Defs

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

Statistics

Metric	Count
Definitions `toModule`, `toModule'`, `add`, `addCommGroup`, `addCommMonoid`, `copy`, `decidableEq`, `inhabited`, `instPartialOrder`, `module'`, `ofClass`, `ofLinearComb`, `setLike`, `smul`, `toAddSubgroup`, `toAddSubmonoid`, `toSubMulAction`, `zero`, `module'`	19
Theorems `addSubgroupClass`, `addSubmonoidClass`, `add_mem`, `add_mem_iff_left`, `add_mem_iff_right`, `carrier_eq_coe`, `coe_add`, `coe_copy`, `coe_eq_zero`, `coe_mem`, `coe_mk`, `coe_neg`, `coe_ofClass`, `coe_ofLinearComb`, `coe_set_mk`, `coe_smul`, `coe_smul_of_tower`, `coe_sub`, `coe_toAddSubgroup`, `coe_toAddSubmonoid`, `coe_toSubMulAction`, `coe_zero`, `copy_eq`, `eta`, `ext`, `ext_iff`, `instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallHSMul`, `isScalarTower`, `isScalarTower'`, `mem_carrier`, `mem_mk`, `mem_toAddSubgroup`, `mem_toAddSubmonoid`, `mk_eq_zero`, `mk_le_mk`, `neg_mem`, `neg_mem_iff`, `nonempty`, `smulMemClass`, `smul_mem`, `smul_mem'`, `smul_mem_iff'`, `smul_mem_iff''`, `smul_mem_iff_of_isUnit`, `smul_of_tower_mem`, `sub_mem`, `sub_mem_iff_left`, `sub_mem_iff_right`, `toAddSubgroup_inj`, `toAddSubgroup_injective`, `toAddSubmonoid_inj`, `toAddSubmonoid_injective`, `toSubMulAction_inj`, `toSubMulAction_injective`, `zero_mem`	55
Total	74

SMulMemClass

Definitions

Name	Category	Theorems
`toModule` 📖	CompOp	—
`toModule'` 📖	CompOp	—

Submodule

Definitions

Name	Category	Theorems
`add` 📖	CompOp	3 math math: `coe_add`, `LinearPMap.map_add`, `LieModule.Cohomology.add_apply_apply`
`addCommGroup` 📖	CompOp	574 math math: `instIsScalarTowerQuotientIdealSubtypeMemTorsionBySetCoe`, `groupCohomology.instEpiModuleCatH2π`, `groupHomology.π_comp_H2Iso_hom_assoc`, `Affine.Simplex.altitudeFoot_restrict`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff`, `AffineSubspace.coe_vsub`, `groupHomology.mapCycles₂_comp_assoc`, `Module.End.isSemisimple_restrict_iff`, `IsIsotypic.linearEquiv_fun`, `groupCohomology.toCocycles_comp_isoCocycles₁_hom`, `Affine.Simplex.orthogonalProjectionSpan_map`, `EuclideanGeometry.vsub_orthogonalProjection_mem_direction_orthogonal`, `groupCohomology.isoCocycles₁_hom_comp_i_apply`, `IsSemisimpleModule.submodule`, `AffineMap.restrict.bijective`, `NumberField.Units.logEmbedding_fundSystem`, `RootPairing.instIsRootSystemSubtypeMemSubmoduleSpanRangeCoeEmbeddingRootCorootRestrictScalars'`, `Affine.Simplex.finiteDimensional_direction_altitude`, `IsIsotypic.isotypicComponents`, `toLinearEquiv_orthogonalDecomposition_symm`, `NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule`, `Affine.Simplex.orthogonalProjection_vadd_smul_vsub_orthogonalProjection`, `covBy_iff_quot_is_simple`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_apply`, `NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_completeFamily_of_ne`, `groupCohomology.toCocycles_comp_isoCocycles₂_hom_apply`, `groupCohomology.cocyclesIso₀_hom_comp_f`, `groupCohomology.H1π_comp_map_assoc`, `groupHomology.mapCycles₁_comp_apply`, `Affine.Simplex.restrict_reindex`, `finiteDimensional_vectorSpan_range`, `RootPairing.restrictScalars_coe_coroot`, `groupCohomology.π_comp_H0Iso_hom`, `Affine.Simplex.faceOppositeCentroid_restrict`, `groupCohomology.π_comp_H1Iso_hom_assoc`, `EuclideanGeometry.orthogonalProjection_congr`, `toLinearEquiv_orthogonalDecomposition`, `groupCohomology.mapCocycles₂_comp_i`, `finiteDimensional_direction_affineSpan_of_finite`, `Module.Basis.ofZLatticeBasis_apply`, `groupCohomology.H0IsoOfIsTrivial_hom`, `bot_lt_isotypicComponent`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_i_hom`, `groupCohomology.coe_mapCocycles₁`, `Affine.Simplex.restrict_map_restrict`, `EuclideanGeometry.orthogonalProjection_contLinear`, `EuclideanGeometry.dist_orthogonalProjection_eq_iff_oangle_eq`, `groupHomology.cyclesMap_comp_isoCycles₂_hom`, `AffineSubspace.coe_subtypeA`, `groupHomology.mapCycles₁_comp_assoc`, `groupHomology.toCycles_comp_isoCycles₂_hom_assoc`, `AffineSubspace.subtype_apply`, `Module.Basis.ofZLatticeComap_apply`, `Module.End.mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo`, `isIsotypicOfType_submodule_iff`, `isSemisimple_torsionBy_of_irreducible`, `eq_isotypicComponent_iff`, `exteriorPower.alternatingMapLinearEquiv_ιMulti`, `groupHomology.δ₀_apply`, `Module.End.eigenspace_restrict_le_eigenspace`, `EuclideanGeometry.reflection_vadd_smul_vsub_orthogonalProjection`, `groupHomology.isoCycles₁_inv_comp_iCycles_apply`, `eq_isotypicComponent_of_le`, `groupCohomology.map_H0Iso_hom_f_apply`, `EuclideanGeometry.inter_eq_singleton_orthogonalProjection`, `groupCohomology.H0IsoOfIsTrivial_inv_apply`, `EuclideanGeometry.Cospherical.inclusion_iff`, `groupHomology.chains₁ToCoinvariantsKer_surjective`, `EuclideanGeometry.Cospherical.inclusion`, `AffineSubspace.linear_topEquiv`, `Affine.Simplex.orthogonalProjectionSpan_eq_point`, `isSimpleModule_iff_isAtom`, `FiniteDimensional.span_of_finite`, `IsIsotypic.submodule_linearEquiv_fun`, `RootPairing.restrictScalars_pairing`, `EuclideanGeometry.orthogonalProjection_map`, `exteriorPower.alternatingMapLinearEquiv_symm_map`, `Affine.Simplex.circumradius_restrict`, `Affine.Simplex.medial_restrict`, `Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_zero_iff`, `coe_orthogonalDecomposition_symm`, `groupCohomology.mapCocycles₂_comp_i_apply`, `groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_apply`, `ZLattice.module_finite`, `EuclideanGeometry.angle_orthogonalProjection_self`, `EuclideanGeometry.orthogonalProjection_apply_mem`, `IsSimpleRing.exists_ringEquiv_matrix_end_mulOpposite`, `EuclideanGeometry.dist_orthogonalProjection_eq_of_oangle_eq`, `groupHomology.cycles₁IsoOfIsTrivial_hom_apply`, `AffineSubspace.coe_subtypeₐᵢ`, `PeriodPair.latticeBasis_zero`, `EuclideanGeometry.dist_orthogonalProjection_eq_iff_angle_eq`, `AdicCompletion.ofPowSMul_ofValEqZero`, `RootPairing.finrank_corootSpanIn_int`, `groupCohomology.cocyclesIso₀_inv_comp_iCocycles`, `groupCohomology.mapCocycles₂_comp_i_assoc`, `ZLattice.rank`, `ZLattice.tsum_norm_rpow_le`, `Subspace.finiteDimensional_quot_dualCoannihilator_iff`, `groupCohomology.H1IsoOfIsTrivial_inv_apply`, `groupHomology.mapCycles₁_id_comp_assoc`, `HahnEmbedding.Partial.toOrderAddMonoidHom_apply`, `goursat`, `groupCohomology.cocycles₁IsoOfIsTrivial_hom_hom_apply_apply`, `groupCohomology.H2π_comp_map_apply`, `groupHomology.mapCycles₁_comp`, `finiteDimensional_vectorSpan_image_of_finite`, `Affine.Simplex.ninePointCircle_restrict`, `groupCohomology.toCocycles_comp_isoCocycles₂_hom`, `length_lt`, `finiteDimensional_finset_sup`, `ModuleCat.imageIsoRange_hom_subtype`, `groupHomology.mapCycles₁_comp_i`, `Affine.Simplex.face_restrict`, `groupCohomology.shortComplexH0_f`, `Affine.Simplex.interior_restrict`, `RootPairing.finrank_rootSpanIn_int`, `EuclideanGeometry.dist_orthogonalProjection_eq_of_two_zsmul_oangle_eq`, `AffineSubspace.inclusion_linear`, `HasStrictFDerivAt.to_implicitFunctionOfComplemented`, `EuclideanGeometry.orthogonalProjection_apply'`, `Affine.Simplex.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq`, `inf_genEigenspace`, `finiteDimensional_sup`, `isIsotypic_submodule_iff`, `finiteDimensional_direction_map`, `groupCohomology.H1IsoOfIsTrivial_H1π_apply_apply`, `ModuleCat.imageIsoRange_inv_image_ι_apply`, `EuclideanGeometry.orthogonalProjection_orthogonalProjection_of_le`, `EuclideanGeometry.Sphere.isTangent_iff_isTangentAt_orthogonalProjection`, `NumberField.mixedEmbedding.fundamentalDomain_idealLattice`, `FiniteDimensional.span_finset`, `ZLattice.tsumNormRPowBound_spec`, `Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_iff`, `IsIsotypic.isotypicComponent`, `LinearMap.restrict_sub`, `Rep.invariantsAdjunction_unit_app`, `groupHomology.mapCycles₂_id_comp`, `ZLattice.comap_equiv_apply`, `Module.Basis.addSubgroupOfClosure_repr_apply`, `NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis`, `Module.length_top`, `groupCohomology.H2π_eq_iff`, `NumberField.det_basisOfFractionalIdeal_eq_absNorm`, `GradedTensorProduct.auxEquiv_comm`, `LinearMap.comap_leq_ker_subToSupQuotient`, `Rep.coindFunctorIso_inv_app_hom_toFun_coe`, `groupCohomology.δ₁_apply`, `LieDerivation.ofGradingSum_of`, `groupHomology.toCycles_comp_isoCycles₁_hom_apply`, `IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp`, `Affine.Simplex.altitude_restrict_eq_comap_subtype`, `AffineSubspace.isometryEquivMap.toAffineMap_eq`, `groupHomology.mapCycles₂_comp_i`, `AffineSubspace.signedInfDist_eq_signedDist_of_mem`, `groupCohomology.map_H0Iso_hom_f`, `Affine.Simplex.signedInfDist_apply_self`, `LinearMap.quotientInfEquivSupQuotient_apply_mk`, `finiteDimensional_vectorSpan_of_finite`, `finiteQuotient_iff`, `groupCohomology.π_comp_H1Iso_hom`, `Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_zero_iff`, `Module.End.eigenspace_restrict_eq_bot`, `AffineSubspace.signedInfDist_eq_signedDist_orthogonalProjection`, `NumberField.mixedEmbedding.span_idealLatticeBasis`, `AffineEquiv.linear_ofEq`, `instPolynormableSpaceSubtypeMemSubmodule`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_assoc_apply`, `Collinear.finiteDimensional_direction_affineSpan`, `finiteDimensional_direction_affineSpan_insert_set`, `Affine.Simplex.orthogonalProjectionSpan_congr`, `Polynomial.adjSylvester_comp_sylveserMap`, `groupHomology.mapCycles₁_id_comp_apply`, `groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_assoc`, `CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe`, `Ideal.quotientToQuotientRangePowQuotSucc_mk`, `Module.End.IsSemisimple.restrict`, `groupCohomology.instEpiModuleCatH1π`, `Coplanar.finiteDimensional_direction_affineSpan`, `IsSemisimpleModule.sup`, `Affine.Simplex.ExcenterExists.touchpointWeights_restrict`, `groupCohomology.H2π_comp_map`, `CharacterModule.intSpanEquivQuotAddOrderOf_apply`, `groupHomology.π_comp_H2Iso_hom`, `AffineSubspace.instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection`, `groupHomology.mapCycles₁_comp_i_apply`, `Rep.FiniteCyclicGroup.groupHomologyπEven_eq_iff`, `groupHomology.mapCycles₂_comp`, `EuclideanGeometry.Sphere.orthogonalProjection_orthRadius_center`, `ModuleCat.kernelIsoKer_inv_kernel_ι_apply`, `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`, `groupCohomology.isoCocycles₂_hom_comp_i`, `EuclideanGeometry.orthogonalProjection_subtype`, `coe_orthogonalDecomposition`, `EuclideanGeometry.two_zsmul_oangle_self_orthogonalProjection`, `groupCohomology.π_comp_H0Iso_hom_apply`, `finiteDimensional_direction_affineSpan_insert`, `Rep.subtype_hom_toFun`, `groupHomology.coe_mapCycles₂`, `groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply`, `groupHomology.H1π_comp_map_apply`, `Affine.Simplex.ExcenterExists.excenter_restrict`, `finiteDimensional_direction_affineSpan_range`, `Polynomial.det_taylorLinearEquiv_toLinearMap`, `EuclideanGeometry.orthogonalProjection_vadd_eq_self`, `groupCohomology.cocyclesMap_comp_isoCocycles₁_hom`, `groupHomology.toCycles_comp_isoCycles₁_hom_assoc`, `Module.length_submodule`, `groupCohomology.cocyclesIso₀_inv_comp_iCocycles_assoc`, `groupCohomology.mapCocycles₁_one`, `EuclideanGeometry.oangle_orthogonalProjection_self`, `Affine.Simplex.map_altitude_restrict`, `Affine.Simplex.faceOpposite_restrict`, `groupHomology.H2π_comp_map_assoc`, `finiteDimensional_vectorSpan_insert_set`, `LinearMap.quotientInfEquivSupQuotient_injective`, `EuclideanGeometry.dist_orthogonalProjection_eq_dist_iff_eq_of_mem`, `GradedTensorProduct.auxEquiv_mul`, `AffineSubspace.abs_signedInfDist_eq_dist_of_mem_affineSpan_insert`, `Affine.Simplex.exradius_restrict`, `finiteDimensional_vectorSpan_singleton`, `finiteDimensional_inf_right`, `groupCohomology.π_comp_H0Iso_hom_assoc`, `ModuleCat.imageIsoRange_hom_subtype_assoc`, `ZLattice.module_free`, `Affine.Simplex.orthogonalProjection_circumcenter`, `groupCohomology.H2π_comp_map_assoc`, `AdicCompletion.ofPowSMul_injective`, `NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm`, `groupHomology.mapCycles₂_comp_apply`, `AffineSubspace.subtypeA_toAffineMap`, `Affine.Simplex.exists_forall_dist_eq_iff_exists_excenterExists_and_eq_excenter`, `instIsSemisimpleModuleSubtypeMemSubmoduleIsotypicComponent`, `HasStrictFDerivAt.to_implicitFunction`, `natAbs_det_equiv`, `Module.End.IsFinitelySemisimple.restrict`, `Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq`, `groupHomology.H2π_eq_iff`, `le_isotypicComponent`, `EuclideanGeometry.oangle_eq_of_dist_orthogonalProjection_eq`, `groupHomology.H1AddEquivOfIsTrivial_single`, `Ideal.quotientToQuotientRangePowQuotSucc_surjective`, `groupHomology.isoCycles₂_hom_comp_i_apply`, `groupCohomology.π_comp_H0IsoOfIsTrivial_hom_apply`, `Module.length_bot`, `Rep.coinvariantsShortComplex_f`, `groupCohomology.toCocycles_comp_isoCocycles₁_hom_assoc`, `Module.End.restrict_eigenspace`, `groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply`, `groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_assoc`, `AffineSubspace.coe_vadd`, `groupCohomology.isoCocycles₁_inv_comp_iCocycles`, `Affine.Simplex.circumcenter_restrict`, `instIsScalarTowerQuotientSpanSingletonSetSubtypeMemTorsionBy`, `AffineMap.restrict.injective`, `Affine.Simplex.height_restrict`, `ModuleCat.imageIsoRange_inv_image_ι`, `instModuleFinite_of_discrete_submodule`, `EuclideanGeometry.orthogonalProjection_affineSpan_singleton`, `groupHomology.cyclesMap_comp_isoCycles₂_hom_assoc`, `groupCohomology.cocyclesMap_comp_isoCocycles₂_hom`, `AffineSubspace.subtype_injective`, `Module.End.isFinitelySemisimple_iff'`, `AffineSubspace.subtype_linear`, `GradedTensorProduct.mulHom_apply`, `EuclideanGeometry.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq`, `AffineSubspace.topEquiv_symm_apply_coe`, `sndL_comp_coe_orthogonalDecomposition`, `IsSemisimpleModule.sSup_simples_le`, `IsSemisimpleModule.exists_simple_submodule`, `groupHomology.mapCycles₂_id_comp_assoc`, `groupHomology.π_comp_H1Iso_hom_apply`, `map_le_isotypicComponent`, `Collinear.finiteDimensional_vectorSpan`, `groupCohomology.map_id_comp_H0Iso_hom_apply`, `groupCohomology.subtype_comp_d₀₁_assoc`, `groupHomology.toCycles_comp_isoCycles₂_hom`, `groupCohomology.map_id_comp_H0Iso_hom`, `groupHomology.cyclesMap_comp_isoCycles₂_hom_apply`, `CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_assoc_apply`, `torsionBy.mk_ideal_smul`, `Affine.Simplex.incenter_restrict`, `groupHomology.mapCycles₁_id_comp`, `AdicCompletion.ofPowSMul_val_apply_eq_zero`, `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`, `groupCohomology.cocycles₁IsoOfIsTrivial_inv_hom_apply_coe`, `EuclideanGeometry.orthogonalProjection_vadd_smul_vsub_orthogonalProjection`, `PeriodPair.latticeBasis_one`, `ZLattice.covolume_eq_det_mul_measureReal`, `Affine.Simplex.orthogonalProjectionSpan_faceOpposite_eq_point_rev`, `EuclideanGeometry.dist_orthogonalProjection_eq_infNndist`, `FiniteDimensional.finiteDimensional_submodule`, `Ideal.toCotangent_apply`, `groupHomology.isoCycles₁_hom_comp_i_apply`, `AffineMap.restrict.surjective`, `Affine.Simplex.inradius_restrict`, `groupCohomology.cocyclesIso₀_hom_comp_f_assoc`, `ZLattice.exists_forall_abs_repr_le_norm`, `AffineMap.restrict.coe_apply`, `ModuleCat.imageIsoRange_inv_image_ι_assoc`, `IsSemisimpleModule.sSup_simples_eq_top`, `groupCohomology.H1π_eq_zero_iff`, `NumberField.fractionalIdeal_rank`, `groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_apply`, `ModuleCat.kernelIsoKer_hom_ker_subtype`, `groupHomology.H1ToTensorOfIsTrivial_H1π_single`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_assoc_apply`, `EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangent`, `NumberField.mixedEmbedding.volume_fundamentalDomain_fractionalIdealLatticeBasis`, `Rep.FiniteCyclicGroup.groupHomologyπEven_eq_zero_iff`, `Ideal.quotientToQuotientRangePowQuotSucc_injective`, `groupCohomology.cocyclesMk₁_eq`, `range_powSMulQuotInclusion`, `Subalgebra.finiteDimensional_toSubmodule`, `EuclideanGeometry.orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem`, `IsSemisimpleModule.exists_end_ringEquiv`, `RootPairing.restrictScalars_coe_root`, `exteriorPower.presentation_relation`, `EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq`, `Affine.Simplex.centroid_restrict`, `groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_apply`, `groupHomology.cyclesMap_comp_isoCycles₁_hom_apply`, `groupHomology.toCycles_comp_isoCycles₂_hom_apply`, `groupCohomology.H2π_eq_zero_iff`, `Coplanar.finiteDimensional_vectorSpan`, `groupCohomology.mapCocycles₁_comp_i_assoc`, `groupCohomology.H1π_comp_map_apply`, `PeriodPair.latticeEquiv_symm_apply`, `groupCohomology.cocyclesMap_cocyclesIso₀_hom_f`, `EuclideanGeometry.dist_orthogonalProjection_eq_infDist`, `EuclideanGeometry.orthogonalProjection_mem`, `groupCohomology.π_comp_H2Iso_hom_assoc`, `Affine.Simplex.orthogonalProjectionSpan_restrict`, `groupCohomology.toCocycles_comp_isoCocycles₂_hom_assoc`, `IsSemisimpleModule.exists_end_algEquiv`, `Affine.Simplex.setInterior_restrict`, `AffineSubspace.finiteDimensional_sup`, `groupHomology.mapCycles₂_hom`, `groupHomology.isoCycles₂_inv_comp_iCycles_apply`, `Affine.Simplex.excenterWeights_restrict`, `groupHomology.isoCycles₂_inv_comp_iCycles_assoc`, `EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff`, `AffineEquiv.ofEq_symm`, `ModuleCat.kernelIsoKer_hom_ker_subtype_apply`, `Rep.indCoindIso_hom_hom_toLinearMap`, `ZLattice.abs_repr_lt_of_norm_lt`, `EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq`, `groupHomology.cyclesMk₂_eq`, `AffineSubspace.linear_equivMapOfInjective`, `groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_assoc`, `EuclideanGeometry.Sphere.IsTangentAt.eq_orthogonalProjection`, `groupHomology.H1π_eq_zero_iff`, `groupHomology.isoCycles₁_inv_comp_iCycles_assoc`, `groupHomology.π_comp_H1Iso_hom_assoc`, `Affine.Simplex.median_restrict`, `groupHomology.H2π_comp_map`, `CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_apply`, `affineSpan_coe_preimage_eq_top`, `Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_iff`, `groupHomology.H1π_comp_map_assoc`, `Ideal.natAbs_det_equiv`, `EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq`, `groupHomology.instEpiModuleCatH1π`, `Affine.Simplex.closedInterior_restrict`, `groupCohomology.isoCocycles₂_inv_comp_iCocycles`, `powSMulQuotInclusion_mk`, `EuclideanGeometry.orthogonalProjection_orthogonalProjection`, `groupCohomology.δ₀_apply`, `EuclideanGeometry.orthogonalProjection_vsub_mem_direction`, `exteriorPower.presentation_R`, `finiteDimensional_direction_affineSpan_image_of_finite`, `groupHomology.instEpiModuleCatH2π`, `NumberField.mem_span_basisOfFractionalIdeal`, `Affine.Simplex.orthogonalProjection_eq_circumcenter_of_dist_eq`, `AffineSubspace.topEquiv_apply`, `groupCohomology.cocyclesMk₂_eq`, `Affine.Simplex.mongePoint_restrict`, `Rep.indCoindIso_inv_hom_toLinearMap`, `mem_isotypicComponents_iff`, `factorPow_comp_powSMulQuotInclusion`, `groupHomology.H1π_comp_map`, `Affine.Simplex.eulerPoint_restrict`, `EuclideanGeometry.vsub_orthogonalProjection_mem_direction`, `AdicCompletion.restrictScalars_range_ofPowSMul_eq_ker_eval`, `AffineSubspace.coe_inclusion_apply`, `groupCohomology.toCocycles_comp_isoCocycles₁_hom_apply`, `Affine.Simplex.ExcenterExists.touchpoint_restrict`, `Affine.Simplex.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection`, `IsSemisimpleRing.exists_ringEquiv_pi_matrix_end_mulOpposite`, `groupHomology.cyclesMk₁_eq`, `groupHomology.mapCycles₂_comp_i_assoc`, `Affine.Simplex.excenterWeightsUnnorm_restrict`, `groupCohomology.isoCocycles₁_hom_comp_i`, `EuclideanGeometry.Sphere.IsTangent.isTangentAt`, `groupCohomology.mapCocycles₁_comp_i_apply`, `groupHomology.mapCycles₂_id_comp_apply`, `Polynomial.det_taylorLinearEquiv`, `IsSemisimpleModule.exists_end_algEquiv_pi_matrix_end`, `groupHomology.H2π_comp_map_apply`, `EuclideanGeometry.reflection_apply'`, `EuclideanGeometry.oangle_self_orthogonalProjection`, `le_isotypicComponent_iff`, `Projectivization.instFiniteDimensionalSubtypeMemSubmoduleSubmodule`, `fg_iff_finiteDimensional`, `Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_iff`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_K`, `LinearMap.coe_quotientInfToSupQuotient`, `MatrixModCat.toModuleCat_map`, `AffineEquiv.coe_ofEq_apply`, `finiteDimensional_bot`, `groupCohomology.π_comp_H2Iso_hom_apply`, `CliffordAlgebra.even.lift.aux_apply`, `instIsSemisimpleModuleSubtypeMemSubmoduleValSetIsotypicComponents`, `AffineMap.restrict.linear`, `Affine.Simplex.abs_signedInfDist_eq_dist_of_mem_affineSpan_range`, `LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces`, `NumberField.mixedEmbedding.fractionalIdealLatticeBasis_apply`, `AffineSubspace.coe_subtype`, `groupHomology.isoCycles₁_hom_comp_i_assoc`, `FractionalIdeal.abs_det_basis_change`, `Rep.invariantsFunctor_map_hom`, `Module.End.iInf_maxGenEigenspace_restrict_map_subtype_eq`, `Affine.Simplex.orthogonalProjectionSpan_reindex`, `groupCohomology.π_comp_H1Iso_hom_apply`, `topologicalAddGroup`, `inf_iInf_maxGenEigenspace_of_forall_mapsTo`, `RootPairing.restrictScalars_toLinearMap_apply_apply`, `fstL_comp_coe_orthogonalDecomposition`, `EuclideanGeometry.orthogonalProjection_linear`, `AffineSubspace.signedInfDist_def`, `IsSemisimpleModule.exists_sSupIndep_sSup_simples_eq_top`, `NumberField.Units.unitLattice_rank`, `CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_apply`, `groupHomology.π_comp_H2Iso_hom_apply`, `instIsSimpleModuleSubtypeMemSubmoduleValSetSetOfNonemptyLinearEquivId`, `MatrixModCat.toModuleCat_obj_isAddCommGroup`, `Affine.Simplex.orthogonalProjectionSpan_eulerPoint_mem_ninePointCircle`, `ZLattice.normBound_spec`, `groupHomology.mapCycles₁_hom`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_π_hom`, `groupHomology.isoCycles₁_inv_comp_iCycles`, `Affine.Simplex.signedInfDist_affineCombination`, `groupHomology.cyclesMap_comp_isoCycles₁_hom_assoc`, `EuclideanGeometry.angle_self_orthogonalProjection`, `groupHomology.coe_mapCycles₁`, `NumberField.Units.logEmbeddingEquiv_apply`, `groupHomology.toCycles_comp_isoCycles₁_hom`, `NumberField.basisOfFractionalIdeal_apply`, `Module.Basis.ofZLatticeComap_repr_apply`, `groupHomology.mapCycles₂_comp_i_apply`, `EuclideanGeometry.orthogonalProjection_eq_self_iff`, `groupCohomology.cocyclesIso₀_hom_comp_f_apply`, `groupCohomology.subtype_comp_d₀₁`, `EuclideanGeometry.reflection_subtype`, `LinearMap.det_eq_det_mul_det`, `FiniteDimensional.instSubtypeMemSubmoduleMap`, `groupHomology.isoCycles₂_hom_comp_i`, `groupHomology.π_comp_H1Iso_hom`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_H`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_assoc_apply`, `groupHomology.isoCycles₂_inv_comp_iCycles`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_apply`, `Module.Basis.ofZLatticeBasis_repr_apply`, `groupHomology.mkH1OfIsTrivial_apply`, `IsIsotypic.linearEquiv_finsupp`, `groupCohomology.isoCocycles₁_inv_comp_iCocycles_assoc`, `groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom`, `AffineSubspace.inclusion_rfl`, `IsSemisimpleModule.eq_bot_or_exists_simple_le`, `NumberField.instFiniteIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule`, `CharacterModule.intSpanEquivQuotAddOrderOf_apply_self`, `AddSubgroup.relIndex_eq_natAbs_det`, `groupHomology.π_comp_H1Iso_inv_apply`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_left`, `sSup_simples_eq_top_iff_isSemisimpleModule`, `exteriorPower.presentation_var`, `groupCohomology.isoCocycles₂_hom_comp_i_apply`, `torsionBySet.mk_smul`, `EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal`, `Module.End.genEigenspace_restrict_eq_top`, `CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_assoc_apply`, `groupHomology.cycles₁IsoOfIsTrivial_inv_apply`, `FiniteDimensional.subalgebra_toSubmodule`, `EuclideanGeometry.eq_orthogonalProjection_of_eq_subspace`, `EuclideanGeometry.orthogonalProjection_apply`, `groupHomology.isoCycles₁_hom_comp_i`, `groupCohomology.cocyclesIso₀_inv_comp_iCocycles_apply`, `AffineSubspace.signedInfDist_apply_self`, `EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection`, `EuclideanGeometry.orthogonalProjection_mem_orthogonal`, `exteriorPower.presentation_G`, `groupCohomology.H1π_comp_map`, `Module.Basis.span_neg`, `groupCohomology.cocyclesMk₀_eq`, `isSemisimpleModule_biSup_of_isSemisimpleModule_submodule`, `EuclideanGeometry.two_zsmul_oangle_orthogonalProjection_self`, `NumberField.instIsLocalizedModuleIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmoduleValFractionalIdealNonZeroDivisorsRestrictScalarsSubtype`, `ContinuousLinearMap.finite_dimensional_eigenspace`, `IsSemisimpleModule.exists_linearEquiv_dfinsupp`, `ModuleCat.mono_as_hom'_subtype`, `AffineSubspace.equivMapOfInjective_toFun`, `exteriorPower.ιMulti_family_span`, `EuclideanGeometry.orthogonalProjection_sup_of_orthogonalProjection_eq`, `powSMulQuotInclusion_injective`, `Ideal.quotientToQuotientRangePowQuotSuccAux_mk`, `ZLattice.sum_piFinset_Icc_rpow_le`, `EuclideanGeometry.coe_orthogonalProjection_eq_iff_mem`, `isFullyInvariant_iff_le_imp_isotypicComponent_le`, `Affine.Simplex.map_subtype_restrict`, `Affine.Simplex.restrict_map_subtype`, `AdicCompletion.ofPowSMul_val_apply`, `Polynomial.sylveserMap_comp_adjSylvester`, `groupCohomology.map_H0Iso_hom_f_assoc`, `ModuleCat.kernelIsoKer_inv_kernel_ι`, `IsIsotypicOfType.isotypicComponent`, `instModuleFree_of_discrete_submodule`, `Ideal.natAbs_det_basis_change`, `ModuleCat.imageIsoRange_hom_subtype_apply`, `finiteDimensional_of_le`, `groupHomology.H1AddEquivOfIsTrivial_symm_tmul`, `IsSemisimpleModule.range`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_right`, `Affine.Simplex.restrict_map_inclusion`, `finiteDimensional_direction_affineSpan_singleton`, `EuclideanGeometry.dist_orthogonalProjection_line_eq_iff_two_zsmul_oangle_eq`, `groupCohomology.map_id_comp_H0Iso_hom_assoc`, `Module.End.genEigenspace_restrict`, `groupCohomology.isoCocycles₂_hom_comp_i_assoc`, `EuclideanGeometry.orthogonalProjection_eq_iff_mem`, `EuclideanGeometry.dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq`, `groupCohomology.coe_mapCocycles₂`, `LinearMap.quotientInfEquivSupQuotient_surjective`, `groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_assoc`, `groupCohomology.H1π_eq_iff`, `Module.Basis.addSubgroupOfClosure_apply`, `groupCohomology.isoCocycles₁_hom_comp_i_assoc`, `IsSemisimpleModule.exists_end_ringEquiv_pi_matrix_end`, `EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangentAt`, `groupHomology.mapCycles₁_quotientGroupMk'_epi`, `groupHomology.mapCycles₁_comp_i_assoc`, `AffineEquiv.ofEq_rfl`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_apply`, `LinearMap.finiteDimensional_range`, `torsionBy.mk_smul`, `groupHomology.π_comp_H2Iso_inv_apply`, `EuclideanGeometry.dist_set_eq_iff_dist_orthogonalProjection_eq`, `IsLocalRing.spanFinrank_eq_finrank_quotient`, `finiteDimensional_inf_left`, `Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_zero_iff`, `groupCohomology.mapCocycles₁_comp_i`, `PeriodPair.finrank_lattice`, `hasFDerivAt_stereoInvFunAux_comp_coe`, `finiteDimensional_vectorSpan_insert`, `groupHomology.isoCycles₂_hom_comp_i_assoc`, `Int.submodule_toAddSubgroup_index_ne_zero_iff`, `groupCohomology.π_comp_H2Iso_hom`, `FiniteDimensional.span_singleton`, `groupHomology.H2π_eq_zero_iff`, `ZLattice.exists_finsetSum_norm_rpow_le_tsum`, `Affine.Simplex.restrict_points_coe`, `groupCohomology.isoCocycles₂_inv_comp_iCocycles_assoc`, `ZLattice.abs_repr_le`, `Affine.Simplex.excenterExists_restrict`, `ZLattice.covolume_eq_det`, `groupHomology.δ₁_apply`, `EuclideanGeometry.orthogonalProjection_mem_subspace_eq_self`, `natAbs_det_basis_change`, `groupHomology.H1π_eq_iff`, `finiteDimensional_iSup`, `Affine.Triangle.dist_orthogonalProjectionSpan_faceOpposite_eq_iff_two_zsmul_oangle_eq`, `isSemisimpleModule_iff_exists_linearEquiv_dfinsupp`, `groupHomology.cyclesMap_comp_isoCycles₁_hom`, `Rep.coindFunctorIso_hom_app_hom_toFun_hom_toFun`, `HahnEmbedding.Partial.toOrderAddMonoidHom_injective`
