective.preimage_image, MeasureTheory.SimpleFunc.integral_eq_sum_of_subset, isUpperSet_preimage_ofDual_iff, OpenPartialHomeomorph.eventually_nhdsWithin', neg_preimage, MeasureTheory.Measure.map_apply_of_aemeasurable, AddMonoidHom.fiberEquiv_apply, image_eq_preimage_of_inverse, mdifferentiableWithinAt_iff, Fin.preimage_addNat_Ici_addNat, Subgroup.coe_unop, preimage_const_add_Iio, TopologicalSpace.IsOpenCover.isOpen_iff_coe_preimage, Finsupp.degree_preimage_nsmul, Int.preimage_Ioo, hom_trivializationAt_source, AddSubmonoid.op_sInf, preimage_const_mul_Ioc₀, WithBot.preimage_coe_Icc, IsCoveringMap.monodromyFunctor_obj, Submonoid.unop_sInf, EReal.preimage_coe_Iio, subset_preimage_val_image_val_iff, ProbabilityTheory.Kernel.compProd_preimage_fst, MulAction.IsTrivialBlock.preimage, UpperSemicontinuous.isOpen_preimage, AddMonoidHom.fiberEquivKer_symm_apply, AddCircle.coe_real_preimage_closedBall_period_zero, ncard_preimage_of_injective_subset_range, IsLUB.dual, PartialEquiv.source_subset_preimage_target, MeasureTheory.MeasurePreserving.measure_preimage_le, tsupport_comp_eq_preimage, MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_memLp, QuotientAddGroup.dense_preimage_mk, Homeomorph.preimage_frontier, UniformEquiv.preimage_symm, LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right, image_preimage_eq_of_subset, preimage_const_sub_Ici, preimage_mul_const_Ico_of_neg, EReal.preimage_coe_Ioo_bot_top, Subtype.image_preimage_val, MeasureTheory.SimpleFunc.eLpNorm'_eq, Homeomorph.preimage_image, preimage_const_sub_uIcc, Euclidean.ball_eq_preimage, mk_preimage_sigma, MeasureTheory.SimpleFunc.measurableSet_preimage, LocallyFinite.preimage_continuous, Subring.unop_closure, Filter.eventuallyConst_preimage, kernImage_union_preimage, Subsemigroup.op_sInf, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_iff, isClosed_sigma_iff, PrimeSpectrum.coe_preimageOrderIsoFiber_apply_asIdeal, Seminorm.ball_comp, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp_assoc, Fin.preimage_natAdd_Iic_natAdd, affineSpan_coe_preimage_eq_top, Metric.preimage_mul_left_eball, MulEquiv.mulDissociated_preimage, DiscreteQuotient.isClosed_preimage, upperSemicontinuous_iff_isClosed_preimage, isLocallyClosed_tfae, preimage_eq_preimage, indicator_preimage, FiberBundle.extChartAt_target, Subgroup.smul_opposite_image_mul_preimage', Metric.preimage_mul_left_closedBall, univ_sigma_preimage_mk, Fin.preimage_succ_uIoc_succ, StructureGroupoid.LocalInvariantProp.right_invariance, BddAbove.dual, IicProdIoc_preimage, ModuleFilterBasis.smul_left, MeasureTheory.measure_preimage_smul_le, AlgebraicGeometry.Spec_zeroLocus, ValuationSubring.coe_comap, MeasureTheory.measure_lintegral_sub_measure, MeasureTheory.mulEquivHaarChar_smul_preimage, MeasureTheory.Measure.addHaar_preimage_linearMap, Topology.IsQuotientMap.trivializationOfVAddDisjoint_source, Int.preimage_Ici, Subfield.closure_preimage_le, MeasureTheory.Measure.measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport, MeasureTheory.measure_preimage_vadd_le, Fin.preimage_val_uIoc_val, Subtype.preimage_eball, Affine.Simplex.closedInterior_restrict, OrderEmbedding.preimage_Ico, disjoint_preimage_iff, AddAction.quotient_preimage_image_eq_union_add, preimage_mul_const_Ico₀, IsCoveringMap.monodromy_bijective, AddGroupFilterBasis.conj, Homeomorph.preimage_closure, WithTop.sInf_eq, AlgebraicGeometry.Scheme.Pullback.range_snd, measurable_measure_prodMk_right, Subalgebra.coe_comap, OrdConnected.preimage_ennreal_ofReal, preimage_const_mul_Ico, OpenPartialHomeomorph.extend_symm_continuousWithinAt_comp_right_iff, Submonoid.unop_sSup, ContinuousAffineEquiv.image_eq_preimage_symm, MeasureTheory.Measure.addHaar_preimage_continuousLinearEquiv, preimage_id_eq, CategoryTheory.GradedObject.CofanMapObjFun.ιMapObj_iso_inv_assoc, piecewise_preimage, image_val_preimage_val_subset_self, AffineEquiv.image_symm, Algebra.QuasiFinite.finite_comap_preimage_singleton, image_sigmaMk_preimage_sigmaMap, NormedAddGroupHom.coe_ker, Ioi_toDual, Pretrivialization.restrictPreimage'_toFun, nhdsWithin_subtype_eq_bot_iff, Manifold.IsImmersionAt.target_subset_preimage_target, Topology.WithLawson.isClosed_preimage_ofLawson, MeasureTheory.ProbabilityMeasure.map_apply, Group.preimage_smul_set, Fin.preimage_val_Icc_val, Diffeomorph.symm_image_eq_preimage, Trivialization.preimageHomeomorph_apply, Subsemigroup.coe_toAddSubsemigroup_symm_apply, Topology.IsClosedEmbedding.isCompact_preimage, ProbabilityTheory.integrable_kernel_prodMk_left, HasFPowerSeriesWithinOnBall.tendstoUniformlyOn, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source, Fin.preimage_rev_Iic, Function.Involutive.preimage, AlgebraicGeometry.Scheme.Hom.coe_preimage, Fin.preimage_rev_Ioo, MonoidWithZeroHom.valueMonoid_eq_closure, preimage_sInter, TopologicalSpace.IsOpenCover.denseRange_iff_restrictPreimage, MeasurableEquiv.image_eq_preimage_symm, MapsTo.range_restrict, ZerothHomotopy.preimage_singleton_eq_pathComponent, NNRat.preimage_cast_Ico, OpenPartialHomeomorph.map_extend_nhdsWithin, mulIndicator_preimage_of_notMem, AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal, AffineSubspace.preimage_coe_affineSpan_singleton, eVariationOn.comp_ofDual, Diffeomorph.uniqueDiffOn_preimage, IsLowerSet.isUpperSet_preimage_coe, MDifferentiableWithinAt.mpullback_vectorField_preimage, StarSubalgebra.coe_comap, PMF.toOuterMeasure_map_apply, Topology.WithLowerSet.isOpen_ofLowerSet_preimage, Trivialization.restrictPreimage_source, Finset.preimage_subset, AddSubgroup.closure_closure_coe_preimage, SeparationQuotient.preimage_mk_interior, CocompactMap.isCompact_preimage, preimage_inl_image_inr, IsLocallyClosed.preimage, EReal.preimage_coe_Iio_top, CategoryTheory.Limits.Types.range_pullbackSnd, AddSubmonoid.coe_toSubmonoid_symm_apply, Trivialization.preimageSingletonHomeomorph_symm_apply, AddMonoidAlgebra.support_divOf, AlgebraicGeometry.Scheme.Hom.finite_preimage_singleton, OpenPartialHomeomorph.preimage_closure, Finset.preimage_inv, IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure, Seminorm.smul_closedBall_preimage, PartialEquiv.image_source_inter_eq, TopologicalSpace.IsTopologicalBasis.continuous_iff, Subsemigroup.unop_sInf, Continuous.preimage_coborder_subset, image_preimage_subset, Topology.WithGeneratedByTopology.isOpen_iff, preimage_mul_sphere, Topology.IsInducing.isCompact_preimage', image_add_right', OrdConnected.dual, IsProperMap.restrictPreimage, AnalyticOnNhd.preimage_mem_codiscreteWithin, MeasureTheory.measure_add_lintegral_eq, prod_subset_preimage_snd, EMetric.preimage_smul_closedBall, Metric.preimage_add_right_closedEBall, UniqueMDiffOn.bundle_preimage, Int.preimage_Icc, Finset.sigma_preimage_mk, WithTop.preimage_coe_Ioi, OrderIso.preimage_Ioi, ContinuousLinearEquiv.contDiffOn_comp_iff, MeasureTheory.SimpleFunc.measurableSet_fiber, Topology.IsInducing.setOf_isOpen, contMDiffWithinAt_iff_source, Topology.IsEmbedding.homeomorphOfSubsetRange_apply_coe, UpperHemicontinuousOn.isInducing_comp, CategoryTheory.GradedObject.mapBifunctorRightUnitorCofan_inj_assoc, MonovaryOn.comp_right, Countable.preimage_cexp, restrictPreimage_isInducing, MeasureTheory.FiniteMeasure.prod_apply_symm, AddSubsemigroup.coe_comap, StarConvex.is_linear_preimage, Filter.preimage_mem_comap, Pi.exists_compact_superset_iff, CategoryTheory.Limits.Types.range_snd_of_isPullback, image_mul_left, MeasureTheory.eq_of_cylinder_eq_of_subset, Fin.preimage_castSucc_Ici_castSucc, MeasureTheory.Measure.ae_measure_lt_top, RingFilterBasis.mul_right', supClosed_preimage_toDual, contMDiffWithinAt_iff_source_of_mem_source, AlgebraicGeometry.Scheme.Pullback.range_fst, Specialization.isUpperSet_ofEquiv_preimage, Dynamics.coverEntropyInfEntourage_image_le, openSegment_translate_preimage, MeasurableEmbedding.comap_preimage, BaireMeasurableSet.preimage, Cardinal.mk_preimage_of_injective_of_subset_range_lift, StrictConcaveOn.translate_left, LowerSemicontinuous.isClosed_preimage, MDifferentiableOn.mpullbackWithin_vectorField_inter, TopologicalSpace.IsTopologicalBasis.induced, sigma_mk_preimage_image', Countable.preimage_circleMap, UpperSemicontinuousOn.isCompact_inter_preimage_Ici, monotone_preimage, ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul, ContMDiffWithinAt.mpullbackWithin_vectorField_inter_of_eq, OpenPartialHomeomorph.trans_of_set', Filter.mem_neg, MeasureTheory.ProbabilityMeasure.map_apply', preimage_val_sUnion, GaloisConnection.upperBounds_l_image, Metric.preimage_add_right_ball, Function.Surjective.preimage_injective, Submonoid.op_closure, Finset.preimage_mul_left_one', MeasureTheory.map_measureReal_apply, Function.LeftInverse.mem_preimage_iff, preimage_const_mul_Ici, Equiv.restrictPreimageFinset_apply_coe, Finset.restrict_preimage, Int.preimage_floor_singleton, Subtype.preimage_emetricBall, IsRetrocompact.preimage_of_isOpenEmbedding, contMDiffWithinAt_iff_of_mem_maximalAtlas, Finset.image_add_left', Subtype.preimage_coe_eq_empty, Nat.preimage_ball, TopologicalSpace.Opens.coe_comap, WithBot.preimage_coe_Iic, AddSubgroup.coe_toSubgroup_apply, MeasureTheory.SimpleFunc.lintegral_eq_of_subset, preimage_const_mul_Ioo_of_neg, Trivialization.restrictPreimage_baseSet, preimage_subtype_val_Ioc, preimage_equiv_eq_image_symm, preimage_add_const_Ico, preimage_univ, MeasurableEquiv.measurableSet_preimage, RelIso.image_eq_preimage_symm, preimage_pi, MeasureTheory.SimpleFunc.setToSimpleFunc_eq_sum_filter, preimage_sUnion, AddMonoidHom.closure_preimage_le, LinearMap.ker_le_iff, Topology.IsQuotientMap.restrictPreimage_isOpen, Subalgebra.coe_unop, ContinuousAffineEquiv.preimage_image, Fin.preimage_addNat_Ioo_addNat, IsPreconnected.preimage_of_isClosedMap, Seminorm.smul_ball_preimage, MvPolynomial.support_divMonomial, Equiv.prod_assoc_preimage, Metric.Snowflaking.preimage_ofSnowflaking_eball, continuousOn_iff_isClosed, MeasureTheory.FiniteMeasure.map_apply', Pretrivialization.restrictPreimage'_source, uIoo_ofDual, update_preimage_pi, TopologicalSpace.Fiber.sigmaIsoHom_surj, ULift.up_sSup, preimage_singleton_true, bddAbove_preimage_ofDual, sigma_subset_preimage_fst, Finset.sigma_preimage_mk_of_subset, MDifferentiableWithinAt.mpullbackWithin_vectorField_inter, preimage_mul_const_Icc_of_neg, preimage_map_fst_pullbackDiagonal, Subsemiring.coe_op, EReal.preimage_coe_Iic, Fin.preimage_castSucc_Icc_castSucc, RingEquiv.image_eq_preimage_symm, Nonempty.subset_preimage_const, Filter.comap_principal, EMetric.preimage_add_left_ball, AddSubmonoid.op_closure, MeasureTheory.IsAddFundamentalDomain.preimage_of_equiv, Subgroup.op_closure, uncurry_preimage_sigma_pi, preimage_nhds_within_coinduced, exists_continuous_one_zero_of_isCompact_of_isGδ, OrderIso.preimage_symm_preimage, MeasureTheory.measure_preimage_smul, ContinuousAlgEquiv.image_symm_eq_preimage, ContMDiffOn.contMDiffOn_tangentMapWithin, image_projection_prod, measurableSet_preimage_up, subset_addCenter_add_units, Topology.WithLower.isOpen_preimage_ofLower, SeparationQuotient.preimage_mk_frontier, SimpleGraph.Subgraph.edgeSet_coe, isClopen_preimage_val, OpenPartialHomeomorph.extend_preimage_mem_nhdsWithin, AlgebraicGeometry.Scheme.Hom.finite_preimage, subset_center_units, Nat.preimage_floor_zero, upperSemicontinuous_iff_isOpen_preimage, Submodule.span_span_coe_preimage, TopologicalSpace.IsOpenCover.isEmbedding_iff_restrictPreimage, Fin.preimage_castSucc_Ioo_castSucc, NumberField.mixedEmbedding.normAtAllPlaces_image_preimage_of_nonneg, IsOpenMap.preimage_closure_subset_closure_preimage, ContinuousLinearEquiv.uniqueDiffOn_preimage_iff, preimage_const_div_Ioo, Algebra.QuasiFinite.isDiscrete_comap_preimage, preimage_image_eq, OrdConnected.preimage_real_toNNReal, Submonoid.coe_toAddSubmonoid_symm_apply, IsClosed.preimage, MeasureTheory.Measure.addHaar_preimage_linearEquiv, Submonoid.coe_toAddSubmonoid_apply, Function.Fiber.eq_fiber_image, Fin.preimage_castLE_Ici_castLE, Balanced.mulActionHom_preimage, MeasureTheory.VectorMeasure.map_apply, preimage_compl_eq_image_compl, Topology.IsClosedEmbedding.restrictPreimage, SubMulAction.map_ofFixingSubgroupUnion_bijective, OrdConnected.preimage_mono, NNRat.preimage_cast_Ioo, subset_image_union, preimage_const_div_Ico, Homeomorph.transOpenPartialHomeomorph_source, ContMDiffOn.mpullback_vectorField_preimage, IsometryEquiv.image_symm, Equiv.preimage_image, ProbabilityTheory.indepFun_iff_indepSet_preimage, preimage_const_sub_Ioc, preimage_mul_left_singleton, center_units_eq, OrdConnected.preimage_coe_nnreal_real, mk_preimage_sigma_eq_empty, Subgroup.unop_sSup, Fin.preimage_castAdd_Ioc_castAdd, SeparationQuotient.comap_mk_nhdsSet, isProperMap_iff_isClosedMap_and_compact_fibers, Measurable.measurableSet_preimage_iff_inter_range, Fin.preimage_natAdd_Ici_natAdd, preimage_smul₀, Topology.IsGeneratedBy.isClosed_iff, IsExtrOn.on_preimage, lowerSemicontinuousOn_iff_preimage_Iic, isClosed_preimage_val, CompleteSublattice.comap_carrier, SeparationQuotient.continuousWithinAt_lift₂, IsLowerSet.toDual, Rat.preimage_cast_Ioo, preimage_boolIndicator_false, AlgebraicGeometry.Scheme.preimage_zeroLocus, preimage_const_div_Icc, QuotientGroup.card_preimage_mk, ArchimedeanClass.FiniteResidueField.ordConnected_preimage_mk, preimage_comp_eq, OpenPartialHomeomorph.IsImage.iff_symm_preimage_eq', preimage_mul_const_Ioi_of_neg, mk_preimage_prod_right, ProbabilityTheory.Kernel.setIntegral_traj_partialTraj, ContinuousAlgEquiv.preimage_symm_preimage, IsIrreducible.preimage, preimage_sub_const_uIcc, IsOpenMap.preimage_frontier_subset_frontier_preimage, TopologicalSpace.Fiber.instCompactSpaceElemValSetMemRangeCoeLocallyConstantPreimageSingleton, mem_nhdsSet_induced, Fin.preimage_addNat_uIoo_addNat, pairwiseDisjoint_fiber, preimage_mul_right_one', sigma_preimage_right, ConvexOn.comp_affineMap, SupClosed.preimage, QuotientGroup.preimage_image_coe, spectrum.preimage_units_mul_comm, preimage_div_const_Ioo, AddSubgroup.toSubgroup'_closure, MeasureTheory.measure_preimage_eq_zero_iff_of_countable, Topology.WithUpper.isOpen_def, Fin.preimage_castAdd_Iic_castAdd, PrimeSpectrum.ConstructibleSetData.toSet_map, Subsemigroup.unop_closure, Matroid.sum'_isBase_iff, AddSubsemigroup.op_closure, MeasureTheory.Measure.nullMeasurableSet_preimage_snd, image_subset_iff, IsSpectralMap.isCompact_preimage_of_isOpen, StrictConvex.preimage_add_right, Fin.preimage_castLE_Ico_castLE, preimage_subset_of_surjOn, extChartAt_target, SeparationQuotient.tendsto_lift_nhdsWithin_mk, Flow.image_eq_preimage_symm, mdifferentiableWithinAt_iff_source_of_mem_source, MDifferentiableWithinAt.mpullback_vectorField_preimage_of_eq, Subring.op_closure, preimage_iUnion, Finset.map_subset_iff_subset_preimage, MeasureTheory.measure_preimage_add_right, NNRat.preimage_cast_Ioc, EReal.preimage_coe_Ioc_bot, MeasureTheory.AnalyticSet.preimage, SimpleGraph.IsVertexCover.preimage, ContinuousLinearEquiv.dimH_preimage, Finite.exists_injOn_of_encard_le, preimage_iInter, preimage_mul_const_Ioc₀, MeasureTheory.Measure.dirac_compProd_apply, Nonempty.preimage, Fin