`addCommMonoid` 📖	CompOp	1351 math math: `DirectSum.IsInternal.ofBijective_coeLinearMap_same`, `TensorProduct.mapInclIsometry_apply`, `coe_sum`, `finrank_sup_add_finrank_inf_eq`, `Ideal.toCotangent_to_quotient_square`, `LinearMap.lTensor_ker_subtype_tensorKerEquiv_symm`, `DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply`, `instIsScalarTowerQuotientIdealSubtypeMemTorsionBySetCoe`, `KaehlerDifferential.kerCotangentToTensor_toCotangent`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inr`, `Rep.invariantsAdjunction_homEquiv_symm_apply_hom`, `TensorProduct.forall_vanishesTrivially_iff_forall_fg_rTensor_injective`, `LieDerivation.IsKilling.ad_mem_ker_killingForm_ad_range_of_mem_orthogonal`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff`, `rTensorOne_symm_apply`, `AddMonoidAlgebra.supportedEquivFinsupp_apply_apply`, `iSupIndep.linearEquiv_symm_apply`, `topEquiv_symm_apply_coe`, `iSupIndep.dfinsupp_lsum_injective`, `FG.rTensor.directedSystem`, `Algebra.idealMap_eq_ofEq_comp_toLocalized₀`, `coe_continuous_linearProjOfClosedCompl'`, `RootPairing.posRootForm_posForm_apply_apply`, `Ideal.selfBasis_def`, `Module.Basis.mem_submodule_iff`, `LinearMap.BilinForm.dualSubmoduleToDual_apply_apply`, `OrthonormalBasis.orthogonalProjection_eq_sum`, `IsIsotypic.linearEquiv_fun`, `finrank_span_finset_eq_card`, `derivationToSquareZeroOfLift_apply`, `finrank_span_set_eq_card`, `linearProjOfIsCompl_apply_left`, `subtype_comp_inclusion`, `Algebra.Extension.Cotangent.span_eq_top_of_span_eq_ker`, `MvPolynomial.DirectSum.coeLinearMap_eq_dfinsuppSum`, `IsSemisimpleModule.exists_submodule_linearEquiv_quotient`, `Ideal.Cotangent.lift_comp_toCotangent`, `LinearMap.BilinForm.dualSubmoduleToDual_injective`, `ExteriorAlgebra.GradedAlgebra.liftι_eq`, `Ideal.tensorCotangentEquiv_tmul`, `RootPairing.root'In_corootSpanMem_eq_pairingIn`, `RootPairing.rootForm_restrict_nondegenerate_of_ordered`, `Polynomial.degreeLT.instFreeSubtypeMemSubmodule`, `finrank_add_finrank_le_of_disjoint`, `orthogonalDecomposition_symm_apply`, `Ideal.finrank_eq_finrank`, `Algebra.PreSubmersivePresentation.cotangentComplexAux_apply`, `LinearMap.subtype_comp_codRestrict`, `exteriorPower.basis_apply`, `LinearMap.IsProj.subtype_comp_codRestrict`, `NumberField.Units.logEmbedding_fundSystem`, `Ideal.exists_smith_normal_form`, `toIsOrderedAddMonoid`, `LinearMap.subtype_comp_restrict`, `LinearMap.IsProj.codRestrict_apply_cod`, `Module.Basis.coe_span_apply`, `Finsupp.restrictDom_apply`, `rank_le_card_isVisible`, `RootPairing.RootPositiveForm.zero_lt_posForm_apply_root`, `exteriorPower.linearMap_ext_iff`, `Ideal.toCotangent_eq`, `RootPairing.finrank_corootSpan_eq'`, `mulMap_tmul`, `TensorProduct.eq_of_fg_of_subtype_eq'`, `comm_trans_lTensorOne`, `isNoetherian_submodule_right`, `Polynomial.toMatrix_sylvesterMap'`, `exists_fg_le_eq_rTensor_inclusion`, `LinearMap.IsRefl.domRestrict`, `TensorProduct.range_mapIncl`, `LinearMap.BilinForm.ker_restrict_eq_of_codisjoint`, `toLinearEquiv_orthogonalDecomposition_symm`, `NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule`, `Module.End.isNilpotent_restrict_genEigenspace_nat`, `LinearMap.injective_restrict_iff_disjoint`, `covBy_iff_quot_is_simple`, `lift_rank_map_le`, `MeasureTheory.norm_condExpL2_le`, `NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_completeFamily_of_ne`, `tensorEquivSpan_apply_tmul`, `linearMap_eq_zero_iff_of_eq_span`, `ContinuousLinearMap.ker_codRestrict`, `Finsupp.supportedEquivFinsupp_symm_single`, `LinearDisjoint.not_linearIndependent_pair_of_flat_left`, `Rep.coindToInd_of_support_subset_orbit`, `re_inner_starProjection_eq_normSq`, `DirectSum.decomposeAlgEquiv_symm_apply`, `LinearMap.le_ker_iff_comp_subtype_eq_zero`, `MeasureTheory.inner_condExpL2_left_eq_right`, `MonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_apply`, `exists_finrank_eq_sigPos_and_posDef`, `sigNeg_isGreatest`, `linearEquiv_of_le_sSup`, `linearProjOfIsCompl_apply_right`, `prodEquivOfIsCompl_symm_apply_snd_eq_zero`, `exteriorPower.coe_basis`, `LinearMap.tensorKer_tmul`, `LinearMap.IsSymmetric.diagonalization_apply_self_apply`, `finrank_vectorSpan_image_finset_le`, `Subspace.dualLift_of_subtype`, `mulMap_one_right_eq`, `exteriorPower.ιMulti_family_linearIndependent_field`, `ContinuousLinearEquiv.snd_equivOfRightInverse`, `span_setOf_mem_eq_top`, `toLinearEquiv_orthogonalDecomposition`, `LinearPMap.mk_apply`, `MeasureTheory.condExpL2_ae_eq_zero_of_ae_eq_zero`, `Subspace.dualRestrict_comp_dualLift`, `dualQuotEquivDualAnnihilator_apply`, `ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm`, `LinearIndependent.linearCombination_repr`, `exteriorPower.basis_repr_ne`, `orthogonalProjection_mem_subspace_eq_self`, `LinearMap.range_domRestrict_le_range`, `Module.Basis.mk_repr`, `LinearMap.submoduleComap_apply_coe`, `Module.Flat.ker_lTensor_eq`, `KaehlerDifferential.cotangentComplexBaseChange_tmul`, `Ideal.primaryComponent_sup`, `Module.Basis.ofZLatticeBasis_apply`, `LinearMap.ofIsCompl_smul`, `quotDualCoannihilatorToDual_apply`, `injective_inclusionSpan`, `IsLattice.rank'`, `dualPairing_apply`, `TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_symm_apply_mk_tmul`, `LinearMap.tensorEqLocusEquiv_apply`, `LinearMap.domRestrict'_apply`, `Rep.coindToInd_indToCoind`, `RootPairing.range_polarization_domRestrict_le_span_coroot`, `RootPairing.finrank_range_polarization_eq_finrank_span_coroot`, `coe_toBaseChange_tmul`, `finrank_vectorSpan_le_iff_not_affineIndependent`, `groupCohomology.coboundariesToCocycles₁_apply`, `LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal`, `Algebra.PreSubmersivePresentation.cotangentComplexAux_zero_iff`, `Module.Basis.ofZLatticeComap_apply`, `le_linearEquiv_of_sSup_eq_top`, `FG.lTensor.directedSystem`, `Module.Basis.coe_mkFinConsOfLE`, `rank_map_le`, `LinearMap.tensorKer_coe`, `LinearMap.BilinForm.restrict_nondegenerate_orthogonal_spanSingleton`, `Module.Baer.iff_surjective`, `FG.exists_rTensor_fg_inclusion_eq`, `Module.End.mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo`, `finrank_quotient_add_finrank`, `AddMonoidAlgebra.supportedEquivFinsupp_apply_support_val`, `isIsotypicOfType_submodule_iff`, `val_mulMap'_tmul`, `SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix`, `RootPairing.coroot'In_rootSpanMem_eq_pairingIn`, `exteriorPower.alternatingMapLinearEquiv_ιMulti`, `KaehlerDifferential.kerToTensor_apply`, `coe_subtypeL'`, `Polynomial.degreeLT.basis_val`, `selfAdjointPart_comp_subtype_skewAdjoint`, `setBasisOfLinearIndependentOfCardEqFinrank_repr_apply`, `IsSemisimpleRing.exists_linearEquiv_ideal_of_isSimpleModule`, `Algebra.Presentation.differentials.hom₁_single`, `LinearMap.rank_eq_of_surjective`, `TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul`, `Algebra.Generators.exists_presentation_of_basis_cotangent`, `lift_rank_range_of_injective`, `Ideal.mem_toCotangent_ker`, `isArtinian_iff_submodule_quotient`, `Module.End.eigenspace_restrict_le_eigenspace`, `LinearMap.IsSymmetric.directSum_decompose_apply`, `linearProjOfIsCompl_comp_surjective_of_exact`, `LinearMap.BilinForm.toLin_restrict_ker_eq_inf_ker`, `dualCopairing_eq`, `Ideal.finsuppTotal_apply_eq_of_fintype`, `rank_span_set`, `RootPairing.RootPositiveForm.zero_lt_posForm_iff`, `quotDualCoannihilatorToDual_nondegenerate`, `rank_bot`, `finrank_add_finrank_orthogonal'`, `LinearMap.submoduleMap_coe_apply`, `Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange`, `AffineSubspace.linear_topEquiv`, `toLinearPMap_range`, `IsNoetherian.injective_of_surjective_of_submodule`, `LieHom.quotKerEquivRange_toFun`, `MeasureTheory.MemLp.condExpL2_ae_eq_condExp`, `sigPos_isGreatest`, `TensorProduct.map₂_eq_range_lift_comp_mapIncl`, `LinearMap.exists_basis_basis_of_span_eq_top_of_mem_algebraMap`, `isArtinian_of_le`, `exteriorPower.ιMulti_span_of_span`, `biSup_comap_subtype_eq_top`, `LinearPMap.adjointAux_unique`, `FractionalIdeal.isNoetherian_coeIdeal`, `RootPairing.injOn_dualMap_subtype_span_root_coroot`, `LinearMap.range_domRestrict_eq_range_iff`, `Ideal.finsuppTotal_apply`, `TensorProduct.exists_finite_submodule_of_finite'`, `biSup_eq_range_dfinsupp_lsum`, `IsOrtho.orthogonalProjection_comp_subtypeL`, `LinearDisjoint.rank_inf_le_one_of_commute_of_flat`, `IsIsotypic.submodule_linearEquiv_fun`, `LieModule.Cohomology.d₂₃_apply`, `LieAlgebra.InvariantForm.restrict_nondegenerate`, `exteriorPower.alternatingMapLinearEquiv_symm_map`, `span_range_inclusion_restrictScalars_eq_top`, `isNoetherian_submodule`, `selfAdjointPart_comp_subtype_selfAdjoint`, `exteriorPower.instFree`, `HahnEmbedding.Partial.truncLT_mem_range_extendFun`, `mulMap_comm`, `Module.Flat.out`, `RootPairing.finrank_rootSpan_map_polarization_eq_finrank_corootSpan`, `LieModule.Cohomology.d₁₂_apply_apply`, `equivSubtypeMap_symm_apply`, `botEquivPUnit_apply`, `LinearMap.ofIsComplProd_apply`, `AffineIndependent.finrank_vectorSpan_add_one`, `coe_orthogonalDecomposition_symm`, `Projectivization.finrank_submodule`, `ker_subtype`, `MeasureTheory.aestronglyMeasurable_condExpL2`, `orthogonalProjection_bot`, `linearProjOfIsCompl_apply_right'`, `ZLattice.module_finite`, `EuclideanGeometry.orthogonalProjection_apply_mem`, `DirectSum.bracket_apply_apply`, `LinearMap.restrict_comp`, `isNoetherian_of_le`, `IsSimpleRing.exists_ringEquiv_matrix_end_mulOpposite`, `coe_isComplEquivProj_symm_apply`, `Representation.card_inv_mul_sum_char_eq_finrank`, `finrank_add_inf_finrank_orthogonal`, `LieAlgebra.bracket_ofTwoCocycle`, `LinearMap.exact_subtype_mkQ`, `Algebra.Extension.Cotangent.val_mk`, `PeriodPair.latticeBasis_zero`, `linearDisjoint_iff`, `GradedTensorProduct.auxEquiv_one`, `toLocalized₀_apply_coe`, `LinearMap.IsSymm.domRestrict`, `Subspace.finrank_dualCoannihilator_eq`, `sup_eq_range`, `MeasureTheory.condExpL2_indicator_ae_eq_smul`, `exteriorPower.ιMultiDual_apply_diag`, `Representation.instIsTrivialSubtypeMemSubgroupSubmoduleInvariantsCompLinearMapIdSubtypeToInvariants`, `RootPairing.finrank_corootSpanIn_int`, `LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_right`, `Algebra.TensorProduct.linearEquivIncludeRange_toLinearMap`, `retractionOfSectionOfKerSqZero_comp_kerToTensor`, `DirectSum.IsInternal.collectedBasis_coe`, `generator_maximal_submoduleImage_dvd`, `rank_quotient_add_rank`, `Ideal.toCotangent_range`, `RootPairing.RootPositiveForm.posForm_apply_root_root_le_zero_iff`, `Module.Invertible.exists_linearEquiv_ideal`, `ZLattice.rank`, `ZLattice.tsum_norm_rpow_le`, `LinearIndependent.repr_eq_single`, `FirstOrder.Field.lift_genericMonicPoly`, `Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange_residueField`, `isNoetherian_iSup`, `rank_submodule_le_one_iff`, `AffineIndependent.card_le_finrank_succ`, `Coplanar.finrank_le_two`, `Algebra.Generators.exists_presentation_of_free_cotangent`, `goursat`, `coe_prodEquivOfIsCompl'`, `Affine.Simplex.finrank_direction_altitude`, `HahnEmbedding.Seed.strictMono_coeff`, `Module.Basis.sumQuot_repr_left`, `LinearEquiv.ofSubmodules_apply`, `LinearMap.lTensor_range`, `topEquiv_apply`, `Polynomial.degreeLT.addLinearEquiv_castAdd`, `subtype_apply`, `Ideal.powQuotSuccInclusion_apply_coe`, `LinearEquiv.toSpanNonzeroSingleton_homothety`, `MeasureTheory.integral_condExpL2_eq`, `IsCompactOperator.restrict'`, `RootPairing.instFiniteSubtypeMemSubmoduleRootSpanOfFinite`, `orzechProperty_iff`, `ExteriorAlgebra.basis_eq_coe_basis`, `LinearMap.tensorKerEquivOfSurjective_symm_tmul`, `Module.End.pow_restrict`, `LinearMap.IsProj.eq_conj_prodMap`, `LinearMap.tensorEqLocus_coe`, `LinearDisjoint.not_linearIndependent_pair_of_flat_right`, `DirectSum.coeLinearMap_lof`, `finrank_lt_finrank_of_lt`, `LinearMap.mem_submoduleImage_of_le`, `Polynomial.degreeLT.addLinearEquiv_apply_fst`, `mulLeftMap_apply`, `LinearMap.restrict_commute`, `MeasureTheory.setIntegral_condExpL2_indicator`, `OrthonormalBasis.orthogonalProjection_apply_eq_sum`, `LinearMap.toKerLocalized_isLocalizedModule`, `submoduleOf_sup_of_le`, `rank_range_le`, `DirectSum.coeLinearMap_of`, `ContinuousLinearMap.coe_equivRange`, `rank_le`, `coe_prodEquivOfIsCompl`, `DirectSum.coeLinearMap_eq_dfinsuppSum`, `RootPairing.polarizationIn_Injective`, `finrank_vectorSpan_insert_le`, `Subspace.dualPairing_nondegenerate`, `ExteriorAlgebra.GradedAlgebra.ι_sq_zero`, `RootPairing.finrank_rootSpanIn_int`, `AddMonoidAlgebra.decomposeAux_coe`, `LinearMap.ker_rangeRestrict`, `LinearMap.IsSymmetric.card_filter_eigenvalues_eq`, `LinearMap.compMultilinearMap_codRestrict`, `Module.Flat.iff_rTensor_injective`, `EuclideanGeometry.orthogonalProjection_apply'`, `MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le`, `Rep.coindVEquiv_symm_apply_coe`, `Ideal.map_includeRight_eq`, `inf_genEigenspace`, `torsionBy_torsionBy_eq_top`, `LinearDisjoint.rank_inf_le_one_of_flat`, `TensorProduct.exists_finite_submodule_left_of_setFinite`, `TensorProduct.quotientTensorEquiv_symm_apply_mk_tmul`, `LinearEquiv.mem_dilatransvections_iff_rank`, `basisOfLinearIndependentOfCardEqFinrank_repr_apply`, `strictMono_comap_prod_map`, `Matrix.cRank_toNat_eq_finrank`, `RootPairing.PolarizationIn_eq`, `NumberField.mixedEmbedding.fundamentalDomain_idealLattice`, `Module.Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul`, `dualRestrict_ker_eq_dualAnnihilator`, `Subalgebra.finrank_toSubmodule`, `ZLattice.tsumNormRPowBound_spec`, `MeasureTheory.norm_condExpL2_coe_le`, `isArtinian_span_of_finite`, `MeasureTheory.setLIntegral_nnnorm_condExpL2_indicator_le`, `toBaseChange_surjective`, `LinearMap.finrank_range_of_inj`, `LinearMap.restrict_sub`, `coe_linearProjOfIsCompl_apply`, `inclusion_apply`, `LinearEquiv.ofLeftInverse_symm_apply`, `LinearMap.range_rangeRestrict`, `isNoetherian_iff_submodule_quotient`, `ZLattice.comap_equiv_apply`, `surjective_tensorToSpan`, `orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero`, `torsion_torsion_eq_top`, `finrank_span_le_card`, `isLocalizedModule`, `range_subtypeL`, `botEquivPUnit_symm_apply`, `Module.Basis.addSubgroupOfClosure_repr_apply`, `NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis`, `Module.Basis.span_repr_eq_single`, `span_range_inclusion_eq_top`, `SchauderBasis.finrank_range_proj`, `GradedTensorProduct.auxEquiv_symm_one`, `le_sigPos_of_posDef`, `NumberField.det_basisOfFractionalIdeal_eq_absNorm`, `GradedTensorProduct.auxEquiv_comm`, `LinearMap.comap_leq_ker_subToSupQuotient`, `QuadraticForm.sigPos_add_sigNeg_add_radical`, `LinearDisjoint.linearIndependent_right_of_flat`, `HilbertBasis.hasSum_orthogonalProjection`, `LinearMap.rTensor_injective_iff_subtype`, `collinear_iff_finrank_le_one`, `LinearMap.subtype_compAlternatingMap_codRestrict`, `LieDerivation.ofGradingSum_of`, `LinearMap.submoduleImage_apply_of_le`, `LinearRecurrence.solSpace_rank`, `LinearEquiv.ofSubmodule'_symm_apply`, `IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp`, `MeasureTheory.lintegral_nnnorm_condExpL2_le`, `exteriorPower.finrank_eq`, `Polynomial.degreeLT.addLinearEquiv_apply_snd`, `LinearMap.finrank_genEigenspace_le`, `Module.Basis.sumQuot_repr_inl`, `LinearMap.IsSymmetric.diagonalization_symm_apply`, `ContinuousLinearMap.coe_codRestrict`, `LieModule.weight_vector_multiplication`, `LinearMap.quotientInfEquivSupQuotient_apply_mk`, `SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux`, `finiteQuotient_iff`, `Ideal.tensorCotangentHom_tmul`, `exteriorPower.map_apply_ιMulti_family`, `NumberField.mixedEmbedding.span_idealLatticeBasis`, `LinearMap.submoduleOf_span_singleton_of_mem`, `norm_orthogonalProjection_apply_le`, `Algebra.Extension.Hom.sub_one_tmul`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_assoc_apply`, `mem_iSup_iff_exists_dfinsupp`, `FDRep.average_char_eq_finrank_invariants`, `isArtinian_range`, `TensorProduct.inner_mapIncl_mapIncl`, `RootPairing.toLinearMap_apply_PolarizationIn`, `exteriorPower.zeroEquiv_naturality`, `KaehlerDifferential.D_apply`, `TensorProduct.exists_finite_submodule_right_of_setFinite`, `HahnEmbedding.IsPartial.truncLT_mem_range`, `LinearIndependent.linearCombinationEquiv_apply_coe`, `Polynomial.adjSylvester_comp_sylveserMap`, `iSup_eq_range_dfinsupp_lsum`, `Module.Basis.span_apply`, `LinearMap.ker_codRestrict`, `CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe`, `RootPairing.instFiniteSubtypeMemSubmoduleCorootSpanOfFinite`, `ContinuousLinearMap.equivRange_symm_apply`, `RootPairing.finrank_corootSpanIn`, `LinearIndependent.linearCombination_comp_repr`, `Ideal.quotientToQuotientRangePowQuotSucc_mk`, `coe_inclusion`, `LinearPMap.inverse_isClosable_iff`, `instLocallyConvexSpaceSubtypeMemSubmodule`, `Polynomial.degreeLT.instFiniteSubtypeMemSubmodule`, `finrank_strictMono`, `Algebra.Extension.Cotangent.mk_eq_zero_iff`, `linearEquiv_det_reflection`, `LinearPMap.apply_comp_inclusion`, `span_val_image_eq_iff`, `LieSubmodule.toEnd_comp_subtype_mem`, `Matrix.rank_eq_finrank_range_toLin`, `Matrix.eRank_toNat_eq_finrank`, `CharacterModule.intSpanEquivQuotAddOrderOf_apply`, `QuadraticMap.Equivalent.rank_radical_eq`, `Ideal.mapCotangent_toCotangent`, `stereoToFun_apply`, `prodEquivOfIsCompl_symm_apply_fst_eq_zero`, `exteriorPower.ιMulti_apply_coe`, `affineIndependent_iff_not_finrank_vectorSpan_le`, `coe_tensorSpanEquivSpan_apply_tmul`, `FG.lTensor.directLimit_apply'`, `dualRestrict_def`, `ContinuousLinearMap.coe_projKerOfRightInverse_apply`, `Rep.coindToInd_apply`, `Module.Basis.SmithNormalForm.snf`, `AlgHom.toKerIsLocalization_apply`, `LinearMap.restrict_eq_codRestrict_domRestrict`, `exists_finrank_eq_sigNeg_and_negDef`, `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`, `HahnEmbedding.Partial.truncLT_mem_range_sSupFun`, `FG.rTensor.directLimit_apply`, `coe_continuous_linearProjOfClosedCompl`, `LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict`, `LieModule.Cohomology.d₁₂_apply_coe_apply_apply`, `coe_orthogonalDecomposition`, `rank_submodule_le_one_iff'`, `QuadraticForm.finrank_radical_of_equiv_weightedSumSquares`, `finrank_orthogonal_span_singleton`, `inclusionSpan_apply_coe`, `Ideal.Cotangent.lift_toCotangent`, `comap_subtype_self`, `exteriorPower.oneEquiv_naturality`, `SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux`, `one_le_finrank_iff`, `IsPrincipal.generator_submoduleImage_dvd_of_mem`, `norm_subtypeL`, `RootPairing.posRootForm_posForm_pos_of_ne_zero`, `Module.Flat.eqLocus_lTensor_eq`, `LinearMap.lift_rank_eq_of_surjective`, `Subspace.dualRestrict_leftInverse`, `TensorProduct.exists_finite_submodule_left_of_finite`, `Module.instFiniteSubtypeMemIdealOfIsNoetherian`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem_ne`, `rank_mono`, `LinearMap.comap_codRestrict`, `ContinuousLinearMap.coe_rangeRestrict`, `setLike.coe_galgebra_toFun`, `norm_sq_eq_add_norm_sq_projection`, `Polynomial.det_taylorLinearEquiv_toLinearMap`, `ContinuousLinearMap.equivRange_symm_toLinearEquiv`, `LinearEquiv.ofInjective_apply`, `comm_trans_rTensorOne`, `DirectSum.decompose_symm_bracket`, `starProjection_apply`, `OrzechProperty.injective_of_surjective_of_submodule`, `FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mul`, `linearIndependent_span`, `Subspace.flip_quotDualCoannihilatorToDual_bijective`, `comap_equiv_self_of_inj_of_le_apply`, `rank_quotient_add_rank_of_isDomain`, `Module.Basis.sumQuot_repr_inl_of_mem`, `exists_fg_le_subset_range_rTensor_subtype`, `DirectSum.coeAlgHom_of`, `TensorProduct.rTensor_injective_of_forall_vanishesTrivially`, `isArtinian_of_fg_of_artinian`, `set_smul_eq_map`, `RingHom.toKerIsLocalization_apply`, `Polynomial.degreeLT.addLinearEquiv_apply`, `fst_orthogonalDecomposition_apply`, `LinearEquiv.mem_dilatransvections_iff_finrank`, `Matrix.rank_eq_finrank_span_row`, `RootPairing.prod_rootForm_smul_coroot_mem_range_domRestrict`, `finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply`, `LinearEquiv.ofSubmodules_symm_apply`, `IsSemisimpleModule.exists_quotient_linearEquiv_submodule`, `LinearMap.kerComplementEquivRange_apply_coe`, `LinearDisjoint.rank_inf_le_one_of_commute_of_flat_right`, `restrictScalarsEquiv_symm_apply`, `LinearMap.quotientInfEquivSupQuotient_injective`, `isArtinian_of_submodule_of_artinian`, `Module.Basis.restrictScalars_repr_apply`, `CliffordAlgebra.GradedAlgebra.ι_apply`, `DirectSum.toBaseChange_bijective`, `LieModule.Cohomology.d₁₂_apply_apply_ofTrivial`, `GradedTensorProduct.auxEquiv_mul`, `lTensorOne'_tmul`, `Module.End.submodule_pow_eq_zero_of_pow_eq_zero`, `RootPairing.CoPolarizationIn_eq`, `rTensorOne'_tmul_one`, `inner_orthogonalProjection_eq_of_mem_right`, `DirectSum.decompose_symm_algebraMap`, `LinearMap.ker_domRestrict`, `rTensorOne'_tmul`, `Finsupp.linearCombinationOn_range`, `Algebra.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex`, `TensorProduct.exists_finite_submodule_right_of_setFinite'`, `realPart_comp_subtype_selfAdjoint`, `projective_units`, `RootPairing.algebraMap_rootFormIn`, `map_subtype_embedding_eq`, `le_linearEquiv_of_le_sSup`, `LinearDisjoint.injective`, `ZLattice.module_free`, `LinearMap.mem_submoduleImage`, `dualCopairing_apply`, `LinearEquiv.toSpanNonzeroSingleton_apply`, `Module.Flat.instSubalgebraToSubmodule`, `ContinuousLinearMap.coe_linearMap_equivRange`, `LinearMap.BilinForm.nondegenerate_restrict_iff_disjoint_ker`, `NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm`, `ker_inclusion`, `AffineIndependent.finrank_vectorSpan_image_finset`, `exteriorPower.alternatingMapLinearEquiv_apply_ιMulti`, `FractionalIdeal.isNoetherian_zero`, `comap_subtype_eq_top`, `LinearIndependent.span_repr_eq`, `prodEquivOfIsCompl_symm_apply`, `injective_tensorToSpan`, `Representation.linHom.invariantsEquivRepHom_apply`, `projective_of_isUnit`, `Module.Basis.restrictScalars_apply`, `exteriorPower.ιMultiDual_apply_ιMulti`, `natAbs_det_equiv`, `Module.Basis.mem_submodule_iff'`, `smithNormalFormBotBasis_def`, `Rep.resCoindHomEquiv_symm_apply`, `map_subtype_span_singleton`, `LinearPMap.adjointDomainMkCLM_apply`, `Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent`, `exteriorPower.pairingDual_ιMulti_ιMulti`, `mulRightMap_eq_mulMap_comp`, `isNoetherian_submodule'`, `QuadraticMap.IsometryEquiv.map_posDef_iff`, `exteriorPower.alternatingMapLinearEquiv_comp_ιMulti`, `rank_span_finset_le`, `Ideal.quotientToQuotientRangePowQuotSucc_surjective`, `MeasureTheory.integral_condExpL2_eq_of_fin_meas_real`, `finrank_map_le`, `HahnEmbedding.Seed.coeff_baseEmbedding`, `finrank_span_eq_card`, `prodComm_trans_prodEquivOfIsCompl`, `DirectSum.decompose_algebraMap`, `LinearMap.compLeftInverse_apply_of_bdd`, `RootPairing.zero_le_posForm`, `finrank_eq_zero`, `rTensorOne_tmul`, `ContinuousLinearEquiv.toSpanNonzeroSingleton_symm_apply`, `MonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_support_val`, `Module.End.restrict_eigenspace`, `exists_fun_fin_finrank_span_eq`, `Ideal.instIsTorsionFreeSubtypeMemSubmodule`, `LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_left`, `Module.Finite.of_isComplemented_domain`, `Polynomial.degreeLT.addLinearEquiv_apply'`, `affineIndependent_iff_le_finrank_vectorSpan`, `Subspace.quotAnnihilatorEquiv_apply`, `skewAdjointPart_comp_subtype_selfAdjoint`, `unitsToPic_apply`, `instIsScalarTowerQuotientSpanSingletonSetSubtypeMemTorsionBy`, `coe_mulMap_comp_eq`, `exteriorPower.oneEquiv_symm_apply`, `instModuleFinite_of_discrete_submodule`, `LinearMap.finrank_range_le`, `isIdempotentElemEquiv_apply_coe`, `TensorProduct.exists_finite_submodule_left_of_finite'`, `sectionOfRetractionKerToTensorAux_prop`, `LinearEquiv.submoduleMap_apply`, `ContinuousLinearEquiv.ofSubmodule'_apply`, `Module.Flat.iff_lTensor_injective`, `LinearPMap.inverse_range`, `LieModule.Cohomology.mem_twoCocycle_iff`, `Rep.coindVEquiv_apply`, `TensorProduct.eq_zero_of_fg_of_subtype_eq_zero`, `exteriorPower.map_comp_ιMulti`, `Ideal.basisSpanSingleton_apply`, `TensorProduct.quotientTensorQuotientEquiv_apply_tmul_mk_tmul_mk`, `DirectSum.decomposeLinearEquiv_apply_coe`, `orthogonalProjection_apply_eq_linearProjOfIsCompl`, `TensorProduct.tensorQuotientEquiv_apply_mk_tmul`, `TensorProduct.exists_finite_submodule_of_finite`, `MeasureTheory.condExpL2_indicator_of_measurable`, `SchauderBasis.RankOneDecomposition.finrank_range`, `derivationToSquareZeroEquivLift_symm_apply_apply_coe`, `GradedTensorProduct.mulHom_apply`, `MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictDegreeOfFinite`, `linearEquiv_of_sSup_eq_top`, `AddMonoidAlgebra.GradesBy.decompose_single`, `ContinuousMultilinearMap.codRestrict_apply_coe`, `coe_isComplEquivProj_apply`, `IsCompactOperator.restrict`, `prodEquivOfIsCompl_symm_apply_left`, `sndL_comp_coe_orthogonalDecomposition`, `RootPairing.PolarizationIn_apply`, `injective_subtype`, `linearProjOfIsCompl_isCompl_projection`, `quotientEquivOfIsCompl_apply_mk_coe`, `MeasureTheory.condExpL2_indicator_eq_toSpanSingleton_comp`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem`, `LinearMap.coe_domRestrict`, `Module.End.isNilpotent_restrict_maxGenEigenspace_sub_algebraMap`, `linearProjOfIsCompl_comp_subtype`, `rank_eq_zero`, `IsLattice.finrank_of_pi`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inl`, `coe_subtypeₗᵢ`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inr_basis`, `MvPolynomial.weightedHomogeneousComponent_directSum`, `groupHomology.π_comp_H1Iso_hom_apply`, `LinearMap.quotKerEquivRange_symm_apply_image`, `Finsupp.restrictDom_comp_subtype`, `Module.Finite.span_singleton`, `MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le_real`, `rTensor_mkQ`, `Representation.coindMap_coe_apply_apply`, `SchauderBasis.finrank_range_succSub_eq_one`, `groupCohomology.map_id_comp_H0Iso_hom_apply`, `exteriorPower.pairingDual_apply_apply_eq_one`, `Module.Flat.tensorProduct_mapIncl_injective_of_left`, `imaginaryPart_comp_subtype_selfAdjoint`, `iSupIndep.linearEquiv_apply`, `ContinuousLinearEquiv.coord_toSpanNonzeroSingleton`, `HasRankNullity.rank_quotient_add_rank`, `Module.Flat.iff_lTensor_injectiveₛ`, `CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_assoc_apply`, `torsionBy.mk_ideal_smul`, `IsLattice.rank_of_pi`, `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`, `norm_orthogonalProjection_apply`, `toENat_rank_span_set`, `torsionBySet_torsionBySet_eq_top`, `LinearMap.singularValues_pos_iff_lt_finrank_range`, `rTensorOne_tmul_one`, `toLinearMap_prodEquivOfIsCompl_symm`, `PeriodPair.latticeBasis_one`, `diffToIdealOfQuotientCompEq_apply`, `finrank_eq_rank`, `ZLattice.covolume_eq_det_mul_measureReal`, `Subspace.finrank_add_finrank_dualCoannihilator_eq`, `DirectSum.IsInternal.toBaseChange_bijective`, `Function.Exact.linearMap_rangeRestrict`, `Module.End.invtSubmodule.map_subtype_mem_of_mem_invtSubmodule`, `Module.Finite.range`, `TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective`, `LinearIndependent.linearCombinationEquiv_symm_apply`, `equivSubtypeMap_apply`, `Algebra.Extension.Hom.mapKer_apply_coe`, `isIdempotentElemEquiv_symm_apply_coe`, `Algebra.Extension.contangentEquiv_tmul`, `det_reflection`, `Representation.linHom.invariantsEquivRepHom_symm_apply_coe`, `LinearMap.quotKerEquivRange_apply_mk`, `Algebra.Generators.cMulXSubOneCotangent_eq`, `rank_span_le`, `ContinuousLinearEquiv.toSpanNonzeroSingleton_coord`, `Ideal.toCotangent_apply`, `inner_orthogonalProjection_eq_of_mem_left`, `LieAlgebra.InvariantForm.restrict_orthogonal_nondegenerate`, `rank_submodule_eq_one_iff`, `TensorProduct.spanFinrank_top_le_of_fg`, `Finsupp.supportedEquivFinsupp_symm_apply_coe_support_val`, `ZLattice.exists_forall_abs_repr_le_norm`, `linearMap_eq_iff_of_eq_span`, `CategoryTheory.ShortComplex.exact_iff_surjective_moduleCatToCycles`, `RootPairing.prod_rootFormIn_smul_coroot_mem_range_PolarizationIn`, `Ideal.toCotangent_eq_zero`, `exteriorPower.ιMulti_family_span_of_span`, `NumberField.fractionalIdeal_rank`, `toLinearMap_orthogonalProjection_eq_linearProjOfIsCompl`, `LinearMap.finrank_range_adjoint`, `mulRightMap_apply`, `LieHom.quotKerEquivRange_invFun`, `IsLattice.free`, `Module.injOn_dualMap_subtype_span_range_range`, `LieSubalgebra.coe_ofLe`, `Real.Convex.dimH_eq_finrank_vectorSpan`, `quotientEquivOfIsCompl_symm_apply`, `exists_smith_normal_form_of_rank_eq`, `GradedTensorProduct.auxEquiv_tmul`, `Ideal.cotangentEquivIdeal_apply`, `exteriorPower.map_injective_field`, `Subspace.dualLift_of_mem`, `LinearEquiv.coe_ofEq_apply`, `Polynomial.degreeLT.addLinearEquiv_symm_apply`, `coord_norm'`, `RootPairing.algebraMap_posRootForm_posForm`, `NumberField.mixedEmbedding.volume_fundamentalDomain_fractionalIdealLatticeBasis`, `Basis.mem_ideal_iff'`, `Ideal.quotientToQuotientRangePowQuotSucc_injective`, `Algebra.idealMap_isLocalizedModule`, `range_powSMulQuotInclusion`, `DirectSum.decomposeLinearEquiv_apply`, `AddMonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_support_val`, `sectionOfRetractionKerToTensor_algebraMap`, `exteriorPower.alternatingMapLinearEquiv_comp`, `collinear_iff_rank_le_one`, `Finsupp.linearCombination_restrict`, `LinearEquiv.ofSubmodule'_toLinearMap`, `finrank_top`, `DirectSum.decomposeLinearEquiv_symm_comp_lof`, `exteriorPower.basis_coord`, `LinearIndependent.repr_range`, `Algebra.Extension.Cotangent.mk_eq_mk_iff_sub_mem`, `IsSemisimpleModule.exists_end_ringEquiv`, `Module.isPrincipal_submodule_iff`, `Ideal.constr_basisSpanSingleton`, `Function.Exact.iff_linearMap_rangeRestrict`, `RootPairing.range_polarizationIn`, `mulMap_one_left_eq`, `groupCohomology.coboundariesToCocycles₂_apply`, `MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictTotalDegreeOfFinite`, `quotDualCoannihilatorToDual_injective`, `MultilinearMap.codRestrict_apply_coe`, `DirectSum.toBaseChange_injective`, `TensorProduct.exists_finite_submodule_of_setFinite'`, `MvPolynomial.DirectSum.coeLinearMap_eq_finsum`, `Subspace.dualRestrict_surjective`, `LinearMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal'`, `torsionBy_isTorsion_nonZeroDivisor`, `ker_subtypeL`, `LinearMap.kerComplementEquivRange_symm_apply`, `LinearMap.lift_rank_range_add_rank_ker`, `mulMap_comm_of_commute`, `TensorAlgebra.GradedAlgebra.ι_apply`, `MeasureTheory.integrable_condExpL2_of_isFiniteMeasure`, `PeriodPair.latticeEquiv_symm_apply`, `HahnEmbedding.ArchimedeanStrata.isInternal_stratum'`, `exteriorPower.instFinite`, `LinearEquiv.coord_self`, `exteriorPower.alternatingMapToDual_apply_ιMulti`, `isArtinian_submodule'`, `KaehlerDifferential.D_tensorProductTo`, `LinearEquiv.ofEq_symm`, `IsSemisimpleModule.exists_end_algEquiv`, `TensorProduct.spanFinrank_top_eq_of_residueField`, `mulMap_eq_mul'_comp_mapIncl`, `DirectSum.decomposeLinearEquiv_symm_apply`, `LinearEquiv.ofLeftInverse_apply`, `setBasisOfTopLeSpanOfCardEqFinrank_repr_apply`, `liftOfDerivationToSquareZero_apply`, `MonoidAlgebra.supportedEquivFinsupp_apply_support_val`, `Algebra.Extension.Cotangent.mk_surjective`, `rank_add_le_rank_add_rank`, `RootPairing.RootPositiveForm.isSymm_posForm`, `FractionalIdeal.isNoetherian_iff`, `mem_smul_top_iff`, `DirectSum.decomposeAlgEquiv_apply`, `AffineIndependent.finrank_vectorSpan`, `AddMonoidAlgebra.grade.decompose_single`, `IsCompl.projection_apply`, `LinearDisjoint.not_linearIndependent_pair_of_flat`, `LinearMap.comp_codRestrict`, `mulLeftMap_eq_mulMap_comp`, `Representation.subrepresentation_apply`, `MeasureTheory.integrable_condExpL2_indicator`, `rank_quotient_add_rank_le`, `LieAlgebra.Extension.d₁₂_oneCochainOfTwoSplitting`, `LinearMap.finrank_range_dualMap_eq_finrank_range`, `Algebra.Extension.Hom.subToKer_apply_coe`, `HahnEmbedding.ArchimedeanStrata.iSupIndep_stratum'`, `LinearMap.IsProj.eq_conj_prod_map'`, `LinearEquiv.submoduleMap_symm_apply`, `LinearMap.map_codRestrict`, `ZLattice.abs_repr_lt_of_norm_lt`, `MonoidAlgebra.supportedEquivFinsupp_apply_apply`, `LinearMap.BilinForm.finrank_orthogonal`, `DirectSum.IsInternal.card_filter_subordinateOrthonormalBasisIndex_eq`, `ContinuousMultilinearMap.codRestrict_toMultilinearMap`, `Polynomial.degreeLT.addLinearEquiv_symm_apply'`, `finrank_add_inf_finrank_orthogonal'`, `LinearDisjoint.rank_le_one_of_flat_of_self`, `ContinuousLinearMap.projKerOfRightInverse_comp_inv`, `TensorProduct.eq_of_fg_of_subtype_eq`, `LinearEquiv.toSpanNonzeroSingleton_symm_apply_smul`, `Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_is_extension`, `instInvertibleSubtypeMemVal`, `AffineSubspace.linear_equivMapOfInjective`, `DirectSum.IsInternal.toBaseChange_injective`, `MeasureTheory.MemLp.condExpL2_ae_eq_condExp'`, `LinearMap.BilinForm.IsSymm.restrict`, `LinearMap.trace_eq_sum_trace_restrict`, `toIsOrderedCancelAddMonoid`, `Ideal.rank_pow_quot_aux`, `isNoetherian_of_fg_of_noetherian`, `CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_apply`, `Subspace.dualEquivDual_def`, `TensorProduct.quotientTensorEquiv_apply_tmul_mk`, `exists_finset_span_eq_linearIndepOn`, `exteriorPower.zeroEquiv_ιMulti`, `LinearMap.surjective_comp_linearProjOfIsCompl`, `LinearMap.exact_subtype_ker_map`, `HahnEmbedding.Seed.truncLT_mem_range_baseEmbedding`, `Polynomial.eval_eq_sum_degreeLTEquiv`, `mulMap'_surjective`, `subtypeL_apply`, `finrank_span_eq_finrank_span`, `Subspace.dualLift_rightInverse`, `Ideal.natAbs_det_equiv`, `finrank_map_subtype_eq`, `ker_orthogonalProjection`, `LinearMap.ofIsCompl_add`, `LinearEquiv.ofSubmodule'_apply`, `FG.directedSystem`, `powSMulQuotInclusion_mk`, `exteriorPower.map_injective`, `Ideal.rank_pow_quot`, `Ideal.subtype_isoBaseOfIsPrincipal_eq_mul`, `LinearDisjoint.rank_le_one_of_commute_of_flat_of_self`, `exteriorPower.basis_repr_apply`, `LinearIndependent.repr_eq`, `mk_quotientEquivOfIsCompl_apply`, `MvPolynomial.weightedDecomposition.decompose'_eq`, `prodEquivOfIsCompl_symm_apply_right`, `coe_mapIic_apply`, `le_sigNeg_of_negDef`, `submoduleOf_eq_top`, `Subspace.dualCopairing_nondegenerate`, `restrictScalars.isScalarTower`, `ContinuousLinearEquiv.submoduleMap_symm_apply`, `NumberField.mem_span_basisOfFractionalIdeal`, `Algebra.Extension.tensorCotangentInvFun_smul_mk`, `eq_span_singleton_of_surjective`, `Basis.mem_ideal_iff`, `LinearEquiv.ofTop_symm_apply`, `rank_span_of_finset`, `finrank_mono`, `mulLeftMap_apply_single`, `ContinuousLinearMap.range_eq_map_coprodSubtypeLEquivOfIsCompl`, `finite_finset_sup`, `LinearIndependent.repr_ker`, `factorPow_comp_powSMulQuotInclusion`, `finrank_add_le_finrank_add_finrank`, `Pi.dim_spanSubset`, `mulMap_map_comp_eq`, `QuadraticMap.restrict_apply`, `Module.Flat.iff_lift_lsmul_comp_subtype_injective`, `Ideal.range_finsuppTotal`, `Module.Finite.top`, `IsSemisimpleRing.exists_ringEquiv_pi_matrix_end_mulOpposite`, `finrank_le_one_iff_isPrincipal`, `Module.End.isNilpotent_restrict_sub_algebraMap`, `FG.lTensor.directLimit_apply`, `finrank_span_eq_finrank`, `LinearMap.IsProj.codRestrict_ker`, `Collinear.finrank_le_one`, `Module.fgSystem.equiv_comp_of`, `RootPairing.posRootForm_posForm_nondegenerate`, `LinearMap.singularValues_finrank_range_self`, `range_inclusion`, `Module.Flat.instSubtypeMemSubmoduleSubmoduleAlgebra`, `ContinuousLinearMap.projKerOfRightInverse_apply_idem`, `spanFinrank_top`, `LinearMap.domRestrict₂_apply`, `Ideal.powQuotSuccInclusion_injective`, `LinearMap.IsProj.trace`, `IsLocalRing.CotangentSpace.map_eq_top_iff`, `rank_range_of_injective`, `HahnEmbedding.ArchimedeanStrata.archimedean_stratum`, `basisOfPid_bot`, `ContinuousLinearMap.coe_codRestrict_apply`, `LinearMap.restrict_smul_one`, `LinearMap.range_domRestrict`, `IsSemisimpleModule.exists_end_algEquiv_pi_matrix_end`, `instIsTorsionFree`, `LinearMap.continuous_domRestrict`, `coe_subtype`, `mulMap_comp_lTensor`, `LinearEquiv.rank_map_eq`, `isNoetherian_span_of_finite`, `RootPairing.algebraMap_coroot'In_apply`, `ContinuousLinearMap.IsPositive.orthogonalProjection_comp`, `LinearMap.linearEquiv_of_ne_zero`, `LinearMap.IsSymm.nondegenerate_restrict_of_isCompl_ker`, `dimH_orthogonalProjection_le`, `LinearMap.range_ofIsCompl`, `RootPairing.rootForm_restrict_nondegenerate_of_isAnisotropic`, `derivationOfSectionOfKerSqZero_apply_coe`, `torsionBy_isTorsionBy`, `Module.Finite.iff_fg`, `lTensorOne_symm_apply`, `LinearMap.tensorEqLocus_tmul`, `exteriorPower.map_comp_ιMulti_family`, `fg_top`, `sectionOfRetractionKerToTensorAux_algebraMap`, `RootPairing.rootFormIn_self_smul_coroot`, `tensorToSpan_apply_tmul`, `exteriorPower.map_id`, `LinearMap.ofIsCompl_left_apply`, `isCompl_comap_subtype_of_isCompl_of_le`, `TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_apply_tmul`, `AddMonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_apply`, `Module.Finite.of_fg`, `HahnEmbedding.Seed.hahnCoeff_apply`, `iSupIndep_iff_dfinsupp_lsum_injective`, `Finsupp.supportedEquivFinsupp_symm_apply_coe`, `LinearEquiv.toSpanNonzeroSingleton_one`, `lTensorOne'_one_tmul`, `LinearMap.compAlternatingMap_codRestrict`, `LinearMap.coe_quotientInfToSupQuotient`, `Subspace.dualLift_injective`, `Algebra.Generators.cotangentRestrict_mk`, `Polynomial.toMatrix_sylvesterMap`, `TensorProduct.exists_finite_submodule_right_of_finite`, `LinearMap.subtype_compMultilinearMap_codRestrict`, `Finsupp.supportedEquivFinsupp_apply_support_val`, `ContinuousLinearMap.exist_extension_of_finiteDimensional_range`, `groupCohomology.π_comp_H2Iso_hom_apply`, `MeasureTheory.hausdorffMeasure_orthogonalProjection_le`, `Algebra.Generators.H1Cotangent.δ_eq_δAux`, `CliffordAlgebra.even.lift.aux_apply`, `finrank_add_finrank_orthogonal`, `map_subtype_top`, `exteriorPower.pairingDual_apply_apply_eq_one_zero`, `lipschitzWith_orthogonalProjection`, `NumberField.mixedEmbedding.fractionalIdealLatticeBasis_apply`, `exists_fg_le_subset_range_rTensor_inclusion`, `derivationToSquareZeroEquivLift_apply_coe_apply`, `span_span_coe_preimage`, `Representation.coind_apply`, `ContinuousLinearEquiv.submoduleMap_apply`, `FractionalIdeal.abs_det_basis_change`, `ClosedComplemented.exists_submodule_equiv_prod`, `Rep.invariantsFunctor_map_hom`, `Module.End.iInf_maxGenEigenspace_restrict_map_subtype_eq`, `finrank_span_singleton`, `liftOfDerivationToSquareZero_mk_apply'`, `ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection`, `Module.Finite.map`, `rank_le_one_iff_isPrincipal`, `subalgebra_top_finrank_eq_submodule_top_finrank`, `stereographic_apply`, `MeasureTheory.inner_condExpL2_eq_inner_fun`, `finite_dualAnnihilator_iff`, `linearProjOfIsCompl_comp_bijective_of_exact`, `RootPairing.RootPositiveForm.algebraMap_apply_eq_form_iff`, `groupCohomology.π_comp_H1Iso_hom_apply`, `MeasureTheory.integrableOn_condExpL2_of_measure_ne_top`, `LinearMap.rTensor_range`, `Module.Basis.SmithNormalForm.repr_comp_embedding_eq_smul`, `TensorProduct.exists_finite_submodule_right_of_finite'`, `IsLocalRing.CotangentSpace.span_image_eq_top_iff`, `AlternatingMap.codRestrict_apply_coe`, `LinearEquiv.coord_apply_smul`, `inf_iInf_maxGenEigenspace_of_forall_mapsTo`, `LinearMap.finrank_maxGenEigenspace_zero_eq`, `fstL_comp_coe_orthogonalDecomposition`, `RootPairing.finrank_corootSpan_eq`, `LinearMap.ofIsCompl_right_apply`, `Polynomial.sylvesterMap_apply_coe`, `Polynomial.degreeLT.addLinearEquiv_natAdd`, `EuclideanGeometry.orthogonalProjection_linear`, `ContinuousLinearEquiv.ofSubmodule'_symm_apply`, `inclusion_injective`, `mulMap_comp_map_inclusion`, `orthogonalProjection_orthogonal_apply_eq_zero`, `LinearMap.submoduleComap_surjective_of_surjective`, `exteriorPower.basis_repr`, `exists_extension_norm_eq`, `NumberField.Units.unitLattice_rank`, `finrank_le`, `groupHomology.π_comp_H2Iso_hom_apply`, `isArtinian_sup`, `lTensorOne_tmul`, `instIsSimpleModuleSubtypeMemSubmoduleValSetSetOfNonemptyLinearEquivId`, `Module.Flat.iff_rTensor_injectiveₛ`, `LinearDisjoint.val_mulMap_tmul`, `LinearMap.trace_restrict_eq_of_forall_mem`, `Module.Dual.finrank_ker_add_one_of_ne_zero`, `Module.Finite.bot`, `Algebra.Extension.Hom.sub_tmul`, `exteriorPower.alternatingMapLinearEquiv_symm_apply`, `Ideal.primaryComponent.map_ker_eq`, `ZLattice.normBound_spec`, `instFreeSubtypeMemIdealOfFiniteOfIsTorsionFree`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_π_hom`, `Rep.indToCoind_coindToInd`, `LinearMap.exists_extend`, `isNoetherian_submodule_left`, `Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisors`, `affineIndependent_iff_finrank_vectorSpan_eq`, `orthogonalProjectionFn_eq`, `RootPairing.finrank_rootSpanIn`, `Ideal.cotangentEquivIdeal_symm_apply`, `Module.flat_iff`, `LinearMap.ker_id_sub_eq_of_proj`, `Polynomial.taylorLinearEquiv_symm`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne`, `exists_fg_le_eq_rTensor_subtype`, `FractionalIdeal.isNoetherian`, `NumberField.Units.logEmbeddingEquiv_apply`, `Module.End.isNilpotent_restrict_of_le`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inl_basis`, `iSup_eq_toSubmodule_range`, `TensorProduct.exists_finite_submodule_of_setFinite`, `TensorProduct.exists_finite_submodule_left_of_setFinite'`, `RootPairing.RootPositiveForm.zero_lt_apply_root_root_iff`, `exteriorPower.map_apply_ιMulti`, `NumberField.basisOfFractionalIdeal_apply`, `ContinuousAlternatingMap.codRestrict_apply_coe`, `Module.Basis.ofZLatticeComap_repr_apply`, `TensorProduct.rTensor_injective_of_forall_fg_rTensor_injective`, `LinearMap.finrank_eq_of_isPerfPair`, `LinearMap.ker_restrict`, `RootPairing.posRootForm_posForm_anisotropic`, `QuadraticMap.IsometryEquiv.map_negDef_iff`, `rank_quotient_add_rank_of_divisionRing`, `Module.End.isNilpotent_restrict_genEigenspace_top`, `groupHomology.boundariesToCycles₂_apply`, `Module.Finite.span_finset`, `Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule_1`, `AddMonoidAlgebra.decomposeAux_single`, `map_range_rTensor_subtype_lid`, `isNoetherian_sup`, `ContinuousLinearEquiv.coord_norm'`, `LinearEquiv.coe_ofTop_symm_apply`, `IsLattice.finite`, `LinearMap.injective_domRestrict_iff`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_H`, `LinearMap.support_singularValues`, `fstEquiv_symm_apply_coe`, `snd_orthogonalDecomposition_apply`, `LinearMap.BilinForm.toLin_restrict_ker_eq_inf_orthogonal`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_apply`, `Module.Basis.ofZLatticeBasis_repr_apply`, `fromModuleCatToModuleCatLinearEquiv_apply`, `Module.Finite.span_of_finite`, `OrzechProperty.injective_of_surjective_of_submodule'`, `IsIsotypic.linearEquiv_finsupp`, `fromModuleCatToModuleCatLinearEquiv_symm_apply_coe`, `LinearMap.surjective_rangeRestrict`, `linearProjOfIsCompl_range`, `NumberField.instFiniteIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule`, `toBaseChange_surjective'`, `CharacterModule.intSpanEquivQuotAddOrderOf_apply_self`, `LinearMap.restrict_coe_apply`, `LieAlgebra.IsKilling.ker_restrict_eq_bot_of_isCartanSubalgebra`, `AlgHom.toKerIsLocalization_isLocalizedModule`, `finite_iSup`, `FractionalIdeal.equivNum_apply`, `AddSubgroup.relIndex_eq_natAbs_det`, `isNoetherian_of_submodule_of_noetherian`, `orthogonalDecomposition_apply`, `map_comap_subtype`, `span_range_subtype_eq_top_iff`, `spanRank_top`, `LinearMap.isCompl_of_proj`, `exteriorPower.map_comp`, `Module.End.isNilpotent.restrict`, `groupHomology.π_comp_H1Iso_inv_apply`, `lift_rank_range_le`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_left`, `exteriorPower.ιMultiDual_apply_nondiag`, `LinearMap.codRestrict_apply`, `AddMonoidAlgebra.decomposeAux_eq_decompose`, `mulMap_op`, `LinearEquiv.ofEq_rfl`, `torsionBySet.mk_smul`, `isNoetherian_top_iff`, `LieModule.Cohomology.mem_twoCocycle_iff_of_trivial`, `MeasureTheory.eLpNorm_condExpL2_le`, `finrank_vectorSpan_range_add_one_le`, `Module.Flat.tensorProduct_mapIncl_injective_of_right`, `ContinuousLinearEquiv.toSpanNonzeroSingleton_apply_coe`, `mem_span_set_iff_exists_finsupp_le_finrank`, `ContinuousLinearEquiv.coord_self`, `Algebra.idealMap_mul`, `LinearMap.linearProjOfIsCompl_of_proj`, `LinearMap.ofIsCompl_subtype_zero_eq`, `Derivation.liftKaehlerDifferential_apply`, `LinearMap.trace_eq_sum_trace_restrict'`, `LinearEquiv.finrank_map_eq`, `LinearEquiv.toLinearMap_ofTop`, `map_subtype_range_inclusion`, `finrank_bot`, `ContinuousLinearEquiv.coord_norm`, `exteriorPower.map_surjective`, `KaehlerDifferential.DLinearMap_apply`, `torsionBySet_ideal_isTorsion_of_noZeroDivisors`, `LinearPMap.mem_adjoint_domain_iff`, `Module.Basis.coe_mkFinCons`, `CliffordAlgebra.GradedAlgebra.lift_ι_eq`, `linearProjOfIsCompl_ker`, `LinearEquiv.lift_rank_map_eq`, `EuclideanGeometry.orthogonalProjection_apply`, `Module.Basis.extendOfIsLattice_apply`, `torsion_isTorsion`, `rank_span`, `norm_orthogonalProjection`, `LinearMap.range_eq_of_proj`, `LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat`, `EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection`, `Polynomial.degreeLT.basisProd_natAdd`, `Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange`, `mem_set_smul`, `goursat_surjective`, `lTensorOne_one_tmul`, `LinearMap.finrank_maxGenEigenspace_eq`, `LinearMap.IsProj.codRestrict_apply`, `DirectSum.decomposeLinearEquiv_symm_lof`, `Representation.coindMap_coe_apply`, `LinearMap.BilinForm.toLin_restrict_range_dualCoannihilator_eq_orthogonal`, `LinearMap.nondegenerate_restrict_of_disjoint_orthogonal`, `rank_le_of_isSMulRegular`, `LinearPMap.coe_vadd`, `Module.Dual.eq_of_preReflection_mapsTo'`, `finrank_lt`, `rank_sup_add_rank_inf_eq`, `MeasureTheory.condExpIndSMul_ae_eq_smul`, `rank_top`, `Module.fgSystem.instDirectedSystemSubtypeSubmoduleFGMemValCoeLinearMapId`, `isNoetherian_range`, `linearProjOfIsCompl_apply_eq_zero_iff`, `coplanar_iff_finrank_le_two`, `LinearMap.tensorKerEquiv_apply`, `Module.Basis.span_neg`, `finite_sup`, `LinearMap.range_codRestrict`, `mem_biSup_iff_exists_dfinsupp`, `TensorProduct.AlgebraTensorModule.range_lTensor_idealMap`, `exteriorPower.ιMulti_family_linearIndependent_ofBasis`, `LieAlgebra.LoopAlgebra.twoCocycleOfBilinear_coe`, `range_subtype`, `NumberField.instIsLocalizedModuleIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmoduleValFractionalIdealNonZeroDivisorsRestrictScalarsSubtype`, `isArtinian_iSup`, `subtype_injective`, `dualRestrict_apply`, `LinearMap.BilinForm.finrank_add_finrank_orthogonal`, `IsSemisimpleModule.exists_linearEquiv_dfinsupp`, `Module.Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix`, `finrank_quotient`, `comapSubtypeEquivOfLe_apply_coe`, `RootPairing.RootPositiveForm.algebraMap_posForm`, `LinearDisjoint.rank_inf_le_one_of_commute_of_flat_left`, `QuadraticForm.sigPos_add_finrank_le_of_nonpos`, `DirectSum.isInternal_biSup_submodule_of_iSupIndep`, `QuadraticMap.Nondegenerate.rank_rad_polar_le`, `LinearMap.ker_le_range_iff`, `Ideal.map_toCotangent_ker`, `LinearMap.ofIsCompl_zero`, `exteriorPower.ιMulti_family_span`, `Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero`, `toLocalized'_apply_coe`, `orthogonalProjection_comp_subtypeL_eq_zero_iff`, `powSMulQuotInclusion_injective`, `dualQuotEquivDualAnnihilator_symm_apply_mk`, `LinearMap.BilinForm.finrank_add_finrank_orthogonal'`, `LieDerivation.IsKilling.killingForm_restrict_range_ad_nondegenerate`, `Ideal.quotTorsionOfEquivSpanSingleton_apply_mk`, `Ideal.quotientToQuotientRangePowQuotSuccAux_mk`, `Module.Basis.sumQuot_inl`, `ZLattice.sum_piFinset_Icc_rpow_le`, `LinearEquiv.ofInjective_symm_apply`, `LieAlgebra.Extension.twoCocycleOf_coe_coe`, `ExteriorAlgebra.GradedAlgebra.ι_apply`, `Algebra.SubmersivePresentation.basisCotangent_apply`, `linearProjOfIsCompl_surjective`, `LinearMap.exists_extend_of_notMem`, `RootPairing.posRootForm_rootFormIn_posDef`, `HahnEmbedding.Partial.truncLT_eval_mem_range_extendFun`, `LinearMap.comp_ker_subtype`, `mulMap_range`, `TensorProduct.tensorQuotientEquiv_symm_apply_tmul_mk`, `Algebra.kerTensorProductMapIdToAlgHomEquiv_symm_apply`, `exteriorPower.oneEquiv_ιMulti`, `Algebra.Extension.Cotangent.map_mk`, `RootPairing.CoPolarizationIn_apply`, `LinearDisjoint.linearIndependent_left_of_flat`, `orthogonalProjection_eq_zero_iff`, `CliffordAlgebra.GradedAlgebra.