.preimage_cast_Ioo, Int.preimage_ball, MeasureTheory.Measure.haar.index_defined, SeparationQuotient.map_mk_nhdsWithin_preimage, MeasureTheory.Measure.measure_prod_null, IsUpperSet.toDual, IsPathConnected.preimage_coe, QuotientAddGroup.preimage_image_mk_eq_iUnion_image, Seminorm.preimage_metric_ball, PrimeSpectrum.coe_preimageHomeomorphFiber_symm_apply_coe_asIdeal, dual_ordConnected_iff, MeasurableEmbedding.integrableOn_iff_comap, Submonoid.toAddSubmonoid'_closure, SubMulAction.val_preimage_orbit, IsConnected.preimage_of_isOpenMap, SetRel.isSymm_preimage, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage, PartialEquiv.trans_ofSet, BooleanSubalgebra.coe_comap, Metric.preimage_add_right_eball, Metric.preimage_smul_ball, preimage_div_const_uIcc, Fin.preimage_rev_uIoo, Metric.Snowflaking.preimage_ofSnowflaking_emetricBall, Int.preimage_ceil_singleton, TopCat.isOpen_iff_of_isColimit_cofork, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux, Subgroup.closure_preimage_eq_top, mulIndicator_one_preimage, MeasurableEmbedding.setIntegral_map, continuousOn_open_iff, GeneralizingMap.restrictPreimage, OrderIso.image_eq_preimage_symm, NonUnitalSubring.coe_comap, TopologicalSpace.IsOpenCover.isLocallyClosed_iff_coe_preimage, Pretrivialization.restrictPreimage_baseSet, EReal.preimage_coe_Ico, isClosed_sum_iff, Fin.preimage_castSucc_Ioc_castSucc, AddSubsemigroup.toSubsemigroup_closure, preimage_invFun_of_notMem, IsAntichain.preimage_relIso, MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm_of_integrable, OrderIso.preimage_image, diag_preimage_prod_self, ProbabilityTheory.Kernel.IndepFun.measure_inter_preimage_eq_mul, Homeomorph.isSimplyConnected_preimage, IsSemilinearSet.preimage, MeasureTheory.Measure.preimage_null_of_map_null, vadd_preimage_set_subsetₛₗ, Fin.preimage_castAdd_uIcc_castAdd, CompleteLatticeHom.coe_setPreimage, IsEvenlyCovered.of_openPartialHomeomorph, Subtype.preimage_coe_compl', volume_preimage_coe, Topology.IsQuotientMap.trivializationOfSMulDisjoint_source, OpenPartialHomeomorph.source_preimage_target, univ_pi_update_univ, ContDiffOn.comp_continuousLinearMap, preimage_add_const_Iic, Ideal.coe_torsionOf, Fin.preimage_val_Ioo_val, preimage_const_div_Iic, PartiallyWellOrderedOn.bddAbove_preimage, MeasureTheory.measure_preimage_vadd, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.preimage_toBase_eq_range_ι, TopCat.isOpen_iff_of_isColimit, ENNReal.tsum_fiberwise, AlgebraicGeometry.locallyQuasiFinite_iff_finite_preimage_singleton, Metric.Snowflaking.isBounded_preimage_toSnowflaking_iff, Matroid.mapEmbedding_isBasis_iff, preimage_add_ball, Fin.preimage_cast_uIcc, Subgroup.coe_subgroupOf, TwoSidedIdeal.coe_unop, OrderEmbedding.preimage_Ioi, OpenPartialHomeomorph.IsImage.preimage_eq, QuotientGroup.preimage_mk_one, AlgebraicGeometry.Scheme.Opens.ι_appLE, MeasureTheory.SimpleFunc.integral_eq, VectorPrebundle.totalSpaceMk_preimage_source, PrimeSpectrum.coe_primesOverOrderIsoFiber_symm_apply_coe, IsAntichain.preimage, preimage_add_left_zero, ProbabilityTheory.IdentDistrib.measure_mem_eq, PartialEquiv.IsImage.preimage_eq, image_preimage_eq_iff, MeasureTheory.MeasurePreserving.setLIntegral_comp_preimage_emb, MeasureTheory.IsAddFundamentalDomain.addProjection_respects_measure_apply, ball_preimage, Subgroup.toAddSubgroup'_closure, Filter.rmap_sets, Submodule.span_preimage_le, Subgroup.coe_op, Equiv.preimage_piEquivPiSubtypeProd_symm_pi, preimage_const_mul_Icc_of_neg, mulTSupport_comp_eq_preimage, continuousOn_iff', DilationEquiv.smulTorsor_preimage_ball, OpenPartialHomeomorph.preimage_eventuallyEq_target_inter_preimage_inter, subset_pi_iff, preimage_symmDiff, MeasureTheory.measure_mul_lintegral_eq, Equiv.sigmaPreimageEquiv_symm_apply_fst, mulIndicator_const_preimage_eq_union, quasispectrum.preimage_algebraMap, WithBot.preimage_coe_Ioc_bot, preimage_subtype_coe_eq_compl, Finset.preimage_map, TwoSidedIdeal.coe_op, preimage_compl, upperHemicontinuous_iff_isClosed_compl_preimage_Iic_compl, preimage_iInter₂, InfClosed.dual, Homeomorph.image_preimage, AddAction.IsBlock.subtype_val_preimage, IsCompl.preimage, MulAction.IsBlock.subtype_val_preimage, preimage_vadd, MeasureTheory.IsFundamentalDomain.preimage_of_equiv, SeparationQuotient.continuousOn_lift₂, Metric.Snowflaking.ediam_preimage_toSnowflaking, Equiv.image_preimage, Ideal.Quotient.quotient_ring_saturate, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_preimage_basicOpen, IsSublattice.dual, ProbabilityTheory.setLIntegral_condDistrib_of_measurableSet, MeasureTheory.SimpleFunc.map_integral, Metric.Snowflaking.preimage_ofSnowflaking_emetricClosedBall, Matroid.comap_isBase_iff, TopCat.colimit_isOpen_iff, MonoidWithZeroHom.mem_valueMonoid_iff, Fin.preimage_castSucc_Ioi_castSucc, WithBot.preimage_coe_Ioi_bot, preimage_div_const_Ici, Continuous.preimage_mem_nhdsSet, ULift.down_sInf, AddHom.mclosure_preimage_le, IsSymmetricRel.preimage_prodMap, IsUniformInducing.basis_uniformity, mulIndicator_comp_right, ProbabilityTheory.Kernel.measurable_kernel_prodMk_left_of_finite, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app', IsSeqClosed.preimage, ValuationSubring.coe_primeSpectrumOrderEquiv_symm_apply_asIdeal, Dynamics.le_coverEntropyInfEntourage_image, ProbabilityTheory.Kernel.compProd_apply, CategoryTheory.Functor.toPreimages_obj, CategoryTheory.HomOrthogonal.matrixDecomposition_symm_apply, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, unitInterval.subtype_Iic_eq_Icc, Subsemiring.unop_sSup, preimage_nhds_coinduced, ContinuousAlgEquiv.preimage_closure, Matroid.sigma_indep_iff, Fin.preimage_apply_01_prod', CategoryTheory.HomOrthogonal.matrixDecompositionAddEquiv_apply, Fin.preimage_castAdd_uIoc_castAdd, Function.Surjective.nonempty_preimage, AddSubmonoid.coe_op, LowerSemicontinuous.isOpen_preimage, ContDiffWithinAt.comp_inter, isSublattice_preimage_toDual, OrderDual.isCobounded_preimage_ofDual, IsUpperSet.isLowerSet_preimage_coe, IsEvenlyCovered.mem_toTrivialization_baseSet, Finset.preimage_add_left_zero', pi_def, IsSymmetricRel.eq, ULift.isClosed_iff, image_toPullbackDiag, preimage_iUnion₂, DiscreteTopology.preimage_of_continuous_injective, Group.preimage_smul_setₛₗ, Finset.restrict_comp_piCongrLeft, continuousAt_def, IsPreconnected.preimage_of_isOpenMap, AddSubgroup.unop_sSup, smul_preimage_set_subsetₛₗ, MeasureTheory.Measure.map_apply₀, AffineEquiv.preimage_symm, preimage_find_eq_disjointed, measurableSet_preimage, sigma_mk_preimage_image_eq_self, Equidecomp.IsDecompOn.comp', Subgroup.comap_normalClosure, WithTop.preimage_coe_Ioo_top, Int.preimage_fract, ProbabilityTheory.setLIntegral_preimage_condDistrib, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_image_preimage_expMapBasis, Topology.IsOpenEmbedding.preimage_closedPoints, RingEquiv.image_symm_eq_preimage, ofDual_sSup, HasFPowerSeriesWithinAt.compContinuousLinearMap, Fin.preimage_cast_Ico, SummationFilter.support_comap, mdifferentiableWithinAt_iff_of_mem_source, Subtype.range_coind, ContinuousLinearEquiv.strictConvex_preimage, preimage_smul_setₛₗ', Submonoid.closure_closure_coe_preimage, StrictConvex.preimage_add_left, image_restrict, IsOpen.isOpen_inter_preimage_of_deriv_eq_zero, Fin.preimage_addNat_Ico_addNat, graphOn_prod_prodMap, Subsemigroup.coe_op, OrderEmbedding.preimage_Ioo, AddSubgroup.vadd_opposite_image_add_preimage, OpenPartialHomeomorph.isOpen_inter_preimage_symm, IsChain.preimage_relEmbedding, ContinuousLinearEquiv.preimage_closure, Function.Surjective.image_preimage, Finset.preimage_add_right_zero, preimage_mul_ball, AddSubmonoid.coe_comap, Subtype.preimage_val_subset_preimage_val_iff, EMetric.preimage_smul_ball, MeasureTheory.Measure.snd_apply, closure_subset_preimage_closure_image, MeasureTheory.SimpleFunc.support_eq, AnalyticOnNhd.preimage_zero_mem_codiscreteWithin, Finset.set_biUnion_preimage_singleton, Function.Embedding.sigmaSet_preimage, VectorField.mlieBracketWithin_apply, Convex.translate_preimage_left, PartialEquiv.target_inter_inv_preimage_preimage, IsOpen.preimage, Finsupp.comapDomain_add_of_injective, Function.Involutive.image_eq_preimage_symm, preimage_const_mul_Ioo, restrictPreimage_injective, Topology.IsOpenEmbedding.coborder_preimage, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp_assoc, preimage_inr_range_inl, MulAction.quotient_preimage_image_eq_union_mul, NFA.append_preimage_acceptsFrom, RingHom.ker_eq, Filter.mem_smul_filter, TopologicalSpace.IsOpenCover.isOpenEmbedding_iff_restrictPreimage, PartialEquiv.trans_target, prod_preimage_right, Filter.HasBasis.tangentConeAt_eq_biInter_closure, lowerSemicontinuous_iff_isOpen_preimage, MeasureTheory.ProbabilityMeasure.prod_apply_symm, Trivialization.restrictPreimage_apply, ProbabilityTheory.lintegral_condKernel_mem, SeparationQuotient.preimage_image_mk_open, OrderIso.isLUB_preimage', Function.Fiber.fiber_nonempty, IsAntichain.preimage_relEmbedding, MeasureTheory.measure_preimage_mul_right, elim_preimage_pi, inv_preimage, VectorField.eventually_contMDiffWithinAt_mpullbackWithin_extChartAt_symm, TopologicalSpace.IsOpenCover.isClosedMap_iff_restrictPreimage, Submonoid.unop_closure, mem_intrinsicInterior, MonoidHom.fiberEquiv_apply, Topology.lawsonOpen_iff_scottOpen_of_isUpperSet, ProbabilityTheory.Kernel.measurable_kernel_prodMk_right, OpenPartialHomeomorph.IsImage.symm_preimage_eq', preimage_sub_const_Iic, Finite.exists_infinite_fiber, Fin.preimage_addNat_Ioc_addNat, CategoryTheory.GradedObject.mapBifunctorLeftUnitorCofan_inj_assoc, MeasureTheory.hausdorffMeasure_homothety_preimage, preimage_add_const_Iio, continuousOn_to_generateFrom_iff, Function.Injective.preimage_pullbackDiagonal, preimage_sub_const_Icc, MeasureTheory.VAddInvariantMeasure.measure_preimage_vadd, isOpen_sigma_iff, Subring.unop_sSup, MeasureTheory.MeasurePreserving.measure_symmDiff_preimage_iterate_le, MeasureTheory.SimpleFunc.restrict_preimage, OrderIso.image_preimage, Polynomial.preimage_eval_singleton, Subgroup.op_sSup, MeasurableSpace.comap_generateFrom, ModelWithCorners.uniqueDiffOn_preimage_source, UpperHemicontinuous.isInducing_comp, ImplicitFunctionData.map_implicitFunction_nhdsWithin_preimage, Function.support_eq_preimage, ProbabilityTheory.Kernel.iIndepFun.cond_iInter, Fin.preimage_rev_uIcc, unitInterval.subtype_Ioi_eq_Ioc, preimage_mul_const_Ioo, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen, MeasureTheory.SimpleFunc.finMeasSupp_iff, Dense.preimage, MeasurableSpace.pi_eq_generateFrom_projections, Function.Semiconj.preimage_dynEntourage, TopologicalSpace.Opens.overEquivalence_counitIso_hom_app, MeasureTheory.measure_preimage_vadd_null, PFun.core_res, Countable.preimage, MeasureTheory.SimpleFunc.map_preimage, RingHom.field_closure_preimage_le, Cardinal.mk_preimage_of_injective_lift, Continuous.isOpen_preimage, Rat.preimage_cast_Ioc, Ici_toDual, contMDiffOn_iff_target, Icc_toDual, Subtype.preimage_coe_eq_preimage_coe_iff, Trivialization.frontier_preimage, OrderDual.isCobounded_preimage_toDual, MeasureTheory.measure_add_right_null, Convex.affine_preimage, OpenPartialHomeomorph.subtypeRestr_source, preimage_mul, ContinuousAffineEquiv.image_symm, Subgroup.coe_comap, preimage_const_mul_Iio, WithTop.preimage_coe_Icc, Equiv.sigmaPreimageEquiv_apply, isClosed_induced, isOpen_sigma_fst_preimage, Matroid.mapSetEquiv_indep_iff, AlgebraicGeometry.Scheme.Hom.isIrreducible_preimage, MeasureTheory.Measure.measure_preimage_inv, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, Finset.preimage_mul_right_singleton, continuous_def, Fin.preimage_castLE_Iio_castLE, IsEvenlyCovered.discreteTopology_fiber, measurableSet_quotient, Seminorm.closedBall_comp, IsLocallyConstant.iff_isOpen_fiber, Nat.card_preimage_of_injective, Function.mulSupport_eq_preimage, Submonoid.coe_unop, image_equiv_eq_preimage_symm, ConvexOn.comp_linearMap, Fin.preimage_cast_Icc, IsLocallyClosed.isOpen_preimage_val_closure, Metric.preimage_vadd_eball, IsCompact.preimage_continuous, ProbabilityTheory.Kernel.indepFun_iff_indepSet_preimage, range_sigmaMk, mk_preimage_prod_right_fn_eq_if, preimage_nhdsWithin_coinduced', MeasureTheory.measure_preimage_snd_singleton_eq_tsum, Subsemigroup.unop_sSup, Fin.preimage_succ_Ici_succ, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec.image_basicOpen_eq_basicOpen, Filter.mem_map, IsMinOn.on_preimage, EMetric.preimage_add_left_closedBall, preimage_subset_image_of_inverse, Isometry.preimage_eball, kernImage_preimage_union, DiscreteQuotient.isOpen_preimage, Stonean.extremallyDisconnected_preimage, Finset.image_mul_right', PrimeSpectrum.BasicConstructibleSetData.toSet_map, ContinuousAlgEquiv.symm_preimage_preimage, Dense.orderDual, MeasureTheory.MeasurePreserving.restrict_preimage_emb, AlgebraicGeometry.Scheme.image_preimage_eq_of_isPullback, Subsemigroup.op_sSup, MeasureTheory.FiniteMeasure.map_apply_of_aemeasurable, totallyBounded_preimage, preimage_image_preimage, preimage_const_of_mem, IsometryEquiv.preimage_symm, Fin.image_cast, MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup, extChartAt_preimage_mem_nhds_of_mem_nhdsWithin, IsEvenlyCovered.to_isEvenlyCovered_preimage, Cardinal.mk_preimage_of_injective, Metric.preimage_add_left_eball, matrix_eq_pi, mk_preimage_sym2, FiberPrebundle.totalSpaceMk_preimage_source, Finset.monotone_preimage, continuous_generateFrom_iff, MeasureTheory.ProbabilityMeasure.prod_apply, Fin.preimage_castLE_uIoo_castLE, Continuous.preimage_mem_nhdsSetWithin, OpenPartialHomeomorph.IsImage.symm_preimage_eq, OpenPartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq, ClosedSubmodule.coe_comap, Homeomorph.isSigmaCompact_preimage, Disjoint.preimage, Finset.image_add_left, MeasureTheory.forall_measure_preimage_add_right_iff, Ideal.coe_piOrderIso_apply, OpenPartialHomeomorph.continuousOn_iff_continuousOn_comp_right, preimage_const_add_Ioc, PartialEquiv.source_inter_preimage_target_inter, Subsemigroup.toAddSubsemigroup'_closure, OpenPartialHomeomorph.trans_target, AlgebraicGeometry.Scheme.Hom.fiberι_fiberHomeo_symm, preimage_const_add_uIcc, preimage_diff, Finset.preimage_add_right_zero', MeasureTheory.OuterMeasure.map_ofFunction_le, Pretrivialization.domExtend_source, IsometryEquiv.preimage_closedBall, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app'_assoc, Topology.IsInducing.closure_eq_preimage_closure_image, Continuous.restrictPreimage, Subtype.preimage_coe_compl, NumberField.mixedEmbedding.fundamentalCone.normLeOne_eq_preimage, EqOn.inter_preimage_eq, bddAbove_preimage_toDual, AddSubgroup.coe_toSubgroup_symm_apply, IsCoveringMap.restrictPreimage, union_preimage_subset, NonUnitalRingHom.closure_preimage_le, AffineSubspace.coe_comap, Topology.IsQuotientMap.continuousOn_isOpen_iff, Equiv.image_symm_eq_preimage, image_val_preimage_restrictPreimage, IsLocallyConstant.iff_isOpen_fiber_apply, preimage_subtype_val_Icc, LocallyConstant.ofIsClopen_fiber_one, ProbabilityTheory.Kernel.setIntegral_traj_partialTraj', Topology.IsInducing.isCompact_preimage_iff, ProbabilityTheory.Kernel.measurable_kernel_prodMk_left, OpenPartialHomeomorph.mapsTo_extend, Equidecomp.restr_target, Topology.IsClosedEmbedding.isLindelof_preimage, IsometryEquiv.preimage_ball, WithTop.preimage_coe_Ico_top, Metric.Snowflaking.preimage_ofSnowflaking_closedEBall, subset_preimage_image, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_zeroLocus, Real.volume_preimage_mul_right, OrderIso.preimage_Ico, AddAction.IsBlock.preimage, ULift.isOpen_iff, preimage_mul_const_Icc, Metric.Snowflaking.preimage_toSnowflaking_emetricBall, isOpen_induced_iff, Fin.preimage_succ_uIoo_succ, mk_preimage_prod_left_eq_if, isClosedMap_iff_upperHemicontinuous, QuotientGroup.sound, surjective_iff_surjective_of_iUnion_eq_univ, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, preimage_rangeSplitting, Isometry.preimage_emetric_closedBall, IsMeagre.preimage_of_isOpenMap, Homeomorph.isOpen_preimage, ContDiffWithinAt.comp_inter_of_eq, IsUnit.preimage_smul_setₛₗ, preimage_const_mul_Ioi, eq_preimage_subtype_val_iff, Trivialization.preimageSingletonHomeomorph_apply, preimage_iUnionLift, Equiv.prod_assoc_symm_preimage, ZLattice.coe_comap, IsLocalization.coe_primeSpectrumOrderIso_apply_coe_asIdeal, AddMonoidAlgebra.mem_gradeBy_iff, WithBot.preimage_coe_Ioo_bot, UniformEquiv.preimage_image, writtenInExtChartAt_comp, Isometry.preimage_emetric_ball, mem_preimage, Continuous.closure_preimage_subset, IsClosedMap.mapClusterPt_iff_lift'_closure, IsCoveringMapOn.continuousOn, NNRat.preimage_cast_Iio, Finset.preimage_inl, IsOpenMap.preimage_frontier_eq_frontier_preimage, Dynamics.le_coverMincard_image, IsClopen.preimage, Fin.preimage_cast_Ioc, MeasureTheory.trim_comap_apply, preimage_const_mul_Iic₀, Fin.preimage_val_Iio_val, preimage_subtype_val_Iio, preimage_const_sub_Iio, MonoidHom.coe_ker, Pretrivialization.target_inter_preimage_symm_source_eq, Topology.IsEmbedding.closure_eq_preimage_closure_image, MeasureTheory.ae_measure_preimage_add_right_lt_top, preimage_mul_const_Iio₀, AlgebraicGeometry.Scheme.Hom.isDiscrete_preimage, IsPreirreducible.preimage, preimage_const_mul_Iic_of_neg, AddSubsemigroup.coe_op, Int.preimage_closedBall, TopCat.GlueData.preimage_image_eq_image', AddSubmonoid.toSubmonoid'_closure, AddSubsemigroup.unop_closure, Trivialization.prod_source, Int.preimage_Ico, DifferentiableWithinAt.comp', IsOpen.iUnion_preimage_smul, isClosed_coinduced, Subsemiring.coe_unop, MeasureTheory.SimpleFunc.restrict_preimage_singleton, MeasureTheory.measure_preimage_add, preimage_mul_left_one', StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart, compact_covered_by_mul_left_translates, Subtype.preimage_emetricClosedBall, IsCoveringMapOn.homeomorph_comp, GroupFilterBasis.inv', MeasureTheory.measure_preimage_fst_singleton_eq_sum, mem_map_indicator_ae_iff_of_zero_notMem, TopCat.pullback_snd_image_fst_preimage, MapsTo.comp_right, RingHom.closure_preimage_le, preimage_const_div_Ioc, Topology.WithGeneratedByTopology.isClosed_iff, EMetric.preimage_vadd_ball, differentiableWithinAtProp_self_target, NonUnitalStarSubalgebra.coe_comap, Topology.IsInducing.isClosed_iff, preimage_mul_const_Ioi₀, preimage_mul_closedBall, preimage_subtype_val_Ici, Matroid.comap_loops, MeasureTheory.MeasurePreserving.measure_preimage, mk_preimage_prod, Metric.preimage_mul_right_closedBall, QuotientAddGroup.preimage_mk_zero, TopologicalSpace.IsOpenCover.generalizingMap_iff_restrictPreimage, AntilipschitzWith.ediam_preimage_le, isGδ_induced, contMDiffWithinAt_iff_of_mem_source, IsClosedMap.comap_nhdsSet_le, Nat.preimage_Ico, MeasureTheory.IsFundamentalDomain.projection_respects_measure_apply, Finset.image_mul_left, preimage_add_const_Ici, IsOpenMap.coborder_preimage_subset, encard_preimage_of_bijective, WithBot.sSup_eq, preimage_restrictPreimage, preimage_subset_iff, OpenPartialHomeomorph.source_inter_preimage_target_inter, PointedCone.coe_comap, Pretrivialization.domExtend_target, Filter.map_inf_principal_preimage, IsSublattice.preimage, ContinuousLinearMap.closure_preimage, IsometryEquiv.preimage_eball, MeasureTheory.SimpleFunc.map_setToSimpleFunc, LinearIsometryEquiv.preimage_sphere, star_preimage, OrdConnected.preimage_monotoneOn, mk_preimage_prod_left_eq_empty, image_compl_preimage, MeasureTheory.Measure.QuasiMeasurePreserving.preimage_mono_ae, Dynamics.dynEntourage_eq_inter_Ico, MeasureTheory.SimpleFunc.map_preimage_singleton, NNRat.preimage_cast_uIoc, Finset.image_mul_right, DiffeologicalSpace.isOpen_iff_preimages_plots, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_normLeOne, TopologicalSpace.IsOpenCover.isOpenMap_iff_restrictPreimage, indicator_preimage_of_notMem, EMetric.preimage_add_right_closedBall, Topology.WithUpperSet.isOpen_ofUpperSet_preimage, Topology.IsCoherentWith.isOpen_iff, SmoothBumpFunction.image_eq_inter_preimage_of_subset_support, Subsemigroup.coe_toAddSubsemigroup_apply, SeminormFamily.basisSets_smul_left, CategoryTheory.Limits.Types.range_fst_of_isPullback, Manifold.LiftSourceTargetPropertyAt.source_subset_preimage_source, image_inter_nonempty_iff, Metric.thickening_eq_preimage_infEdist, eval_preimage', SimpleGraph.Subgraph.coe_deleteEdges_eq, upperSemicontinuousOn_iff_preimage_Ici, preimage_swap_prod, Subsemiring.coe_comap, RingFilterBasis.SubmodulesBasis.smul, IsTopologicalBasis.iInf_induced, mem_intrinsicClosure, Ico_ofDual, mk_preimage_tprod, OpenPartialHomeomorph.IsImage.iff_preimage_eq, Matroid.comapOn_preimage_eq, MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton, MeasureTheory.forall_measure_preimage_mul_right_iff, continuous_discrete_rng, PMF.toMeasure_map_apply, StarConvex.linear_preimage, Nat.preimage_ceil_zero, OrderIso.preimage_Iic, Filter.HasBasis.coprod, preimage_mul_const_Ioc, Convex.linear_preimage, Complex.countable_preimage_exp, Subalgebra.unop_sInf, Subsingleton.preimage, FiberPrebundle.continuousOn_of_comp_right, Trivialization.restrictPreimage_target, Dynamics.coverEntropyInf_restrict_subset, ModelWithCorners.extChartAt_transContinuousLinearEquiv_target, IsFundamentalDomain.AddQuotientMeasureEqMeasurePreimage_vaddAddHaarMeasure, LinearIsometryEquiv.preimage_closedBall, NonUnitalSubsemiring.coe_comap, Finsupp.sym2_support_eq_preimage_support_mul, Fin.preimage_castLE_uIcc_castLE, MeasureTheory.Measure.QuasiMeasurePreserving.preimage_null, ChartedSpace.liftPropWithinAt_iff', HasSum.tsum_fiberwise, SubAddAction.map_ofFixingAddSubgroupUnion_def, EMetric.preimage_vadd_closedBall, ProbabilityTheory.Kernel.iIndepFun_iff_measure_inter_preimage_eq_mul, WithBot.preimage_coe_Ioo, Trivialization.source_eq
prod 📖	CompOp	5 math math: SSet.Subcomplex.prod_obj, ModelWithCorners.boundary_of_boundaryless_left, ModelWithCorners.boundary_of_boundaryless_right, ModelWithCorners.boundary_prod, boundary_product
range 📖	CompOp	2047 math math: range_cfcₙ_nnreal, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_left_iff, MeasureTheory.SimpleFunc.coe_range, Finsupp.range_mapRange, Monoid.exponent_ne_zero_iff_range_orderOf_finite, Affine.Simplex.isTangentAt_insphere_iff_eq_touchpoint, TopCat.binaryCofan_isColimit_iff, Matrix.isTotallyUnimodular_iff, PrimeSpectrum.closure_range_comap, iUnion_eq_range_sigma, Real.range_sin, Finsupp.range_subset_insert_frange, GrpCat.SurjectiveOfEpiAuxs.τ_apply_fromCoset, Matrix.range_cons, pairwiseDisjoint_range_iff, CompleteLatticeHom.range_coe, mapClusterPt_atTop_pow_tfae, rangeFactorization_coe, Order.isMax_or_mem_range_pred_or_isPredLimit, MeasureTheory.OuterMeasure.map_iInf, Affine.Simplex.faceOppositeCentroid_mem_affineSpan_face, GromovHausdorff.hausdorffDist_optimal, Int.ceil_eq_self_iff_mem, bddAbove_range_prod, range_sigma_eq_iUnion_range, ClosureOperator.setOf_isClosed_eq_range_closure, Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_left_iff, Algebra.PreSubmersivePresentation.jacobiMatrix_naive, exists_subset_range_and_iff, mem_range_self, RootPairing.posRootForm_posForm_apply_apply, SetTheory.PGame.grundyValue_eq_sInf_moveLeft, MonoidHom.coe_mrange, range_eq_univ, integralClosure_le_span_dualBasis, MeasureTheory.measurableSet_range_of_continuous_injective, Profinite.NobelingProof.GoodProducts.sum_to_range, lp.isLUB_norm, Affine.Simplex.orthogonalProjectionSpan_map, compl_range_inr, contMDiffAt_iff_of_mem_source, Fintype.coe_image_univ, vectorSpan_range_eq_span_range_vsub_left_ne, Nat.range_succ, Real.range_log, OpenPartialHomeomorph.extend_preimage_inter_eq, HahnSeries.SummableFamily.embDomain_apply, IsLUB.range_of_tendsto, CategoryTheory.Limits.Types.binaryCofan_isColimit_iff, range_comp, linearIndependent_option, IsMulIndecomposable.pairwise_baseOf_div_notMem, Affine.Simplex.ExcenterExists.excenter_notMem_affineSpan_face, algebraicIndependent_adjoin, Ordinal.sInf_compl_lt_lift_ord_succ, ModelWithCorners.range_prod, tendsto_tsum_div_pow_atTop_integral, MulHom.coe_srange, Topology.IsClosedEmbedding.continuousOn_comap_finiteMeasure, LocalSubring.range_toSubring, linearIndependent_iff_card_le_finrank_span, range_comp_subset_range, NumberField.canonicalEmbedding.mem_span_latticeBasis, Affine.Simplex.sSameSide_excenter_singleton_point, AlgebraicGeometry.Scheme.isEmpty_pullback_iff, iUnion_range_eq_sUnion, Ideal.mem_span_iff_exists_isHomogeneous, CommRingCat.closure_range_union_range_eq_top_of_isPushout, ContDiff.dense_compl_range_of_finrank_lt_finrank, Ordinal.mem_range_typein_iff, PrimeSpectrum.residueField_specComap, Submodule.mem_ideal_smul_span_iff_exists_sum, Subgroup.closure_image_isMulIndecomposable_baseOf, SmoothBumpFunction.support_eq_symm_image, isSeparable_range_derivWithin, RootPairing.instIsRootSystemSubtypeMemSubmoduleSpanRangeCoeEmbeddingRootCorootRestrictScalars', Algebra.Presentation.naive_toGenerators, range_nonempty_iff_nonempty, PrimeSpectrum.localization_specComap_range, Isometry.ediam_range, Function.Fiber.map_eq_image, infinite_range_of_injective, IsometryEquiv.range_eq_univ, preimage_fst_singleton_eq_range, image_univ, Module.Basis.range_ofVectorSpace, AlgebraicGeometry.Scheme.affineBasisCover_is_basis, MDifferentiableWithinAt.differentiableWithinAt_mpullbackWithin_vectorField, Algebra.PreSubmersivePresentation.ofHasCoeffs_map, isLUB_of_tendsto_atBot, Finset.mem_range_iff_mem_finset_range_of_mod_eq, ProbabilityTheory.Kernel.borelMarkovFromReal_apply', SpecializingMap.stableUnderSpecialization_range, AlgebraicGeometry.range_eq_range_of_surjective, Ordinal.mem_range_lift_of_le, PrimeSpectrum.iUnion_range_specComap_comp_evalRingHom, Filter.not_bddAbove_of_tendsto_atTop, AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraic, instNonemptyRange, orthonormal_subtype_range, Module.Basis.mem_span, Module.Relations.Solution.surjective_π_iff_span_eq_top, Affine.Simplex.ExcenterExists.excenter_notMem_affineSpan_faceOpposite, AffineBasis.tot, range_circleMap, Metric.isBounded_range_of_tendsto_cofinite, span_fourier_closure_eq_top, mdifferentiableWithinAt_iff_of_mem_maximalAtlas, range_inclusion, GeneralizingMap.stableUnderGeneralization_range, mem_range, DoubleCentralizer.range_toProdMulOpposite, Polynomial.Sequence.span, ModelWithCorners.symm_continuousWithinAt_comp_right_iff, Affine.Simplex.circumcenter_mem_affineSpan, range_IciExtend, finiteDimensional_vectorSpan_range, PrincipalSeg.isLowerSet_range, AEMeasurable.exists_ae_eq_range_subset, RootPairing.restrictScalars_coe_coroot, Affine.Simplex.excenter_singleton_mem_affineSpan_range, Finset.pairwiseDisjoint_range_singleton, AlgebraicGeometry.Scheme.Hom.range_fiberι, AddSubgroup.mem_closure_range_iff_of_fintype, range_list_map_coe, IntermediateField.coe_bot, OrderIso.range_eq, LocallyFinite.on_range, AlgebraicGeometry.Scheme.Cover.iUnion_range, LowerAdjoint.closed_eq_range_close, OrdinalApprox.gfp_mem_range_gfpApprox, RootPairing.Base.sub_notMem_range_coroot, contMDiffAt_iff_source, Complex.range_cos, eventually_enorm_mfderivWithin_symm_extChartAt_lt, Finset.ordConnected_range_coe, StarSubalgebra.range_le, Matrix.GeneralLinearGroup.mem_center_iff_val_mem_range_scalar, range_comap_of_surjective, AlgebraicGeometry.isCompact_iff_exists, RootPairing.coreflection_image_eq, Continuous.range_subset_closure_image_dense, extChartAt_target_eventuallyEq, TangentBundle.symmL_trivializationAt, IsMulIndecomposable.pairwise_div_notMem_range, Function.LeftInverse.image_eq, Fin.range_natAdd_eq_Ici, ContinuousWithinAt.extChartAt_symm_preimage_inter_range_eventuallyEq, CategoryTheory.Limits.Types.range_pullbackFst, map_extChartAt_symm_nhdsWithin, EquicontinuousOn.isClosed_range_pi_of_uniformOnFun, Real.range_sin_infinite, Finsupp.instCountableSubtypeMemSubmoduleSpanRange, Seminorm.bddAbove_range_iff, LinearIndependent.linearCombination_repr, ordConnected_range, TensorAlgebra.adjoin_range_ι, PrincipalSeg.mem_range_iff_rel, range_eq_singleton_iff, AntilipschitzWith.isClosed_range, AffineIsometryEquiv.range_eq_univ, Fintype.card_range_le, Fin.range_succ, InnerProductSpace.span_gramSchmidtNormed_range, LinearIsometryEquiv.range_eq_univ, Metric.isBounded_range_of_tendsto, ExteriorAlgebra.ιMulti_span_fixedDegree, Cardinal.range_ofENat, InitialSeg.mem_range_of_le, Module.Relations.Solution.injective_fromQuotient_iff_ker_π_eq_span, Ideal.span_pow_eq_map_homogeneousSubmodule, SimpleGraph.Walk.range_getVert_eq_range_support_getElem, IsArtinianRing.primeSpectrum_asIdeal_range_eq, Path.range_subpath_of_ge, RootPairing.zero_notMem_range_root, BoxIntegral.unitPartition.mem_smul_span_iff, RootPairing.weylGroup_toSubmonoid, Ideal.span_eq_map_homogeneousSubmodule, ODE.FunSpace.range_toContinuousMap, RootPairing.root_sub_root_mem_of_pairingIn_pos, Function.Injective.mem_range_iff_existsUnique, WithBot.range_coe, sUnion_range, insert_none_range_some, Preorder.conePt_mem_lowerBounds, IsOpen.exists_smooth_support_eq, AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C, Function.Injective.right_inv_of_invOfMemRange, LocallyConstant.range_finite, MvPolynomial.image_comap_C_basicOpen, ContinuousAlternatingMap.isClosed_range_toContinuousMultilinearMap, AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_f, AlgebraicIndependent.repr_ker, Complex.range_sin, MvPolynomial.mem_range_map_iff_coeffs_subset, ModelWithCorners.mdifferentiableOn_symm, MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_enorm, range_specComap_of_surjective, Function.Embedding.sumSet_range, UnitAddTorus.mFourierSubalgebra_coe, Complex.range_re, image_union_image_compl_eq_range, Path.range_subpathAux, PrincipalSeg.mem_range_iff_rel', tangentCoordChange_def, Order.mem_range_succ_or_isSuccPrelimit, ModelWithCorners.image_eq, SemilinearIsometryClass.ediam_range, Preorder.hasColimit_iff_hasLUB, RootPairing.invtRootSubmodule.eq_top_iff, finrank_vectorSpan_le_iff_not_affineIndependent, Tendsto.isLindelof_insert_range_of_coLindelof, RootPairing.range_weylGroup_coweightHom, AddAction.orbit.pairwiseDisjoint, CliffordAlgebra.range_lift, Affine.Simplex.wSameSide_affineSpan_faceOpposite_iff, Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_right_iff, AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin, ZSpan.smul, AlgebraicGeometry.Scheme.Hom.range_asFiberHom, map_extChartAt_symm_nhdsWithin', isPiSystem_Ioi, InitialSeg.mem_range_of_rel', Equiv.ofLeftInverse_apply_coe, toFinset_range, MeasureTheory.analyticSet_iff_exists_polishSpace_range, GrpCat.SurjectiveOfEpiAuxs.