ι_sq_scalar`, `AdicCompletion.ofPowSMul_val_apply`, `Polynomial.sylveserMap_comp_adjSylvester`, `exteriorPower.ιMulti_span`, `map_equivMapOfInjective_symm_apply`, `Finsupp.supportedEquivFinsupp_symm_apply_coe_apply`, `orthogonalProjection_starProjection_of_le`, `Real.exists_extension_norm_eq`, `LinearMap.finrank_range_add_finrank_ker`, `exteriorPower.toTensorPower_apply_ιMulti`, `coe_orthogonalProjection_apply`, `torsion'_torsion'_eq_top`, `instModuleFree_of_discrete_submodule`, `Ideal.natAbs_det_basis_change`, `Module.Flat.iff_rTensor_injective'`, `skewAdjointPart_comp_subtype_skewAdjoint`, `KaehlerDifferential.fromIdeal_surjective`, `Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op`, `Polynomial.degreeLT.basisProd_castAdd`, `finrank_add_eq_of_isCompl`, `Finsupp.supportedEquivFinsupp_apply_apply`, `FG.rTensor.directLimit_apply'`, `LinearMap.restrict_eq_domRestrict_codRestrict`, `LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure`, `LieModule.Cohomology.d₂₃_comp_d₁₂`, `Subspace.quotDualCoannihilatorToDual_bijective`, `Ideal.toCotangent_surjective`, `lTensor_mkQ`, `norm_subtypeL_le`, `ContinuousLinearEquiv.fst_equivOfRightInverse`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_right`, `exteriorPower.zeroEquiv_symm_apply`, `isAtom_iff_finrank_eq_one`, `finsetBasisOfLinearIndependentOfCardEqFinrank_repr_apply`, `basis_of_pid_aux`, `torsion'_isTorsion'`, `HahnEmbedding.Partial.apply_of_mem_stratum`, `sndEquiv_apply`, `RingHom.toKerIsLocalization_isLocalizedModule`, `LinearPMap.adjointAux_inner`, `sndEquiv_symm_apply_coe`, `Algebra.idealMap_apply_coe`, `comapSubtypeEquivOfLe_symm_apply`, `Module.End.genEigenspace_restrict`, `Matrix.rank_eq_finrank_span_cols`, `LinearMap.trace_eq_sum_trace_restrict_of_eq_biSup`, `injective_lTensor_quotient_iff_inf_eq_mul`, `coe_subtypeL`, `orthogonalProjection_coe_eq_linearProjOfIsCompl`, `quotient_prod_linearEquiv`, `LinearMap.domRestrict₁₂_apply`, `Ideal.primaryComponent_torsionBySet_of_isCoprime`, `Ideal.map_includeLeft_eq`, `orthogonalProjection_norm_le`, `ContinuousLinearEquiv.equivOfRightInverse_symm_apply`, `LinearMap.injective_rangeRestrict_iff`, `LinearDisjoint.rank_inf_le_one_of_flat_right`, `LinearMap.quotientInfEquivSupQuotient_surjective`, `LinearPMap.graph_map_snd_eq_range`, `ModuleCat.toKernelSubobject_arrow`, `orthogonalProjection_orthogonal`, `LinearMap.finrank_eigenspace_le`, `Module.Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord`, `LinearMap.domRestrict_apply`, `LinearMap.surjective_comp_subtype_of_isComplemented`, `Polynomial.taylorLinearEquiv_apply_coe`, `MeasureTheory.norm_condExpL2_le_one`, `Module.finrank_quotient_add_finrank_le`, `Module.Flat.iff_lTensor_injective'`, `Module.Basis.addSubgroupOfClosure_apply`, `MeasureTheory.condExpL2_comp_continuousLinearMap`, `LinearEquiv.ofTop_apply`, `mulMap_comp_rTensor`, `Algebra.Extension.Cotangent.ker_mk`, `torsionBySet_isTorsionBySet`, `orthogonalProjection_orthogonalComplement_singleton_eq_zero`, `Polynomial.degreeLT.basis_repr`, `IsSemisimpleModule.exists_end_ringEquiv_pi_matrix_end`, `Module.End.pos_finrank_genEigenspace_of_hasEigenvalue`, `LinearDisjoint.rank_inf_le_one_of_flat_left`, `Ideal.primaryComponent_torsionBySet_eq_inf`, `orthogonalProjection_eq_linearProjOfIsCompl`, `LinearPMap.adjoint_apply_of_dense`, `RootPairing.range_polarizationIn_le_span_coroot`, `exists_smith_normal_form_of_le`, `MeasureTheory.condExpL2_const_inner`, `LinearPMap.toFun_eq_coe`, `map_subtype_le`, `coe_prodEquivOfClosedCompl`, `Module.Basis.restrictScalars_toMatrix`, `TensorProduct.exists_of_fg`, `torsionBy.mk_smul`, `Subalgebra.rank_toSubmodule`, `groupHomology.π_comp_H2Iso_inv_apply`, `LinearMap.BilinForm.restrict_apply`, `DirectSum.range_coeLinearMap`, `submoduleOf_self`, `IsLocalRing.spanFinrank_eq_finrank_quotient`, `Finsupp.range_restrictDom`, `LinearMap.BilinForm.nondegenerate_restrict_of_disjoint_orthogonal`, `linearIndepOn_iff_linearCombinationOnₛ`, `EuclideanGeometry.Sphere.finrank_orthRadius`, `ContinuousLinearEquiv.ofSubmodule'_toContinuousLinearMap`, `LinearMap.rank_range_add_rank_ker`, `finrank_vectorSpan_insert_le_set`, `coe_equivMapOfInjective_apply`, `disjoint_iff_comap_eq_bot`, `LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm`, `Subspace.finrank_add_finrank_dualAnnihilator_eq`, `TensorProduct.toLinearMap_mapInclIsometry`, `lsum_comp_mapRange_toSpanSingleton`, `RootPairing.algebraMap_root'In_apply`, `groupHomology.boundariesToCycles₁_apply`, `OrthogonalFamily.projection_directSum_coeAddHom`, `TensorProduct.range_mapIncl_mono`, `Module.Finite.of_isComplemented_codomain`, `MatrixModCat.fromMatrixLinear_apply_coe`, `TensorProduct.spanRank_top_le`, `PeriodPair.finrank_lattice`, `linearIndepOn_iff_linearCombinationOn`, `Module.Flat.instInvertibleSubtypeMemSubmoduleSubmoduleAlgebra`, `LinearMap.toKerIsLocalized_apply_coe`, `LinearMap.submoduleMap_surjective`, `LinearMap.finrank_le_of_isSMulRegular`, `LinearPMap.inverse_domain`, `exteriorPower.basis_repr_self`, `MeasureTheory.condExpL2_indicator_nonneg`, `flip_quotDualCoannihilatorToDual_injective`, `coe_prodEquivOfClosedCompl_symm`, `Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule`, `Algebra.Extension.cotangentComplex_mk`, `mulRightMap_apply_single`, `Subspace.dualPairing_eq`, `subalgebra_top_rank_eq_submodule_top_rank`, `Int.submodule_toAddSubgroup_index_ne_zero_iff`, `fstEquiv_apply`, `Ideal.to_quotient_square_comp_toCotangent`, `retractionOfSectionOfKerSqZero_tmul_D`, `DirectSum.IsInternal.subordinateOrthonormalBasisIndex_def`, `finrank_vectorSpan_range_le`, `GeneralSchauderBasis.finrank_range_proj`, `ContinuousLinearEquiv.ofSubmodules_apply`, `LinearMap.ofIsCompl_eq_add`, `span_range_inclusionSpan`, `LinearMap.surjective_domRestrict_iff`, `ZLattice.exists_finsetSum_norm_rpow_le_tsum`, `Subspace.dualEquivDual_apply`, `instIsLocalizedModuleSubtypeMemLocalized₀ToLocalized₀`, `Ideal.isoBaseOfIsPrincipal_apply`, `ZLattice.abs_repr_le`, `LinearMap.singularValues_eq_zero_iff_le_finrank_range`, `DirectSum.IsInternal.collectedBasis_repr_of_mem`, `MvPolynomial.decomposition.decompose'_eq`, `restrictScalarsEquiv_apply`, `ZLattice.covolume_eq_det`, `rank_range_of_surjective`, `LinearMap.restrict_apply`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply`, `natAbs_det_basis_change`, `nonempty_basis_of_pid`, `comap_subtype_le_iff`, `Ideal.Cotangent.lift_surjective_iff`, `isSemisimpleModule_iff_exists_linearEquiv_dfinsupp`, `ContinuousLinearEquiv.ofSubmodules_symm_apply`, `LinearMap.card_support_singularValues`
`copy` 📖	CompOp	2 math math: `coe_copy`, `copy_eq`
`decidableEq` 📖	CompOp	29 math math: `isInternal_prime_power_torsion_of_pid`, `Ideal.exists_minimal_isPrimary_decomposition_of_isPrimary_decomposition`, `Ideal.Factors.fact_ramificationIdx_neZero`, `Ideal.count_normalizedFactors_eq`, `isInternal_prime_power_torsion_of_is_torsion_by_ideal`, `Ideal.IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count`, `Ideal.Factors.isScalarTower`, `count_span_normalizedFactors_eq_of_normUnit`, `irreducible_pow_sup`, `IsMinimalPrimaryDecomposition.minimal`, `Ideal.Factors.liesOver`, `count_le_of_ideal_ge`, `Ideal.decomposition_erase_inf`, `IsMinimalPrimaryDecomposition.comap_localized₀_eq_ite`, `Ideal.Factors.finrank_pow_ramificationIdx`, `count_span_normalizedFactors_eq`, `isInternal_prime_power_torsion`, `Ideal.IsDedekindDomain.ramificationIdx_eq_factors_count`, `exists_minimal_isPrimary_decomposition_of_isPrimary_decomposition`, `IsDedekindDomain.HeightOneSpectrum.maxPowDividing_eq_pow_multiset_count`, `Ideal.eq_prime_pow_mul_coprime`, `Ideal.Factors.finiteDimensional_quotient_pow`, `count_associates_factors_eq`, `Ideal.Factors.isPrime`, `sup_eq_prod_inf_factors`, `Ideal.Factors.piQuotientEquiv_mk`, `decomposition_erase_inf`, `IsDedekindDomain.quotientEquivPiFactors_mk`, `Ideal.Factors.piQuotientEquiv_map`
`inhabited` 📖	CompOp	—
`instPartialOrder` 📖	CompOp	1127 math math: `comap_lt_comap_iff_of_surjective`, `Ideal.exists_pow_le_of_le_radical_of_fg_radical`, `IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal`, `isArtinian_iff`, `Ideal.quotientEquivAlgOfEq_coe_eq_factorₐ`, `MvPolynomial.vanishingIdeal_anti_mono`, `Polynomial.not_ker_le_map_C_of_surjective_of_quasiFiniteAt`, `Ideal.radical_mono`, `toAddSubgroup_toZModSubmodule`, `toAddSubmonoid_toNatSubmodule`, `comap_le_comap_iff_of_le_range`, `isNoetherian_iff'`, `AffineSubspace.direction_inf`, `FG.rTensor.directedSystem`, `Ideal.comap_fiberIsoOfBijectiveResidueField_apply`, `disjoint_span_singleton'`, `LinearMap.IsIdempotentElem.comp_eq_right_iff`, `LinearMap.ker_le_of_iterateMapComap_eq_succ`, `Ideal.span_le`, `Module.End.isSemisimple_restrict_iff`, `FiniteArchimedeanClass.ball_lt_closedBall`, `Ideal.comap_bot_le_of_injective`, `LinearMap.iterateRange_succ`, `Ideal.dvd_iff_le`, `MvPolynomial.radical_le_vanishingIdeal_zeroLocus`, `integralClosure_le_span_dualBasis`, `Ideal.subset_union`, `Valuation.leSubmodule_v_le_of_mem`, `Algebra.Extension.Cotangent.span_eq_top_of_span_eq_ker`, `LinearMap.map_le_map_iff`, `Valuation.ltIdeal_le_leIdeal`, `Valuation.supp_quot_supp`, `orthogonalBilin_le`, `set_smul_le_iff`, `le_annihilator_iff`, `Subalgebra.toSubmodule_injective`, `Ideal.span_singleton_mul_le_span_singleton_mul`, `IdealFilter.IsUniform.colon_mem`, `span_sInf_le`, `groupHomology.boundaries₂_le_cycles₂`, `Ideal.le_comap_of_map_le`, `TensorProduct.eq_of_fg_of_subtype_eq'`, `IdealFilter.ringFilterBasis_sets`, `exists_fg_le_eq_rTensor_inclusion`, `Polynomial.disjoint_ker_aeval_of_isCoprime`, `le_of_le_smul_of_le_jacobson_bot`, `Module.End.isNilpotent_restrict_genEigenspace_nat`, `IsLocalization.AtPrime.coe_orderIsoOfPrime_apply_coe`, `IsFractionRing.coeSubmodule_strictMono`, `map_dualCoannihilator_le`, `Module.End.mem_invtSubmodule`, `LinearMap.injective_restrict_iff_disjoint`, `Ideal.mul_le_right`, `covBy_iff_quot_is_simple`, `RingEquiv.idealComapOrderIso_apply`, `topologicalClosure_minimal`, `MonoidAlgebra.Submodule.exists_isCompl`, `IsLocalRing.map_maximalIdeal_le`, `RingHom.eq_liftOfRightInverse`, `ImplicitFunctionData.isCompl_ker`, `WithIdealFilter.mem_nhds_zero_iff`, `LinearMap.ofIsCompl_symm`, `LieSubmodule.toSubmodule_le_toSubmodule`, `Ideal.le_span_singleton_mul_iff`, `LinearMap.le_ker_iff_comp_subtype_eq_zero`, `LinearEquiv.ker_le_fixedSubmodule_transvection`, `IdealFilter.isPFilter_gabrielComposition`, `PrimeSpectrum.equivSubtype_apply_coe`, `DividedPowers.IsDPMorphism.ideal_comp`, `AffineSubspace.direction_le`, `mulMap_one_right_eq`, `LieAlgebra.IsKilling.isCompl_ker_weight_span_coroot`, `minimalPrimes_eq_minimals`, `Ring.isField_iff_isSimpleOrder_ideal`, `idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors`, `restrictScalars_monotone`, `Ideal.exists_minimal_isPrimary_decomposition_of_isPrimary_decomposition`, `RootPairing.rootSpan_dualAnnihilator_le_ker_rootForm`, `ringKrullDim_eq_one_iff_of_isLocalRing_isDomain`, `Ideal.instCanLiftTwoSidedIdealCoeOrderHomAsIdealIsTwoSided`, `MvPolynomial.le_vanishingIdeal_zeroLocus`, `LinearMap.range_domRestrict_le_range`, `PrimeSpectrum.equivSubtype_symm_apply_asIdeal`, `Polynomial.degreeLT_mono`, `orthogonal_disjoint`, `PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical`, `Module.End.independent_genEigenspace`, `MvPolynomial.weightedHomogeneousSubmodule_mul`, `codisjoint_span_image_of_codisjoint`, `Valuation.leSubmodule_monotone`, `mul_one_div_le_one`, `Module.End.invtSubmodule.disjoint_mk_iff`, `LieModule.exists_genWeightSpace_zero_le_ker_of_isNoetherian`, `PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical`, `bot_lt_isotypicComponent`, `ValuationRing.le_total_ideal`, `map₂_le_map₂`, `RingHom.liftOfSurjective_comp`, `PrimeSpectrum.gc`, `exists_spanRank_le_and_le_height_of_le_height`, `PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff`, `RootPairing.range_polarization_domRestrict_le_span_coroot`, `ContinuousLinearMap.ker_le_ker_iff_range_le_range`, `Module.End.genEigenspace_mem_invtSubmodule`, `Ideal.subset_pointwise_smul_iff`, `Ideal.IsPrime.prod_le`, `Localization.localRingHom_mk'`, `Ideal.bot_lt_of_maximal`, `ClosedComplemented.exists_isClosed_isCompl`, `AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp`, `UnitAddTorus.mFourierSubalgebra_coe`, `Module.mem_support_iff_of_finite`, `exists_le_ker_of_lt_top`, `set_smul_mono_left`, `DividedPowers.SubDPIdeal.le_iff`, `LieSubalgebra.submodule_span_le_lieSpan`, `RootPairing.invtRootSubmodule.eq_top_iff`, `nilradical_le_prime`, `LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal`, `ValuationRing.TFAE`, `Module.Basis.flag_le_ker_dual`, `LieModule.lowerCentralSeries_one_inf_center_le_ker_traceForm`, `IsArtinian.eventuallyConst_of_isArtinian`, `le_linearEquiv_of_sSup_eq_top`, `FG.lTensor.directedSystem`, `Ideal.Filtration.smul_le`, `comap_mono`, `TwoSidedIdeal.instIsTwoSidedCoeOrderHomIdealAsIdeal`, `one_le_traceDual_one`, `comap_le_comap_smul`, `FractionalIdeal.coe_le_coe`, `FG.exists_rTensor_fg_inclusion_eq`, `Ideal.relNorm_mono`, `exists_le_isAssociatedPrime_of_isNoetherianRing`, `map_span_le`, `IsSemisimpleModule.jacobson_le_annihilator`, `AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_comap_ideal`, `Module.End.genEigenspace_directed`, `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply`, `LinearMap.ker_inf_lt_ker_inf_of_map_eq_of_lt`, `Subspace.dualAnnihilator_dualAnnihilator_eq`, `le_dualAnnihilator_dualCoannihilator`, `PrimeSpectrum.denseRange_comap_iff_minimalPrimes`, `iSup_map_single_le`, `Module.jacobson_lt_top`, `Ideal.map_under_le_under_map`, `RingHom.liftOfSurjective_comp_apply`, `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply`, `smul_mono`, `Ideal.spanNorm_mono`, `one_mem_div`, `Module.End.maxGenEigenspace_eq`, `Subalgebra.mulMap_toLinearMap`, `PointedCone.maxTensorProduct_mono`, `Module.End.eigenspace_restrict_le_eigenspace`, `dualAnnihilator_map_dualMap_le`, `le_comap_range_lTensor`, `Ideal.Quotient.factor_comp_apply`, `IsSimpleRing.tfae`, `Submonoid.adjoin_eq_span_of_eq_span`, `LieSubmodule.disjoint_toSubmodule`, `linearIndepOn_id_union_iff`, `TwoSidedIdeal.bot_asIdeal`, `le_of_localization_maximal`, `IsLocalRing.maximalIdeal_le_jacobson`, `Ring.jacobson_smul_top_le`, `Ideal.exists_pow_le_of_le_radical_of_fg`, `idealFactorsFunOfQuotHom_comp`, `Ring.DimensionLEOne.not_lt_lt`, `isSimpleModule_iff_isAtom`, `Module.End.maxGenEigenspace_eq_genEigenspace_finrank`, `Module.End.mapsTo_genEigenspace_of_comm`, `pow_smul_top_le`, `PrimeSpectrum.vanishingIdeal_anti_mono_iff`, `RootPairing.disjoint_corootSpan_ker_corootForm`, `TensorProduct.exists_finite_submodule_of_finite'`, `IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe`, `IsIsotypic.submodule_linearEquiv_fun`, `Ideal.subset_union_prime'`, `Ideal.jacobson_bot_polynomial_le_sInf_map_maximal`, `QuadraticMap.le_radical_iff`, `Ideal.quotientEquivAlgOfEq_coe_eq_factor`, `Representation.Coinvariants.le_comap_ker`, `le_self_mul_one_div`, `LieIdeal.toInvtRootSubmodule_mono`, `AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι`, `le_div_iff_mul_le`, `Valuation.ltIdeal_v_le_of_mem`, `Ideal.le_of_localization_maximal`, `Ideal.toTwoSided_asIdeal`, `Finsupp.iInf_ker_lapply_le_bot`, `LieSubmodule.isCompl_toSubmodule`, `LinearMap.iInf_ker_proj_le_iSup_range_single`, `map_lt_map_of_le_of_sup_lt_sup`, `coe_isComplEquivProj_symm_apply`, `Ideal.le_radical`, `Ideal.Filtration.Stable.bounded_difference`, `LieSubalgebra.toSubmodule_le_toSubmodule`, `PreValuationRing.iff_ideal_total`, `LinearMap.map_le_map_iff'`, `Ideal.comap_le_map_of_inverse`, `LieSubmodule.toSubmodule_orderEmbedding_apply`, `prod_mono`, `Algebra.span_le_adjoin`, `LinearMap.range_le_bot_iff`, `Ideal.factors_decreasing`, `span_le_of_C_coeff_mem`, `Ideal.prod_mono_left`, `DividedPowers.SubDPIdeal.isSubideal`, `Module.mapEvalEquiv_apply`, `Module.Basis.flag_mono`, `IsLocalRing.exists_maximalIdeal_pow_le_of_finite_quotient`, `torsionBySet_le_torsionBySet_of_subset`, `span_smul_le`, `Ideal.pointwise_smul_subset_iff`, `Finsupp.span_le_supported_biUnion_support`, `LinearPMap.le_graph_iff`, `le_orthogonal_orthogonal`, `Ideal.ramificationIdx_ne_one_iff`, `Module.Dual.isCompl_ker_of_disjoint_of_ne_bot`, `PrimeSpectrum.zeroLocus_subset_zeroLocus_iff`, `Ideal.instCovariantClassHSMulLe`, `Ideal.prod_le_inf`, `LinearMap.isIdempotentElem_iff_eq_isCompl_projection_range_ker`, `OrderIso.setIsotypicComponents_apply`, `LinearMap.isCompl_range_inl_inr`, `Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective`, `IsPrincipal.contentIdeal_le_span_iff_dvd`, `Subalgebra.mem_toSubmodule`, `rank_submodule_le_one_iff`, `Subalgebra.range_isScalarTower_toAlgHom`, `orthogonal_orthogonal_monotone`, `LinearMap.disjoint_inl_inr`, `prod_le_iff`, `Profinite.NobelingProof.GoodProducts.spanFin`, `dualAnnihilator_anti`, `Algebra.toSubmodule_bot`, `map_le_map_iff_of_injective`, `LinearIndependent.disjoint_span_image`, `ContinuousLinearMap.isClosed_genEigenspace`, `nilradical_le_jacobson`, `RootPairing.exist_set_root_not_disjoint_and_le_ker_coroot'_of_invtSubmodule`, `LinearMap.span_preimage_le`, `AlgebraicGeometry.IsAffineOpen.ideal_le_iff`, `finite_span_isCompactElement`, `codisjoint_map`, `AddValuation.self_le_supp_comap`, `topologicalClosure_map`, `LieSubmodule.coe_map_toEnd_le`, `LinearMap.BilinForm.isCompl_orthogonal_iff_disjoint`, `PowerSeries.map_constantCoeff_le_self_of_X_mem`, `Ideal.IsPrime.inf_le`, `AddSubmonoid.toNatSubmodule_toAddSubmonoid`, `RingHom.eq_liftOfSurjective`, `Ideal.Quotient.factorₐ_refl`, `Ideal.map_le_comap_of_inv_on`, `LinearMap.bot_lt_ker_of_det_eq_zero`, `ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal`, `Ideal.IsPrime.inf_le'`, `AddSubmonoid.toNatSubmodule_closure`, `Algebra.sInf_toSubmodule`, `LinearMap.isCompl_iSup_ker_pow_iInf_range_pow`, `groupCohomology.coboundaries₂_le_cocycles₂`, `Representation.invtSubmodule.coe_top`, `traceDual_top'`, `Ideal.sup_pow_add_le_pow_sup_pow`, `AddSubgroup.toZModSubmodule_toAddSubgroup`, `Ideal.relNorm_le_comap`, `inf_genEigenspace`, `smul_le_self_of_tower`, `span_le_restrictScalars`, `instPosMulStrictMonoIdeal`, `linearIndependent_sum`, `Ideal.pow_succ_lt_pow`, `Module.End.genEigenspace_top_eq_maxUnifEigenspaceIndex`, `exists_le_ker_of_notMem`, `IsDedekindDomain.primesOverEquivPrimesOver_inertiagDeg_eq`, `height_strictMono`, `Ideal.exists_ideal_over_prime_of_isIntegral_of_isPrime`, `LinearMap.IsProj.isCompl`, `Ring.DimensionLeOne.prime_le_prime_iff_eq`, `strictMono_comap_prod_map`, `Module.End.map_genEigenrange_le`, `mem_annihilator'`, `ValuationSubring.idealOfLE_le_of_le`, `Ideal.IsPrime.multiset_prod_map_le`, `le_traceDual_traceDual`, `Subalgebra.finrank_toSubmodule`, `PrimeSpectrum.sup_vanishingIdeal_le`, `TwoSidedIdeal.orderIsoIsTwoSided_symm_apply`, `IsLocalization.coeSubmodule_mono`, `restrictScalars_lt`, `IsNoetherian.wf`, `Finsupp.supported_comap_lmapDomain`, `Subalgebra.isIdempotentElem_toSubmodule`, `LinearMap.IsPerfectCompl.isCompl_right`, `Ideal.span_singleton_le_span_singleton`, `StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span`, `span_coeff_minpolyDiv`, `LieModule.exists_genWeightSpace_le_ker_of_isNoetherian`, `Module.Basis.addSubgroupOfClosure_repr_apply`, `toAddSubmonoid_strictMono`, `LinearMap.comap_leq_ker_subToSupQuotient`, `Ideal.le_of_dvd`, `Ideal.exists_le_prime_disjoint`, `disjoint_span_singleton`, `Ideal.disjoint_map_primeCompl_iff_comap_le`, `Module.End.invtSubmodule.isCompl_iff`, `Algebra.top_toSubmodule`, `nonPrincipals_zorn`, `one_le_one_div`, `Module.coe_length`, `Polynomial.map_under_lt_comap_of_weaklyQuasiFiniteAt`, `Ideal.exists_ideal_le_liesOver_of_le`, `Ideal.homogeneousHull.gc`, `LinearMap.finrank_genEigenspace_le`, `codisjoint_iff_exists_add_eq`, `Module.Basis.isMaxChain_range_flag`, `LieModule.weight_vector_multiplication`, `Module.End.iSup_genEigenspace_eq`, `IsSimpleModule.isAtom`, `Ideal.comp_quotientMap_eq_of_comp_eq`, `Polynomial.gaussNorm_lt_one_iff_contentIdeal_le`, `Module.Basis.flag_le_ker_coord`, `Function.Exact.exact_mapQ_iff`, `iSup_dualAnnihilator_le_iInf`, `Module.End.genEigenspace_inf_le_add`, `CliffordAlgebra.range_ι_le_evenOdd_one`, `Module.End.eigenspace_le_genEigenspace`, `AddSubgroup.toIntSubmodule_symm`, `PointedCone.minTensorProduct_mono`, `RootPairing.invtRootSubmodule.le_ker_coroot'`, `differentialIdeal_le_iff`, `HahnEmbedding.ArchimedeanStrata.disjoint_ball_stratum`, `singleton_span_isCompactElement`, `span_prod_le`, `LinearMap.disjoint_ker`, `PointedCone.lineal_eq_sSup`, `Ideal.nonPrincipals_zorn`, `DividedPowers.isDPMorphism_iff`, `restrictScalars_le`, `toAddSubgroup_mono`, `Module.length_eq_top_iff_infiniteDimensionalOrder`, `dualCoannihilator_anti`, `lt_of_le_of_finrank_lt_finrank`, `toSubalgebra_toSubmodule`, `IsDedekindDomain.primesOverEquivPrimesOver_apply`, `isCoatom_map_of_ker_le`, `traceForm_dualSubmodule_adjoin`, `FractionalIdeal.adjoinIntegral_coe`, `CliffordAlgebra.one_le_evenOdd_zero`, `AlgebraicGeometry.Scheme.IdealSheafData.le_ofIdeals_iff`, `Subalgebra.mul_toSubmodule`, `LinearPMap.HasCore.le_domain`, `Algebra.adjoin_nonUnitalSubalgebra_eq_span`, `ArchimedeanClass.ball_le_closedBall`, `atom_iff_nonzero_span`, `instMulLeftMono`, `PointedCone.maxTensorProduct_map_le`, `Ideal.le_pow_ramificationIdx`, `AlgebraicGeometry.AffineSpace.spec_le_iff`, `finrank_strictMono`, `LinearPMap.apply_comp_inclusion`, `LinearMap.ker_prod_ker_le_ker_coprod`, `pi_mono`, `mk_mem_projectivization_iff`, `Ideal.mul_mono`, `Module.jacobson_pi_le`, `Ideal.exists_ideal_comap_le_prime`, `jacobson_smul_lt_top`, `Finsupp.codisjoint_supported_supported`, `LinearPMap.sSup_apply`, `map_strictMono_of_injective`, `isNoetherian_iff_fg_wellFounded`, `PointedCone.dual_antitone`, `Algebra.Generators.sq_ker_comp_le_ker_compLocalizationAwayAlgHom`, `TensorAlgebra.ι_range_disjoint_one`, `isCoatom_comap_iff`, `FG.lTensor.directLimit_apply'`, `DividedPowers.le_equalizer_of_isDPMorphism`, `Ideal.IsPrime.le_of_pow_le`, `FG.rTensor.directLimit_apply`, `PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal`, `Ideal.le_map_of_comap_le_of_surjective`, `LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict`, `IsLocalization.bot_lt_comap_prime`, `Ideal.pi_le_pi_iff`, `rank_submodule_le_one_iff'`, `PrimeSpectrum.vanishingIdeal_anti_mono`, `ker_liftQ_eq_bot'`, `orderIsoMapComap_apply'`, `FractionalIdeal.isFractional_adjoin_integral`, `ExteriorAlgebra.