τ_symm_apply_infinity, Matrix.mem_range_scalar_iff_commute_transvectionStruct, Function.Periodic.image_Icc, Equiv.Set.rangeInl_symm_apply_coe, HomogeneousLocalization.range_awayMapAux_subset, isCompact_range_of_support_subset_isCompact, Function.Embedding.exists_of_card_le_finset, Real.range_arctan, RootPairing.Iic_chainTopCoeff_eq, RootPairing.coe_chainBotCoeff_eq_sSup, FreeGroup.closure_range_of, MeasureTheory.StronglyMeasurable.isSeparable_range, CategoryTheory.ObjectProperty.isLocalization_isColocal, map_nhdsSet_induced_eq, Finsupp.mem_range_mapDomain_iff, IsOpenMap.continuousOn_range_of_leftInverse, RootPairing.Base.cartanMatrix_apply_eq_zero_iff, tsum_range, Order.not_isPredPrelimit_iff', isCompl_range_some_none, setBasisOfLinearIndependentOfCardEqFinrank_repr_apply, TopCat.range_prod_map, Setoid.classes_ker_subset_fiber_set, MulArchimedeanClass.range_mk, OrderedFinpartition.disjoint, OpenPartialHomeomorph.extend_target_mem_nhdsWithin, Language.start_toDFA, nonempty_iInter_Iic_iff, PrimeSpectrum.denseRange_comap_iff_minimalPrimes, range_vecMulLinear, Flow.orbit_eq_range, Nat.mem_range_succ, CategoryTheory.Types.mem_jointlySurjectivePrecoverage_iff, AlgebraicGeometry.isImmersion_eq_inf, GrpCat.SurjectiveOfEpiAuxs.τ_apply_infinity, borel_eq_generateFrom_Ici, TopologicalSpace.NonemptyCompacts.isClosed_in_closeds, Finset.ordConnected_range_val, CompleteType.isTopologicalBasis_range_typesWith, map_extChartAt_symm_nhdsWithin_range', RootPairing.span_coroot'_eq_top, Besicovitch.exist_disjoint_covering_families, rangeFactorization_bijective, Algebra.mem_adjoin_iff, Subgroup.isComplement_range_right, TopologicalSpace.Fiber.sigmaIsoHom_inj, extChartAt_preimage_inter_eq, RootPairing.RootPositiveForm.zero_lt_posForm_iff, AddSubgroup.coe_zmultiples, AddSubmonoid.mem_closure_range_iff, mem_image_univ_iff_mem_range, RootPairing.setOf_root_add_zsmul_mem_eq_Icc, MeasureTheory.ProbabilityMeasure.range_toFiniteMeasure, OnePoint.compl_infty, IsGalois.mem_range_algebraMap_iff_fixed, Matrix.isTotallyUnimodular_iff_fintype, Algebra.FormallySmooth.adjoin_of_algebraicIndependent, Ordinal.enumOrd_range, map_extChartAt_symm_nhdsWithin_range, PrimeSpectrum.vanishingIdeal_range_comap, isLUB_iSup, exists_smooth_tsupport_subset, Module.Basis.ofSpan_subset, map_extChartAt_nhdsWithin', spectrum_diagonal, sphere_subset_range_iff_surjective, range_eq_image_mulTSupport_or, Nucleus.restrict_toFun, range_zero, Subsemigroup.mem_corner_iff_mem_range_mul_right, CategoryTheory.ObjectProperty.isColocal_eq_inverseImage_isomorphisms, range_unique, Affine.Simplex.orthogonalProjectionSpan_eq_point, Equiv.Perm.mclosure_swap_castSucc_succ, countable_iff_exists_subset_range, Pairwise.range_pairwise, AlgHom.coe_range, comap_nhdsWithin_range, mem_vectorSpan_iff_eq_weightedVSub, RootPairing.mapsTo_reflection_root, SpecialLinearGroup.mem_range_toGeneralLinearGroup_iff, Affine.Simplex.sSameSide_incenter_point, MeasureTheory.diracProba_diracProbaInverse, StarOrderedRing.lt_iff, inTangentCoordinates_eq_mfderiv_comp, exteriorPower.ιMulti_span_of_span, range_list_getElem?, Algebra.PreSubmersivePresentation.ofHasCoeffs_relation, MeasureTheory.Measure.comap_preimage, IsometryClass.ediam_range, StrictMono.orderIso_apply, RootPairing.injOn_dualMap_subtype_span_root_coroot, mdifferentiableAt_iff, Filter.Tendsto.isCompact_insert_range_of_cofinite, Affine.Simplex.affineSpan_face_le, AffineBasis.convexHull_eq_nonneg_coord, Equiv.ofLeftInverse_symm_apply, RootPairing.restrictScalars_pairing, Ordinal.range_omega, subset_range_enumerate, RootPairing.zero_notMem_range_coroot, Cardinal.le_range_of_union_finset_eq_top, finprod_mem_range, Submodule.span_range_inclusion_restrictScalars_eq_top, OnePoint.isOpen_range_coe, preimage_singleton_nonempty, Affine.Simplex.ExcenterExists.touchpoint_notMem_affineSpan_of_ne, dimH_range_le_of_locally_holder_on, AlgebraicGeometry.Scheme.exists_affine_mem_range_and_range_subset, PrincipalSeg.mem_range_of_rel_top, restrict_extend_range, FiniteDimensional.mem_span_of_iInf_ker_le_ker, Preorder.coconePt_mem_upperBounds, IsEmbedding.isDiscrete_range, Real.range_logb, Sum.elim_range, AffineIndependent.finrank_vectorSpan_add_one, MeasureTheory.SimpleFunc.mem_range, MDifferentiableWithinAt.differentiableWithinAt_writtenInExtChartAt, RootPairing.GeckConstruction.cartanSubalgebra_eq_lieSpan, OnePoint.notMem_range_coe_iff, Profinite.NobelingProof.GoodProducts.span_sum, CategoryTheory.Abelian.isoModSerre_kernel_eq_isLocal_of_rightAdjoint, BoxIntegral.unitPartition.integralSum_eq_tsum_div, AlgebraicGeometry.IsOpenImmersion.isOpen_range, contMDiffOn_isOpenEmbedding_symm, ExteriorAlgebra.ιMulti_span, bddBelow_range_pi, Fintype.card_range, AlgebraicIndependent.aeval_comp_repr, CategoryTheory.Presieve.ofArrows_mem_comap_jointlySurjectivePrecoverage_iff, isEmbedding_sumElim, Ordinal.card_sInf_range_compl_le_lift, kernImage_preimage_eq_iff, finite_range_iff, range_eq_singleton, CentroidHom.centroid_eq_centralizer_mulLeftRight, Matrix.center_eq_range, Dilation.ediam_range, RingHom.rangeS_toSubmonoid, Polynomial.finite_range_coeff, preimage_invFun_of_mem, finprod_emb_domain, PrimeSpectrum.iSup_basicOpen_eq_top_iff, tendsto_card_div_pow_atTop_volume, affineSpan_eq_affineSpan_lineMap_units, CategoryTheory.Localization.LeftBousfield.W_eq_inverseImage_isomorphisms, AffineBasis.interior_convexHull, AddMonoidAlgebra.supDegree_mem_range, AffineBasis.tot', Filter.map_top, Module.Basis.extendLe_subset, Module.Basis.mk_range_eq_rank, preimage_inl_range_inr, MeasurableEmbedding.measurableSet_preimage, AlgebraicGeometry.Scheme.presieve₀_mem_precoverage_iff, range_quot_lift, Algebra.mem_bot, Module.Basis.le_span, StarOrderedRing.pos_iff, iInf_range', piecewise_range_comp, Monoid.Coprod.closure_range_inl_union_inr, Algebra.PreSubmersivePresentation.ofHasCoeffs_σ', extChartAt_preimage_mem_nhdsWithin', Ordinal.brange_bfamilyOfFamily', spectralValueTerms_bddAbove, MeasureTheory.continuous_diracProbaEquiv, SimpleGraph.Copy.range_toSubgraph, ZSpan.quotientEquiv_apply_mk, subsingleton_range, Equiv.coe_ofInjective_symm, vectorSpan_range_eq_span_range_vsub_right, AlgebraicGeometry.IsOpenImmersion.range_pullbackSnd, PrimeSpectrum.exists_range_eq_of_isConstructible, infinite_range_iff, AffineIndependent.card_le_finrank_succ, range_restrictPreimage, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι_eq_support_inter, Affine.Simplex.range_mkOfPoint_points, exists_retractionCantorSet, ZeroAtInftyContinuousMap.isClosed_range_toBCF, ModelWithCorners.uniqueDiffOn_preimage, AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right, StarSubalgebra.mem_bot, Topology.IsEmbedding.isLocallyClosed_iff, Module.Basis.reindexRange_repr', Affine.Simplex.isTangentAt_insphere_touchpoint, mem_range_mulIndicator, KaehlerDifferential.span_range_map_derivation_of_isLocalization, Real.range_cos, ENNReal.range_coe, RootPairing.Base.exists_mem_span_pairingIn_ne_zero_and_pairwise_ne, preimage_snd_singleton_eq_range, Profinite.NobelingProof.GoodProducts.spanFin, AddAction.period_bounded_of_exponent_pos, NormedField.denselyOrdered_range_norm, extChartAt_target_mem_nhdsWithin, Function.range_subset_insert_image_support, Affine.Simplex.ExcenterExists.affineSpan_faceOpposite_eq_orthRadius, eventually_norm_mfderivWithin_symm_extChartAt_lt, mapClusterPt_atTop_nsmul_tfae, Filter.map_comap, Fin.Embedding.exists_embedding_disjoint_range_of_add_le_Nat_card, MeasureTheory.SimpleFunc.finite_range, AlgebraicGeometry.Scheme.range_fromSpecStalk, range_add_eq_image_Ici, Directed.directedOn_range, Function.Semiconj.bijOn_range, RootPairing.neg_mem_range_coroot_iff, AlgebraicIndependent.coe_range, Module.Basis.reindexRange_apply, Algebra.coe_bot, IsOpen.exists_contMDiff_support_eq_aux, iInf_range, Flow.forwardOrbit_eq_range_nonneg, extChartAt_target_union_compl_range_mem_nhds_of_mem, ContinuousMap.spectrum_eq_preimage_range, LinearIndependent.span_eq_top_of_card_eq_finrank', Ctop.Realizer.is_basis, Ordinal.card_sInf_range_compl_le, mapsTo_extChartAt, Affine.Simplex.affineSpan_faceOpposite_eq_orthRadius_insphere, surjective_codRestrict, LinearMap.BilinForm.dualSubmodule_span_of_basis, Algebra.Presentation.naive_relation, isLUB_hasProd, NumberField.Units.span_basisOfIsMaxRank, SignType.range_eq, StrictMono.not_bddAbove_range_of_isSuccArchimedean, preimage_inter_range, preimage_extChartAt_eventuallyEq_compl_singleton, Sigma.univ, Function.Periodic.image_uIcc, range_stereographic_symm, AddMonoidHom.isClosed_range_coe, isPreconnected_range, Nat.zero_union_range_succ, MvPolynomial.adjoin_range_X, IsNoetherian.range_finsetBasis, MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto, CategoryTheory.Localization.LeftBousfield.W_adj_unit_app, SeparationQuotient.range_mk, AlgebraicGeometry.Scheme.Pullback.range_fst_comp, BoundedContinuousFunction.isBounded_range, MeasureTheory.OuterMeasure.map_le_restrict_range, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, linearIndependent_sum, ENNReal.iSup_coe_lt_top, basisOfLinearIndependentOfCardEqFinrank_repr_apply, range_diff_image, Matroid.mapEmbedding_isBase_iff, Isometry.diam_range, Filter.atTop_eq_generate_Ici, preimage_eq_univ_iff, isClosed_range_inl, Order.isMin_or_mem_range_succ_or_isSuccLimit, Cardinal.mk_range_eq_of_injective, mem_affineSpan_iff_eq_affineCombination, AbstractCompletion.closure_range, Ideal.span_single_eq_top, Module.Relations.range_map, Matrix.cRank_toNat_eq_finrank, Finite.Set.finite_range, RootPairing.setOf_root_sub_zsmul_mem_eq_Icc, isClopen_range_inl, Monoid.CoprodI.mclosure_iUnion_range_of, continuous_rangeFactorization_iff, PresentedMonoid.closure_range_of, RootPairing.corootSpan_map_flip_toPerfPair, OpenPartialHomeomorph.extend_image_target_mem_nhds, convexHull_basis_eq_stdSimplex, OnePoint.compl_range_coe, AffineBasis.affineSpan_eq_top_of_toMatrix_left_inv, Ordinal.mem_range_omega_iff, eqOn_range, BoundedContinuousFunction.bddAbove_range_norm_comp, vadd_set_range, Diffeomorph.range_comp, Metric.dense_iUnion_range_toInductiveLimit, Nat.range_rec, Ordinal.mem_range_lift_of_card_le, Filter.bddBelow_range_of_tendsto_atTop_atTop, affineCombination_mem_affineSpan, AlgebraicIndependent.sumElim_iff, ZSpan.repr_ceil_apply, range_inter_ssubset_iff_preimage_ssubset, contMDiffAt_iff_source_of_mem_source, EReal.range_coe_ennreal, Ideal.mem_span_range_self, image2_range, MeasureTheory.OuterMeasure.map_comap, span_coeff_minpolyDiv, Affine.Simplex.ExcenterExists.isTangentAt_touchpoint, AlgebraicGeometry.Scheme.Cover.covers, spectralValueTerms_finite_range, ContinuousMultilinearMap.range_toUniformOnFun, IsometryClass.diam_range, ZFSet.isOrdinal_iff_mem_range_toZFSet, NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis, linearIndependent_fin_cons, SymmetricPower.span_tprod_eq_top, LinearMap.BilinForm.dualSubmodule_dualSubmodule_flip_of_basis, Submodule.span_range_inclusion_eq_top, finsum_emb_domain, Function.Embedding.optionEmbeddingEquiv_apply_snd_coe, image_preimage_eq_range_inter, Finsupp.finite_range, range_cfcₙ, Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_left_iff, RootPairing.root_add_root_mem_of_pairingIn_neg, surjective_onto_range, Finite.fin_param, Filter.Tendsto.bddAbove_range, EReal.range_coe_eq_Ioo, DoubleCentralizer.range_toProd, ContinuousAddMonoidHom.range_toContinuousMap, range_subset_insert_image_mulTSupport, Module.Basis.range_extendLe, extChartAt_target_mem_nhdsWithin_of_mem, IsPathConnected.exists_path_through_family, biUnion_range, Order.mem_range_pred_or_isPredPrelimit, IsDenseInducing.closure_range, Topology.IsInducing.image_mem_nhdsWithin, Module.Basis.reindexRange_repr_self, Submodule.fg_iff_exists_finite_generating_family, AlgebraicGeometry.isBasis_basicOpen, SmoothBumpFunction.tsupport_subset_symm_image_closedBall, iSup_range, MDifferentiableAt.differentiableWithinAt_writtenInExtChartAt, Module.Basis.subset_extendLe, AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C', SimpleGraph.dist_eq_sInf, Algebra.PreSubmersivePresentation.ofHasCoeffs_algebra_algebraMap_apply, Fin.snoc_injective_iff, KaehlerDifferential.span_range_derivation, AlgebraicGeometry.IsImmersion.isLocallyClosed_range, NumberField.InfinitePlace.Completion.isClosed_image_extensionEmbedding, Function.Semiconj.mapsTo_range, CliffordAlgebra.adjoin_range_ι, Module.Basis.isMaxChain_range_flag, Affine.Simplex.signedInfDist_apply_self, TangentBundle.trivializationAt_apply, PiTensorProduct.bddBelow_projectiveSemiNormAux, rangeSplitting_injective, TopCat.range_pullback_map, Filter.comap_inf_principal_range, NNReal.range_coe, exists_mem_interior_convexHull_affineBasis, iUnion_singleton_eq_range, ciInf_mem, Polynomial.idealSpan_range_update_divByMonic, memℓp_infty_iff, MonoidWithZeroHom.valueGroup_eq_range, LinearIsometryEquiv.reflections_generate, Equiv.Perm.perm_mapsTo_inl_iff_mapsTo_inr, NumberField.mixedEmbedding.span_idealLatticeBasis, RootPairing.coe_chainTopCoeff_eq_sSup, frontier_range_modelWithCornersEuclideanHalfSpace, ExteriorAlgebra.ιMulti_range, Affine.Simplex.wOppSide_affineSpan_faceOpposite_iff, isLUB_hasProd', OpenPartialHomeomorph.continuousWithinAt_writtenInExtend_iff, Antitone.lowerBounds_range_comp_tendsto_atTop, Affine.Simplex.ExcenterExists.sOppSide_excenter_point_iff, MeasurableEmbedding.equivRange_apply, Affine.Simplex.incenter_notMem_affineSpan_face, GrpCat.SurjectiveOfEpiAuxs.fromCoset_eq_of_mem_range, AlgebraicIndependent.lift_reprField, Finset.card_preimage, range_const_subset, AlgebraicGeometry.Scheme.IdealSheafData.range_subschemeι, range_inl_union_range_inr, LinearMap.isClosed_range_coe, isInducing_sumElim, Equiv.embeddingFinSucc_snd, ZFSet.coe_range, exists_subtype_range_iff, NNReal.bddAbove_range_agmSequences_fst, Submodule.span_range_eq_iSup, MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton, AlgebraicGeometry.Scheme.Hom.isLocallyClosed_range, CategoryTheory.ObjectProperty.isLocalization_isLocal, range_enumerateCountable_of_mem, Finset.coe_image_subset_range, Algebra.adjoin_eq_ring_closure, AlgebraicGeometry.IsClosedImmersion.iff_isPreimmersion, PrimeSpectrum.iUnion_range_comap_comp_evalRingHom, range_smul_range, iUnion_eq_range_psigma, LinearIndependent.linearCombinationEquiv_apply_coe, IsTranscendenceBasis.isAlgebraic_field, RootPairing.rootSpan_map_toPerfPair, Module.Relations.Solution.range_π, MeasurableEmbedding.measurable_rangeSplitting, OpenPartialHomeomorph.map_extend_symm_nhdsWithin, Affine.Simplex.mongePlane_def, Order.not_isSuccPrelimit_iff', Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_right_iff, iSup_range', Module.Basis.span_apply, Monoid.PushoutI.NormalWord.cons_head, Algebra.Presentation.naive_relation_apply, mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm, LinearMap.toMatrix_reindexRange, SSet.horn_obj, ModelWithCorners.range_eq_univ_of_not_isRCLikeNormedField, linearIndepOn_id_range_iff, RootPairing.setOf_root_add_zsmul_eq_Icc_of_linearIndependent, SetTheory.PGame.grundyValue_eq_sInf_moveRight, exteriorPower.ιMulti_span_fixedDegree_of_span_eq_top, measurableSet_range_inl, algebraicIndependent_subtype_range, Module.range_piEquiv, finprod_mem_range', LinearIndependent.linearCombination_comp_repr, AddHom.isClosed_range_coe, Complex.isOpen_compl_range_intCast, CoxeterSystem.subgroup_closure_range_simple, AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some, RootPairing.corootSpan_dualAnnihilator_map_eq, AddAction.IsBlock.pairwiseDisjoint_range_vadd, Matroid.mapEmbedding_indep_iff, Cardinal.mem_range_lift_of_le, StrictMono.not_bddAbove_range_of_wellFoundedLT, Valuation.Integers.not_denselyOrdered_of_isPrincipalIdealRing, image_subset_range, MeasureTheory.SimpleFunc.memLp_approxOn_range, InitialSeg.mem_range_of_rel, MaximalSpectrum.range_asIdeal, StrictMono.not_bddBelow_range_of_isPredArchimedean, ContinuousFunctionalCalculus.exists_cfc_of_predicate, range_piMap, OpenPartialHomeomorph.extend_target, FreeMonoid.closure_range_of, Urysohns.CU.bddAbove_range_approx, Topology.IsInducing.disjoint_of_sumElim_aux, EquicontinuousOn.isClosed_range_pi_of_uniformOnFun', Affine.Simplex.altitudeFoot_mem_affineSpan, Matrix.eRank_toNat_eq_finrank, Module.Basis.ofZLatticeBasis_span, powersetCard.coe_ofFinEmb, tendsto_nhdsWithin_range, NumberField.mixedEmbedding.mem_span_latticeBasis, Valuation.Integers.isPrincipalIdealRing_iff_not_denselyOrdered, AddGroupSeminorm.add_bddBelow_range_add, affineIndependent_iff_not_finrank_vectorSpan_le, PowerBasis.mem_span_pow', range_list_get, AddSubgroup.isComplement_range_left, OpenPartialHomeomorph.contDiffWithinAt_extend_coord_change', MeasureTheory.diracProba_comp_diracProbaEquiv_symm_eq_val, Matrix.IsTotallyUnimodular.apply, OpenPartialHomeomorph.extend_coord_change_source_mem_nhdsWithin', rightInverse_rangeSplitting, Matrix.GeneralLinearGroup.fixpointPolynomial_eq_zero_iff, LinearIndependent.linearIndepOn_id, SmoothBumpFunction.ball_subset, OpenPartialHomeomorph.extend_coord_change_source_mem_nhdsWithin, Nat.range_mul_add, MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto_aux, unitInterval.range_sigmoid, AlgebraicGeometry.isCompl_range_inl_inr, AffineIndependent.vectorSpan_eq_top_of_card_eq_finrank_add_one, Function.range_subset_insert_image_mulSupport, finite_range, Equiv.Set.rangeInr_apply_inr, Ordinal.zero_or_succ_or_isSuccLimit, MonoidWithZeroHom.range_nontrivial, Bool.range_eq, MulAction.IsPartition.of_orbits, Subgroup.mem_closure_range_iff_of_fintype, GromovHausdorff.ghDist_le_hausdorffDist, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, AlgebraicGeometry.Scheme.Pullback.range_map, codRestrict_range_surjective, RootPairing.bijOn_reflection_root, Nucleus.coe_toSublocale, PrincipalSeg.mem_range_of_rel, eq_pos_convex_span_of_mem_convexHull, Matrix.mem_range_scalar_iff_commute_single, ENNReal.coe_range_mem_nhds, Affine.Simplex.points_mem_affineSpan_face, Preorder.hasLimit_iff_hasGLB, RootPairing.chainTopCoeff_eq_zero_iff, RootPairing.GeckConstruction.span_range_h'_eq_top, RootPairing.isValuedIn_iff_mem_range, AlgebraicGeometry.mem_range_iff_of_surjective, RootPairing.posRootForm_posForm_pos_of_ne_zero, Ideal.Quotient.mk_span_range, StrictMono.denselyOrdered_range, HolderWith.dimH_range_le, Algebra.PreSubmersivePresentation.naive_toPresentation, VectorField.mlieBracketWithin_def, Module.Basis.range_extend, isUniformInducing_rangeFactorization_iff, Algebra.adjoin_range_eq_range_freeAlgebra_lift, Fin.Embedding.exists_embedding_disjoint_range_of_add_le_ENat_card, Affine.Simplex.range_face_points, Affine.Simplex.points_mem_affineSpan_faceOpposite, isClopen_range_inr, finiteDimensional_direction_affineSpan_range, Isometry.isometryEquivOnRange_toEquiv, Affine.Simplex.circumsphere_unique_dist_eq, ENat.iSup_coe_ne_top, Finset.sum_preimage', MeasureTheory.SimpleFunc.finite_range', Path.image_extend_of_subset, Cardinal.bddAbove_range, OmegaCompletePartialOrder.Chain.range_pair, isOpen_range_inr, stronglyMeasurable_iff_measurable_separable, Filter.IsBoundedUnder.bddBelow_range, linearIndependent_span, AddHom.srange_mk, Int.ceil_eq_floor_add_one_iff_notMem, WithSeminorms.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup, span_range_eq_top_iff_surjective_fintypeLinearCombination, Surreal.bddBelow_range_of_small, Finset.range_orderEmbOfFin, Sym2.range_diag, IsFractionRing.closure_range_algebraMap, ContinuousLinearMap.isClosed_range_iff_antilipschitz_of_injective, contMDiffAt_iff, Subalgebra.adjoin_eq_span_basis, Continuous.bddBelow_range_of_hasCompactSupport, LinearMap.coe_range, setOf_isPreconnected_eq_of_ordered, Algebra.Presentation.span_range_relation_eq_ker, ModelWithCorners.range_eq_target, UniformContinuous.rangeFactorization, ContinuousAlternatingMap.range_toContinuousMultilinearMap, LinearMap.mem_span_iff_continuous_of_finite, kernImage_empty, range_subset_range_iff_exists_comp, Subtype.range_val, HilbertBasis.dense_span, Matrix.rank_eq_finrank_span_row, equicontinuousAt_iff_range, Subgroup.mem_closure_range_iff, ContinuousLinearEquiv.nhds, range_inr_union_range_inl, finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply, EReal.range_coe, WithTop.iSup_coe_lt_top, Rat.addSubmonoid_closure_range_pow, Module.Basis.restrictScalars_repr_apply, TopCat.GlueData.range_fromOpenSubsetsGlue, finrank_range_le_card, OpenPartialHomeomorph.map_extend_nhds, extChartAt_preimage_mem_nhdsWithin, range_quotient_lift, ModelWithCorners.symm_map_nhdsWithin_range, extChartAt_target_mem_nhdsWithin', NumberField.Units.regOfFamily_of_isMaxRank, HasCountableSeparatingOn.range_Ioi, Language.isRegular_iff_finite_range_leftQuotient, HasCountableSeparatingOn.range_Iio, EquicontinuousOn.isClosed_range_uniformOnFun_iff_pi, Finset.univ_filter_mem_range, subset_range_iff_exists_image_eq, NumberField.Units.regOfFamily_div_regulator, NumberField.InfinitePlace.Completion.isClosed_image_extensionEmbeddingOfIsReal, TopologicalSpace.Fiber.sigmaIsoHom_apply, IsUniformInducing.rangeFactorization, range_cfcₙ_eq_range_cfcₙHom, powersetCard.mem_ofFinEmb_iff_mem_range, compl_range_some, IsOpenMap.isDiscrete_range, isClopen_range_sigmaMk, contMDiffWithinAt_extChartAt_symm_range_self, EquivLike.range_eq_univ, FiniteDimensional.range_basisSingleton, exists_root_adjoin_eq_top_of_isCyclic, finite_range_findGreatest, Subtype.range_coe_subtype, FreeGroup.range_lift_eq_closure, AddSubgroup.mem_closure_range_iff, ZSpan.coe_floor_self, isTranscendenceBasis_subtype_range, CategoryTheory.ObjectProperty.isColocal_adj_counit_app, closedBall_subset_range_iff_surjective, Module.infinite_range_reflection_reflection_iterate_iff, Valuation.IsRankOneDiscrete.generator_zpowers_eq_range, directedOn_range, Matrix.range_mulVecLin, Path.extend_range, instIsZLatticeRealSpan, Affine.Simplex.orthogonalProjection_circumcenter, Affine.Simplex.affineSpan_faceOpposite_le, PiNat.exists_lipschitz_retraction_of_isClosed, coe_comp_rangeFactorization, Fin.range_snoc, equicontinuous_iff_range, PrimeSpectrum.residueField_comap, preimage_range, Quotient.range_mk'', WithBot.range_eq, OpenPartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq_aux, FirstOrder.Language.Substructure.fg_iff_exists_fin_generating_family, range_quot_mk, Monotone.lowerBounds_range_comp_tendsto_atBot, contMDiffWithinAt_iff_source_of_mem_maximalAtlas, LieHom.coe_range, centroid_mem_affineSpan_of_nonempty, range_inl, PartialOrder.mem_range_nerve_σ_iff, Matrix.range_toLin', RootPairing.GeckConstruction.span_range_h_le_range_diagonal, AddSubgroup.IsComplement.rightQuotientEquiv_apply, LinearIndependent.span_repr_eq, AddSubgroup.closure_image_isAddIndecomposable_baseOf, card_range_of_injective, NumberField.IsCMField.closure_realFundSystem_sup_torsion, Complex.range_arg, Module.Basis.restrictScalars_apply, AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none, NormedSpace.equicontinuous_TFAE, Function.Surjective.range_comp, Profinite.NobelingProof.GoodProducts.span, HasCompactMulSupport.isCompact_range, StarSubalgebra.coe_bot, range_inr, Subtype.range_coe, range_two_mul, IsLocalization.Away.span_range_mulNumerator_eq_top, Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq, Int.floor_eq_self_iff_mem, range_restrict, Real.range_exp, linearIndepOn_range_iff, range_insert, sigma_univ_range_eq, iUnion_range_eq_iUnion, EuclideanGeometry.Sphere.sOppSide_faceOpposite_secondInter_of_mem_interior_faceOpposite, ZSpan.isAddFundamentalDomain', linearIndependent_fin_succ', image_preimage_eq_inter_range, Affine.Simplex.touchpoint_mem_affineSpan, ModelWithCorners.nonempty_interior, biUnion_range_preimage_singleton, LieModuleHom.coe_range, Path.range_subpath, LinearMap.injective_range_liftQ_of_exact, GromovHausdorff.ghDist_eq_hausdorffDist, TopologicalSpace.Opens.set_range_inclusion', exists_msmooth_support_eq_eq_one_iff, image_eq_range, Nucleus.range_subset_range, AlgebraicGeometry.Scheme.Pullback.range_snd_comp, Ordinal.principal_add_iff_zero_or_omega0_opow, RootPairing.bijOn_coreflection_coroot, MDifferentiableAt.mfderiv, range_const_eq_diagonal, nonempty_iInter_Ici_iff, Topology.IsInducing.multipliable_iff_tprod_comp_mem_range, Affine.Simplex.direction_mongePlane, range_extend_subset, Ordinal.unbounded_range_of_le_iSup, Equiv.Perm.IsCycleOn.range_pow, Affine.Simplex.range_faceOpposite_reindex, finrank_span_eq_card, inv_range, AddAction.IsBlock.isBlockSystem, LinearMap.BilinForm.dualSubmodule_dualSubmodule_of_basis, RootPairing.IsRootSystem.span_root_eq_top, Equiv.ofInjective_symm_apply, AlgebraicGeometry.range_eq_univ, RootPairing.zero_le_posForm, PowerBasis.mem_span_pow, hasFDerivWithinAt_tangentCoordChange, Ordinal.bddAbove_range, ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range, Submodule.exists_fun_fin_finrank_span_eq, range_cfcₙ_nnreal_eq_image_cfcₙ_real, MDifferentiableWithinAt.mfderivWithin, Finset.mem_range_iff_mem_finset_range_of_mod_eq', Continuous.rangeFactorization, NonUnitalRingHom.coe_range, NumberField.Units.closure_fundSystem_sup_torsion_eq_top, EuclideanGeometry.Sphere.sOppSide_faceOpposite_secondInter_of_mem_interior, contMDiffWithinAt_iff, affineIndependent_iff_le_finrank_vectorSpan, mem_range_indicator, Real.range_arcsin, OpenPartialHomeomorph.contDiffWithinAt_extend_coord_change, range_euclideanHalfSpace, KaehlerDifferential.kerTotal_map, MulHom.isClosed_range_coe, MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable, AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply, Monotone.isChain_range, IntermediateField.adjoin.range_algebraMap_subset, Cardinal.mk_range_le, exists_msmooth_zero_iff_one_iff_of_isClosed, Cardinal.mk_range_inr, ModelWithCorners.map_nhds_eq, range_list_map, Path.range_segment, RootPairing.root_sub_nsmul_mem_range_iff_le_chainBotCoeff, isCusp_SL2Z_iff, Orthonormal.toSubtypeRange, extChartAt_target_eventuallyEq', Metric.mem_closure_range_iff, AlgebraicIndependent.option_iff_transcendental, isClosed_range_sigmaMk, ZSpan.span_top, RootPairing.Base.span_int_root_support, MulAction.period_bounded_of_exponent_pos, RootPairing.iInf_ker_root'_eq, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left, PartialEquiv.target_subset_range, NumberField.mixedEmbedding.span_latticeBasis, ArchimedeanClass.range_mk, Equiv.Set.rangeSplittingImageEquiv_symm_apply_coe, PrimeSpectrum.isTopologicalBasis_basic_opens, Matrix.mem_range_scalar_iff_commute_single', AlgebraicIndependent.algebraMap_aevalEquiv, linearIndependent_fin_snoc, NonUnitalAlgHom.coe_range, LieHom.range_coe, Affine.Simplex.sOppSide_excenter_singleton_point, Filter.Tendsto.bddBelow_range, preimage_singleton_eq_empty, SSet.Subcomplex.PairingCore.equivI_apply_coe, Fintype.range_linearCombination, tangentBundleCore_coordChange, equicontinuousOn_iff_range, ContinuousLinearMap.range_coeFn_eq, Algebra.adjoin_range_eq_range_aeval, Affine.Simplex.ExcenterExists.sOppSide_point_excenter_iff, countable_range, ZSpan.map, PrimeSpectrum.mem_range_comap_iff, PrincipalSeg.subrelIso_symm_apply, isLUB_iUnion_iff_of_isLUB, exists_equiv_range_eq_graphOn_univ, contDiffOn_fderiv_coord_change, ZSpan.exist_unique_vadd_mem_fundamentalDomain, SmoothBumpFunction.isClosed_symm_image_closedBall, Ordinal.typein_surj, ModelWithCorners.image_mem_nhdsWithin, Topology.IsEmbedding.toHomeomorph_apply_coe, Isometry.isometryEquivOnRange_apply, CategoryTheory.Functor.ranges_directed, setOf_isPreconnected_subset_of_ordered, Finite.toFinset_range, centroid_mem_affineSpan_of_cast_card_ne_zero, range_some_inter_none, ProbabilityTheory.iSup_partitionFiltration_eq_generateFrom_range, Function.Injective.injOn_range, Surreal.bddAbove_range_of_small, ENNReal.iSup_coe_eq_top, Algebra.Presentation.tensorModelOfHasCoeffsEquiv_tmul, range_codRestrict, Function.Surjective.range_eq, extChartAt_target_subset_range, range_diff_image_subset, QuotientGroup.range_mk, SimpleGraph.edist_eq_sInf, NumberField.InfinitePlace.Completion.isClosed_image_extensionEmbedding_of_isReal, MeasurableSpace.generateFrom_natGeneratingSequence, isCompact_range, Function.Embedding.right_inv_of_invOfMemRange, Rat.addSubmonoid_closure_range_mul_self, Types.monoOverEquivalenceSet_functor_map, Matrix.SpecialLinearGroup.range_toGL, range_subtype_map, Ordinal.mem_range_veblen_iff_le_invVeblen₁, Topology.isClosedEmbedding_iff, Equiv.asEmbedding_range, range_vadd_range, IsDenseInducing.nhdsWithin_neBot, Topology.IsOpenEmbedding.isOpen_range, Function.Periodic.image_Ioc, range_cfc_eq_range_cfcHom, IsUniformInducing.isComplete_range, MeasureTheory.Martingale.bddAbove_range_iff_bddBelow_range, Affine.Simplex.sSameSide_affineSpan_faceOpposite_iff, map_extChartAt_nhds', RootPairing.iInf_ker_coroot'_eq, Matrix.IsHermitian.spectrum_eq_image_range, Affine.Simplex.sSameSide_affineSpan_faceOpposite_of_sign_eq, Perfect.exists_nat_bool_injection, mdifferentiableWithinAt_iff', Matroid.mapSetEmbedding_indep_iff, Int.fract_ne_zero_iff, DFinsupp.add_closure_iUnion_range_single, Affine.Simplex.orthogonalProjectionSpan_faceOpposite_eq_point_rev, range_vadd, Order.mem_range_pred_of_not_isPredPrelimit, OpenPartialHomeomorph.extend_target_eventuallyEq, ZSpan.instDiscreteTopologySubtypeMemSubmoduleIntSpanRangeCoeBasisRealOfFinite, exists_linearIndependent', IntermediateField.mem_adjoin_range_iff, range_add, range_one, LinearIndependent.linearCombinationEquiv_symm_apply, NNRat.addSubmonoid_closure_range_pow, ZSpan.isAddFundamentalDomain, finite_range_const, PrimeSpectrum.range_comap_snd, Algebra.Presentation.tensorModelOfHasCoeffsHom_tmul, Module.Relations.Solution.isPresentation_iff, ModelWithCorners.target_eq, range_prodMap, rangeFactorization_surjective, Affine.Simplex.ExcenterExists.isTangentAt_exsphere_iff_eq_touchpoint, Function.Embedding.sigmaSet_range, FreeAddGroup.closure_range_of, IsLocalDiffeomorph.isOpen_range, Function.Injective.left_inv_of_invOfMemRange, Finite.fin_embedding, TopCat.Opens.