ι_range_disjoint_one`, `Ideal.exists_radical_pow_le_of_fg`, `SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux`, `FractionalIdeal.coeIdeal_le_coeIdeal'`, `smul_le_smul`, `mem_integralClosure_iff_mem_fg`, `Subrepresentation.submoduleSubrepresentationOrderIso_apply`, `nonzero_span_atom`, `smul_top_inf_eq_smul_of_isSMulRegular_on_quot`, `ModuleCat.span_exact`, `toIntSubmodule_toAddSubgroup`, `RingOfIntegers.not_dvd_exponent_iff`, `toAddSubmonoid_mono`, `TwoSidedIdeal.asIdeal_jacobson`, `le_topologicalClosure`, `IsArtinian.monotone_stabilizes`, `traceDual_le_span_map_traceDual`, `orthogonal_gc`, `Ideal.mul_le`, `mem_colon_iff_le`, `TwoSidedIdeal.mem_fromIdeal`, `Module.length_submodule`, `factor_comp`, `linearIndepOn_iff_disjoint`, `Subalgebra.adjoin_eq_span_basis`, `isCoatom_comap_or_eq_top`, `comap_strictMono_of_surjective`, `Algebra.fg_trans`, `Module.mem_support_iff_exists_annihilator`, `ConvexCone.IsGenerating.top_le_span`, `LinearMap.annihilator_le_of_injective`, `AlgebraicGeometry.Scheme.Hom.ideal_ker_le`, `Ring.KrullDimLE.mem_minimalPrimes_iff_le_of_isPrime`, `Finsupp.isCompl_range_lmapDomain_span`, `TensorProduct.range_map_mono`, `AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono`, `isOrtho_iff_le`, `disjoint_ker_of_finrank_le`, `Ideal.pow_le_self`, `TensorProduct.exists_finite_submodule_right_of_setFinite'`, `MvPolynomial.restrictSupport_mono`, `TwoSidedIdeal.mem_asIdealOpposite`, `map_subtype_embedding_eq`, `le_linearEquiv_of_le_sSup`, `Module.jacobson_le_of_eq_bot`, `Ideal.exists_le_prime_notMem_of_isIdempotentElem`, `Subalgebra.mul_toSubmodule_le`, `LinearMap.iterateKer_coe`, `le_comap_mkQ`, `Ideal.IsIntegralClosure.comap_lt_comap`, `Module.Flat.instSubalgebraToSubmodule`, `Module.End.disjoint_genEigenspace`, `LinearMap.BilinForm.nondegenerate_restrict_iff_disjoint_ker`, `localized₀_le_localized₀_of_smul_le`, `disjoint_map`, `IsDedekindDomain.primesOverEquivPrimesOver_ramificationIdx_eq`, `Ideal.pow_right_mono`, `RootPairing.isCompl_rootSpan_ker_rootForm`, `Algebra.adjoin_toSubmodule_le`, `RootPairing.GeckConstruction.span_range_h_le_range_diagonal`, `comap_subtype_eq_top`, `inf_iSup_genEigenspace`, `IsLocalization.coeSubmodule_le_coeSubmodule`, `prodEquivOfIsCompl_symm_apply`, `isCompl_orthogonal_of_hasOrthogonalProjection`, `Ideal.IsPrime.multiset_prod_le`, `RootPairing.invtRootSubmodule.eq_bot_iff`, `toAddSubgroup_toIntSubmodule`, `Ideal.setOf_isPrincipal_wellFoundedOn_gt`, `Profinite.NobelingProof.GoodProducts.span`, `MvPolynomial.le_coeffsIn_pow`, `Module.End.genEigenspace_eq_genEigenspace_maxUnifEigenspaceIndex_of_le`, `le_comap_map`, `instCovariantClassHSMulLe`, `Module.End.isFinitelySemisimple_iff`, `closure_le`, `Ideal.le_mul_of_no_prime_factors`, `set_has_minimal_iff_artinian`, `Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent`, `vectorSpan_mono`, `Valuation.ltSubmodule_monotone`, `le_isotypicComponent`, `IsNoetherian.wellFoundedGT`, `differentialIdeal_le_fractionalIdeal_iff`, `Module.End.disjoint_iInf_maxGenEigenspace`, `WittVector.ker_map_le_ker_mk_comp_ghostComponent`, `Ideal.le_comap_of_ramificationIdx_ne_zero`, `comap_covBy_of_surjective`, `Ideal.comap_mono`, `Ideal.isSimpleOrder`, `AddSubgroup.toZModSubmodule_symm`, `prodComm_trans_prodEquivOfIsCompl`, `Ideal.span_singleton_le_iff_mem`, `AlgHom.IsArithFrobAt.le_comap`, `Ring.jacobson_le_of_isMaximal`, `Ideal.le_spanNorm_spanNorm`, `Module.Basis.flag_covBy`, `Ideal.idealProdEquiv_symm_apply`, `Ideal.homogeneousHull_mono`, `toConvexCone_le_toConvexCone`, `orderIsoMapComap_symm_apply'`, `orderIsoMapComapOfBijective_symm_apply`, `Module.End.mem_genEigenspace_top`, `InnerProductSpace.nullSubmodule_le_ker_toDualMap_left`, `RingEquiv.idealComapOrderIso_symm_apply`, `DirectSum.IsInternal.isCompl`, `Module.End.injOn_genEigenspace`, `Ideal.map_le_comap_of_inverse`, `MvPolynomial.le_zeroLocus_iff_le_vanishingIdeal`, `TensorProduct.exists_finite_submodule_left_of_finite'`, `Module.End.genEigenspace_one`, `smul_le_right`, `Ideal.Filtration.antitone`, `restrictScalars_mono`, `Ideal.prod_mono_right`, `TensorProduct.eq_zero_of_fg_of_subtype_eq_zero`, `Algebra.inf_toSubmodule`, `Ideal.IsRadical.radical_le_iff`, `AddSubmonoid.toNatSubmodule_symm`, `LinearMap.hasEigenvalue_zero_tfae`, `Ideal.comap_le_comap_iff_of_surjective`, `Algebra.adjoin_eq_span_of_subset`, `PointedCone.minTensorProduct_le_maxTensorProduct`, `annihilator_mono`, `Ideal.isCoprime_tfae`, `Ideal.map_radical_le`, `Ideal.quotEquivOfEq_eq_factor`, `IsAdicComplete.le_jacobson_bot`, `DividedPowers.IsSubDPIdeal.isSubideal`, `coe_isComplEquivProj_apply`, `Module.Basis.flag_lt_flag`, `Ideal.le_toIdeal_homogeneousHull`, `le_traceDual`, `Module.End.invtSubmodule.mk_eq_top_iff`, `Subalgebra.linearDisjoint_iff`, `Valuation.leIdeal_v_le_of_mem`, `IsFractionRing.coeSubmodule_le_coeSubmodule`, `goursatFst_prod_goursatSnd_le`, `IsSemisimpleModule.sSup_simples_le`, `comapMkQOrderEmbedding_eq`, `Ideal.exists_ideal_lt_liesOver_of_lt`, `PointedCone.gc_ofSubmodule_lineal`, `RingHom.liftOfRightInverse_comp_apply`, `IsCompl.projection_add_projection_eq_self`, `PrimeSpectrum.exists_primeSpectrum_prod_le`, `Module.FaithfullyFlat.range_le_ker_of_exact_rTensor`, `Ideal.IsPrime.mul_le`, `map_le_isotypicComponent`, `Ideal.prod_mono`, `le_span_singleton_iff`, `pi_liftQ_eq_liftQ_pi`, `span_le_span_iff`, `EuclideanGeometry.Sphere.direction_orthRadius_le_iff`, `FG.stabilizes_of_iSup_eq`, `map_strict_mono_of_ker_inf_eq`, `Subalgebra.map_toSubmodule`, `instCanonicallyOrderedAdd`, `DividedPowers.DPMorphism.ideal_comp`, `Ideal.pow_le_pow_right`, `orderIsoMapComap_symm_apply`, `Ideal.radical_eq_sInf`, `Ideal.IsLocal.le_jacobson`, `DividedPowers.isDPMorphism_def`, `IsSemisimpleModule.jacobson_le_ker`, `toLinearMap_prodEquivOfIsCompl_symm`, `LinearMap.IsIdempotentElem.comp_eq_left_iff`, `le_of_isLocalized_span`, `Ideal.eq_jacobson_iff_notMem`, `Ideal.exists_minimalPrimes_le`, `le_traceDual_comm`, `LieAlgebra.IsKilling.disjoint_ker_weight_corootSpace`, `span_lt_top_of_card_lt_finrank`, `span_range_natDegree_eq_adjoin`, `AddSubgroup.coe_toZModSubmodule`, `idealFactorsEquivOfQuotEquiv_symm`, `AffineSubspace.direction_iInf`, `Ideal.ker_le_comap`, `IsOrtho.le`, `AlgebraicClosure.le_maxIdeal`, `Finsupp.disjoint_supported_supported`, `IsPRadical.ker_le`, `disjoint_def'`, `Ideal.map_le_iff_le_comap`, `IsMinimalPrimaryDecomposition.minimal`, `rank_submodule_eq_one_iff`, `Ideal.homogeneousCore'_le`, `LinearMap.range_prod_le`, `orderIsoMapComap_apply`, `topologicalClosure_mono`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_descH_hom`, `Ideal.mem_normalizedFactors_iff`, `Polynomial.span_of_finite_le_degreeLT`, `IsIntegralClosure.range_le_span_dualBasis`, `Ideal.matrix_jacobson_le`, `toAddSubgroup_strictMono`, `Module.End.genEigenspace_eq_iSup_genEigenspace_nat`, `Ideal.Filtration.pow_smul_le_pow_smul`, `Module.Finite.instIsCoatomicSubmodule`, `AdicCompletion.pow_smul_top_le_ker_eval`, `Module.Basis.SmithNormalForm.le_ker_coord_of_notMem_range`, `top_le_span_range_iff_forall_exists_fun`, `map₂_le`, `Algebra.iInf_toSubmodule`, `isSimpleModule_iff`, `LinearMap.eventually_isCompl_ker_pow_range_pow`, `fourierSubalgebra_coe`, `Module.End.eventually_disjoint_ker_pow_range_pow`, `monotone_stabilizes_iff_noetherian`, `Polynomial.not_ker_le_map_C_of_surjective_of_weaklyQuasiFiniteAt`, `dualAnnihilator_gc`, `Profinite.NobelingProof.spanFinBasis.span`, `Subalgebra.finiteDimensional_toSubmodule`, `gc_map_comap`, `mem_projectivization_iff_submodule_le`, `instCovariantClassHSMulLe_1`, `sup_dualAnnihilator_le_inf`, `linearIndepOn_union_iff`, `Subalgebra.toSubmodule_toSubalgebra`, `inf_comap_le_comap_add`, `AddMonoidHom.coe_toIntLinearMap_ker`, `RingHom.pNilradical_le_ker_of_perfectRing`, `Algebra.adjoin_union_coe_submodule`, `Ideal.iInf_ker_le`, `Ideal.exists_maximal_not_isPrincipal`, `mulMap_one_left_eq`, `LinearMap.range_le_iff_comap`, `toAddSubmonoid_le`, `toIdeal_le_toIdeal_iff`, `PrimeSpectrum.coe_primesOverOrderIsoFiber_apply_asIdeal`, `Ideal.span_singleton_mul_left_mono`, `Module.AEval.mem_mapSubmodule_symm_apply`, `Valuation.ltSubmodule_le_leSubmodule`, `TensorProduct.exists_finite_submodule_of_setFinite'`, `Valuation.leIdeal_mono`, `Ideal.comap_lt_comap_of_integral_mem_sdiff`, `LinearMap.disjoint_ker_iff_injOn`, `LieSubalgebra.disjoint_toSubmodule`, `Module.Basis.flag_wcovBy`, `le_orthogonal_iff_le_orthogonal`, `AddSubgroup.coe_toIntSubmodule`, `torsionBySet_le_torsionBySet_pow`, `ProjectiveSpectrum.gc_ideal`, `fg_adjoin_of_finite`, `Ring.jacobson_le_of_eq_bot`, `lt_map_of_comap_lt_of_surjective`, `Ideal.jacobson_mono`, `Module.End.genEigenspace_le_genEigenspace_maxUnifEigenspaceIndex`, `comap_lt_of_lt_map_of_injective`, `AffineSubspace.direction_lt_of_nonempty`, `disjoint_span_singleton_of_notMem`, `ClosedSubmodule.toSubmodule_le_toSubmodule`, `Ideal.subset_union_prime_finite`, `le_of_map_mkQ_le_map_mkQ_of_le_jacobson_bot`, `Ideal.IsMaximal.out`, `RingHom.liftOfRightInverse_comp`, `Ideal.mul_le_left`, `Algebra.adjoin_eq_span`, `Ideal.IsPrime.pow_le_iff`, `ContinuousMap.ideal_gc`, `isCompl_iff_disjoint`, `Ideal.radical_le_radical_iff`, `ValuationRing.iff_ideal_total`, `CliffordAlgebra.even_toSubmodule`, `Module.End.genEigenspace_zero_nat`, `Algebra.toSubmodule_eq_top`, `Representation.le_comap_invariants`, `Algebra.Generators.disjoint_ker_toKaehler_of_linearIndependent`, `Finsupp.disjoint_lsingle_lsingle`, `TwoSidedIdeal.top_asIdeal`, `eq_iSup_inf_genEigenspace`, `IsLocalization.orderIsoOfPrime_symm_apply_coe`, `TwoSidedIdeal.mem_asIdeal`, `SubmodulesRingBasis.leftMul`, `Module.End.mem_invtSubmodule_symm_iff_le_map`, `Finsupp.codisjoint_supported_supported_iff`, `CategoryTheory.ShortComplex.moduleCat_exact_iff_ker_sub_range`, `Module.End.genEigenspace_top`, `Module.End.invtSubmodule.isCompl_mk_iff`, `Ideal.asIdeal_toTwoSided`, `RingHom.IsIntegral.quotient`, `Ideal.decomposition_erase_inf`, `LieAlgebra.InvariantForm.orthogonal_isCompl_toSubmodule`, `Ideal.span_singleton_mul_right_mono`, `Ideal.Filtration.Stable.exists_forall_le`, `toSubMulAction_strictMono`, `LinearMap.map_le_range`, `Ideal.isCompactElement_top`, `Subalgebra.pi_toSubsemiring`, `AffineMap.restrict.linear_aux`, `set_smul_le_of_le_le`, `RingFilterBasis.SubmodulesBasis.inter`, `Ring.KrullDimLE.mem_minimalPrimes_iff`, `Module.fgSystem.instIsDirectedOrderSubtypeSubmoduleFG`, `TensorProduct.eq_of_fg_of_subtype_eq`, `disjoint_iff_add_eq_zero`, `LinearMap.fixedSubmodule_comp_inf_fixedSubmodule_le`, `Ideal.Filtration.pow_smul_le`, `LinearMap.disjoint_ker_of_nondegenerate_restrict`, `Polynomial.sup_ker_aeval_le_ker_aeval_mul`, `Finsupp.supported_mono`, `SubmodulesRingBasis.inter`, `Ideal.IntegralClosure.comap_lt_comap`, `Ring.map_jacobson_le`, `colon_mono`, `LinearMap.range_comp_le_range`, `AffineSubspace.direction_affineSpan_pair_le_iff_exists_smul`, `Ideal.homogeneousCore'_eq_sSup`, `IsSemisimpleModule.isCoatomic_submodule`, `Module.End.invtSubmodule.codisjoint_iff`, `Valuation.ltSubmodule_v_le_of_mem`, `AddMonoidHom.coe_toIntLinearMap_map`, `ClosedComplemented.isCompl_complement`, `Ideal.ne_top_iff_exists_maximal`, `torsion_gc`, `Ideal.FG.isNilpotent_iff_le_nilradical`, `AffineSubspace.direction_sInf`, `FG.directedSystem`, `IsLocalRing.le_maximalIdeal`, `span_lt_of_subset_of_card_lt_finrank`, `IsOrtho.disjoint`, `LinearMap.IsIdempotentElem.isCompl`, `idealFactorsEquivOfQuotEquiv_is_dvd_iso`, `map_lt_map_iff_of_injective`, `neg_eq_self_iff_neg_le`, `IsAssociatedPrime.annihilator_le`, `coe_mapIic_apply`, `submoduleOf_eq_top`, `ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus`, `AddMonoidHom.coe_toIntLinearMap_range`, `Subalgebra.coe_pi`, `instIsOrderedRing`, `exists_isCompl`, `AddMonoidAlgebra.supported_mono`, `PointedCone.minTensorProduct_map_le`, `span_mono`, `Module.End.eigenspace_le_maxGenEigenspace`, `Ideal.le_comap_map`, `ModuleCat.disjoint_span_sum`, `CliffordAlgebra.evenOdd_mul_le`, `top_le_span_of_exact_of_retraction`, `Ideal.Filtration.mono`, `Module.End.generalized_eigenvec_disjoint_range_ker`, `le_div_iff`, `Module.Basis.flag_le_iff`, `Ideal.radical_le_jacobson`, `lt_sup_iff_notMem`, `Module.length_eq_coheight`, `Module.End.isNilpotent_restrict_sub_algebraMap`, `FG.lTensor.directLimit_apply`, `Module.fgSystem.equiv_comp_of`, `PointedCone.dual_le_dual`, `Subrepresentation.subrepresentationSubmoduleOrderIso_symm_apply`, `RootPairing.coe_bot`, `comap_le_comap_iff_of_surjective`, `map_mono`, `FractionalIdeal.instPosMulReflectLEIdealOfIsDedekindDomain`, `Ideal.comap_le_map_of_inv_on`, `Ideal.ringJacobson_le_jacobson`, `Module.End.mem_genEigenspace_one`, `smul_comap_le_comap_smul`, `MvPolynomial.zeroLocus_vanishingIdeal_galoisConnection`, `Ideal.pointwise_smul_le_pointwise_smul_iff`, `Subspace.map_le_dualAnnihilator_dualAnnihilator`, `Ideal.radical_iInf_le`, `factor_comp_apply`, `Ideal.le_comap_pow`, `Subalgebra.adjoin_eq_span_of_eq_span`, `AddMonoidHom.coe_toIntLinearMap_comap`, `Ideal.map_inf_le`, `AffineSubspace.sup_direction_le`, `Module.End.IsFinitelySemisimple.genEigenspace_eq_eigenspace`, `LinearMap.isCoatom_ker_of_surjective`, `idealFactorsFunOfQuotHom_id`, `LinearMap.BilinForm.le_orthogonal_orthogonal`, `Ideal.mul_le_inf`, `smul_le`, `instMulPosStrictMonoIdeal`, `LinearMap.ker_sup_ker_le_ker_comp_of_commute`, `instIsSimpleOrderIdeal`, `Module.Basis.flag_le_flag`, `Module.End.genEigenspace_div`, `Ideal.span_mono`, `mul_le`, `IsLocalRing.exists_maximalIdeal_pow_le_of_isArtinianRing_quotient`, `LinearMap.iterateRange_coe`, `le_orthogonalBilin_orthogonalBilin`, `Profinite.NobelingProof.GoodProducts.span_iff_products`, `isCompl_comap_subtype_of_isCompl_of_le`, `le_isotypicComponent_iff`, `IsDedekindDomain.primesOverEquivPrimesOver_symm_apply`, `Ideal.le_ker_atPrime_of_forall_exists_eq_mul`, `mk_le_mk`, `AlgebraicGeometry.Scheme.IdealSheafData.ideal_mono`, `LinearMap.ker_le_iff`, `Module.End.genEigenspace_eq_genEigenspace_finrank_of_le`, `map_strict_mono_or_ker_sup_lt_ker_sup`, `LinearMap.le_comap_isotypicComponent`, `Ideal.coe_piOrderIso_symm_apply`, `orthogonal_le`, `RootPairing.isIrreducible_iff_invtRootSubmodule`, `IsCompl.projection_eq_self_sub_projection`, `LieSubmodule.codisjoint_toSubmodule`, `smul_le_of_le_smul_of_le_jacobson_bot`, `Module.length_ne_top_iff_finiteDimensionalOrder`, `is_simple_module_of_finrank_eq_one`, `Module.Basis.flag_le_ker_coord_iff`, `eventuallyConst_of_isNoetherian`, `TwoSidedIdeal.gc`, `AddSubmonoid.coe_toNatSubmodule`, `Ideal.pow_right_strictAnti`, `exists_fg_le_subset_range_rTensor_inclusion`, `Module.End.genEigenspace_nat`, `Ideal.spanNorm_le_comap`, `exists_minimal_isPrimary_decomposition_of_isPrimary_decomposition`, `IdealFilter.IsGabriel.gabriel_closed`, `Ideal.span_singleton_lt_span_singleton`, `LinearMap.map_span_le`, `Ideal.homogeneousCore'_mono`, `Ideal.matrix_strictMono_of_nonempty`, `Polynomial.map_under_lt_comap_of_quasiFiniteAt`, `Subalgebra.coe_toSubmodule`, `LinearMap.le_eqLocus`, `Ideal.exists_le_maximal`, `FG.jacobson_smul_lt`, `LinearMap.range_dualMap_le_dualAnnihilator_ker`, `bot_lt_isotypicComponents`, `fg_iff_compact`, `ContinuousLinearMap.ker_prod_ker_le_ker_coprod`, `TensorProduct.exists_finite_submodule_right_of_finite'`, `AddSubgroup.toIntSubmodule_toAddSubgroup`, `Module.End.genEigenspace_le_maximal`, `LinearMap.BilinForm.orthogonal_le`, `Ideal.map_mono`, `monotone_stabilizes_iff_artinian`, `LinearMap.ker_le_ker_comp`, `RootPairing.coe_top`, `PrimeSpectrum.exist_mem_one_of_mem_maximal_ideal`, `Module.End.mem_genEigenspace_nat`, `le_comap_single_pi`, `FiniteArchimedeanClass.submodule_strictAnti`, `IsSimpleModule.toIsSimpleOrder`, `Subspace.dualAnnihilator_le_dualAnnihilator_iff`, `LinearMap.fixedSubmodule_inf_fixedSubmodule_le_comp`, `Module.Relations.Solution.span_relation_le_ker_π`, `Ideal.mul_mono_left`, `Module.Baer.ExtensionOf.le`, `Module.End.isSemisimple_iff'`, `Ideal.quotientMap_injective`, `wellFoundedGT`, `IsSMulRegular.isSMulRegular_on_quot_iff_smul_top_inf_eq_smul`, `mem_iSup_of_chain`, `Subalgebra.restrictScalars_toSubmodule`, `ContinuousLinearMap.HasLeftInverse.isCompl_complement`, `orderIsoMapComapOfBijective_apply`, `le_dualAnnihilator_iff_le_dualCoannihilator`, `Ideal.spanNorm_mul_spanNorm_le`, `Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisors`, `StarAlgebra.adjoin_eq_span`, `Polynomial.span_le_degreeLE_of_finite`, `AlgebraicGeometry.Scheme.IdealSheafData.ideal_ofIdeals_le`, `IsOrtho.ge`, `Module.length_quotient`, `Module.length_eq_height`, `DividedPowers.SubDPIdeal.sup_carrier_def`, `QuadraticMap.radical_le_ker_polarBilin`, `Ideal.map_le_of_le_comap`, `iSup_eq_toSubmodule_range`, `TensorProduct.exists_finite_submodule_left_of_setFinite'`, `Ideal.le_pow_of_le_ramificationIdx`, `isSimpleModule_iff_isCoatom`, `LinearMap.iSup_range_single_le_iInf_ker_proj`, `Ideal.pow_lt_self`, `exists_multiset_prod_cons_le_and_prod_not_le`, `Localization.le_comap_primeCompl_iff`, `Module.Basis.mem_toFlag`, `LinearPMap.le_graph_of_le`, `LinearMap.disjoint_single_single`, `LinearMap.eventually_codisjoint_ker_pow_range_pow`, `Ideal.IsMaximal.lt_top`, `MonoidAlgebra.supported_mono`, `RootPairing.corootSpan_dualAnnihilator_le_ker_rootForm`, `Module.End.isNilpotent_restrict_genEigenspace_top`, `Ideal.exists_comap_eq_of_mem_minimalPrimes`, `Ideal.subset_union_prime`, `idealFactorsFunOfQuotHom_coe_coe`, `orthogonal_le_orthogonal_iff`, `Ideal.exists_ideal_over_prime_of_isIntegral`, `orthogonal_le_iff_orthogonal_le`, `span_singleton_le_iff_mem`, `map_comap_le`, `IsIntegral.fg_adjoin_singleton`, `LinearMap.isCompl_span_singleton_orthogonal`, `FiniteArchimedeanClass.ball_strictAnti`, `Ideal.multiset_prod_le_inf`, `PrimeSpectrum.coe_primesOverOrderIsoFiber_symm_apply_coe`, `torsionBy_le_torsionBy_of_dvd`, `TwoSidedIdeal.orderIsoIsTwoSided_apply_coe`, `TwoSidedIdeal.asIdeal_matrix`, `starProjection_le_starProjection_iff`, `IsSemisimpleModule.eq_bot_or_exists_simple_le`, `disjoint_def`, `Ideal.le_comap_sup`, `IsLocalization.coeSubmodule_strictMono`, `set_has_maximal_iff_noetherian`, `Module.End.ker_pow_le_ker_pow_finrank`, `span_preimage_le`, `groupCohomology.coboundaries₁_le_cocycles₁`, `Polynomial.contentIdeal_le_span_content`, `WithIdealFilter.mem_nhds_iff`, `range_ker_disjoint`, `le_traceDual_mul_iff`, `Ideal.homogeneousCore_eq_sSup`, `Finsupp.lsingle_range_le_ker_lapply`, `AddSubgroup.relIndex_eq_natAbs_det`, `MvPolynomial.range_mapAlgHom`, `PrimeSpectrum.asIdeal_lt_asIdeal`, `span_singleton_smul_le`, `IsArtinian.set_has_minimal`, `Ideal.comap_fiberIsoOfBijectiveResidueField_symm`, `isCompl_span_singleton_of_isCoatom_of_notMem`, `Ideal.Quotient.factorₐ_comp`, `LinearMap.isCompl_of_proj`, `le_map_of_comap_le_of_surjective`, `Valuation.self_le_supp_comap`, `Module.Basis.toFlag_carrier`, `instIsAtomisticSubmodule`, `LinearMap.annihilator_le_of_surjective`, `Ideal.exists_spanRank_eq_and_height_eq`, `span_monotone`, `finset_span_isCompactElement`, `Ideal.homogeneousCore_mono`, `DividedPowers.SubDPIdeal.lt_iff`, `map_le_iff_le_comap`, `Module.End.genEigenspace_restrict_eq_top`, `Module.map_jacobson_le`, `Ideal.Quotient.factor_comp`, `isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule`, `one_le`, `map_covBy_of_injective`, `FiniteDimensional.subalgebra_toSubmodule`, `AddSubgroup.toIntSubmodule_closure`, `LinearMap.IsSymmetricProjection.le_iff_range_le_range`, `OrderIso.setIsotypicComponents_symm_apply`, `AddSubgroup.mem_toZModSubmodule`, `IsLocalRing.le_maximalIdeal_of_isPrime`, `HomogeneousIdeal.toIdeal_irrelevant_le`, `RootPairing.disjoint_rootSpan_ker_rootForm`, `Module.End.hasGenEigenvector_iff`, `Ideal.map_comap_le`, `Ideal.matrix_monotone`, `CharP.quotient_iff_le_ker_natCast`, `Module.End.mem_genEigenspace_zero`, `Valuation.ltIdeal_mono`, `restrictScalarsEmbedding_apply`, `Polynomial.contentIdeal_mul_le_mul_contentIdeal`, `Polynomial.degreeLE_mono`, `le_prod_iff`, `isPRadical_iff`, `MvPolynomial.homogeneousSubmodule_mul`, `Module.fgSystem.instDirectedSystemSubtypeSubmoduleFGMemValCoeLinearMapId`, `isIntegral_quotientMap_iff`, `IsLocalFrameOn.generating`, `ModuleCat.span_rightExact`, `Ideal.mul_mono_right`, `Representation.invtSubmodule.coe_bot`, `LinearMap.disjoint_ker'`, `IsLocalization.orderIsoOfPrime_apply_coe`, `map₂_le_map₂_left`, `Ideal.IsHomogeneous.radical_eq`, `Polynomial.contentIdeal_le_contentIdeal_of_dvd`, `Algebra.HasGoingDown.exists_ideal_le_liesOver_of_lt`, `Module.End.genEigenspace_le_genEigenspace_finrank`, `instIsOrderedAddMonoid`, `NumberField.mixedEmbedding.disjoint_span_commMap_ker`, `Ring.le_comap_jacobson`, `Ideal.span_prod_le`, `Module.End.mem_genEigenspace`, `Ideal.dvdNotUnit_iff_lt`, `Subalgebra.pointwise_smul_toSubmodule`, `instMulRightMono`, `groupHomology.boundaries₁_le_cycles₁`, `Ideal.Quotient.factor_eq`, `LinearMap.ker_le_range_iff`, `isSMulRegular_on_quot_iff_lsmul_comap_le`, `Ideal.isCoprime_iff_codisjoint`, `LinearMap.range_le_ker_iff`, `IdealFilter.isTorsionQuot_def`, `DividedPowers.dpow_span_isSubideal`, `IsPRadical.ker_le'`, `Subalgebra.prod_toSubmodule`, `smul_top_le_comap_smul_top`, `Module.End.invtSubmodule.mk_eq_bot_iff`, `smul_le_span`, `Ideal.toIdeal_homogeneousCore_le`, `closedComplemented_iff_isClosed_exists_isClosed_isCompl`, `Ideal.IsPrime.radical_le_iff`, `RootPairing.isCompl_corootSpan_ker_corootForm`, `Ideal.le_comap_mul`, `Module.End.genEigenspace_zero`, `Finsupp.disjoint_supported_supported_iff`, `isFullyInvariant_iff_le_imp_isotypicComponent_le`, `map_add_le`, `LieSubmodule.submodule_span_le_lieSpan`, `covBy_span_singleton_sup`, `IsLocalization.comap_le_comap_iff`, `map_le_localized₀`, `LinearIndependent.isCompl_span_image`, `Ideal.homogeneousHull_eq_sInf`, `toAddSubgroup_le`, `Ideal.span_singleton_absNorm_le`, `LinearMap.le_ker_iff_map`, `wcovBy_span_singleton_sup`, `SubmodulesBasis.inter`, `Module.mem_support_iff_of_span_eq_top`, `LinearMap.iterateMapComap_le_succ`, `IsLocalRing.finite_quotient_iff`, `Ideal.leadingCoeffNth_mono`, `decomposition_erase_inf`, `Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op`, `Module.End.mem_invtSubmodule_iff_map_le`, `Module.End.invtSubmodule.codisjoint_mk_iff`, `AlgebraicGeometry.Scheme.IdealSheafData.strictMono_ideal`, `MvPolynomial.coeffsIn_le`, `Finsupp.lmapDomain_disjoint_ker`, `Module.Basis.isChain_range_flag`, `height_lt_top`, `FG.rTensor.directLimit_apply'`, `CliffordAlgebra.evenOdd_isCompl`, `PointedCone.lineal_le`, `Module.End.genEigenspace_le_smul`, `IsLocalRing.local_hom_TFAE`, `isNoetherian_iff`, `HenselianRing.jac`, `Subrepresentation.subrepresentationSubmoduleOrderIso_apply`, `AlgHom.equalizer_toSubmodule`, `Ideal.coe_piOrderIso_apply`, `Polynomial.mul_contentIdeal_le_radical_contentIdeal_mul`, `set_smul_le`, `isAtom_iff_finrank_eq_one`, `Module.AEval.mem_mapSubmodule_apply`, `basis_of_pid_aux`, `Module.Basis.flag_strictMono`, `DirectSum.isInternal_submodule_iff_isCompl`, `Module.le_comap_jacobson`, `Subalgebra.pi_toSubmodule`, `AddValuation.supp_quot_supp`, `localized'_le_localized'_of_smul_le`, `PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain`, `Module.End.genEigenspace_restrict`, `le_comap_range_rTensor`, `le_traceDual_iff_map_le_one`, `pNilradical_le_nilradical`, `ArchimedeanClass.ball_antitone`, `Ring.jacobson_lt_top`, `mem_invtSubmodule_reflection_iff`, `disjoint_torsionBySet_ideal`, `LieModule.range_traceForm_le_span_weight`, `lieModule_lcs_map_le`, `LinearMap.IsPerfectCompl.isCompl_left`, `Module.annihilator_le_of_mem_support`, `span_le`, `instCovariantClassSetHSMulLe`, `LinearMap.BilinForm.le_flip_dualSubmodule`, `LinearMap.ker_le_ker_of_range`, `LinearPMap.domain_mono`, `AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty`, `Module.Basis.addSubgroupOfClosure_apply`, `Module.End.invtSubmodule.disjoint_iff`, `Module.End.isSemisimple_iff`, `Subspace.isCompl_dualAnnihilator`, `IsDedekindDomain.HeightOneSpectrum.ideal_ne_top_iff_exists`, `Module.End.pos_finrank_genEigenspace_of_hasEigenvalue`, `LinearMap.BilinForm.isCompl_orthogonal_of_restrict_nondegenerate`, `IsLocalRing.map_maximalIdeal_lt_top`, `NumberField.Ideal.primesOverSpanEquivMonicFactorsMod_symm_apply`, `Ideal.comap_lt_comap_of_root_mem_sdiff`, `Ideal.le_jacobson`, `PointedCone.dual_anti`, `RootPairing.range_polarizationIn_le_span_coroot`, `Ideal.gc_map_comap`, `map_inf_le`, `PrimeSpectrum.le_vanishingIdeal_zeroLocus`, `map_subtype_le`, `le_dualCoannihilator_dualAnnihilator`, `Subalgebra.rank_toSubmodule`, `lt_top_of_finrank_lt_finrank`, `Subalgebra.fg_bot_toSubmodule`, `Projectivization.Subspace.mem_submodule_iff`, `TwoSidedIdeal.coe_asIdeal`, `LieModule.coe_lowerCentralSeries_ideal_le`, `disjoint_iff_comap_eq_bot`, `LinearMap.range_add_le`, `MvPolynomial.restrictTotalDegree_le_restrictDegree`, `ContinuousMap.idealOpensGI_choice`, `Ideal.comap_jacobson`, `TensorProduct.range_mapIncl_mono`, `AlgebraicGeometry.Scheme.IdealSheafData.le_def`, `Ideal.instIsCoatomic`, `isGLB_sInf`, `InnerProductSpace.nullSubmodule_le_ker_toDualMap_right`, `neg_le_neg`, `Ideal.comap_le_iff_le_map`, `IdealFilter.isUniform_iff_exists_ringFilterBasis`, `toSubMulAction_mono`, `Subrepresentation.submoduleSubrepresentationOrderIso_symm_apply`, `Ideal.map_eq_bot_iff_le_ker`, `Ideal.le_comap_pow_ramificationIdx`, `smul_mono_left`, `LinearMap.ofIsCompl_eq_add`, `IsCoprime.codisjoint`, `Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top`, `coe_iSup_of_chain`, `Module.mapEvalEquiv_symm_apply`, `surjOn_iff_le_map`, `map₂_le_map₂_right`, `map_le_smul_top`, `Ideal.homogeneousCore.gc`, `Ideal.span_singleton_mul_le_iff`, `LinearMap.ker_le_comap`, `exists_sub_one_mem_and_smul_le_of_fg_of_le_sup`, `Ideal.le_colon`, `LinearMap.range_lt_top_of_det_eq_zero`, `LinearMap.comap_le_comap_iff`, `FractionalIdeal.coeIdeal_le_coeIdeal`, `LinearMap.BilinForm.isCompl_span_singleton_orthogonal`, `comap_subtype_le_iff`, `ZModModule.exists_submodule_subset_card_le`, `disjoint_span_singleton''`, `PrimeSpectrum.asIdeal_le_asIdeal`, `Polynomial.isIntegral_isLocalization_polynomial_quotient`, `le_span_of_map_mkQ_le_map_mkQ_span_of_le_jacobson_bot`, `le_comap_pow_of_le_comap`, `neg_le`, `Ideal.isMaximal_def`, `Algebra.ZariskisMainProperty.exists_fg_and_exists_notMem_and_awayMap_bijective`
`module'` 📖	CompOp	56 math math: `Ideal.selfBasis_def`, `derivationToSquareZeroOfLift_apply`, `Ideal.Cotangent.lift_comp_toCotangent`, `Ideal.exists_smith_normal_form`, `injective_inclusionSpan`, `TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_symm_apply_mk_tmul`, `Ideal.powQuotSuccInclusion_apply_coe`, `isLocalizedModule`, `Algebra.Extension.Hom.sub_one_tmul`, `Ideal.quotientToQuotientRangePowQuotSucc_mk`, `AlgHom.toKerIsLocalization_apply`, `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`, `inclusionSpan_apply_coe`, `Ideal.Cotangent.lift_toCotangent`, `Module.instFiniteSubtypeMemIdealOfIsNoetherian`, `RingHom.toKerIsLocalization_apply`, `Ideal.quotientToQuotientRangePowQuotSucc_surjective`, `Ideal.instIsTorsionFreeSubtypeMemSubmodule`, `Ideal.basisSpanSingleton_apply`, `derivationToSquareZeroEquivLift_symm_apply_apply_coe`, `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`, `diffToIdealOfQuotientCompEq_apply`, `Algebra.Extension.Hom.mapKer_apply_coe`, `Ideal.cotangentEquivIdeal_apply`, `Basis.mem_ideal_iff'`, `Ideal.quotientToQuotientRangePowQuotSucc_injective`, `Algebra.idealMap_isLocalizedModule`, `Ideal.constr_basisSpanSingleton`, `IsSemisimpleModule.exists_end_algEquiv`, `liftOfDerivationToSquareZero_apply`, `Algebra.Extension.Hom.subToKer_apply_coe`, `Ideal.rank_pow_quot_aux`, `Ideal.rank_pow_quot`, `Algebra.Extension.tensorCotangentInvFun_smul_mk`, `Basis.mem_ideal_iff`, `Ideal.powQuotSuccInclusion_injective`, `IsSemisimpleModule.exists_end_algEquiv_pi_matrix_end`, `derivationOfSectionOfKerSqZero_apply_coe`, `TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_apply_tmul`, `derivationToSquareZeroEquivLift_apply_coe_apply`, `liftOfDerivationToSquareZero_mk_apply'`, `Algebra.Extension.Hom.sub_tmul`, `instFreeSubtypeMemIdealOfFiniteOfIsTorsionFree`, `Ideal.cotangentEquivIdeal_symm_apply`, `Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule_1`, `AlgHom.toKerIsLocalization_isLocalizedModule`, `Algebra.idealMap_mul`, `TensorProduct.AlgebraTensorModule.range_lTensor_idealMap`, `finrank_quotient`, `toLocalized'_apply_coe`, `Ideal.quotientToQuotientRangePowQuotSuccAux_mk`, `RingHom.toKerIsLocalization_isLocalizedModule`, `Algebra.idealMap_apply_coe`, `Module.Finite.of_isComplemented_codomain`, `span_range_inclusionSpan`, `Ideal.Cotangent.lift_surjective_iff`