coverDense_iff_isBasis, range_eq_empty_iff, Matrix.range_empty, Fin.range_natAdd, Monotone.upperBounds_range_comp_tendsto_atTop, Equiv.Set.rangeSplittingImageEquiv_apply_coe_coe, linearIndependent_fin_succ, Cardinal.range_aleph, IsOpen.exists_msmooth_support_eq_aux, DenseRange.nhdsWithin_neBot, isClosed_range_of_spaced_out, Finset.image_preimage, IsIntegralClosure.range_le_span_dualBasis, Path.range_reparam, range_quotient_mk', GrpCat.SurjectiveOfEpiAuxs.h_apply_fromCoset', ModelWithCorners.convex_range, TopCat.range_pullback_to_prod, DenseRange.closure_range, exists_contMDiff_zero_iff_one_iff_of_isClosed, TopologicalSpace.NonemptyCompacts.range_map, contDiffWithinAtProp_self_target, Filter.IsBoundedUnder.bddBelow_range_of_cofinite, Filter.Tendsto.bddBelow_range_of_cofinite, Function.Periodic.compact_of_continuous, LinearIndependent.span_eq_top_of_card_eq_finrank, cfcₙHom_map_quasispectrum, FreeAddMonoid.closure_range_of, AlgebraicGeometry.isImmersion_iff, Submodule.top_le_span_range_iff_forall_exists_fun, GromovHausdorff.hausdorffDist_optimal_le_HD, mem_affineSpan_iff_eq_weightedVSubOfPoint_vadd, CategoryTheory.Abelian.isoModSerre_kernel_eq_leftBousfield_W_of_rightAdjoint, Subtype.range_val_subtype, RestrictedProduct.range_coe_principal, range_inr_inter_range_inl, fourierSubalgebra_coe, Types.monoOverEquivalenceSet_functor_obj, RootPairing.GeckConstruction.diagonal_elim_mem_span_h_iff, AlgHom.coe_fieldRange, Function.Fiber.mem_iff_eq_image, EMetric.NonemptyCompacts.isClosed_in_closeds, Submodule.span_range_update_sub_smul, RootPairing.range_weylGroupToPerm, Setoid.range_kerLift_eq_range, VectorField.eventuallyEq_mpullback_mpullbackWithin_extChartAt, Matrix.GeneralLinearGroup.mem_center_iff_val_eq_scalar, Ideal.exists_subset_radical_span_sup_of_subset_radical_sup, RootPairing.algebraMap_posRootForm_posForm, MulAction.IsBlock.isBlockSystem, FreeAlgebra.adjoin_range_ι, PrimeSpectrum.stableUnderSpecialization_range_iff, Equiv.ofInjective_apply, KaehlerDifferential.submodule_span_range_eq_ideal, map_extChartAt_nhdsWithin, spec_cfcₙAux, Profinite.NobelingProof.spanFinBasis.span, RootPairing.span_root'_eq_top, AlgebraicIndependent.liftAlgHom_comp_reprField, EquivLike.range_comp, comp_rangeSplitting, MeasurableSpace.generateFrom_memPartition_le_range, ENNReal.range_coe', Filter.not_bddBelow_of_tendsto_atBot, RingHom.coe_fieldRange, Function.periodicPts_subset_range, Ordinal.range_familyOfBFamily', Affine.Simplex.span_eq_top, PrincipalSeg.mem_range_of_le, compl_range_inl, ModelWithCorners.uniqueMDiffOn, IccRightChart_extend_top_mem_frontier, FreeMonoid.mrange_lift, Fin.range_castLE, Path.source_mem_range, LinearIndependent.repr_range, range_diag, range_deriv_subset_closure_span_image, range_cfc_nnreal, MvPolynomial.ker_eval₂Hom_universalFactorizationMap, Matrix.mem_range_scalar_of_commute_single, Path.target_mem_range, Polynomial.coeff_divModByMonicAux_mem_span_pow_mul_span, Cardinal.mk_range_eq_lift, FreeAddMonoid.mrange_lift, RootPairing.restrictScalars_coe_root, InfiniteGalois.mem_range_algebraMap_iff_fixed, Submodule.mem_span_range_iff_exists_fun, range_nnnorm, NonarchAddGroupSeminorm.add_bddBelow_range_add, Nat.range_cast_int, range_toPullbackDiag, Isometry.hausdorffMeasure_preimage, AlgebraicGeometry.Proj.isBasis_basicOpen, AlgebraicIndependent.aeval_repr, finite_range_ite, PathConnectedSpace.exists_path_through_family, bddAbove_range_pi, CategoryTheory.Subfunctor.range_obj, Finsupp.range_linearCombination, mapsTo_range_iff, Polynomial.lifts_iff_coeff_lifts, StarSubalgebra.range_subset, isSeparable_range_deriv, AddMonoidHom.mclosure_range, NumberField.IsCMField.ringOfIntegersComplexConj_eq_self_iff, Module.Relations.ker_toQuotient, RestrictedProduct.range_coe, ContinuousAlternatingMap.range_toAlternatingMap, OrderEmbedding.maximal_apply_mem_inter_range_iff, Affine.Simplex.sOppSide_point_excenter_singleton, Nat.range_nth_subset, MeasurableEmbedding.measurableSet_range, StrictMono.range_inj, Sym2.isDiag_iff_mem_range_diag, ProjectiveSpectrum.isTopologicalBasis_basic_opens, isGLB_of_tendsto_atBot, Subgroup.isComplement_range_left, Cardinal.mk_range_inl, Topology.IsOpenEmbedding.toOpenPartialHomeomorph_target, vectorSpan_range_eq_span_range_vsub_right_ne, restrict_extend_compl_range, Fin.range_cons, Affine.Simplex.ninePointCircle_center_mem_affineSpan, range_id', StrictAnti.not_bddBelow_range_of_isSuccArchimedean, finsum_mem_range, range_subset_singleton, range_nonempty, AddAction.isBlock_iff_pairwiseDisjoint_range_vadd, Composition.mem_range_embedding, CategoryTheory.ObjectProperty.isColocal_iff_isIso_map, Preorder.isLUB_of_isColimit, infClosed_range, AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left, range_projIic, LipschitzWith.dimH_range_le, Filter.comap_eq_bot_iff_compl_range, Function.Embedding.invFun_restrict, AlgebraicGeometry.Scheme.AffineCover.covers, Module.Basis.range_ofSpan, Affine.Simplex.affineCombination_mem_affineSpan_faceOpposite_iff, Rat.isLocalizationIsInteger_iff, rangeFactorization_eq_iff, NumberField.canonicalEmbedding.mem_rat_span_latticeBasis, isLUB_hasSum', OmegaCompletePartialOrder.ωScottContinuous.isLUB, SubAddAction.image_inclusion, uniqueMDiffWithinAt_iff_inter_range, Submodule.fg_iff_exists_fin_generating_family, IsMaxChain.range_fin_of_covBy, Bundle.TotalSpace.range_mk, MeasureTheory.SimpleFunc.integrable_approxOn_range, measurableSet_range_inr, AddMonoid.exponent_eq_zero_iff_range_addOrderOf_infinite, Seminorm.bddBelow_range_add, Ordinal.range_familyOfBFamily, Function.Injective.convexIndependent_iff_set, TensorAlgebra.range_lift, isCompl_range_inl_range_inr, Ideal.mem_span_range_iff_exists_fun, Finite.bddBelow_range, IsZLattice.isCompact_range_of_periodic, setBasisOfTopLeSpanOfCardEqFinrank_repr_apply, span_fourierLp_closure_eq_top, Ordinal.mem_range_veblen, weightedVSub_mem_vectorSpan, isConnected_range, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, GromovHausdorff.HD_below_aux2, Composition.mem_range_embedding_iff', Algebra.Generators.disjoint_ker_toKaehler_of_linearIndependent, RootPairing.neg_root_mem, Submodule.span_range_update_add_smul, Polynomial.image_comap_C_basicOpen, Affine.Simplex.ExcenterExists.excenter_mem_affineSpan_range, range_euclideanQuadrant, RootPairing.RootPositiveForm.isSymm_posForm, nhdsWithin_extChartAt_target_eq_of_mem, Ideal.span_range_pow_eq_top, TopCat.GlueData.preimage_range, AffineIndependent.finrank_vectorSpan, range_quotient_mk, IsGLB.range_of_tendsto, isLindelof_range, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right, exists_range_eq_graphOn_univ, Ordinal.brange_bfamilyOfFamily, MonoidHom.isClosed_range_coe, preimage_range_inter, smul_set_range, VectorField.contMDiffWithinAt_mpullbackWithin_extChartAt_symm, preimage_eq_empty_iff, Measurable.measurableSet_preimage_iff_preimage_val, Int.fract_eq_zero_iff, mdifferentiableWithinAt_iff_source_of_mem_maximalAtlas, AffineIsometry.diam_range, isClosed_range_inr, RootPairing.chainBotCoeff_eq_sSup, Ordinal.veblen_mem_range_opow, AddSubmonoid.coe_multiples, range_list_getI, Option.image_elim_range_some_eq_range, mem_range_of_mem_image, mdifferentiableWithinAt_iff, Affine.Simplex.sOppSide_affineSpan_faceOpposite_of_pos_of_neg, range_eq_image_tsupport_or, isPiSystem_Iio, MeasurableEmbedding.integrableOn_range_iff_comap, IsIntegral.mem_span_pow, AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one, finsum_mem_range', denseRange_iff_closure_range, range_cfc_nnreal_eq_image_cfc_real, Ordinal.mem_range_gamma, Affine.Simplex.range_face_reindex, mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm', Filter.Tendsto.isVonNBounded_range, Int.addSubmonoid_closure_range_pow, polynomialFunctions_coe, Topology.IsInducing.summable_iff_tsum_comp_mem_range, mem_span_of_iInf_ker_le_ker, OpenPartialHomeomorph.interior_extend_target_subset_interior_range, LinearIsometry.diam_range, Topology.IsInducing.map_nhdsSet_eq, ContinuousMap.spectrum_eq_range, OrderedFinpartition.cover, IsClosed.exists_nat_bool_injection_of_not_countable, SetTheory.Game.bddAbove_range_of_small, powersetCard.mem_ofFinEmbEquiv_iff_mem_range, Affine.Simplex.affineSpan_range_medial, Measurable.rangeFactorization, ENat.iSup_coe_eq_top, biInter_range, Affine.Simplex.points_notMem_affineSpan_faceOpposite, AffineIndependent.indicator_extend_eq_of_affineCombination_comp_embedding_eq_of_fintype, StarOrderedRing.le_iff, Submodule.spanRank_span_range_of_linearIndependent, CauchySeq.isBounded_range, Equiv.embeddingFinSucc_fst, AddUnits.range_embedProduct, writtenInExtChartAt_sumSwap_eventuallyEq_id, RelSeries.mem_def, interior_range_modelWithCornersEuclideanHalfSpace, SimpleGraph.isClique_range_copy_top, RelEmbedding.acc_iff_isEmpty_subtype_mem_range, Equiv.Set.rangeInl_apply_inl, Equiv.Set.rangeInr_symm_apply_coe, Algebra.Presentation.coeffs_subset_range, closure_ordConnected_inter_rat, Nucleus.mem_range, Equiv.Perm.IsCycleOn.range_zpow, wellFoundedOn_range, Finite.bddAbove_range, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, Metric.isBounded_range_of_cauchy_map_cofinite, Fin.range_succAbove, Matrix.IsHermitian.spectrum_real_eq_range_eigenvalues, TopCat.GlueData.image_inter, Fin.range_castAdd, Algebra.Presentation.instFinitePresentationModelOfHasCoeffsOfFinite, RootPairing.root_add_zsmul_mem_range_iff, range_quotient_lift_on', ZSpan.quotientEquiv.symm_apply, Sum.range_eq, NumberField.mem_span_integralBasis, IsLocalRing.span_eq_top_of_tmul_eq_basis, mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt', IntermediateField.adjoin_eq_range_algebraMap_adjoin, MaximalSpectrum.toPrimeSpectrum_range, Path.trans_range, Affine.Simplex.incenter_mem_affineSpan_range, AlgebraicGeometry.Scheme.Pullback.range_snd, Option.injective_iff, upperBounds_range_partialSups, OpenPartialHomeomorph.extend_symm_continuousWithinAt_comp_right_iff, Nat.Subtype.ofNat_range, Matrix.range_cons_cons_empty, CategoryTheory.Presieve.uncurry_ofArrows, IsTranscendenceBasis.to_subtype_range, intermediate_value_univ, Function.Embedding.optionEmbeddingEquiv_symm_apply, Algebra.SubmersivePresentation.basisKaehler_apply, OrderEmbedding.orderIso_apply, RestrictedProduct.range_inclusion, SetTheory.PGame.bddBelow_range_of_small, Fin.range_fin_succ, AlgebraicGeometry.Scheme.ofArrows_mem_precoverage_iff, Prod.range_fst, CategoryTheory.Limits.Types.eq_or_eq_of_isPushout', GrpCat.SurjectiveOfEpiAuxs.h_apply_fromCoset, ModelWithCorners.range_eq_closure_interior, ModelWithCorners.range_subset_closure_interior, Finsupp.range_single_subset, ModelWithCorners.convex_range', AddMonoid.Coprod.mclosure_range_inl_union_inr, Path.range_coe, Equiv.coe_embeddingFinSucc_symm, equicontinuousWithinAt_iff_range, affineCombination_mem_convexHull, uniformEquicontinuousOn_iff_range, CategoryTheory.Presieve.mem_comap_jointlySurjectivePrecoverage_iff, cosetToCuspOrbit_apply_mk, CategoryTheory.ObjectProperty.isLocal_iff_isIso_map, NumberField.mem_span_basisOfFractionalIdeal, AffineIsometry.ediam_range, MonoidWithZeroHom.valueMonoid_eq_closure, Affine.Simplex.orthogonalProjection_eq_circumcenter_of_dist_eq, MapsTo.range_restrict, ModelWithCorners.transContinuousLinearEquiv_range, OpenPartialHomeomorph.map_extend_nhdsWithin, mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt, uniformEquicontinuous_iff_range, MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite, Finsupp.mem_span_range_iff_exists_finsupp, BoundedContinuousFunction.isBounded_range_integral, exists_contMDiff_support_eq_eq_one_iff, MulAction.orbit.pairwiseDisjoint, mapsTo_range, OrderEmbedding.minimal_apply_mem_inter_range_iff, Topology.IsEmbedding.isTotallyDisconnected_range, Submodule.range_map_nonempty, Fin.range_castSucc, InnerProductSpace.span_gramSchmidt, ModuleCat.disjoint_span_sum, Option.range_elim, LinearIndependent.repr_ker, affineCombination_mem_affineSpan_of_nonempty, CategoryTheory.Limits.Types.range_pullbackSnd, HasCompactSupport.isCompact_range, StarOrderedRing.nonneg_iff, Subalgebra.range_subset, ZSpan.instDiscreteTopologySubtypeMemAddSubgroupToAddSubgroupIntSpanRangeCoeBasisRealOfFinite, Real.not_bddBelow_coe, Scott.isωSup_iff_isLUB, range_id, OnePoint.range_coe_union_infty, Primcodable.mem_range_encode, Ordinal.nonempty_compl_range, Ideal.range_finsuppTotal, AlgebraicIndependent.aevalEquivField_apply_coe, UniformEquiv.range_coe, Language.mem_accept_toDFA, MeasurableEmbedding.equivRange_symm_apply_mk, Metric.mem_closure_range_iff_nat, zero_mem_lowerBounds_smoothingSeminormSeq_range, CauchySeq.totallyBounded_range, WithTop.iSup_coe_eq_top, BoundedContinuousFunction.dist_extend_extend, StrictAnti.not_bddBelow_range_of_isPredArchimedean, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_right_iff, ProbabilityTheory.Kernel.borelMarkovFromReal_apply, Fin.card_range_le, supClosed_range, AlgebraicIndependent.aevalEquiv_apply_coe, contMDiffWithinAt_iff_source, RootPairing.posRootForm_posForm_nondegenerate, exists_range_iff, uniformContinuous_rangeFactorization_iff, range_projIci, SSet.mem_skeletonOfMono_obj_iff_of_nonDegenerate, sInter_range, OpenPartialHomeomorph.nhdsWithin_extend_target_eq, nhdsWithin_extChartAt_target_eq, Affine.Triangle.orthocenter_mem_affineSpan, Fin.range_val, Module.Basis.localizationLocalization_span, CategoryTheory.Limits.Types.range_snd_of_isPullback, Finsupp.range_single_one, Affine.Simplex.mongePoint_mem_affineSpan, CategoryTheory.Functor.isMittagLeffler_iff_subset_range_comp, SubMulAction.coe_one, pairwiseDisjoint_range_singleton, AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_X, contMDiffWithinAt_iff_source_of_mem_source, AlgebraicGeometry.Scheme.Pullback.range_fst, Function.Embedding.toEquivRange_symm_apply_self, isSigmaCompact_range, MeasurableEmbedding.comap_preimage, smoothingSeminormSeq_bddBelow, Ordinal.invVeblen₁_lt_iff, IsOpen.exists_contDiff_support_eq, PresentedGroup.closure_range_of, ContinuousLinearEquiv.isOpen, Real.not_bddAbove_coe, AffineBasis.isUnit_toMatrix_iff, MeasureTheory.OuterMeasure.isometry_map_mkMetric, Ctop.toTopsp_isTopologicalBasis, AddSubmonoid.mem_closure_range_iff_of_fintype, ContinuousMonoidHom.range_toContinuousMap, IntermediateField.set_range_subset, Path.refl_range, contMDiffWithinAt_iff_of_mem_maximalAtlas, Algebra.Presentation.tensorModelOfHasCoeffsEquiv_symm_tmul, Metric.isBounded_range_of_tendsto_cofinite_uniformity, RootPairing.chainTopCoeff_eq_sSup, Submonoid.mem_closure_range_iff, RootPairing.ext_iff, affineIndependent_iff_not_collinear, Function.Injective.hasSum_range_iff, Fin.cons_injective_iff, AlgebraicGeometry.Scheme.Pullback.range_diagonal_subset_diagonalCoverDiagonalRange, QuaternionAlgebra.eq_re_iff_mem_range_coe, MeasureTheory.isCountablySpanning_spanningSets, Function.LeftInverse.isClosed_range, Profinite.NobelingProof.GoodProducts.span_iff_products, AlgebraicIndependent.option_iff, mdifferentiableAt_iff_source_of_mem_source, RootPairing.neg_coroot_mem, modelProd_range_prod_id, MeasureTheory.tendsto_diracProbaEquivSymm_iff_tendsto, mdifferentiableAt_iff_of_mem_source, RootPairing.Iic_chainBotCoeff_eq, Isometry.map_hausdorffMeasure, FirstOrder.Language.Hom.range_coe, PresentedAddMonoid.closure_range_of, Function.Injective.compl_image_eq, compl_range_subset_kernImage, FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize, Complex.range_norm, span_flip_eq_top_iff_linearIndependent, ModelWithCorners.uniqueDiffWithinAt_image, SSet.mem_degenerate_iff, ConvexIndependent.range, Dilation.diam_range, Language.step_toDFA, Int.floor_lt_self_iff, Submodule.span_singleton_eq_range, LinearMap.BilinForm.dualSubmodule_flip_dualSubmodule_of_basis, TopologicalSpace.Fiber.sigmaIsoHom_surj, isOpen_range_sigmaMk, MeasurableSpace.generateFrom_iUnion_memPartition, instWellFoundedGTElemRangeBound, Ordinal.mem_range_deriv, Composition.disjoint_range, AlgebraicGeometry.Scheme.Opens.range_ι, AlgebraicGeometry.IsAffineOpen.range_fromSpec, Algebra.Presentation.HasCoeffs.coeffs_subset_range, MulAction.IsBlock.pairwiseDisjoint_range_smul, MulAction.isBlock_iff_pairwiseDisjoint_range_smul, summable_iff_cauchySeq_finset_and_tsum_mem, CategoryTheory.PreGaloisCategory.autEmbedding_range, mdifferentiableWithinAt_extChartAt_symm, contMDiffWithinAt_extChartAt_symm_range, range_piecewise, contDiffWithinAt_ext_coord_change, PrimeSpectrum.range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk, Ordinal.mem_range_veblenWith, AlgebraicGeometry.Scheme.Hom.range_subset_ker_support, CategoryTheory.Subfunctor.toRangeSheafify_app_coe, eventually_norm_mfderivWithin_symm_extChartAt_comp_lt, ball_subset_range_iff_surjective, Filter.isTopologicalBasis_Iic_principal, OpenPartialHomeomorph.extend_preimage_mem_nhdsWithin, rangeFactorization_eq, GrpCat.SurjectiveOfEpiAuxs.h_apply_fromCoset_nin_range, LinearIsometry.ediam_range, Prod.range_snd, Ordinal.sInf_compl_lt_ord_succ, MvPolynomial.homogeneousSubmodule_one_eq_span_X, span_range_eq_top_iff_surjective_finsuppLinearCombination, Function.Periodic.isBounded_of_continuous, Affine.Simplex.orthogonalProjectionSpan_reindex, Function.Fiber.eq_fiber_image, InitialSeg.eq_or_principal, Equiv.range_eq_univ, KaehlerDifferential.span_range_eq_ideal, Submonoid.mem_closure_range_iff_of_fintype, Fin.range_natAdd_eq_Ioi, RootPairing.RootPositiveForm.algebraMap_apply_eq_form_iff, OnePoint.range_coe_inter_infty, SetTheory.Game.bddBelow_range_of_small, Affine.Simplex.touchpoint_empty_notMem_affineSpan_of_ne, ContinuousLinearMap.bddAbove_rayleighQuotient, RootPairing.restrictScalars_toLinearMap_apply_apply, Polynomial.isGreatest_supNorm, iInter_Iic_eq_empty_iff, IsWellFounded.mem_range_rank_of_le, Language.IsRegular.finite_range_leftQuotient, range_list_getD, NonUnitalRingHom.coe_srange, Nat.range_of_succ, Affine.Simplex.sSameSide_point_excenter_singleton, CategoryTheory.ObjectProperty.isLocal_adj_unit_app, isPiSystem_Ici, leftInverse_rangeSplitting, Function.Embedding.invOfMemRange_surjective, ZeroAtInftyContinuousMap.isBounded_range, TopologicalSpace.Fiber.instCompactSpaceElemValSetMemRangeCoeLocallyConstantPreimageSingleton, prod_range_range_eq, borel_eq_generateFrom_Ioi, isPathConnected_range, SSet.degenerate_eq_iUnion_range_σ, AlgebraicGeometry.Scheme.Hom.support_ker, rangeFactorization_injective, insert_image_compl_eq_range, iSupIndep.sSupIndep_range, Filter.Tendsto.isCompact_insert_range_of_cocompact, Module.Relations.Solution.span_relation_le_ker_π, UnitAddTorus.span_mFourier_closure_eq_top, ContDiff.dimH_range_le, IsLocalHomeomorph.discreteTopology_range_iff, SetTheory.PGame.bddAbove_range_of_small, Affine.Simplex.orthogonalProjectionSpan_eulerPoint_mem_ninePointCircle, extChartAt_target, mdifferentiableWithinAt_iff_source_of_mem_source, Ordinal.lsub_notMem_range, Monoid.Coprod.mclosure_range_inl_union_inr, IntermediateField.adjoin_subset_adjoin_iff, MeasurableEmbedding.iff_comap_eq, AntilipschitzWith.isComplete_range, Algebra.Presentation.HasCoeffs.relation_mem_range_map, ModelWithCorners.isClosed_range, centroid_mem_affineSpan_of_card_eq_add_one, Antitone.upperBounds_range_comp_tendsto_atBot, range_smul, dimH_range_le_of_locally_lipschitzOn, Matrix.range_cons_empty, Affine.Simplex.reindex_range_points, affineIndependent_iff_finrank_vectorSpan_eq, Cardinal.range_natCast, MonoidHom.coe_range, MeasureTheory.AnalyticSet_def, ContinuousMap.isOpen_setOf_range_subset, ZSpan.discreteTopology_pi_basisFun, AddMonoid.Coprod.closure_range_inl_union_inr, Affine.Simplex.signedInfDist_affineCombination, disjoint_image_inl_range_inr, CompleteLatticeHom.toOrderIsoRangeOfInjective_apply, Ordinal.mem_range_preOmega_iff, small_range, Module.Basis.range_reindex, Matroid.IsCircuit.strong_multi_elimination, Types.monoOverEquivalenceSet_unitIso, IsLocalHomeomorph.isTopologicalBasis, NumberField.Units.isMaxRank_iff_closure_finiteIndex, Nat.orderEmbeddingOfSet_range, borel_eq_generateFrom_Iio, NormedField.denselyOrdered_range_nnnorm, MeasureTheory.continuous_diracProbaEquivSymm, RootPairing.posRootForm_posForm_anisotropic, subset_range_of_surjective, Finsupp.mem_ideal_span_range_iff_exists_finsupp, OrderedFinpartition.range_extendLeft_zero, IccExtend_range, infinite_range_add_nsmul_iff, range_const, isGLB_iUnion_iff_of_isLUB, Encodable.decode₂_ne_none_iff, Filter.comap_neBot_iff_frequently, MulHom.srange_mk, Subalgebra.setRange_algebraMap, isGLB_iInf, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.preimage_toBase_eq_range_ι, PrimeSpectrum.exists_maximal_notMem_range_sigmaToPi_of_infinite, range_mul, AlgebraicIndependent.aevalEquivField_algebraMap_apply_coe, RootPairing.root_sub_zsmul_mem_range_iff, OnePoint.infty_notMem_range_coe, Subring.coe_bot, Quaternion.eq_re_iff_mem_range_coe, LinearMap.mem_span_iff_bound, MeasurableEmbedding.map_comap, Filter.range_mem_map, Matroid.mapEmbedding_isBasis_iff, Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_right_iff, IsLocallyConstant.range_finite, tendsto_card_div_pow_atTop_volume', OrdinalApprox.lfp_mem_range_lfpApprox, Algebra.IsStandardSmooth.iff_exists_basis_kaehlerDifferential, range_injOn_strictMono, Complex.range_exp, Nat.range_nth_of_finite, AlgebraicGeometry.IsOpenImmersion.range_pullbackFst, image_preimage_eq_iff, sSup_range, AlgebraicGeometry.Scheme.affineBasisCover_map_range, PiTensorProduct.span_tprod_eq_top, TopologicalSpace.isSeparable_range, PrimeSpectrum.isClosed_range_comap_of_surjective, Module.Basis.ofIsLocalizedModule_span, Filter.comap_neBot_iff_compl_range, convexHull_rangle_single_eq_stdSimplex, Function.Injective.encard_range, OnePoint.insert_infty_range_coe, linearIndependent_option', AddAction.IsPartition.of_orbits, AffineBasis.centroid_mem_interior_convexHull, MeasureTheory.OuterMeasure.map_top, PairwiseDisjoint.range, Function.Semiconj.surjOn_range, Dense.lowerBounds_image, NNRat.addSubmonoid_closure_range_mul_self, Submodule.range_ker_disjoint, AffineIndependent.range, neg_range, range_image, isCompact_range_of_mulSupport_subset_isCompact, RootPairing.Base.sub_notMem_range_root, AkraBazziRecurrence.one_mem_range_sumCoeffsExp, Finsupp.range_lmapDomain, RootPairing.rootSpan_dualAnnihilator_map_eq, range_modelWithCornersEuclideanHalfSpace, infinite_range_add_smul_iff, SSet.Subcomplex.PairingCore.equivII_apply_coe, RootPairing.neg_mem_range_root_iff, Submodule.span_range_subtype_eq_top_iff, isLUB_of_tendsto_atTop, Continuous.bddBelow_range_of_hasCompactMulSupport, MonoidWithZeroHom.mem_valueMonoid_iff, OpenPartialHomeomorph.MDifferentiable.range_mfderiv_eq_univ, OpenPartialHomeomorph.extend_target_subset_range, AffineEquiv.range_eq, Module.Basis.toFlag_carrier, CategoryTheory.GrothendieckTopology.W_eq_isLocal_range_sheafToPresheaf_obj, contMDiffOn_model_symm, ZSpan.vadd_mem_fundamentalDomain, map_nhds_induced_eq, OrderEmbedding.range_inj, LinearOrderedField.cutMap_self, Sublocale.range_toNucleus, HasCountableSeparatingOn.range_Ici, IsSublattice_range, ZSpan.fract_apply, Nat.range_nth_of_infinite, seminormFromConst_bddBelow, IsTranscendenceBasis.isAlgebraic, exists_contDiff_tsupport_subset, Equiv.self_comp_ofInjective_symm, Function.LeftInverse.rightInvOn_range, Affine.Simplex.setInterior_subset_affineSpan, Affine.Simplex.touchpoint_mem_affineSpan_simplex, finrank_vectorSpan_range_add_one_le, range_subset_iff, RootPairing.chainBotCoeff_eq_zero_iff, Algebra.adjoin_toSubsemiring, Polynomial.lifts_iff_set_range, Filter.Tendsto.bddAbove_range_of_cofinite, Countable.exists_eq_range, range_eq_iUnion, range_eq_empty, TopologicalSpace.Compacts.range_map, RootPairing.nsmul_notMem_range_root, RootPairing.Base.exists_eq_sum_and_forall_sum_mem_of_isPos, summable_iff_summable_compl_and_tsum_mem, Int.addSubmonoid_closure_range_mul_self, Filter.atBot_eq_generate_Iic, CoxeterSystem.submonoid_closure_range_simple, Subgroup.IsComplement.rightQuotientEquiv_apply, AlgebraicGeometry.ExistsHomHomCompEqCompAux.range_g_subset, Subalgebra.range_le, Algebra.IsAlgebraic.range_eval_eq_rootSet_minpoly, mdifferentiableWithinAt_iff_of_mem_source, Subtype.range_coind, graphOn_univ_eq_range, StrictAnti.range_inj, AffineIndependent.existsUnique_dist_eq, rank_span, range_IicExtend, Module.Basis.span_eq, Ordinal.mem_range_derivFamily, Function.range_eq_image_or_of_mulSupport_subset, Filter.bddAbove_range_of_tendsto_atTop_atBot, Algebra.PreSubmersivePresentation.ofHasCoeffs_val, Finsupp.mem_frange, IsAlgClosed.isAlgClosure_of_transcendence_basis, range_injOn_strictAnti, Matrix.mem_range_scalar_of_commute_transvectionStruct, OpenPartialHomeomorph.map_extend_symm_nhdsWithin_range, IntermediateField.adjoin_toSubfield, Filter.IsBoundedUnder.bddAbove_range, StrictMono.not_bddBelow_range_of_wellFoundedGT, isInvertible_mfderivWithin_extChartAt_symm, Module.surjective_piEquiv_apply_iff, borel_eq_generateFrom_Iic, VectorField.mlieBracketWithin_apply, Algebra.PreSubmersivePresentation.ofHasCoeffs_algebra_smul, ZSpan.repr_floor_apply, Function.range_eq_image_or_of_support_subset, Algebra.IsAlgebraic.range_eval_eq_rootSet_minpoly_of_splits, Topology.IsEmbedding.toHomeomorph_symm_apply, Subgroup.IsComplement.leftQuotientEquiv_apply, Path.range_subpath_of_le, LieAlgebra.IsKilling.span_weight_eq_top, CategoryTheory.GrothendieckTopology.W_eq_W_range_sheafToPresheaf_obj, MonoidHom.mclosure_range, Finite.card_range_le, IsLocalFrameOn.generating, PiNat.exists_retraction_of_isClosed, preimage_inr_range_inl, PrimeSpectrum.localization_comap_range, range_eval, range_derivWithin_subset_closure_span_image, IsOpenMap.range_mem_nhds, LinearMap.toMatrixAlgEquiv_reindexRange, AlgebraicGeometry.Scheme.range_fromSpecResidueField, Matroid.mapSetEmbedding_ground, Flag.rangeFin_carrier, indicator_range_comp, Ordinal.range_enumOrd, OmegaCompletePartialOrder.Chain.isChain_range, range_comp', ClassGroup.equivPic_apply, NumberField.mixedEmbedding.disjoint_span_commMap_ker, Function.Fiber.fiber_nonempty, TopCat.exists_mem_zeroHypercover_range, isLUB_hasSum, Field.Emb.Cardinal.adjoin_basis_eq_top, Function.fixedPoints_subset_range, VectorField.eventually_contMDiffWithinAt_mpullbackWithin_extChartAt_symm, disjoint_range_inl_image_inr, Fin.cycleIcc_def_le, NumberField.Embeddings.range_eval_eq_rootSet_minpoly, RootPairing.range_weylGroup_weightHom, MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd, RootPairing.RootPositiveForm.algebraMap_posForm, closure_range_zpow_eq_pow, powersetCard.mem_range_ofFinEmbEquiv_symm_iff_mem, RootPairing.EmbeddedG2.allRoots_subset_range_root, PrimeSpectrum.range_comap_fst, IsOpenMap.isOpen_range, Module.Basis.constr_range, seminormFromBounded_bddAbove_range, isPiSystem_Iic, Filter.IsBoundedUnder.bddAbove_range_of_cofinite, RelIso.range_eq, AlgebraicGeometry.Scheme.image_zeroLocus, ModelWithCorners.uniqueDiffOn_preimage_source, Function.Injective.hasProd_range_iff, Homeomorph.range_coe, ContinuousMap.range_toUniformOnFunIsCompact, centroid_mem_affineSpan_of_card_ne_zero, SmoothBumpFunction.nhdsWithin_range_basis, Pi.mem_span_range_single_inl_iff, FirstOrder.Language.Substructure.cg_iff_empty_or_exists_nat_generating_family, UpperHalfPlane.range_coe, exteriorPower.ιMulti_family_span, RootPairing.IsRootSystem.span_coroot_eq_top, ZSpan.fract_eq_fract, GrpCat.SurjectiveOfEpiAuxs.g_apply_fromCoset, exteriorPower.ιMulti_span_fixedDegree, forall_subtype_range_iff, extChartAt_target_eventuallyEq_of_mem, iUnion_Iio_eq_univ_iff, ModelWithCorners.rightInvOn, Function.Embedding.toEquivRange_apply, completeSpace_iff_isComplete_range, AlgebraicGeometry.Scheme.JointlySurjective.exists_eq, range_some_union_none, cfcₙHom_of_cfcHom_map_quasispectrum, Ideal.mem_span_pow_iff_exists_isHomogeneous, Order.mem_range_succ_of_not_isSuccPrelimit, GrpCat.SurjectiveOfEpiAuxs.τ_symm_apply_fromCoset, measurableSet_bddBelow_range, mapsTo_univ_iff_range_subset, ZFSet.toSet_range, Affine.Simplex.map_subtype_restrict, RootPairing.Base.span_int_coroot_support, Nat.Subtype.coe_comp_ofNat_range, IntermediateField.mem_bot, Affine.Simplex.restrict_map_subtype, Monoid.exponent_eq_zero_iff_range_orderOf_infinite, MeasureTheory.tendsto_integral_norm_approxOn_sub, Dense.upperBounds_image, List.mem_ofFn', SemilinearIsometryClass.diam_range, Module.Basis.mk_eq_spanRank, Equiv.apply_ofInjective_symm, LinearMap.range_coe, exteriorPower.