`ofClass` 📖	CompOp	1 math math: `coe_ofClass`
`ofLinearComb` 📖	CompOp	1 math math: `coe_ofLinearComb`
`setLike` 📖	CompOp	3677 math math: `DirectSum.IsInternal.ofBijective_coeLinearMap_same`, `Ideal.mem_bot`, `IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal`, `exists_lieSubmodule_coe_eq_iff`, `TensorProduct.mapInclIsometry_apply`, `Polynomial.coeff_prod_mem_ideal_pow_tsub`, `coe_sum`, `ZLattice.summable_norm_zpow`, `finrank_sup_add_finrank_inf_eq`, `Ideal.toCotangent_to_quotient_square`, `LinearMap.lTensor_ker_subtype_tensorKerEquiv_symm`, `DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply`, `instIsScalarTowerQuotientIdealSubtypeMemTorsionBySetCoe`, `KaehlerDifferential.kerCotangentToTensor_toCotangent`, `PrimeSpectrum.closure_range_comap`, `groupCohomology.instEpiModuleCatH2π`, `HasStrictFDerivAt.map_implicitFunctionOfComplemented_eq`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inr`, `NumberField.mixedEmbedding.fundamentalDomain_integerLattice`, `groupHomology.π_comp_H2Iso_hom_assoc`, `IsAdjoinRoot.repr_add_sub_repr_add_repr_mem_span`, `Rep.invariantsAdjunction_homEquiv_symm_apply_hom`, `LinearMap.eqOn_span'`, `Affine.Simplex.altitudeFoot_restrict`, `TensorProduct.forall_vanishesTrivially_iff_forall_fg_rTensor_injective`, `LieDerivation.IsKilling.ad_mem_ker_killingForm_ad_range_of_mem_orthogonal`, `Ideal.mem_image_of_mem_map_of_surjective`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff`, `rTensorOne_symm_apply`, `AddMonoidAlgebra.supportedEquivFinsupp_apply_apply`, `iSupIndep.linearEquiv_symm_apply`, `LinearMap.IsProj.map_mem`, `mem_colon_span_singleton`, `DividedPowers.dpow_mem`, `topEquiv_symm_apply_coe`, `Ideal.Filtration.submodule_closure_single`, `Algebra.exists_unramified_of_isUnramifiedAt`, `iSupIndep.dfinsupp_lsum_injective`, `LinearMap.BilinForm.dualSubmoduleParing_spec`, `FG.rTensor.directedSystem`, `AffineSubspace.coe_vsub`, `quotientRel_def`, `disjoint_span_singleton'`, `Algebra.idealMap_eq_ofEq_comp_toLocalized₀`, `Quotient.eq`, `Module.End.pow_apply_mem_of_forall_mem`, `continuousOn_stereoToFun`, `coe_continuous_linearProjOfClosedCompl'`, `Ideal.span_le`, `groupHomology.mapCycles₂_comp_assoc`, `Module.End.isSemisimple_restrict_iff`, `ContinuousLinearMap.IsIdempotentElem.isClosed_range`, `Irreducible.maximalIdeal_eq_setOf_le_v_coe`, `RootPairing.posRootForm_posForm_apply_apply`, `Ideal.selfBasis_def`, `Ideal.Filtration.submodule_eq_span_le_iff_stable_ge`, `Module.Basis.mem_submodule_iff`, `LinearMap.BilinForm.dualSubmoduleToDual_apply_apply`, `coe_add`, `AlgebraicGeometry.Scheme.zeroLocus_iInf`, `supIndep_torsionBySet_ideal`, `OrthonormalBasis.orthogonalProjection_eq_sum`, `exists_finset_of_mem_iSup`, `IsIsotypic.linearEquiv_fun`, `LinearPMap.mem_domain_sSup_iff`, `finrank_span_finset_eq_card`, `derivationToSquareZeroOfLift_apply`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_fst`, `finrank_span_set_eq_card`, `linearProjOfIsCompl_apply_left`, `Ideal.subset_union`, `groupCohomology.toCocycles_comp_isoCocycles₁_hom`, `ValuationSubring.valuation_lt_one_iff`, `subtype_comp_inclusion`, `Algebra.Extension.Cotangent.span_eq_top_of_span_eq_ker`, `MvPolynomial.DirectSum.coeLinearMap_eq_dfinsuppSum`, `IsSemisimpleModule.exists_submodule_linearEquiv_quotient`, `Module.mem_annihilator`, `mem_of_span_eq_top_of_smul_pow_mem`, `Affine.Simplex.orthogonalProjectionSpan_map`, `Ideal.Cotangent.lift_comp_toCotangent`, `LinearMap.BilinForm.dualSubmoduleToDual_injective`, `EuclideanGeometry.vsub_orthogonalProjection_mem_direction_orthogonal`, `ExteriorAlgebra.GradedAlgebra.liftι_eq`, `groupCohomology.isoCocycles₁_hom_comp_i_apply`, `Ideal.tensorCotangentEquiv_tmul`, `Ideal.pow_multiset_sum_mem_span_pow`, `isScalarTower`, `Module.support_quotient`, `PadicInt.lift_sub_val_mem_span`, `IsSemisimpleModule.submodule`, `smul_def`, `AffineMap.restrict.bijective`, `Finsupp.mem_supported_support`, `isConnected_compl_of_one_lt_codim`, `nonzero_mem_of_bot_lt`, `bot_coe`, `Polynomial.mem_image_comap_C_basicOpen`, `groupCohomology.mem_cocycles₂_def`, `mem_toAffineSubspace`, `RootPairing.root'In_corootSpanMem_eq_pairingIn`, `set_smul_le_iff`, `RootPairing.rootForm_restrict_nondegenerate_of_ordered`, `Polynomial.degreeLT.instFreeSubtypeMemSubmodule`, `algebraMap_mem`, `linearIndependent_option`, `span_subset_span`, `finrank_add_finrank_le_of_disjoint`, `ValuationSubring.image_maximalIdeal`, `GradedAlgHom.toGradedRingHom_ofClass`, `ZLattice.summable_norm_sub_inv_pow`, `tendsto_tsum_div_pow_atTop_integral`, `NumberField.canonicalEmbedding.mem_span_latticeBasis`, `orthogonalDecomposition_symm_apply`, `Ideal.finrank_eq_finrank`, `Algebra.PreSubmersivePresentation.cotangentComplexAux_apply`, `PeriodPair.hasSumLocallyUniformly_weierstrassP`, `vadd_def`, `LinearMap.subtype_comp_codRestrict`, `Ideal.span_singleton_mul_le_span_singleton_mul`, `IsHilbertSum.mkInternal`, `exteriorPower.basis_apply`, `Ideal.mem_span_iff_exists_isHomogeneous`, `LinearMap.IsProj.subtype_comp_codRestrict`, `NumberField.Units.logEmbedding_fundSystem`, `Ideal.exists_smith_normal_form`, `toIsOrderedAddMonoid`, `LinearMap.subtype_comp_restrict`, `linearIndepOn_iff_notMem_span`, `mem_ideal_smul_span_iff_exists_sum`, `LinearMap.IsProj.codRestrict_apply_cod`, `LinearMap.BilinForm.isSkewAdjoint_bracket`, `Module.Basis.coe_span_apply`, `RootPairing.instIsRootSystemSubtypeMemSubmoduleSpanRangeCoeEmbeddingRootCorootRestrictScalars'`, `Finsupp.restrictDom_apply`, `span_span_of_tower`, `Valuation.mem_maximalIdeal_iff`, `rank_le_card_isVisible`, `RootPairing.RootPositiveForm.zero_lt_posForm_apply_root`, `exteriorPower.linearMap_ext_iff`, `PrimeSpectrum.localization_specComap_range`, `vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints`, `Ideal.toCotangent_eq`, `Ideal.subset_span`, `RootPairing.finrank_corootSpan_eq'`, `mulMap_tmul`, `AddMonoidAlgebra.of'_mem_span`, `PeriodPair.coeff_weierstrassPExceptSeries`, `Valuation.mem_leSubmodule_iff`, `subsingleton_iff_eq_bot`, `TensorProduct.eq_of_fg_of_subtype_eq'`, `comm_trans_lTensorOne`, `isNoetherian_submodule_right`, `Polynomial.toMatrix_sylvesterMap'`, `Valuation.mem_supp_iff`, `IdealFilter.ringFilterBasis_sets`, `LinearMap.injOn_iff_surjOn`, `iUnion_ssubset_of_forall_ne_top_of_card_lt`, `Ideal.single_mem_pi`, `exists_fg_le_eq_rTensor_inclusion`, `Ideal.mem_span_singleton'`, `LinearMap.IsRefl.domRestrict`, `Ideal.mem_ofPolynomial`, `Affine.Simplex.finiteDimensional_direction_altitude`, `TensorProduct.range_mapIncl`, `LinearMap.BilinForm.ker_restrict_eq_of_codisjoint`, `IsIsotypic.isotypicComponents`, `toLinearEquiv_orthogonalDecomposition_symm`, `NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule`, `IsAdjoinRoot.mem_ker_map`, `isInternal_prime_power_torsion_of_pid`, `Module.End.isNilpotent_restrict_genEigenspace_nat`, `IsLocalization.AtPrime.coe_orderIsoOfPrime_apply_coe`, `Affine.Simplex.orthogonalProjection_vadd_smul_vsub_orthogonalProjection`, `mem_localized'`, `smoothSheafCommRing.nonunits_stalk`, `RootPairing.EmbeddedG2.mem_span_of_mem_allRoots`, `LinearMap.injective_restrict_iff_disjoint`, `PeriodPair.derivWeierstrassPExcept_sub`, `IsLocalization.smul_mem_finsetIntegerMultiple_span`, `mem_reesAlgebra_iff_support`, `Ideal.isHomogeneous_iff_subset_iInter`, `covBy_iff_quot_is_simple`, `lift_rank_map_le`, `PrimeSpectrum.zeroLocus_nilradical`, `MeasureTheory.norm_condExpL2_le`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_apply`, `ZSpan.floor_eq_self_of_mem`, `Ideal.exists_mem_span_singleton_map_residueField_eq`, `NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_completeFamily_of_ne`, `instStrictConvexSpace`, `AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk`, `tensorEquivSpan_apply_tmul`, `groupCohomology.toCocycles_comp_isoCocycles₂_hom_apply`, `linearMap_eq_zero_iff_of_eq_span`, `MvPowerSeries.mem_hasEvalIdeal_iff`, `Rep.resCoindToHom_hom_apply_coe`, `coe_mapAlgEquiv_symm_apply`, `ContinuousLinearMap.ker_codRestrict`, `Matrix.coeff_charpoly_mem_ideal_pow`, `PrimeSpectrum.zeroLocus_eq_univ_iff`, `Ideal.exists_nonzero_mem_of_ne_bot`, `StrongDual.polarSubmodule_eq_setOf`, `Algebra.Extension.H1Cotangent.equiv_apply`, `groupCohomology.cocyclesIso₀_hom_comp_f`, `PeriodPair.ω₁_div_two_notMem_lattice`, `LinearMap.IsProj.mem_iff_map_id`, `Module.Basis.mem_span`, `Finsupp.supportedEquivFinsupp_symm_single`, `vsub_mem_vectorSpan`, `LinearDisjoint.not_linearIndependent_pair_of_flat_left`, `PointedCone.coe_ofSubmodule`, `Rep.coindToInd_of_support_subset_orbit`, `mem_sup'`, `groupCohomology.H1π_comp_map_assoc`, `addSubgroupClass`, `Ideal.le_span_singleton_mul_iff`, `re_inner_starProjection_eq_normSq`, `mem_pairSelfAdjointMatricesSubmodule`, `groupHomology.mapCycles₁_comp_apply`, `DirectSum.decomposeAlgEquiv_symm_apply`, `Ring.coe_jacobson_quotient`, `Affine.Simplex.restrict_reindex`, `Ideal.mem_leadingCoeffNth_zero`, `finiteDimensional_vectorSpan_range`, `LinearMap.le_ker_iff_comp_subtype_eq_zero`, `MeasureTheory.inner_condExpL2_left_eq_right`, `LieAlgebra.LoopAlgebra.twoCochainOfBilinear_apply_apply`, `MonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_apply`, `RootPairing.restrictScalars_coe_coroot`, `groupCohomology.π_comp_H0Iso_hom`, `exists_finrank_eq_sigPos_and_posDef`, `addSubmonoidClass`, `ChevalleyThm.chevalley_polynomialC`, `Affine.Simplex.faceOppositeCentroid_restrict`, `sigNeg_isGreatest`, `Matrix.isSkewAdjoint_bracket`, `GradedTensorProduct.tmul_zero_coe_mul_coe_tmul`, `linearEquiv_of_le_sSup`, `linearProjOfIsCompl_apply_right`, `groupCohomology.π_comp_H1Iso_hom_assoc`, `prodEquivOfIsCompl_symm_apply_snd_eq_zero`, `exteriorPower.coe_basis`, `LinearMap.tensorKer_tmul`, `LinearMap.IsSymmetric.diagonalization_apply_self_apply`, `dense_iff_topologicalClosure_eq_top`, `ProjectiveSpectrum.zeroLocus_mul_ideal`, `finrank_vectorSpan_image_finset_le`, `Ideal.coeff_zero_mem_comap_of_root_mem_of_eval_mem`, `EuclideanGeometry.orthogonalProjection_congr`, `Subspace.dualLift_of_subtype`, `mem_colon'`, `Ideal.map_pi`, `mulMap_one_right_eq`, `LinearIndepOn.notMem_span`, `PadicInt.appr_spec`, `exteriorPower.ιMulti_family_linearIndependent_field`, `vsub_mem_vectorSpan_pair`, `ContinuousLinearEquiv.snd_equivOfRightInverse`, `AffineIsometryEquiv.ofTop_apply`, `mem_of_isSMulRegular_quotient_of_smul_mem`, `span_setOf_mem_eq_top`, `IsLocalization.AtPrime.mk'_mem_maximal_iff`, `toLinearEquiv_orthogonalDecomposition`, `AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical`, `DirectSum.IsInternal.baseChange`, `LinearPMap.mk_apply`, `MeasureTheory.condExpL2_ae_eq_zero_of_ae_eq_zero`, `Subspace.dualRestrict_comp_dualLift`, `Ideal.mem_colon_span_singleton`, `range_comap_of_surjective`, `groupCohomology.mapCocycles₂_comp_i`, `PrimeSpectrum.isRetrocompact_zeroLocus_compl_of_fg`, `LieSubalgebra.mem_map_submodule`, `Orthonormal.codRestrict`, `Ideal.intNorm_mem_spanNorm`, `HahnEmbedding.Partial.orderTop_eq_archimedeanClassMk`, `dualQuotEquivDualAnnihilator_apply`, `ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm`, `Finsupp.instCountableSubtypeMemSubmoduleSpanRange`, `coe_toAddSubmonoid`, `LinearIndependent.linearCombination_repr`, `mem_of_localization_maximal`, `exteriorPower.basis_repr_ne`, `Ideal.mem_sSup_of_mem`, `IsLocalizedModule.exists_subsingleton_away`, `orthogonalProjection_mem_subspace_eq_self`, `LinearPMap.supSpanSingleton_apply_of_mem`, `Subbimodule.smul_mem'`, `LinearMap.range_domRestrict_le_range`, `RootPairing.Base.root_mem_span_int`, `RingPreordering.coe_support`, `MvPolynomial.homogeneousComponent_mem`, `ContinuousLinearMap.completeSpace_eqLocus`, `smul_mem_span_smul`, `DirectSum.isInternal_ne_bot_iff`, `Module.isTorsionBySet_quotient_ideal_smul`, `Ideal.isHomogeneous_iff_forall_subset`, `Module.Basis.mk_repr`, `PowerSeries.coeff_mul_mem_ideal_of_coeff_right_mem_ideal'`, `Valuation.mem_ltIdeal_iff`, `LinearMap.submoduleComap_apply_coe`, `finiteDimensional_direction_affineSpan_of_finite`, `Module.Flat.ker_lTensor_eq`, `KaehlerDifferential.cotangentComplexBaseChange_tmul`, `Ideal.primaryComponent_sup`, `Module.Basis.ofZLatticeBasis_apply`, `DividedPowers.OfInvertibleFactorial.dpow_apply`, `Ideal.mem_pointwise_smul_iff_inv_smul_mem`, `mul_mem_mul`, `groupCohomology.H0IsoOfIsTrivial_hom`, `PadicInt.sub_zmodRepr_mem`, `LinearMap.ofIsCompl_smul`, `OrthonormalBasis.orthogonalProjection_eq_sum_rankOne`, `mem_orthogonal_singleton_iff_inner_right`, `NumberField.mixedEmbedding.mem_idealLattice`, `PointedCone.dual_sup`, `BoxIntegral.unitPartition.mem_smul_span_iff`, `quotDualCoannihilatorToDual_apply`, `OrthogonalFamily.of_pairwise`, `injective_inclusionSpan`, `ContinuousMap.notMem_idealOfSet`, `IsLattice.rank'`, `sub_mem_iff_left`, `Span.repr_def`, `LinearMap.mem_isPairSelfAdjointSubmodule`, `bot_lt_isotypicComponent`, `dualPairing_apply`, `starProjection_singleton`, `TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_symm_apply_mk_tmul`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_i_hom`, `mem_iSup_iff_exists_finsupp`, `AlgebraicGeometry.Scheme.zeroLocus_mul`, `LinearMap.tensorEqLocusEquiv_apply`, `PrimeSpectrum.gc`, `groupCohomology.coe_mapCocycles₁`, `LinearMap.domRestrict'_apply`, `groupHomology.mem_cycles₂_iff`, `Subrepresentation.mem_ofSubmodule'_iff`, `Fintype.mem_span_image_iff_exists_fun`, `Affine.Simplex.restrict_map_restrict`, `RootPairing.invtRootSubmodule.eq_span_root`, `Valuation.Integers.isPrincipal_iff_exists_isGreatest`, `EuclideanGeometry.orthogonalProjection_contLinear`, `RootPairing.range_polarization_domRestrict_le_span_coroot`, `isUnit_iff_notMem_of_isAdicComplete_maximal`, `RootPairing.finrank_range_polarization_eq_finrank_span_coroot`, `EuclideanGeometry.dist_orthogonalProjection_eq_iff_oangle_eq`, `Module.FinitePresentation.exists_notMem_bijective`, `groupHomology.cyclesMap_comp_isoCycles₂_hom`, `mem_smul_span_singleton`, `ClosedComplemented.exists_isClosed_isCompl`, `range_specComap_of_surjective`, `coe_toBaseChange_tmul`, `ModuleCat.instSmallSubtypeForallCarrierObjMemSubmoduleSectionsSubmodule`, `Algebra.IsSmoothAt.exists_notMem_isStandardSmooth`, `AffineSubspace.coe_subtypeA`, `LinearMap.IsSymmetric.orthogonalFamily_eigenspaces`, `RootPairing.invtRootSubmodule.eq_top_iff`, `finrank_vectorSpan_le_iff_not_affineIndependent`, `groupCohomology.coboundariesToCocycles₁_apply`, `groupHomology.mapCycles₁_comp_assoc`, `PointedCone.DualFG.dual_dual_flip`, `eq_top_iff'`, `mem_span_image_iff_exists_fun`, `mem_iInf`, `LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal`, `PeriodPair.eqOn_deriv_weierstrassPExcept_derivWeierstrassPExcept`, `PeriodPair.weierstrassP_coe`, `groupHomology.toCycles_comp_isoCycles₂_hom_assoc`, `Ideal.notMem_of_isUnit`, `AffineSubspace.subtype_apply`, `Algebra.PreSubmersivePresentation.cotangentComplexAux_zero_iff`, `DividedPowers.OfInvertibleFactorial.dpow_mem`, `Module.Basis.ofZLatticeComap_apply`, `le_linearEquiv_of_sSup_eq_top`, `FG.lTensor.directedSystem`, `IsLocalization.mem_span_map`, `Ideal.mem_toTwoSided`, `Module.Basis.coe_mkFinConsOfLE`, `sub_mem_orthogonal_of_inner_left`, `Ideal.ne_top_iff_one`, `Ideal.coe_connectedComponentOfZero`, `Ideal.IsPrime.mem_pow_mul`, `rank_map_le`, `subtypeₗᵢ_toContinuousLinearMap`, `LinearMap.tensorKer_coe`, `LinearMap.BilinForm.restrict_nondegenerate_orthogonal_spanSingleton`, `Module.Baer.iff_surjective`, `FG.exists_rTensor_fg_inclusion_eq`, `Module.End.mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo`, `finrank_quotient_add_finrank`, `AddMonoidAlgebra.supportedEquivFinsupp_apply_support_val`, `isIsotypicOfType_submodule_iff`, `map_span_le`, `closure_subset_topologicalClosure_span`, `val_mulMap'_tmul`, `Finsupp.mem_supported`, `isSemisimple_torsionBy_of_irreducible`, `Ideal.exists_eq_mul_of_pure`, `SimpleGraph.card_connectedComponent_eq_finrank_ker_toLin'_lapMatrix`, `Ideal.map_spanIntNorm`, `eq_isotypicComponent_iff`, `groupHomology.mem_cycles₁_of_comp_eq_d₂₁`, `RootPairing.coroot'In_rootSpanMem_eq_pairingIn`, `PointedCone.dual_hull`, `Ideal.IsPrime.pow_mem_iff_mem`, `exteriorPower.alternatingMapLinearEquiv_ιMulti`, `KaehlerDifferential.kerToTensor_apply`, `Subspace.forall_mem_dualAnnihilator_apply_eq_zero_iff`, `coe_subtypeL'`, `DirectSum.isInternal_submodule_iff_iSupIndep_and_iSup_eq_top`, `Finsupp.mem_span_iff_linearCombination`, `HenselianLocalRing.is_henselian`, `coe_sup`, `Polynomial.degreeLT.basis_val`, `selfAdjointPart_comp_subtype_skewAdjoint`, `setBasisOfLinearIndependentOfCardEqFinrank_repr_apply`, `IsSemisimpleRing.exists_linearEquiv_ideal_of_isSimpleModule`, `Algebra.Presentation.differentials.hom₁_single`, `LinearMap.rank_eq_of_surjective`, `RingOfIntegers.exponent_eq_sInf`, `RingHom.SurjectiveOnStalks.exists_mul_eq_tmul`, `TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul`, `Algebra.zariskisMainProperty_iff`, `Algebra.Generators.exists_presentation_of_basis_cotangent`, `lift_rank_range_of_injective`, `Ideal.mem_toCotangent_ker`, `groupCohomology.mem_cocycles₁_of_addMonoidHom`, `groupHomology.δ₀_apply`, `groupCohomology.cocycles₂.d₂₃_apply`, `ProperCone.mem_map`, `one_mem_div`, `isArtinian_iff_submodule_quotient`, `PeriodPair.hasSum_derivWeierstrassPExcept`, `IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal`, `Ideal.Filtration.mem_submodule`, `Module.End.eigenspace_restrict_le_eigenspace`, `LinearMap.IsSymmetric.directSum_decompose_apply`, `mem_torsion_iff`, `linearProjOfIsCompl_comp_surjective_of_exact`, `sum_smul_mem`, `LinearMap.BilinForm.toLin_restrict_ker_eq_inf_ker`, `dualCopairing_eq`, `EuclideanGeometry.reflection_vadd_smul_vsub_orthogonalProjection`, `Nat.exists_mem_span_nat_finset_of_ge`, `LieSubmodule.mem_mk_iff'`, `mem_span_singleton_trans`, `ringKrullDim_quotient`, `Ideal.finsuppTotal_apply_eq_of_fintype`, `groupHomology.isoCycles₁_inv_comp_iCycles_apply`, `reflection_singleton_apply`, `rank_span_set`, `PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical`, `RootPairing.RootPositiveForm.zero_lt_posForm_iff`, `eq_zero_of_coe_mem_of_disjoint`, `DividedPowers.SubDPIdeal.dpow_mem_span_of_mem_span`, `IsCompl.projection_apply_left`, `SetLike.Homogeneous.smul`, `eq_isotypicComponent_of_le`, `groupCohomology.map_H0Iso_hom_f_apply`, `LieModule.Cohomology.twoCochain_alt`, `EuclideanGeometry.inter_eq_singleton_orthogonalProjection`, `PointedCone.coe_map`, `groupCohomology.H0IsoOfIsTrivial_inv_apply`, `ContinuousLinearMap.mkOfIsCompactOperator_mem_compactOperator`, `LinearMap.mem_skewAdjointSubmodule`, `quotDualCoannihilatorToDual_nondegenerate`, `rank_bot`, `LinearPMap.mem_inverse_graph`, `Ideal.mem_iff_of_associated`, `LieSubmodule.mem_toSubmodule`, `mem_dualAnnihilator`, `finrank_add_finrank_orthogonal'`, `mem_ideal_smul_span_iff_exists_sum'`, `EuclideanGeometry.Cospherical.inclusion_iff`, `coeff_minpolyDiv_sub_pow_mem_span`, `groupHomology.chains₁ToCoinvariantsKer_surjective`, `ZLattice.comap_discreteTopology`, `adjoin_monomial_eq_reesAlgebra`, `InnerProductSpace.gramSchmidt_def`, `LinearMap.submoduleMap_coe_apply`, `ContinuousLinearMap.isComplete_ker`, `Module.End.mem_invtSubmodule_iff_mapsTo`, `IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_mem`, `EuclideanGeometry.Cospherical.inclusion`, `biUnion_associatedPrimes_eq_compl_regular`, `ContinuousMap.mem_idealOfSet`, `Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange`, `AffineSubspace.linear_topEquiv`, `prod_coe`, `DirectSum.decompose_eq_mul_idempotent`, `toLinearPMap_range`, `IsNoetherian.injective_of_surjective_of_submodule`, `Affine.Simplex.orthogonalProjectionSpan_eq_point`, `coe_dualCoannihilator_span`, `LieHom.quotKerEquivRange_toFun`, `isSimpleModule_iff_isAtom`, `FiniteDimensional.span_of_finite`, `MeasureTheory.MemLp.condExpL2_ae_eq_condExp`, `HahnEmbedding.Partial.archimedeanClassMk_eq_iff`, `sigPos_isGreatest`, `mem_vectorSpan_iff_eq_weightedVSub`, `TensorProduct.map₂_eq_range_lift_comp_mapIncl`, `Module.End.mapsTo_genEigenspace_of_comm`, `LinearMap.exists_basis_basis_of_span_eq_top_of_mem_algebraMap`, `isArtinian_of_le`, `exteriorPower.ιMulti_span_of_span`, `DirectSum.IsInternal.collectedBasis_mem`, `biSup_comap_subtype_eq_top`, `MonoidAlgebra.of_mem_span_of_iff`, `IsLocalRing.residue_eq_zero_iff`, `PointedCone.subset_dual_dual`, `Polynomial.mem_iff_eq_smul_annIdealGenerator`, `IsLinearTopology.hasBasis_open_submodule`, `LinearPMap.adjointAux_unique`, `FractionalIdeal.isNoetherian_coeIdeal`, `RootPairing.injOn_dualMap_subtype_span_root_coroot`, `MvPolynomial.mem_pow_idealOfVars_iff`, `LinearMap.range_domRestrict_eq_range_iff`, `LinearMap.sub_mem_ker_iff`, `Ideal.finsuppTotal_apply`, `TensorProduct.exists_finite_submodule_of_finite'`, `biSup_eq_range_dfinsupp_lsum`, `IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe`, `groupCohomology.mem_cocycles₁_def`, `IsOrtho.orthogonalProjection_comp_subtypeL`, `Ideal.exists_mul_mem_of_mem_minimalPrimes`, `LinearDisjoint.rank_inf_le_one_of_commute_of_flat`, `IsIsotypic.submodule_linearEquiv_fun`, `RootPairing.restrictScalars_pairing`, `EuclideanGeometry.orthogonalProjection_map`, `Ideal.subset_union_prime'`, `Algebra.adjoin_span`, `Ideal.prod_mem`, `LieModule.Cohomology.d₂₃_apply`, `LieAlgebra.InvariantForm.restrict_nondegenerate`, `Ideal.mem_map_of_mem`, `LinearEquiv.fixedReduce_eq_one`, `exteriorPower.alternatingMapLinearEquiv_symm_map`, `span_range_inclusion_restrictScalars_eq_top`, `isNoetherian_submodule`, `selfAdjointPart_comp_subtype_selfAdjoint`, `PointedCone.subset_hull`, `AffineSubspace.vsub_mem_direction`, `HasStrictFDerivAt.eq_implicitFunctionOfComplemented`, `exteriorPower.instFree`, `Affine.Simplex.circumradius_restrict`, `HahnEmbedding.Partial.truncLT_mem_range_extendFun`, `DividedPowers.mem_dpEqualizer_iff`, `PrimeSpectrum.union_zeroLocus`, `stabilizer_coe`, `mulMap_comm`, `HahnEmbedding.Seed.baseEmbedding_strictMono`, `AlgebraicGeometry.Scheme.zeroLocus_biInf`, `Module.Flat.out`, `RootPairing.finrank_rootSpan_map_polarization_eq_finrank_corootSpan`, `LieModule.Cohomology.d₁₂_apply_apply`, `Affine.Simplex.medial_restrict`, `Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_zero_iff`, `FiniteDimensional.mem_span_of_iInf_ker_le_ker`, `equivSubtypeMap_symm_apply`, `mem_prod`, `botEquivPUnit_apply`, `LinearMap.ofIsComplProd_apply`, `AffineIndependent.finrank_vectorSpan_add_one`, `coe_orthogonalDecomposition_symm`, `groupCohomology.mapCocycles₂_comp_i_apply`, `Projectivization.finrank_submodule`, `ker_subtype`, `groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_apply`, `MeasureTheory.aestronglyMeasurable_condExpL2`, `Ideal.absNorm_mem`, `orthogonalProjection_bot`, `BoxIntegral.unitPartition.integralSum_eq_tsum_div`, `linearProjOfIsCompl_apply_right'`, `groupHomology.single_one_snd_sub_single_one_fst_mem_boundaries₂`, `ZLattice.module_finite`, `EuclideanGeometry.angle_orthogonalProjection_self`, `mem_pi`, `EuclideanGeometry.orthogonalProjection_apply_mem`, `DirectSum.bracket_apply_apply`, `LinearMap.restrict_comp`, `isNoetherian_of_le`, `PointedCone.dual_ker`, `IsSimpleRing.exists_ringEquiv_matrix_end_mulOpposite`, `coe_isComplEquivProj_symm_apply`, `Representation.card_inv_mul_sum_char_eq_finrank`, `EuclideanGeometry.dist_orthogonalProjection_eq_of_oangle_eq`, `finrank_add_inf_finrank_orthogonal`, `groupHomology.cycles₁IsoOfIsTrivial_hom_apply`, `exists_add_mem_mem_orthogonal`, `LieAlgebra.bracket_ofTwoCocycle`, `LinearMap.exact_subtype_mkQ`, `coe_torsion'`, `AffineSubspace.coe_subtypeₐᵢ`, `Algebra.Extension.Cotangent.val_mk`, `isClosed_topologicalClosure`, `PowerSeries.IsWeierstrassDivisorAt.coeff_seq_succ_sub_seq_mem`, `PeriodPair.latticeBasis_zero`, `linearDisjoint_iff`, `PointedCone.DualFG.exists_fg_dual`, `PrimeSpectrum.coe_preimageHomeomorphFiber_apply_asIdeal`, `Algebra.IsUnramifiedAt.exists_hasStandardEtaleSurjectionOn`, `GradedTensorProduct.auxEquiv_one`, `mem_bot`, `tendsto_card_div_pow_atTop_volume`, `toLocalized₀_apply_coe`, `MvPolynomial.weightedDecomposition.decompose'_apply`, `EuclideanGeometry.dist_orthogonalProjection_eq_iff_angle_eq`, `LinearMap.IsSymm.domRestrict`, `Module.End.mem_eigenspace_intrinsicStar_iff`, `Subspace.finrank_dualCoannihilator_eq`, `apply_mem_span_image_iff_mem_span`, `sup_eq_range`, `AdicCompletion.ofPowSMul_ofValEqZero`, `mem_map`, `mem_div_iff_forall_mul_mem`, `MeasureTheory.condExpL2_indicator_ae_eq_smul`, `exteriorPower.