ιMulti_span, sInf_range, Topology.isOpenEmbedding_iff, range_sigmaMk, writtenInExtChartAt_sumInr_eventuallyEq_id, PeriodPair.lattice_eq_span_range_basis, range_norm, IsAddIndecomposable.pairwise_sub_notMem_range, IsLocalDiffeomorph.image_coe, Types.monoOverEquivalenceSet_counitIso, bddAbove_range_partialSups, range_mem_nhds_isInteriorPoint, ModelWithCorners.nonempty_interior', Module.Finite.exists_fin, Function.Injective.invFun_restrict, TopCat.pullback_snd_range, vectorSpan_range_eq_span_range_vsub_left, AddMonoidHom.coe_range, Stonean.extremallyDisconnected_preimage, UnitAddTorus.span_mFourierLp_closure_eq_top, Affine.Simplex.altitudeFoot_mem_affineSpan_faceOpposite, finsum_emb_domain', range_projIcc, Cardinal.mk_range_eq, CategoryTheory.Functor.mem_eventualRange_iff, finprod_emb_domain', Monotone.rangeFactorization, Language.accepts_toDFA, isOpen_range_inl, Cardinal.mem_range_aleph_iff, Module.Basis.isChain_range_flag, Complex.range_exp_mul_I, mem_range_of_exists_le_of_exists_ge, Units.range_embedProduct, disjoint_range_iff, Complex.range_im, OpenPartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq, Affine.Simplex.affineSpan_pair_eq_altitude_iff, Affine.Simplex.closedInterior_subset_affineSpan, Subsemigroup.mem_corner_iff_mem_range_mul_left, PrimeSpectrum.localization_away_comap_range, range_cfc, RootPairing.GeckConstruction.coe_genWeightSpace_zero_eq_span_range_u, prod_range_univ_eq, Affine.Simplex.direction_altitude, finsetBasisOfLinearIndependentOfCardEqFinrank_repr_apply, Topology.IsClosedEmbedding.isClosed_range, cfcHom_map_spectrum, ModelWithCorners.isBoundaryPoint_iff, Path.symm_range, IsClosedMap.isClosed_range, range_inl_inter_range_inr, GroupSeminorm.mul_bddBelow_range_add, Topology.IsEmbedding.map_nhds_eq, Composition.mem_range_embedding_iff, Matrix.rank_eq_finrank_span_cols, MulAction.IsBlockSystem.of_normal, Ordinal.invVeblen₁_eq_iff, apply_rangeSplitting, Module.Basis.mem_span_iff_repr_mem, TrivSqZeroExt.range_inlAlgHom_sup_adjoin_range_inr, Module.Relations.Solution.IsPresentation.ker_π, range_singleton, MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp, KaehlerDifferential.kerTotal_map', range_two_mul_add_one, Real.range_sigmoid, prod_univ_range_eq, OpenPartialHomeomorph.mapsTo_extend, OnePoint.isCompl_range_coe_infty, range_ite_subset', Affine.Simplex.sOppSide_affineSpan_faceOpposite_iff, measurableSet_bddAbove_range, LieModule.range_traceForm_le_span_weight, Affine.Simplex.range_faceOpposite_points, CategoryTheory.PreZeroHypercover.shrink_I₀, isTranscendenceBasis_iff_algebraicIndependent_isAlgebraic, collinear_iff_not_affineIndependent, Complex.range_normSq, AddMonoid.exponent_ne_zero_iff_range_addOrderOf_finite, ModelWithCorners.range_eq_univ, image_compl_eq_range_diff_image, Filter.Tendsto.isCompact_insert_range, Ordinal.IsInitial.mem_range_preOmega, ncard_range_of_injective, pow_mem_range_pow_of_coprime, PrimeSpectrum.isBasis_basic_opens, SubMulAction.image_inclusion, CategoryTheory.Types.ofArrows_mem_jointlySurjectivePrecoverage_iff, Module.Basis.is_basis_iff_det, writtenInExtChartAt_sumInl_eventuallyEq_id, iUnion_insert_eq_range_union_iUnion, Polynomial.lifts_iff_coeffs_subset_range, preimage_rangeSplitting, LinearMap.mem_span_iff_continuous, UniqueMDiffOn.uniqueDiffWithinAt_range_inter, MeasureTheory.analyticSet_range_of_polishSpace, Affine.Simplex.incenter_notMem_affineSpan_faceOpposite, GromovHausdorff.eq_toGHSpace_iff, closure_range_zsmul_eq_nsmul, Subgroup.coe_zpowers, ZSpan.setFinite_inter, Real.range_cos_infinite, Ordinal.range_preOmega, Continuous.bddAbove_range_of_hasCompactSupport, TopCat.pullback_fst_range, CategoryTheory.Localization.LeftBousfield.isLocalization, boundedSpace_induced_iff, AddMonoidHom.coe_mrange, range_enumerateCountable_subset, Nat.range_casesOn, Nat.subset_range_nth, writtenInExtChartAt_comp, tprod_range, QuotientAddGroup.range_mk, InitialSeg.isLowerSet_range, WithTop.range_coe, Algebra.Presentation.HasCoeffs.coeffs_relation_mem_range, Valuation.IsRankOneDiscrete.generator_mem_range, range_ite_subset, range_extend, PrimeSpectrum.nontrivial_iff_mem_rangeComap, RootPairing.mapsTo_coreflection_coroot, Affine.Simplex.centroid_mem_affineSpan, GrpCat.SurjectiveOfEpiAuxs.τ_apply_fromCoset', CauchySeq.mul_norm_bddAbove, CategoryTheory.Localization.LeftBousfield.W_iff_isIso_map, Module.Basis.restrictScalars_toMatrix, Path.truncate_range, LinearPMap.mem_range_iff, map_extChartAt_nhds, Cardinal.mk_range_le_lift, Metric.isBounded_range_iff, RingHom.coe_rangeS, IccLeftChart_extend_bot_mem_frontier, CategoryTheory.Functor.eventualRange_eq_iff, Submonoid.coe_powers, Function.Embedding.left_inv_of_invOfMemRange, range_pair_subset, convexHull_range_eq_exists_affineCombination, IsAddIndecomposable.pairwise_baseOf_sub_notMem, SimpleGraph.cliqueSet_one, TopologicalSpace.NonemptyCompacts.range_toCompacts, ModelWithCorners.uniqueDiffOn, AddHom.coe_srange, Affine.Simplex.ExcenterExists.sSameSide_excenter_point_iff, ENat.iSup_coe_lt_top, differentiableWithinAtProp_self_target, range_eq_iff, RingHom.coe_range, ENat.range_natCast, AddSubgroup.IsComplement.leftQuotientEquiv_apply, orthonormal_span, isGLB_of_tendsto_atTop, CategoryTheory.ObjectProperty.isLocal_eq_inverseImage_isomorphisms, contMDiffWithinAt_iff_of_mem_source, CategoryTheory.Functor.isMittagLeffler_iff_eventualRange, Algebra.Presentation.mem_ker_naive, Monoid.PushoutI.NormalWord.cons_toList, AlgebraicGeometry.Scheme.Hom.coe_opensRange, ConnectedComponents.range_coe, bddBelow_range_prod, Nat.card_range_of_injective, Preorder.isGLB_of_isLimit, Ordinal.eq_enumOrd, nhdsWithin_extChartAt_target_eq', NNReal.bddAbove_range_natCast_iff, HasCountableSeparatingOn.range_Iic, MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_eLpNorm, FreeAddGroup.range_lift_eq_closure, Ordinal.principal_mul_iff_le_two_or_omega0_opow_opow, NumberField.mixedEmbedding.mem_rat_span_latticeBasis, RootPairing.root_add_nsmul_mem_range_iff_le_chainTopCoeff, Fin.range_rev, image_compl_preimage, finrank_vectorSpan_range_le, Affine.Simplex.altitude_def, TrivSqZeroExt.range_liftAux, IndexedPartition.range_piecewise, ModelWithCorners.Boundaryless.range_eq_univ, Option.range_eq, BoxIntegral.unitPartition.tag_mem_smul_span, IsProperMap.isClosed_range, Submodule.span_range_inclusionSpan, SmoothBumpFunction.image_eq_inter_preimage_of_subset_support, Memℓp.bddAbove, CategoryTheory.Limits.Types.range_fst_of_isPullback, ContinuousMapZero.range_toContinuousMap, WithTop.range_eq, Submodule.unitsQuotEquivRelPic_apply_coe, PrimeSpectrum.isConstructible_range_comap, GromovHausdorff.HD_below_aux1, Ordinal.veblenWith_mem_range, Function.Injective.invOfMemRange_surjective, RootPairing.reflection_image_eq, isCyclic_tfae, SmoothBumpFunction.closedBall_subset, IsSeqCompact.range, tangentBundleCore_coordChange_achart, AddSubgroup.isComplement_range_right, OmegaCompletePartialOrder.isLUB_range_ωSup, isLinearSet_iff_exists_fin_addMonoidHom, range_subset_insert_image_tsupport, mulIndicator_range_comp, AddAction.IsBlockSystem.of_normal, PrimeSpectrum.range_asIdeal, SmoothBumpFunction.isCompact_symm_image_closedBall, Fin.range_natAdd_castLEEmb, Subsemiring.coe_bot, Affine.Simplex.sSameSide_point_incenter, Continuous.bddAbove_range_of_hasCompactMulSupport, Complex.isClosed_range_intCast, Module.Relations.Solution.ofQuotient_π, RestrictedProduct.range_structureMap, NonUnitalStarAlgHom.coe_range, Finset.prod_preimage', linearIndependent_iff_card_eq_finrank_span, CauchySeq.norm_bddAbove, ball_zero_subset_range_iff_surjective, Topology.IsInducing.map_nhds_eq, Finset.coe_map_subset_range, Measurable.measurable_comp_iff_restrict, CategoryTheory.PreGaloisCategory.autEmbedding_range_isClosed, IsAddIndecomposable.pairwise_sub_notMem_range', AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAt, SmoothBumpFunction.ball_inter_range_eq_ball_inter_target, AlgebraicIndependent.to_subtype_range, SurjOn.subset_range, Function.Embedding.optionEmbeddingEquiv_apply_fst, BijOn.subset_range, MeasureTheory.SimpleFunc.approxOn_range_nonneg, Affine.Simplex.ExcenterExists.sSameSide_point_excenter_iff, Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_left_iff, Algebra.TensorProduct.closure_range_union_range_eq_top, IsMulIndecomposable.pairwise_div_notMem_range', TopologicalSpace.NoetherianSpace.range, Algebra.Presentation.tensorModelOfHasCoeffsInv_aeval_val, IndexedPartition.range_piecewise_subset
rangeFactorization 📖	CompOp	26 math math: rangeFactorization_coe, rangeFactorization_bijective, Function.MulExact.iff_rangeFactorization, Nucleus.restrict_toFun, Profinite.NobelingProof.GoodProducts.equiv_toFun_eq_eval, Function.Exact.rangeFactorization, continuous_rangeFactorization_iff, surjective_onto_range, rightInverse_rangeSplitting, isUniformInducing_rangeFactorization_iff, UniformContinuous.rangeFactorization, IsUniformInducing.rangeFactorization, coe_comp_rangeFactorization, Continuous.rangeFactorization, Function.MulExact.rangeFactorization, rangeFactorization_surjective, rangeFactorization_eq_iff, Measurable.rangeFactorization, uniformContinuous_rangeFactorization_iff, rangeFactorization_eq, Function.Exact.iff_rangeFactorization, leftInverse_rangeSplitting, rangeFactorization_injective, rangeFactorization_eq_rangeFactorization_iff, Monotone.rangeFactorization, preimage_rangeSplitting
rangeSplitting 📖	CompOp	10 math math: Equiv.coe_ofInjective_symm, rangeSplitting_injective, MeasurableEmbedding.measurable_rangeSplitting, rightInverse_rangeSplitting, Equiv.Set.rangeSplittingImageEquiv_symm_apply_coe, Equiv.Set.rangeSplittingImageEquiv_apply_coe_coe, comp_rangeSplitting, leftInverse_rangeSplitting, apply_rangeSplitting, preimage_rangeSplitting
restrictPreimage 📖	CompOp	61 math math: TopologicalSpace.IsOpenCover.isInducing_iff_restrictPreimage, restrictPreimage_surjective, Topology.IsInducing.restrictPreimage, Pretrivialization.restrictPreimage_target, restrictPreimage_isClosedEmbedding, IsOpenMap.restrictPreimage, restrictPreimage_isOpenEmbedding, restrictPreimage_bijective, image_restrictPreimage, restrictPreimage_isEmbedding, range_restrictPreimage, restrictPreimage_coe, Pretrivialization.restrictPreimage'_target, TopologicalSpace.IsOpenCover.isClosedEmbedding_iff_restrictPreimage, Trivialization.restrictPreimage'_source, AlgebraicGeometry.morphismRestrict_base, TopologicalSpace.IsOpenCover.isHomeomorph_iff_restrictPreimage, IsClosedMap.restrictPreimage, Pretrivialization.restrictPreimage_toFun, ContinuousAt.restrictPreimage, IsCoveringMapOn.isCoveringMap_restrictPreimage, ContinuousMap.restrictPreimage_apply, Trivialization.restrictPreimage'_baseSet, IsCoveringMapOn.restrictPreimage, Function.Surjective.restrictPreimage, Topology.IsOpenEmbedding.restrictPreimage, injective_iff_injective_of_iUnion_eq_univ, Pretrivialization.restrictPreimage'_baseSet, Pretrivialization.restrictPreimage_source, Topology.IsEmbedding.restrictPreimage, Function.Bijective.restrictPreimage, IsEvenlyCovered.restrictPreimage, Function.Injective.restrictPreimage, Trivialization.restrictPreimage'_target, Trivialization.restrictPreimage'_apply, bijective_iff_bijective_of_iUnion_eq_univ, Pretrivialization.restrictPreimage'_toFun, TopologicalSpace.IsOpenCover.denseRange_iff_restrictPreimage, Trivialization.restrictPreimage_source, IsProperMap.restrictPreimage, restrictPreimage_isInducing, Trivialization.restrictPreimage_baseSet, Topology.IsQuotientMap.restrictPreimage_isOpen, Pretrivialization.restrictPreimage'_source, TopologicalSpace.IsOpenCover.isEmbedding_iff_restrictPreimage, Topology.IsClosedEmbedding.restrictPreimage, GeneralizingMap.restrictPreimage, Pretrivialization.restrictPreimage_baseSet, restrictPreimage_mk, restrictPreimage_injective, TopologicalSpace.IsOpenCover.isOpenEmbedding_iff_restrictPreimage, Trivialization.restrictPreimage_apply, TopologicalSpace.IsOpenCover.isClosedMap_iff_restrictPreimage, Continuous.restrictPreimage, IsCoveringMap.restrictPreimage, image_val_preimage_restrictPreimage, surjective_iff_surjective_of_iUnion_eq_univ, TopologicalSpace.IsOpenCover.generalizingMap_iff_restrictPreimage, preimage_restrictPreimage, TopologicalSpace.IsOpenCover.isOpenMap_iff_restrictPreimage, Trivialization.restrictPreimage_target
seq 📖	CompOp	17 math math: prod_image_seq_comm, Finite.Set.finite_seq, seq_singleton, seq_eq_set_seq, Filter.mem_seq_iff, seq_subset, image2_eq_seq, seq_seq, prod_eq_seq, mem_seq_iff, seq_eq_image2, seq_mono, image_seq, Finite.seq, seq_def, Filter.seq_mem_seq, singleton_seq
term_''_ 📖	CompOp	—
«term_⁻¹'_» 📖	CompOp	—

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
apply_rangeSplitting 📖	mathematical	—	rangeSplitting Set instMembership range	—	—
comp_rangeSplitting 📖	mathematical	—	Elem range rangeSplitting Set instMembership	—	apply_rangeSplitting
diff_eq 📖	mathematical	—	Set instSDiff instInter Compl.compl instCompl	—	—
eq_mem_setOf 📖	mathematical	—	Set instMembership setOf	—	—
mapsTo_image 📖	mathematical	—	MapsTo image	—	mem_image_of_mem
mapsTo_preimage 📖	mathematical	—	MapsTo preimage	—	—
mem_compl_iff 📖	mathematical	—	Set instMembership Compl.compl instCompl	—	—
mem_diagonal 📖	mathematical	—	Set instMembership diagonal	—	—
mem_diagonal_iff 📖	mathematical	—	Set instMembership diagonal	—	—
mem_diff 📖	mathematical	—	Set instMembership instSDiff	—	—
mem_diff_of_mem 📖	mathematical	Set instMembership	instSDiff	—	—
mem_image 📖	mathematical	—	Set instMembership image	—	—
mem_image2 📖	mathematical	—	Set instMembership image2	—	—
mem_image2_of_mem 📖	mathematical	Set instMembership	image2	—	—
mem_image_of_mem 📖	mathematical	Set instMembership	image	—	—
mem_offDiag 📖	mathematical	—	Set instMembership offDiag	—	—
mem_pi 📖	mathematical	—	Set instMembership pi	—	—
mem_preimage 📖	mathematical	—	Set instMembership preimage	—	—
mem_prod 📖	mathematical	—	Set instMembership SProd.sprod instSProd	—	—
mem_prod_eq 📖	mathematical	—	Set instMembership SProd.sprod instSProd	—	—
mem_range 📖	mathematical	—	Set instMembership range	—	—
mem_range_self 📖	mathematical	—	Set instMembership range	—	—
mem_seq_iff 📖	mathematical	—	Set instMembership seq	—	—
mem_setOf 📖	mathematical	—	Set instMembership setOf	—	—
mem_setOf_eq 📖	mathematical	—	Set instMembership setOf	—	—
mem_univ 📖	mathematical	—	Set instMembership univ	—	—
mem_univ_pi 📖	mathematical	—	Set instMembership pi univ	—	—
mk_mem_prod 📖	mathematical	Set instMembership	SProd.sprod instSProd	—	—
notMem_setOf_iff 📖	mathematical	—	Set instMembership setOf	—	—
prodMk_mem_set_prod_eq 📖	mathematical	—	Set instMembership SProd.sprod instSProd	—	—
prod_eq 📖	mathematical	—	SProd.sprod Set instSProd instInter preimage	—	—
prod_image_left 📖	mathematical	—	SProd.sprod Set instSProd image	—	ext
prod_image_right 📖	mathematical	—	SProd.sprod Set instSProd image	—	ext
rangeFactorization_bijective 📖	mathematical	—	Function.Bijective Elem range rangeFactorization	—	—
rangeFactorization_eq_iff 📖	mathematical	—	rangeFactorization Set instMembership range	—	mem_range_self rangeFactorization.eq_1 Subtype.coe_eta
rangeFactorization_eq_rangeFactorization_iff 📖	mathematical	—	rangeFactorization	—	mem_range_self
rangeFactorization_injective 📖	mathematical	—	Elem range rangeFactorization	—	mem_range_self
rangeFactorization_surjective 📖	mathematical	—	Elem range rangeFactorization	—	—
restrictPreimage_coe 📖	mathematical	—	Set instMembership restrictPreimage preimage	—	—
seq_eq_image2 📖	mathematical	—	seq image2	—	—
setOf_mem_eq 📖	mathematical	—	setOf Set instMembership	—	—
subset_kernImage_iff 📖	mathematical	—	Set instHasSubset kernImage preimage	—	—

Set.MapsTo

Definitions

Name	Category	Theorems
restrict 📖	CompOp	32 math math: iterate_restrict, Dynamics.coverEntropy_restrict, SimpleGraph.coe_induceHom, LipschitzOnWith.mapsToRestrict, coe_restrict, coe_iterate_restrict, ContinuousOn.restrict_mapsTo, Continuous.restrict, val_restrict_apply, OpenPartialHomeomorph.homeomorphOfImageSubsetSource_symm_apply, Set.surjective_mapsTo_image_restrict, IsClosedMap.mapsToRestrict, Dynamics.coverEntropy_restrict_subset, ContractingWith.restrict, restrict_eq_codRestrict, Topology.IsEmbedding.restrict, CategoryTheory.Functor.toPreimages_map, LipschitzOnWith.to_restric_mapsTo, IsOpenMap.mapsToRestrict, OpenPartialHomeomorph.homeomorphOfImageSubsetSource_apply, CategoryTheory.Functor.toEventualRanges_map, range_restrict, lipschitzOnWith_iff_restrict, Set.codRestrict_restrict, Topology.IsOpenEmbedding.restrict, Set.BijOn.bijective, ContinuousOn.mapsToRestrict, restrict_commutes, restrict_surjective_iff, restrict_inj, ContinuousAt.restrict, Dynamics.coverEntropyInf_restrict_subset

Set.Subtype

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
range_coind 📖	mathematical	—	Set.range Subtype.coind Set.preimage	—	—

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