ιMultiDual_apply_diag`, `Representation.instIsTrivialSubtypeMemSubgroupSubmoduleInvariantsCompLinearMapIdSubtypeToInvariants`, `RootPairing.finrank_corootSpanIn_int`, `sub_mem_sup`, `LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_right`, `groupCohomology.cocyclesIso₀_inv_comp_iCocycles`, `Algebra.TensorProduct.linearEquivIncludeRange_toLinearMap`, `SModEq.zero`, `IsLocalRing.mem_maximalIdeal`, `retractionOfSectionOfKerSqZero_comp_kerToTensor`, `PadicInt.exists_mem_range`, `groupCohomology.cocycles₂_map_one_fst`, `MvPolynomial.monomial_mul_mem_coeffsIn`, `Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_eq`, `groupCohomology.mapCocycles₂_comp_i_assoc`, `DirectSum.IsInternal.collectedBasis_coe`, `generator_maximal_submoduleImage_dvd`, `rank_quotient_add_rank`, `Ideal.toCotangent_range`, `Ideal.mem_sup_left`, `HenselianRing.is_henselian`, `FractionalIdeal.mem_coeSubmodule`, `PrimeSpectrum.zeroLocus_subset_zeroLocus_iff`, `RootPairing.RootPositiveForm.posForm_apply_root_root_le_zero_iff`, `Module.Invertible.exists_linearEquiv_ideal`, `ZLattice.rank`, `ZLattice.tsum_norm_rpow_le`, `Subspace.finiteDimensional_quot_dualCoannihilator_iff`, `groupCohomology.H1IsoOfIsTrivial_inv_apply`, `LinearIndependent.repr_eq_single`, `PeriodPair.ω₂_div_two_notMem_lattice`, `IsLocalization.AtPrime.coe_primeSpectrumOrderIso_apply_coe_asIdeal`, `LieSubalgebra.mem_toSubmodule`, `coe_toConvexCone`, `Module.Baer.ExtensionOf.is_extension`, `Localization.exists_finite_awayMapₐ_of_surjective_awayMapₐ`, `FirstOrder.Field.lift_genericMonicPoly`, `smul_mem_pointwise_smul`, `GradedTensorProduct.tmul_coe_mul_one_tmul`, `Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective`, `Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange_residueField`, `SubmodulesBasis.smul`, `isNoetherian_iSup`, `PeriodPair.hasSum_derivWeierstrassP`, `groupHomology.mapCycles₁_id_comp_assoc`, `Subalgebra.mem_toSubmodule`, `PrimeSpectrum.coe_preimageOrderIsoFiber_symm_apply_coe_asIdeal`, `rank_submodule_le_one_iff`, `PeriodPair.compl_lattice_diff_singleton_mem_nhds`, `AffineIndependent.card_le_finrank_succ`, `HahnEmbedding.Partial.toOrderAddMonoidHom_apply`, `Coplanar.finrank_le_two`, `LocalizedModule.mem_ker_mkLinearMap_iff`, `Algebra.Generators.exists_presentation_of_free_cotangent`, `goursat`, `Ideal.iSup_eq_span`, `coe_prodEquivOfIsCompl'`, `Affine.Simplex.finrank_direction_altitude`, `IsCompactOperator.codRestrict`, `PeriodPair.hasSumLocallyUniformly_derivWeierstrassP`, `groupCohomology.mem_cocycles₁_of_comp_eq_d₀₁`, `HahnEmbedding.Seed.strictMono_coeff`, `subtypeₗᵢ_toLinearMap`, `groupCohomology.cocycles₁IsoOfIsTrivial_hom_hom_apply_apply`, `mem_span_finset`, `LinearMap.BilinMap.apply_apply_mem_of_mem_span`, `Module.Basis.sumQuot_repr_left`, `mem_skewAdjointMatricesSubmodule'`, `LinearEquiv.ofSubmodules_apply`, `LinearMap.lTensor_range`, `groupCohomology.H2π_comp_map_apply`, `groupHomology.mapCycles₁_comp`, `topEquiv_apply`, `Module.isTorsionBySet_annihilator`, `PrimeSpectrum.zeroLocus_span`, `Polynomial.degreeLT.addLinearEquiv_castAdd`, `subtype_apply`, `GradedTensorProduct.tmul_coe_mul_coe_tmul`, `Ideal.powQuotSuccInclusion_apply_coe`, `LinearEquiv.toSpanNonzeroSingleton_homothety`, `RootPairing.Base.exists_mem_span_pairingIn_ne_zero_and_pairwise_ne`, `MeasureTheory.integral_condExpL2_eq`, `IsCompactOperator.restrict'`, `Representation.Coinvariants.sub_mem_ker`, `isInternal_prime_power_torsion_of_is_torsion_by_ideal`, `finiteDimensional_vectorSpan_image_of_finite`, `RootPairing.instFiniteSubtypeMemSubmoduleRootSpanOfFinite`, `prod_mem_ideal_map_of_mem_conductor`, `OrthogonalFamily.isInternal_iff_of_isComplete`, `LieAlgebra.hasCentralRadical_and_of_isIrreducible_of_isFaithful`, `orzechProperty_iff`, `ExteriorAlgebra.basis_eq_coe_basis`, `ContinuousLinearMap.isClosed_genEigenspace`, `Affine.Simplex.ninePointCircle_restrict`, `Algebra.Extension.H1Cotangent.val_zero`, `Nat.subset_span_setGcd`, `Ideal.IsPrime.mul_mem_pow`, `exists_forall_notMem_of_forall_ne_top`, `IsAdic.hasBasis_nhds`, `submodule_eq_sSup_le_nonzero_spans`, `LinearMap.tensorKerEquivOfSurjective_symm_tmul`, `RootPairing.Base.coroot_mem_span_int`, `Module.End.pow_restrict`, `groupCohomology.toCocycles_comp_isoCocycles₂_hom`, `LinearMap.IsProj.eq_conj_prodMap`, `LinearMap.tensorEqLocus_coe`, `LinearDisjoint.not_linearIndependent_pair_of_flat_right`, `mem_comap`, `DirectSum.coeLinearMap_lof`, `length_lt`, `PointedCone.tmul_mem_maxTensorProduct`, `PrimeSpectrum.isClosed_iff_zeroLocus_radical_ideal`, `finrank_lt_finrank_of_lt`, `LinearIsometry.isComplete_map_iff`, `LinearMap.mem_submoduleImage_of_le`, `Polynomial.degreeLT.addLinearEquiv_apply_fst`, `mulLeftMap_apply`, `ZLattice.covolume.tendsto_card_div_pow`, `finiteDimensional_finset_sup`, `LinearMap.restrict_commute`, `MeasureTheory.setIntegral_condExpL2_indicator`, `IsLocalization.isMaximal_iff_isMaximal_disjoint`, `PeriodPair.mem_lattice`, `ModuleCat.imageIsoRange_hom_subtype`, `IsLinearTopology.hasBasis_submodule`, `stereographic_apply_neg`, `instCountable_of_discrete_submodule`, `OrthonormalBasis.orthogonalProjection_apply_eq_sum`, `groupHomology.mapCycles₁_comp_i`, `LinearMap.toKerLocalized_isLocalizedModule`, `submoduleOf_sup_of_le`, `Affine.Simplex.face_restrict`, `rank_range_le`, `LieSubalgebra.engel_carrier`, `ClosedSubmodule.mem_toSubmodule_iff`, `DirectSum.coeLinearMap_of`, `ContinuousLinearMap.coe_equivRange`, `rank_le`, `coe_prodEquivOfIsCompl`, `groupCohomology.shortComplexH0_f`, `DirectSum.coeLinearMap_eq_dfinsuppSum`, `HahnEmbedding.Partial.orderTop_eq_iff`, `RootPairing.polarizationIn_Injective`, `finrank_vectorSpan_insert_le`, `LinearIsometryEquiv.coe_ofEq_apply`, `norm_eq_iInf_iff_inner_eq_zero`, `PeriodPair.analyticOnNhd_derivWeierstrassPExcept`, `groupCohomology.cocyclesOfIsCocycle₁_coe`, `Subspace.dualPairing_nondegenerate`, `Affine.Simplex.interior_restrict`, `Ideal.map_relNorm`, `GradedAlgebra.algebraMap_apply`, `vsub_rev_mem_vectorSpan_pair`, `monomial_mem_adjoin_monomial`, `Polynomial.mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero`, `ExteriorAlgebra.GradedAlgebra.ι_sq_zero`, `ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal`, `DirectSum.IsInternal.subordinateOrthonormalBasis_subordinate`, `mul_mem_smul_iff`, `AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff_of_mem`, `Ideal.span_singleton_mul_eq_span_singleton_mul`, `RootPairing.finrank_rootSpanIn_int`, `AddMonoidAlgebra.decomposeAux_coe`, `LinearMap.ker_rangeRestrict`, `CliffordAlgebra.evenOdd.gradedMonoid`, `smul_subset_smul`, `LinearMap.IsSymmetric.card_filter_eigenvalues_eq`, `FiniteDimensional.RCLike.properSpace_submodule`, `Ideal.isTorsionBySet_cotangent`, `EuclideanGeometry.dist_orthogonalProjection_eq_of_two_zsmul_oangle_eq`, `range_stereographic_symm`, `AffineSubspace.inclusion_linear`, `Ideal.IsPrime.mem_of_pow_mem`, `ProjectiveSpectrum.union_zeroLocus`, `AddMonoidAlgebra.gradeBy.isInternal`, `LinearMap.compMultilinearMap_codRestrict`, `MvPolynomial.mul_X_mem_coeffsIn`, `Module.Flat.iff_rTensor_injective`, `ChevalleyThm.chevalley_mvPolynomialC`, `HasStrictFDerivAt.to_implicitFunctionOfComplemented`, `EuclideanGeometry.orthogonalProjection_apply'`, `LinearEquiv.mem_fixedSubmodule_transvection_iff`, `CliffordAlgebra.involute_mem_evenOdd_iff`, `Affine.Simplex.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq`, `isOpenEmbedding_stereographic_symm`, `HahnEmbedding.Partial.extendFun_strictMono`, `TensorProduct.Algebra.exists_of_fg`, `MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le`, `Set.Finite.submoduleSpan`, `Rep.coindVEquiv_symm_apply_coe`, `exists_extension_of_le_sublinear`, `IsNoetherianRing.isClosed_ideal`, `EuclideanGeometry.Sphere.coe_secondInter`, `Ideal.map_includeRight_eq`, `inf_genEigenspace`, `finiteDimensional_sup`, `isIsotypic_submodule_iff`, `finiteDimensional_direction_map`, `PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical`, `AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι`, `torsionBy_torsionBy_eq_top`, `LinearDisjoint.rank_inf_le_one_of_flat`, `smul_mem_span_smul_of_mem`, `sum_mem`, `coe_span_smul`, `TensorProduct.exists_finite_submodule_left_of_setFinite`, `TensorProduct.quotientTensorEquiv_symm_apply_mk_tmul`, `groupCohomology.H1IsoOfIsTrivial_H1π_apply_apply`, `LinearEquiv.mem_dilatransvections_iff_rank`, `ModuleCat.imageIsoRange_inv_image_ι_apply`, `basisOfLinearIndependentOfCardEqFinrank_repr_apply`, `mem_pairSelfAdjointMatricesSubmodule'`, `ContinuousLinearMap.isClosed_ker`, `MvPolynomial.X_mul_mem_coeffsIn`, `EuclideanGeometry.orthogonalProjection_orthogonalProjection_of_le`, `Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomial`, `IsCyclotomicExtension.Rat.p_mem_span_zeta_sub_one`, `strictMono_comap_prod_map`, `Matrix.cRank_toNat_eq_finrank`, `ContinuousMap.idealOfSet_closed`, `mem_annihilator'`, `HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorphOfComplemented_target`, `groupCohomology.coboundaries₂.val_eq_coe`, `RootPairing.PolarizationIn_eq`, `EuclideanGeometry.Sphere.isTangent_iff_isTangentAt_orthogonalProjection`, `adjoint_subtypeL`, `NumberField.mixedEmbedding.fundamentalDomain_idealLattice`, `instIsTwoSidedColonCoe`, `ext_iff`, `Ideal.disjoint_powers_iff_notMem`, `Module.Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul`, `Pi.mem_spanSubset_iff`, `dualRestrict_ker_eq_dualAnnihilator`, `LinearIsometry.equivRange_apply_coe`, `mem_left_iff_eq_zero_of_disjoint`, `FiniteDimensional.span_finset`, `reflection_orthogonalComplement_singleton_eq_neg`, `lipschitzGroup.conjAct_smul_ι_mem_range_ι`, `Subalgebra.finrank_toSubmodule`, `groupHomology.single_one_fst_sub_single_one_snd_mem_boundaries₂`, `ZLattice.tsumNormRPowBound_spec`, `LinearIsometryEquiv.reflections_generate_dim_aux`, `sub_mem_pNilradical_iff_pow_expChar_pow_eq`, `MvPolynomial.IsHomogeneous.HomogeneousSubmodule.gcommSemiring`, `Affine.Triangle.dist_circumcenter_reflection_orthocenter`, `MeasureTheory.norm_condExpL2_coe_le`, `isArtinian_span_of_finite`, `MeasureTheory.setLIntegral_nnnorm_condExpL2_indicator_le`, `Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_iff`, `IsIsotypic.isotypicComponent`, `toBaseChange_surjective`, `LinearMap.finrank_range_of_inj`, `MeasureTheory.mem_lpMeas_iff_aestronglyMeasurable`, `LinearMap.restrict_sub`, `LinearMap.toPMap_apply`, `groupCohomology.d₁₂_apply_mem_cocycles₂`, `Rep.invariantsAdjunction_unit_app`, `groupHomology.mapCycles₂_id_comp`, `coe_linearProjOfIsCompl_apply`, `inclusion_apply`, `UniqueDiffWithinAt.dense_tangentConeAt`, `LinearMap.linearProjOfIsCompl_apply_right`, `ArchimedeanClass.mem_ball_iff`, `groupCohomology.d₀₁_apply_mem_cocycles₁`, `mem_smul_iff_inv_mul_mem`, `LinearEquiv.ofLeftInverse_symm_apply`, `LinearMap.range_rangeRestrict`, `isNoetherian_iff_submodule_quotient`, `ZSpan.repr_ceil_apply`, `PowerSeries.coe_hasEvalIdeal`, `RingHom.mem_ker`, `ZLattice.comap_equiv_apply`, `surjective_tensorToSpan`, `PrimeSpectrum.mem_compl_zeroLocus_iff_notMem`, `PrimeSpectrum.zeroLocus_radical`, `mem_torsionBySet_iff`, `PrimeSpectrum.zeroLocus_minimalPrimes`, `orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero`, `Ideal.mem_span_range_self`, `torsion_torsion_eq_top`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'`, `LinearIsometryEquiv.ofTop_toLinearEquiv`, `Ideal.single_mem_jacobson_matrix`, `finrank_span_le_card`, `coe_ofClass`, `isLocalizedModule`, `range_subtypeL`, `PrimeSpectrum.isIrreducible_zeroLocus_iff`, `botEquivPUnit_symm_apply`, `Module.Basis.addSubgroupOfClosure_repr_apply`, `isSMulRegular_submodule_iff_right_eq_zero_of_smul`, `NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis`, `groupCohomology.subtype_comp_d₀₁_apply`, `Module.Basis.span_repr_eq_single`, `PeriodPair.isClosed_lattice`, `linearIndependent_fin_cons`, `Module.length_top`, `span_range_inclusion_eq_top`, `IsDedekindDomain.exists_forall_sub_mem_ideal`, `NumberField.mixedEmbedding.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIdealLattice`, `IsDedekindDomain.HeightOneSpectrum.valuation_def`, `contDiffOn_stereoToFun`, `SchauderBasis.finrank_range_proj`, `spinGroup.conjAct_smul_ι_mem_range_ι`, `image_span_subset_span`, `ValuativeRel.supp_def`, `MonoidAlgebra.mem_span_support`, `GradedTensorProduct.auxEquiv_symm_one`, `groupCohomology.H2π_eq_iff`, `le_sigPos_of_posDef`, `NumberField.det_basisOfFractionalIdeal_eq_absNorm`, `GradedTensorProduct.auxEquiv_comm`, `LinearMap.comap_leq_ker_subToSupQuotient`, `QuadraticForm.sigPos_add_sigNeg_add_radical`, `Rep.coindFunctorIso_inv_app_hom_toFun_coe`, `LinearDisjoint.linearIndependent_right_of_flat`, `HilbertBasis.hasSum_orthogonalProjection`, `smul_coe_torsionBy`, `LinearMap.rTensor_injective_iff_subtype`, `Ideal.exists_le_prime_disjoint`, `MeasureTheory.mem_lpMeas_indicatorConstLp`, `stereoInvFun_apply`, `Ideal.image_subset_nonunits_valuationSubring`, `HahnEmbedding.IsPartial.strictMono`, `collinear_iff_finrank_le_one`, `mem_annihilator`, `LinearMap.subtype_compAlternatingMap_codRestrict`, `groupCohomology.cocycles₂_map_one_snd`, `mem_set_smul_def`, `Ideal.disjoint_map_primeCompl_iff_comap_le`, `groupCohomology.δ₁_apply`, `Polynomial.exists_monic_aeval_eq_zero_forall_mem_pow_of_isIntegral`, `LinearIsometryEquiv.ofTop_symm_apply_coe`, `LieDerivation.ofGradingSum_of`, `LinearMap.submoduleImage_apply_of_le`, `LinearRecurrence.solSpace_rank`, `LieSubalgebra.coe_toSubmodule`, `LinearEquiv.ofSubmodule'_symm_apply`, `groupHomology.toCycles_comp_isoCycles₁_hom_apply`, `AffineIsometryEquiv.coe_ofEq_apply`, `IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp`, `closed_of_finiteDimensional`, `Representation.mem_invariants_iff_of_forall_mem_zpowers`, `spinGroup.involute_act_ι_mem_range_ι`, `Ideal.IsMaximal.isClosed`, `Affine.Simplex.altitude_restrict_eq_comap_subtype`, `MeasureTheory.lintegral_nnnorm_condExpL2_le`, `PeriodPair.periodic_derivWeierstrassP`, `exteriorPower.finrank_eq`, `ContinuousLinearMap.eqOn_closure_span`, `AffineSubspace.isometryEquivMap.toAffineMap_eq`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff`, `groupHomology.mapCycles₂_comp_i`, `AffineSubspace.signedInfDist_eq_signedDist_of_mem`, `Polynomial.degreeLT.addLinearEquiv_apply_snd`, `IsLocalization.mk'_mem_iff`, `LinearPMap.supSpanSingleton_apply_mk_of_mem`, `ClassGroup.mkMMem_surjective`, `Finsupp.mem_submodule_iff`, `groupCohomology.map_H0Iso_hom_f`, `mem_span_singleton`, `LinearMap.finrank_genEigenspace_le`, `codisjoint_iff_exists_add_eq`, `Module.Basis.sumQuot_repr_inl`, `LinearMap.IsSymmetric.diagonalization_symm_apply`, `CategoryTheory.ShortComplex.moduleCat_pOpcycles_eq_zero_iff`, `AddMonoidAlgebra.mem_ideal_span_of'_image`, `ContinuousLinearMap.coe_codRestrict`, `groupHomology.boundariesOfIsBoundary₁_coe`, `LieModule.weight_vector_multiplication`, `Affine.Simplex.signedInfDist_apply_self`, `Representation.mem_invariants`, `LinearMap.quotientInfEquivSupQuotient_apply_mk`, `WittVector.mem_span_p_pow_iff_le_coeff_eq_zero`, `AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal`, `SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux`, `Ideal.mem_of_dvd`, `finiteDimensional_vectorSpan_of_finite`, `Nat.finite_setOf_setGcd_dvd_and_mem_span`, `finiteQuotient_iff`, `Ideal.IsPrime.one_notMem`, `groupCohomology.π_comp_H1Iso_hom`, `LieAlgebra.mem_ker_killingForm_of_mem_rootSpace_of_forall_rootSpace_neg`, `RootPairing.corootSpanMem_reflectionPerm_self`, `Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_zero_iff`, `Ideal.mem_of_liesOver`, `Module.isTorsionBySet_span_singleton_iff`, `smoothSheafCommRing.isUnit_stalk_iff`, `Module.End.eigenspace_restrict_eq_bot`, `Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes`, `LinearIsometryEquiv.reflections_generate`, `AffineSubspace.signedInfDist_eq_signedDist_orthogonalProjection`, `Ideal.tensorCotangentHom_tmul`, `Subrepresentation.mem_asSubmodule'_iff`, `exteriorPower.map_apply_ιMulti_family`, `NumberField.mixedEmbedding.span_idealLatticeBasis`, `coe_mk`, `HomogeneousSubmodule.mem_toSubmodule_iff`, `RingPreordering.mem_support`, `ExteriorAlgebra.ιMulti_range`, `sum_mem_iSup`, `AffineEquiv.linear_ofEq`, `LinearMap.submoduleOf_span_singleton_of_mem`, `norm_orthogonalProjection_apply_le`, `instPolynormableSpaceSubtypeMemSubmodule`, `HahnEmbedding.Partial.exists_sub_mem_ball`, `HasStrictFDerivAt.map_implicitFunction_eq`, `AffineSubspace.coe_direction_eq_vsub_set`, `Algebra.Extension.Hom.sub_one_tmul`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_assoc_apply`, `IsLinearTopology.hasBasis_open_ideal`, `GradedAlgebra.proj_recompose`, `mem_span_iff_exists_finset_subset`, `AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff`, `mem_span_mul_finite_of_mem_mul`, `apply_coe_mem_map`, `OrthonormalBasis.starProjection_eq_sum_rankOne`, `Ideal.isTwoSided_iff`, `LinearMap.BilinForm.apply_apply_same_eq_zero_iff`, `PointedCone.mem_dual`, `mem_iSup_iff_exists_dfinsupp`, `AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff`, `FDRep.average_char_eq_finrank_invariants`, `DividedPowers.IsSubDPIdeal.dpow_mem`, `Algebra.zariskisMainProperty_iff'`, `groupCohomology.cocycles₂_ρ_map_inv_sub_map_inv`, `AffineIsometryEquiv.ofTop_symm_apply_coe`, `LinearIndepOn.subset_span_extend`, `isArtinian_range`, `TopologicalSpace.IsSeparable.span`, `PointedCone.DualFG.dual_flip_dual`, `ValuationSubring.primeSpectrumOrderEquiv_apply_coe_carrier`, `TensorProduct.inner_mapIncl_mapIncl`, `Ideal.mem_radical_of_pow_mem`, `IsDedekindDomain.HeightOneSpectrum.valuation_div_le_one_iff`, `RootPairing.toLinearMap_apply_PolarizationIn`, `IsDiscreteValuationRing.isOpen_iff`, `exteriorPower.zeroEquiv_naturality`, `instNonUnitalSubsemiringClassIdeal`, `Collinear.finiteDimensional_direction_affineSpan`, `KaehlerDifferential.D_apply`, `finiteDimensional_direction_affineSpan_insert_set`, `TensorProduct.exists_finite_submodule_right_of_setFinite`, `HahnEmbedding.IsPartial.truncLT_mem_range`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply_ker`, `groupHomology.single_one_fst_sub_single_one_fst_mem_boundaries₂`, `Affine.Simplex.orthogonalProjectionSpan_congr`, `LinearIndependent.linearCombinationEquiv_apply_coe`, `Polynomial.adjSylvester_comp_sylveserMap`, `subset_span_trans`, `IsLocalization.algebraMap_mem_map_algebraMap_iff`, `mapsTo_smul_closure`, `LieModule.lie_mem_maxGenEigenspace_toEnd`, `iSup_eq_range_dfinsupp_lsum`, `Module.Basis.span_apply`, `Ideal.eq_span_singleton_mul`, `LieModule.Cohomology.twoCochain_skew`, `Ideal.ker_piRingHom_atPrime_eq_of_pure`, `norm_eq_iInf_iff_real_inner_eq_zero`, `LinearMap.ker_codRestrict`, `AffineSubspace.mem_direction_iff_eq_vsub`, `groupCohomology.cocycles₂.coe_mk`, `mem_conductor_iff`, `MeasureTheory.lpMeasToLpTrimLie_symm_indicator`, `ContinuousLinearMap.closed_complemented_range_of_isCompl_of_ker_eq_bot`, `groupHomology.mapCycles₁_id_comp_apply`, `groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_assoc`, `LinearMap.IsSymmetric.restrict_invariant`, `Associates.le_singleton_iff`, `CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe`, `RootPairing.instFiniteSubtypeMemSubmoduleCorootSpanOfFinite`, `AffineIsometryEquiv.ofEq_symm`, `ContinuousLinearMap.equivRange_symm_apply`, `Ideal.IsPrime.mul_mem_right_iff`, `RootPairing.finrank_corootSpanIn`, `LinearIndependent.linearCombination_comp_repr`, `Ideal.quotientToQuotientRangePowQuotSucc_mk`, `Ideal.mem_map_of_equiv`, `coe_restrictScalars`, `LinearPMap.mkSpanSingleton'_apply_self`, `Representation.coe_coindV`, `coe_inclusion`, `LinearIsometry.extend_apply`, `NumberField.FinitePlace.norm_lt_one_iff_mem`, `LinearPMap.inverse_isClosable_iff`, `Module.support_of_algebra`, `instLocallyConvexSpaceSubtypeMemSubmodule`, `Polynomial.degreeLT.instFiniteSubtypeMemSubmodule`, `KaehlerDifferential.one_smul_sub_smul_one_mem_ideal`, `PointedCone.FG.dual_dualfg`, `IsSMulRegular.notMem_of_mem_minimalPrimes`, `finrank_strictMono`, `Module.End.IsSemisimple.restrict`, `exists_linearIndepOn_extension`, `comap_coe`, `Algebra.Extension.Cotangent.mk_eq_zero_iff`, `Rat.valuation_le_one_iff_den`, `OrthogonalFamily.isInternal_iff`, `LinearMap.BilinForm.mem_dualSubmodule`, `asymptoticCone_affineSubspace`, `linearEquiv_det_reflection`, `groupCohomology.instEpiModuleCatH1π`, `Polynomial.coeff_modByMonic_mem_pow_natDegree_mul`, `PeriodPair.differentiableOn_derivWeierstrassP`, `LinearPMap.apply_comp_inclusion`, `PeriodPair.weierstrassPExcept_def`, `span_val_image_eq_iff`, `LieSubmodule.toEnd_comp_subtype_mem`, `GradedTensorProduct.tmul_one_mul_coe_tmul`, `Algebra.QuasiFiniteAt.exists_basicOpen_eq_singleton`, `mk_mem_projectivization_iff`, `coe_pointwise_smul`, `Coplanar.finiteDimensional_direction_affineSpan`, `IsSemisimpleModule.sup`, `Matrix.rank_eq_finrank_range_toLin`, `Affine.Simplex.ExcenterExists.touchpointWeights_restrict`, `PrimeSpectrum.mem_zeroLocus`, `groupCohomology.H2π_comp_map`, `Matrix.eRank_toNat_eq_finrank`, `LinearPMap.sSup_apply`, `CharacterModule.intSpanEquivQuotAddOrderOf_apply`, `GradedAlgHom.instGradedFunLikeSubmodule`, `QuadraticMap.Equivalent.rank_radical_eq`, `Ideal.mapCotangent_toCotangent`, `groupHomology.π_comp_H2Iso_hom`, `AffineSubspace.instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection`, `stereoToFun_apply`, `uniqueDiffWithinAt_iff`, `Ideal.add_mem`, `IsPrincipal.mem_iff_eq_smul_generator`, `sub_starProjection_mem_orthogonal`, `LinearPMap.snd_apply`, `prodEquivOfIsCompl_symm_apply_fst_eq_zero`, `IsLocalization.isPrime_iff_isPrime_disjoint`, `PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff`, `NumberField.mixedEmbedding.mem_span_latticeBasis`, `exteriorPower.ιMulti_apply_coe`, `affineIndependent_iff_not_finrank_vectorSpan_le`, `coe_tensorSpanEquivSpan_apply_tmul`, `annihilator_map_mkQ_eq_colon`, `LieSubmodule.mem_baseChange_iff`, `SModEq.sub_mem`, `FG.lTensor.directLimit_apply'`, `PowerBasis.mem_span_pow'`, `PrimeSpectrum.subset_vanishingIdeal_zeroLocus`, `dualRestrict_def`, `ContinuousLinearMap.coe_projKerOfRightInverse_apply`, `Rep.coindToInd_apply`, `groupHomology.mapCycles₁_comp_i_apply`, `IsLocalization.AtPrime.to_map_mem_maximal_iff`, `Module.Basis.SmithNormalForm.snf`, `Rep.FiniteCyclicGroup.groupHomologyπEven_eq_iff`, `groupHomology.mapCycles₂_comp`, `mem_span_C_coeff`, `EuclideanGeometry.Sphere.orthogonalProjection_orthRadius_center`, `AlgHom.toKerIsLocalization_apply`, `LinearMap.restrict_eq_codRestrict_domRestrict`, `exists_finrank_eq_sigNeg_and_negDef`, `SetLike.homogeneous_zero_submodule`, `ModuleCat.kernelIsoKer_inv_kernel_ι_apply`, `AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop`, `Ideal.mem_sInf`, `Nat.mem_maximalIdeal_iff`, `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`, `MvPolynomial.DirectSum.coeAddMonoidHom_eq_support_sum`, `HahnEmbedding.Partial.truncLT_mem_range_sSupFun`, `Ideal.IsPrime.prod_mem_iff`, `Ideal.mem_quotient_iff_mem`, `FG.rTensor.directLimit_apply`, `PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal`, `AlgebraicGeometry.Scheme.zeroLocus_iInf_of_nonempty`, `groupCohomology.isoCocycles₂_hom_comp_i`, `coe_continuous_linearProjOfClosedCompl`, `EuclideanGeometry.orthogonalProjection_subtype`, `LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict`, `PrimeSpectrum.exist_mem_one_of_mem_two`, `HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorphOfComplemented_source`, `LieModule.Cohomology.d₁₂_apply_coe_apply_apply`, `coe_orthogonalDecomposition`, `EuclideanGeometry.two_zsmul_oangle_self_orthogonalProjection`, `rank_submodule_le_one_iff'`, `mem_singleton_set_smul`, `groupCohomology.π_comp_H0Iso_hom_apply`, `IsSMulRegular.submodule`, `finiteDimensional_direction_affineSpan_insert`, `Rep.subtype_hom_toFun`, `QuadraticForm.finrank_radical_of_equiv_weightedSumSquares`, `StrongDual.polarSubmodule_eq_polar`, `finrank_orthogonal_span_singleton`, `inclusionSpan_apply_coe`, `groupHomology.coe_mapCycles₂`, `Ideal.Cotangent.lift_toCotangent`, `coe_topologicalNilradical`, `Ideal.add_pow_mem_of_pow_mem_of_le`, `small_span_singleton`, `comap_subtype_self`, `exteriorPower.oneEquiv_naturality`, `NumberField.Units.dirichletUnitTheorem.unitLattice_span_eq_top`, `SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux`, `one_le_finrank_iff`, `IsLocalizedModule.mem_ker_iff`, `groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply`, `PrimeSpectrum.exist_ltSeries_mem_one_of_mem_last`, `groupHomology.H1π_comp_map_apply`, `MvPolynomial.mem_pow_idealOfVars_iff'`, `Affine.Simplex.ExcenterExists.excenter_restrict`, `LinearMap.mem_eqLocus`, `IsPrincipal.generator_submoduleImage_dvd_of_mem`, `PointedCone.basis_coord_mem_dual`, `norm_subtypeL`, `RootPairing.posRootForm_posForm_pos_of_ne_zero`, `mem_orthogonal'`, `biUnion_associatedPrimes_eq_zero_divisors`, `LinearPMap.sub_apply`, `Module.Flat.eqLocus_lTensor_eq`, `LinearMap.lift_rank_eq_of_surjective`, `Subspace.dualRestrict_leftInverse`, `TensorProduct.exists_finite_submodule_left_of_finite`, `Ideal.mem_jacobson_iff`, `IsCompl.projection_eq_self_iff`, `Module.instFiniteSubtypeMemIdealOfIsNoetherian`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem_ne`, `AffineSubspace.toContinuousAffineMap_subtypeₐᵢ`, `rank_mono`, `LieModule.Cohomology.mem_twoCochain_iff`, `Ideal.polynomial_mem_ideal_of_coeff_mem_ideal`, `LinearMap.comap_codRestrict`, `ContinuousLinearMap.coe_rangeRestrict`, `setLike.coe_galgebra_toFun`, `norm_sq_eq_add_norm_sq_projection`, `finiteDimensional_direction_affineSpan_range`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul`, `Polynomial.det_taylorLinearEquiv_toLinearMap`, `traceDual_le_span_map_traceDual`, `linearDepOn_iff`, `ContinuousLinearMap.equivRange_symm_toLinearEquiv`, `MvPolynomial.decompose'_apply`, `EuclideanGeometry.orthogonalProjection_vadd_eq_self`, `Ideal.mul_le`, `mem_colon_iff_le`, `groupCohomology.cocyclesMap_comp_isoCocycles₁_hom`, `TwoSidedIdeal.mem_fromIdeal`, `groupHomology.toCycles_comp_isoCycles₁_hom_assoc`, `LinearEquiv.ofInjective_apply`, `Module.length_submodule`, `FiniteArchimedeanClass.mem_ball_iff`, `comm_trans_rTensorOne`, `Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_aux₂`, `DirectSum.decompose_symm_bracket`, `starProjection_apply`, `OrzechProperty.injective_of_surjective_of_submodule`, `card_quotient_mul_card_quotient`, `AlgebraicGeometry.Scheme.zeroLocus_radical`, `FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mul`, `HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorph_target`, `coe_finsetInf`, `ZLattice.summable_norm_rpow`, `Ideal.IsPrime.mul_notMem`, `StandardEtalePair.existsUnique_hasMap_of_hasMap_quotient_of_sq_eq_bot`, `linearIndependent_span`, `WittVector.mem_ker_truncate`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_self`, `mem_span_set`, `MatrixModCat.toModuleCat_obj_carrier`, `Subspace.flip_quotDualCoannihilatorToDual_bijective`, `PrimeSpectrum.zeroLocus_ideal_mem_irreducibleComponents`, `groupCohomology.cocyclesIso₀_inv_comp_iCocycles_assoc`, `apply_mem_map₂`, `PowerSeries.coeff_mul_mem_ideal_of_coeff_right_mem_ideal`, `comap_equiv_self_of_inj_of_le_apply`, `Ideal.mem_prime_of_mul_mem_pow`, `rank_quotient_add_rank_of_isDomain`, `pinGroup.conjAct_smul_ι_mem_range_ι`, `groupCohomology.cocycles₁_map_inv`, `Module.Basis.sumQuot_repr_inl_of_mem`, `exists_fg_le_subset_range_rTensor_subtype`, `DirectSum.coeAlgHom_of`, `TensorProduct.rTensor_injective_of_forall_vanishesTrivially`, `isArtinian_of_fg_of_artinian`, `set_smul_eq_map`, `Ideal.mul_add_mem_pow_succ_unique`, `Polynomial.annIdealGenerator_mem`, `Affine.Triangle.dist_orthocenter_reflection_circumcenter`, `groupCohomology.mapCocycles₁_one`, `Ideal.symm_apply_mem_of_equiv_iff`, `IsDedekindDomain.HeightOneSpectrum.intAdicAbv_lt_one_iff`, `Polynomial.aeval_apply_smul_mem_of_le_comap`, `RingHom.toKerIsLocalization_apply`, `LinearMap.coe_range`, `EuclideanGeometry.oangle_orthogonalProjection_self`, `Affine.Simplex.map_altitude_restrict`, `LinearMap.mem_span_iff_continuous_of_finite`, `MvPolynomial.monomial_mem_pow_idealOfVars_iff`, `Ideal.mem_leadingCoeff`, `Affine.Simplex.faceOpposite_restrict`, `groupHomology.H2π_comp_map_assoc`, `neg_mem_iff`, `Polynomial.degreeLT.addLinearEquiv_apply`, `PrimeSpectrum.zeroLocus_iSup`, `PeriodPair.summable_weierstrassPSummand`, `Ideal.ResidueField.exists_smul_eq_tmul_one`, `fst_orthogonalDecomposition_apply`, `LinearEquiv.mem_dilatransvections_iff_finrank`, `DividedPowers.RatAlgebra.dpow_apply`, `Matrix.rank_eq_finrank_span_row`, `RootPairing.prod_rootForm_smul_coroot_mem_range_domRestrict`, `smul_starProjection_singleton`, `Polynomial.mem_annIdeal_iff_aeval_eq_zero`, `PeriodPair.hasSumLocallyUniformly_weierstrassPExcept`, `finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply`, `PointedCone.convex`, `LinearEquiv.ofSubmodules_symm_apply`, `IsSemisimpleModule.exists_quotient_linearEquiv_submodule`, `Ideal.homogeneousHull_eq_iSup`, `Matrix.isRepresentation.toEnd_exists_mem_ideal`, `LinearMap.kerComplementEquivRange_apply_coe`, `mem_mul_span_singleton`, `LinearDisjoint.rank_inf_le_one_of_commute_of_flat_right`, `mem_span_finset'`, `restrictScalarsEquiv_symm_apply`, `mem_span_singleton_self`, `finiteDimensional_vectorSpan_insert_set`, `LinearMap.quotientInfEquivSupQuotient_injective`, `map₂_eq_span_image2`, `IsAdic.hasBasis_nhds_zero`, `isArtinian_of_submodule_of_artinian`, `Module.Basis.restrictScalars_repr_apply`, `CliffordAlgebra.GradedAlgebra.ι_apply`, `orthogonalFamily_self`, `EuclideanGeometry.dist_orthogonalProjection_eq_dist_iff_eq_of_mem`, `DirectSum.toBaseChange_bijective`, `LieModule.Cohomology.d₁₂_apply_apply_ofTrivial`, `GradedTensorProduct.auxEquiv_mul`, `IsSelfAdjoint.dense_domain`, `AlgebraicGeometry.Scheme.zeroLocus_inf`, `lTensorOne'_tmul`, `Module.End.submodule_pow_eq_zero_of_pow_eq_zero`, `image_comap_zeroLocus_eq_zeroLocus_comap`, `RootPairing.CoPolarizationIn_eq`, `affineSpan_subset_span`, `AffineSubspace.abs_signedInfDist_eq_dist_of_mem_affineSpan_insert`, `Ideal.mul_homogeneous_element_mem_of_mem`, `Affine.Simplex.exradius_restrict`, `Subrepresentation.mem_asSubmodule_iff`, `rTensorOne'_tmul_one`, `inner_orthogonalProjection_eq_of_mem_right`, `DirectSum.decompose_symm_algebraMap`, `LinearMap.ker_domRestrict`, `Ideal.IsTwoSided.mul_mem_of_left`, `ContinuousMap.adjoin_id_eq_span_one_union`, `ContinuousMap.AlgHom.closure_ker_inter`, `Subbimodule.coe_mk`, `finiteDimensional_vectorSpan_singleton`, `reesAlgebra.monomial_mem`, `PrimeSpectrum.zeroLocus_eq_iff`, `exists_max_ideal_of_mem_nonunits`, `LieDerivation.IsKilling.ad_mem_orthogonal_of_mem_orthogonal`, `Ideal.IsPrime.notMem_of_isCoprime_of_mem`, `rTensorOne'_tmul`, `isScalarTower'`, `finiteDimensional_inf_right`, `toSubalgebra_mk`, `Finsupp.linearCombinationOn_range`, `Algebra.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex`, `colon_span`, `LinearPMap.coe_smul`, `coe_mem`, `TensorProduct.exists_finite_submodule_right_of_setFinite'`, `Ideal.isCompact_of_fg`, `realPart_comp_subtype_selfAdjoint`, `groupCohomology.π_comp_H0Iso_hom_assoc`, `ZSpan.coe_floor_self`, `projective_units`, `RootPairing.algebraMap_rootFormIn`, `Ideal.mem_matrix`, `TwoSidedIdeal.mem_asIdealOpposite`, `RingPreordering.one_notMem_support`, `map_subtype_embedding_eq`, `le_linearEquiv_of_le_sSup`, `Ideal.exists_le_prime_notMem_of_isIdempotentElem`, `MvPolynomial.mem_restrictDegree`, `ModuleCat.imageIsoRange_hom_subtype_assoc`, `groupCohomology.mem_cocycles₂_iff`, `FractionalIdeal.mem_coeIdeal`, `LinearDisjoint.injective`, `baseChange_eq_span`, `PointedCone.coe_lineal`, `ZLattice.module_free`, `LinearMap.mem_submoduleImage`, `dualCopairing_apply`, `ClosedComplemented.isClosed_complement`, `LinearEquiv.toSpanNonzeroSingleton_apply`, `IsLocalization.mk'_mem_map_algebraMap_iff`, `Affine.Simplex.orthogonalProjection_circumcenter`, `mem_span_insert'`, `finite_iff_ideal_quotient`, `Ideal.coe_restrictScalars`, `Module.Flat.instSubalgebraToSubmodule`, `neg_mem`, `ContinuousLinearMap.coe_linearMap_equivRange`, `Ideal.mk_mem_cotangentIdeal`, `LinearPMap.map_zero`, `MvPolynomial.indicator_mem_restrictDegree`, `coe_mapAlgEquiv_apply`, `LinearMap.BilinForm.nondegenerate_restrict_iff_disjoint_ker`, `LinearPMap.map_add`, `LinearIndependent.notMem_span`, `LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'`, `LinearIsometryEquiv.completeSpace_map`, `groupCohomology.H2π_comp_map_assoc`, `AdicCompletion.ofPowSMul_injective`, `HahnEmbedding.ArchimedeanStrata.nontrivial_stratum`, `conductor_subset_adjoin`, `NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm`, `Ideal.IsMaximal.mem_pow_mul`, `ker_inclusion`, `AffineIndependent.finrank_vectorSpan_image_finset`, `exteriorPower.alternatingMapLinearEquiv_apply_ιMulti`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_liftK_hom`, `FractionalIdeal.isNoetherian_zero`, `groupHomology.mapCycles₂_comp_apply`, `DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top`, `Algebra.adjoin_toSubmodule_le`, `LinearIsometryEquiv.toSpanUnitSingleton_apply`, `comap_subtype_eq_top`, `AffineSubspace.subtypeA_toAffineMap`, `Ideal.isOpen_pow_of_isMaximal`, `LinearIndependent.span_repr_eq`, `Affine.Simplex.exists_forall_dist_eq_iff_exists_excenterExists_and_eq_excenter`, `Ideal.sum_mem`, `norm_inner_eq_norm_tfae`, `LinearPMap.mkSpanSingleton'_apply`, `RootPairing.rootForm_self_eq_zero_iff`, `prodEquivOfIsCompl_symm_apply`, `Module.End.hasEigenvector_iff`, `injective_tensorToSpan`, `Representation.linHom.invariantsEquivRepHom_apply`, `Ideal.mem_leadingCoeffNth`, `projective_of_isUnit`, `PeriodPair.derivWeierstrassP_sub_coe`, `Module.Basis.restrictScalars_apply`, `RootPairing.invtRootSubmodule.eq_bot_iff`, `instIsSemisimpleModuleSubtypeMemSubmoduleIsotypicComponent`, `HasStrictFDerivAt.to_implicitFunction`, `exteriorPower.ιMultiDual_apply_ιMulti`, `natAbs_det_equiv`, `Algebra.zariskisMainProperty_iff_exists_saturation_eq_top`, `Module.Basis.mem_submodule_iff'`, `smithNormalFormBotBasis_def`, `Ideal.exists_not_mem_forall_mem_of_ne_of_liesOver`, `Rep.resCoindHomEquiv_symm_apply`, `MvPolynomial.pointToPoint_zeroLocus_le`, `Module.End.IsFinitelySemisimple.restrict`, `map_subtype_span_singleton`, `Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq`, `AdicCompletion.exists_smodEq_pow_add_one_smul`, `Ideal.isMaximal_iff`, `LinearPMap.adjointDomainMkCLM_apply`, `LinearPMap.mem_domain_iff`, `PointedCone.mem_map`, `LocalizedModule.exists_subsingleton_away`, `mem_set_smul_of_mem_mem`, `MatrixModCat.mem_toModuleCatObj`, `starProjection_bot`, `Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent`, `linearIndependent_fin_succ'`, `IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_mem`, `LinearIndepOn.notMem_span_of_insert`, `exteriorPower.ιMulti_family_eq_coe_comp`, `exteriorPower.pairingDual_ιMulti_ιMulti`, `mem_skewAdjointMatricesLieSubalgebra`, `groupHomology.H2π_eq_iff`, `span_eq`, `mulRightMap_eq_mulMap_comp`, `mod_mem_iff`, `le_isotypicComponent`, `LinearMap.IsSymmetric.orthogonalFamily_eigenspace_inf_eigenspace`, `Ideal.mul_mem_mul`, `ZLattice.covolume.tendsto_card_le_div'`, `LinearMap.mem_range_self`, `isNoetherian_submodule'`, `QuadraticMap.IsometryEquiv.map_posDef_iff`, `mem_inf`, `exteriorPower.alternatingMapLinearEquiv_comp_ιMulti`, `EuclideanGeometry.oangle_eq_of_dist_orthogonalProjection_eq`, `rank_span_finset_le`, `PointedCone.coe_closure`, `groupHomology.H1AddEquivOfIsTrivial_single`, `Int.absNorm_under_mem`, `mem_span_insert_exchange`, `isLinearTopology_iff_hasBasis_ideal`, `PadicInt.zmodRepr_spec`, `groupCohomology.mem_cocycles₁_iff`, `Ideal.quotientToQuotientRangePowQuotSucc_surjective`, `coe_iInf`, `MvPolynomial.monomial_mem_coeffsIn`, `Ideal.span_eq`, `IsAdjoinRoot.repr_zero_mem_span`, `coe_subset_nonunits`, `MeasureTheory.integral_condExpL2_eq_of_fin_meas_real`, `Ideal.Filtration.submodule_span_single`, `finrank_map_le`, `HahnEmbedding.Seed.coeff_baseEmbedding`, `finrank_span_eq_card`, `starProjection_unit_singleton`, `groupCohomology.mem_cocycles₂_of_comp_eq_d₁₂`, `LinearIsometryEquiv.ofTop_apply`, `prodComm_trans_prodEquivOfIsCompl`, `groupHomology.isoCycles₂_hom_comp_i_apply`, `LinearPMap.fst_apply`, `PointedCone.tmul_subset_maxTensorProduct`, `Ideal.span_singleton_le_iff_mem`, `DirectSum.decompose_algebraMap`, `LinearMap.compLeftInverse_apply_of_bdd`, `groupCohomology.π_comp_H0IsoOfIsTrivial_hom_apply`, `RootPairing.zero_le_posForm`, `finrank_eq_zero`, `Module.length_bot`, `CliffordAlgebra.ι_mul_ι_mem_evenOdd_zero`, `PowerBasis.mem_span_pow`, `rTensorOne_tmul`, `ContinuousLinearEquiv.toSpanNonzeroSingleton_symm_apply`, `MonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_support_val`, `annihilator_top_inter_nonZeroDivisors`, `Rep.coinvariantsShortComplex_f`, `groupCohomology.toCocycles_comp_isoCocycles₁_hom_assoc`, `Module.End.restrict_eigenspace`, `reflection_sub`, `exists_fun_fin_finrank_span_eq`, `EuclideanGeometry.Sphere.mem_inter_orthRadius_iff_radius_nonneg_and_vsub_mem_and_norm_sq`, `groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply`, `Ideal.hasBasis_nhds_zero_adic`, `groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_assoc`, `AffineSubspace.coe_vadd`, `Ideal.add_eq_one_iff`, `Ideal.instIsTorsionFreeSubtypeMemSubmodule`, `LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_left`, `Ideal.add_pow_add_pred_mem_of_pow_mem_of_commute`, `Module.Finite.of_isComplemented_domain`, `Polynomial.degreeLT.addLinearEquiv_apply'`, `affineIndependent_iff_le_finrank_vectorSpan`, `Subspace.quotAnnihilatorEquiv_apply`, `skewAdjointPart_comp_subtype_selfAdjoint`, `groupCohomology.isoCocycles₁_inv_comp_iCocycles`, `unitsToPic_apply`, `Affine.Simplex.circumcenter_restrict`, `Ideal.mem_span_singleton_sup`, `Module.isTorsionBySet_iff_subseteq_ker_lsmul`, `instIsScalarTowerQuotientSpanSingletonSetSubtypeMemTorsionBy`, `coe_zero`, `AffineMap.restrict.injective`, `FractionalIdeal.exists_mem_algebraMap_eq`, `Module.End.mem_genEigenspace_top`, `coe_mulMap_comp_eq`, `Affine.Simplex.height_restrict`, `exteriorPower.oneEquiv_symm_apply`, `ModuleCat.imageIsoRange_inv_image_ι`, `instModuleFinite_of_discrete_submodule`, `LinearMap.finrank_range_le`, `isIdempotentElemEquiv_apply_coe`, `mem_toAddSubgroup`, `TensorProduct.exists_finite_submodule_left_of_finite'`, `LinearPMap.add_apply`, `EuclideanGeometry.orthogonalProjection_affineSpan_singleton`, `Rat.HeightOneSpectrum.natGenerator_dvd_iff`, `sectionOfRetractionKerToTensorAux_prop`, `LinearEquiv.submoduleMap_apply`, `ContinuousLinearEquiv.ofSubmodule'_apply`, `mem_of_span_top_of_smul_mem`, `Module.Flat.iff_lTensor_injective`, `LinearPMap.inverse_range`, `groupHomology.cyclesMap_comp_isoCycles₂_hom_assoc`, `Module.Basis.mem_span_image`, `LieModule.Cohomology.mem_twoCocycle_iff`, `Rep.coindVEquiv_apply`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_notMem`, `groupCohomology.cocyclesMap_comp_isoCocycles₂_hom`, `exists_isInternal_prime_power_torsion_of_pid`, `TensorProduct.eq_zero_of_fg_of_subtype_eq_zero`, `PeriodPair.weierstrassP_add_coe`, `mem_span_mul_finite_of_mem_span_mul`, `Ideal.natCast_mem_of_charP_quotient`, `Ideal.Quotient.eq_zero_iff_mem`, `closure_subset_span`, `exteriorPower.map_comp_ιMulti`, `AffineSubspace.subtype_injective`, `PrimeSpectrum.isClosed_iff_zeroLocus_ideal`, `groupCohomology.coboundariesOfIsCoboundary₁_coe`, `Module.End.isFinitelySemisimple_iff'`, `AffineSubspace.subtype_linear`, `Ideal.basisSpanSingleton_apply`, `TensorProduct.quotientTensorQuotientEquiv_apply_tmul_mk_tmul_mk`, `colon_eq_top_iff_subset`, `AdicCompletion.val_apply_mem_smul_top_iff`, `PeriodPair.instProperSpaceSubtypeComplexMemSubmoduleIntLattice`, `coe_inner`, `Ideal.pow_mem_of_mem`, `torsionBySet_ideal_span_singleton_eq`, `Subrepresentation.apply_mem_toSubmodule`, `DirectSum.decomposeLinearEquiv_apply_coe`, `exists_smul_notMem_of_rank_lt`, `orthogonalProjection_apply_eq_linearProjOfIsCompl`, `ZSpan.span_top`, `TensorProduct.tensorQuotientEquiv_apply_mk_tmul`, `isSMulRegular_quotient_iff_mem_of_smul_mem`, `IsLattice.span_eq_top`, `ExteriorAlgebra.instGradedMonoidNatSubmoduleExteriorPower`, `Algebra.Presentation.relation_mem_ker`, `TensorProduct.exists_finite_submodule_of_finite`, `MvPolynomial.mem_map_C_iff`, `MeasureTheory.condExpL2_indicator_of_measurable`, `LinearDisjoint.linearIndependent_mul_of_flat_right`, `SchauderBasis.RankOneDecomposition.finrank_range`, `derivationToSquareZeroEquivLift_symm_apply_apply_coe`, `GradedTensorProduct.mulHom_apply`, `EuclideanGeometry.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq`, `MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictDegreeOfFinite`, `Ideal.exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff`, `exists_mem_ne_zero_of_ne_bot`, `closure_coe_iSup_map_single`, `Ideal.isCoprime_tfae`, `MvPolynomial.mem_restrictSupport_iff`, `LinearMap.eqOn_span_iff`, `linearEquiv_of_sSup_eq_top`, `AddMonoidAlgebra.GradesBy.decompose_single`, `ContinuousMultilinearMap.codRestrict_apply_coe`, `AffineSubspace.topEquiv_symm_apply_coe`, `coe_isComplEquivProj_apply`, `AffineSubspace.isometryEquivMap.coe_apply`, `smul_mem_iff_of_isUnit`, `Subbimodule.coe_toSubbimoduleInt`, `IsCompactOperator.restrict`, `prodEquivOfIsCompl_symm_apply_left`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem`, `linearIndependent_fin_snoc`, `sndL_comp_coe_orthogonalDecomposition`, `RootPairing.PolarizationIn_apply`, `injective_subtype`, `LinearIndepOn.mem_span_iff_id`, `linearProjOfIsCompl_isCompl_projection`, `stereographic_neg_apply`, `LinearMap.mem_ker`, `quotientEquivOfIsCompl_apply_mk_coe`, `InnerProductSpace.gramSchmidt_def'`, `MeasureTheory.condExpL2_indicator_eq_toSpanSingleton_comp`, `groupCohomology.coboundaries₂.coe_mk`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem`, `LinearMap.coe_domRestrict`, `NumberField.Units.dirichletUnitTheorem.unitLattice_inter_ball_finite`, `Ideal.exists_forall_sub_mem_ideal`, `localized'_eq_span`, `Module.End.isNilpotent_restrict_maxGenEigenspace_sub_algebraMap`, `linearProjOfIsCompl_comp_subtype`, `Polynomial.coeff_mem_pow_of_mem_adjoin_C_mul_X`, `Ring.HasFiniteQuotients.finite_setOf_mem`, `Ideal.mem_of_one_mem`, `torsionBySet_isInternal`, `mem_span_finite_of_mem_span`, `IsSemisimpleModule.sSup_simples_le`, `Representation.averageMap_invariant`, `Ideal.IsPrime.mem_or_mem`, `AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact`, `rank_eq_zero`, `IsLattice.finrank_of_pi`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inl`, `coe_subtypeₗᵢ`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inr_basis`, `LinearMap.IsSymmetric.directSum_isInternal_of_commute`, `LinearMap.iInf_invariant`, `MvPolynomial.weightedHomogeneousComponent_directSum`, `IsSemisimpleModule.exists_simple_submodule`, `smul_closure_subset`, `MeasureTheory.instCompleteSpaceSubtypeAEEqFunMemAddSubgroupLpSubmoduleLpMeasOfFactLeMeasurableSpace`, `AddMonoidAlgebra.mem_grade_iff'`, `Ideal.coe_toTwoSided`, `ZSpan.exist_unique_vadd_mem_fundamentalDomain`, `groupHomology.mapCycles₂_id_comp_assoc`, `groupHomology.π_comp_H1Iso_hom_apply`, `LinearMap.quotKerEquivRange_symm_apply_image`, `Finsupp.restrictDom_comp_subtype`, `Module.Finite.span_singleton`, `span_eq_iUnion_nat`, `MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le_real`, `Module.isTorsionBySet_iff_is_torsion_by_span`, `PointedCone.tmul_subset_minTensorProduct`, `rTensor_mkQ`, `Representation.coindMap_coe_apply_apply`, `map_le_isotypicComponent`, `Collinear.finiteDimensional_vectorSpan`, `Polynomial.Monic.leadingCoeff_notMem`, `Valuation.Integers.maximalIdeal_eq_setOf_le_v_algebraMap`, `SchauderBasis.finrank_range_succSub_eq_one`, `groupCohomology.map_id_comp_H0Iso_hom_apply`, `groupCohomology.cocycles₁_map_mul_of_isTrivial`, `exteriorPower.pairingDual_apply_apply_eq_one`, `Algebra.IsSmoothAt.exists_notMem_smooth`, `groupCohomology.subtype_comp_d₀₁_assoc`, `complete_of_finiteDimensional`, `Module.Flat.tensorProduct_mapIncl_injective_of_left`, `imaginaryPart_comp_subtype_selfAdjoint`, `iSupIndep.linearEquiv_apply`, `ZLattice.summable_norm_pow_inv`, `Localization.exists_awayMap_bijective_of_residueField_surjective`, `groupHomology.toCycles_comp_isoCycles₂_hom`, `ContinuousLinearEquiv.coord_toSpanNonzeroSingleton`, `HasRankNullity.rank_quotient_add_rank`, `groupCohomology.map_id_comp_H0Iso_hom`, `convex`, `Module.Flat.iff_lTensor_injectiveₛ`, `groupHomology.cyclesMap_comp_isoCycles₂_hom_apply`, `coe_sub`, `mem_smul_pointwise_iff_exists`, `IsPrimary.radical_colon_singleton_eq_ite`, `CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_assoc_apply`, `torsionBy.mk_ideal_smul`, `Span.finsupp_linearCombination_repr`, `SetLike.algebraMap_mem_graded`, `mem_span_iff_of_fintype`, `EuclideanGeometry.eq_or_eq_reflection_of_dist_eq`, `Affine.Simplex.incenter_restrict`, `PrimeSpectrum.zeroLocus_bot`, `LinearPMap.mem_graph_iff`, `isClosed_orthogonal`, `IsLattice.rank_of_pi`, `mem_vectorSpan_pair_rev`, `groupHomology.mapCycles₁_id_comp`, `SetLike.GradeZero.coe_algebraMap`, `AdicCompletion.ofPowSMul_val_apply_eq_zero`, `InnerProductGeometry.angle_eq_angle_add_angle_iff`, `coe_torsionBySet`, `mem_dualCoannihilator`, `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`, `PointedCone.subset_span`, `tmul_mem_baseChange_of_mem`, `norm_orthogonalProjection_apply`, `toENat_rank_span_set`, `iSup_eq_span`, `CliffordAlgebra.reverse_mem_evenOdd_iff`, `LieModule.Cohomology.add_apply_apply`, `groupCohomology.cocycles₁IsoOfIsTrivial_inv_hom_apply_coe`, `EuclideanGeometry.orthogonalProjection_vadd_smul_vsub_orthogonalProjection`, `torsionBySet_torsionBySet_eq_top`, `LinearMap.singularValues_pos_iff_lt_finrank_range`, `MeasureTheory.condExpL1CLM_lpMeas`, `rTensorOne_tmul_one`, `Ideal.prod_mem_prod`, `toLinearMap_prodEquivOfIsCompl_symm`, `LinearEquiv.fixedReduce_eq_smul_iff`, `Ideal.Quotient.eq`, `Ideal.IsHomogeneous.isPrime_iff`, `PeriodPair.latticeBasis_one`, `Subbimodule.coe_toSubbimoduleNat`, `ClosedSubmodule.mem_orthogonal_toSubmodule_iff`, `diffToIdealOfQuotientCompEq_apply`, `finrank_eq_rank`, `ZLattice.covolume_eq_det_mul_measureReal`, `Ideal.eq_jacobson_iff_notMem`, `LinearPMap.map_neg`, `Subspace.finrank_add_finrank_dualCoannihilator_eq`, `DirectSum.IsInternal.toBaseChange_bijective`, `MvPolynomial.mem_weightedHomogeneousSubmodule`, `RingHom.surjectiveOnStalks_iff_forall_maximal'`, `mem_span_singleton_mul`, `multiple_mem_span_of_mem_localization_span`, `groupCohomology.cocyclesOfIsMulCocycle₂_coe`, `Ideal.IsPrime.multiset_prod_mem_iff_exists_mem`, `Function.Exact.linearMap_rangeRestrict`, `PeriodPair.weierstrassP_sub_coe`, `Module.End.invtSubmodule.map_subtype_mem_of_mem_invtSubmodule`, `Ideal.primaryComponent_map_mem`, `Affine.Simplex.orthogonalProjectionSpan_faceOpposite_eq_point_rev`, `Module.Finite.range`, `ZSpan.instDiscreteTopologySubtypeMemSubmoduleIntSpanRangeCoeBasisRealOfFinite`, `TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective`, `Valued.integer.coe_span_singleton_eq_closedBall`, `Ideal.IsPrime.mem_or_mem_of_mul_eq_zero`, `PadicInt.mem_span_pow_iff_le_valuation`, `LinearDisjoint.linearIndependent_mul_of_flat_left`, `LinearIndependent.linearCombinationEquiv_symm_apply`, `equivSubtypeMap_apply`, `ClosedComplemented.isClosed`, `ZSpan.isAddFundamentalDomain`, `Algebra.Extension.Hom.mapKer_apply_coe`, `NumberField.discr_mem_differentIdeal`, `EuclideanGeometry.dist_orthogonalProjection_eq_infNndist`, `isIdempotentElemEquiv_symm_apply_coe`, `AddSubgroup.coe_toZModSubmodule`, `mem_skewAdjointMatricesSubmodule`, `MvPolynomial.mem_coeffsIn`, `FiniteDimensional.finiteDimensional_submodule`, `PrimeSpectrum.range_comap_snd`, `abs_signedDist_eq_dist_iff_vsub_mem_span`, `groupCohomology.coboundariesOfIsMulCoboundary₁_coe`, `Algebra.Extension.contangentEquiv_tmul`, `IsLinearTopology.hasBasis_submodule'`, `coe_inf`, `Ideal.add_mem_iff_right`, `NumberField.Units.instDiscrete_unitLattice`, `LinearMap.IsSymmetric.direct_sum_isInternal`, `det_reflection`, `Representation.linHom.invariantsEquivRepHom_symm_apply_coe`, `LinearPMap.map_sub`, `LinearMap.quotKerEquivRange_apply_mk`, `sum_mem_biSup`, `mem_of_isLocalized_span`, `Algebra.Generators.cMulXSubOneCotangent_eq`, `rank_span_le`, `FractionalIdeal.coeToSet_coeToSubmodule`, `NumberField.mixedEmbedding.euclidean.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIntegerLattice`, `ValuationSubring.coe_mem_nonunits_iff`, `Profinite.NobelingProof.GoodProducts.finsuppSum_mem_span_eval`, `ContinuousLinearEquiv.toSpanNonzeroSingleton_coord`, `Ideal.toCotangent_apply`, `inner_orthogonalProjection_eq_of_mem_left`, `LieAlgebra.InvariantForm.restrict_orthogonal_nondegenerate`, `rank_submodule_eq_one_iff`, `Module.End.mem_iInf_maxGenEigenspace_iff`, `TensorProduct.spanFinrank_top_le_of_fg`, `groupHomology.isoCycles₁_hom_comp_i_apply`, `Polynomial.isEisensteinAt_iff`, `starConvex`, `PowerSeries.IsWeierstrassDivisorAt.coeff_seq_mem`, `AffineMap.restrict.surjective`, `Affine.Simplex.inradius_restrict`, `Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq`, `Finsupp.supportedEquivFinsupp_symm_apply_coe_support_val`, `groupCohomology.cocyclesIso₀_hom_comp_f_assoc`, `PointedCone.ext_iff`, `AffineSubspace.subtypeₐᵢ_linear`, `ZLattice.exists_forall_abs_repr_le_norm`, `linearIndependent_fin_succ`, `AffineMap.restrict.coe_apply`, `linearMap_eq_iff_of_eq_span`, `CategoryTheory.ShortComplex.exact_iff_surjective_moduleCatToCycles`, `Algebra.IsEtaleAt.exists_isStandardEtale`, `ValuationSubring.mem_nonunits_iff_exists_mem_maximalIdeal`, `PrimeSpectrum.zeroLocus_mul`, `RootPairing.prod_rootFormIn_smul_coroot_mem_range_PolarizationIn`, `ModuleCat.imageIsoRange_inv_image_ι_assoc`, `Ideal.eq_top_iff_one`, `AlgebraicGeometry.ProjectiveSpectrum.Proj.mk_mem_toSpec_base_apply`, `IsSemisimpleModule.sSup_simples_eq_top`, `groupCohomology.H1π_eq_zero_iff`, `AddMonoidAlgebra.single_mem_grade`, `PeriodPair.hasSum_weierstrassP`, `mem_right_iff_eq_zero_of_disjoint`, `DirectSum.IsInternal.subordinateOrthonormalBasis_def`, `ProjectiveSpectrum.zeroLocus_bot`, `PeriodPair.coeff_weierstrassPSeries`, `AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine`, `Algebra.Extension.h1Cotangentι_ext_iff`, `PeriodPair.isOpen_compl_lattice_diff`, `Ideal.coe_comap`, `Ideal.toCotangent_eq_zero`, `exteriorPower.ιMulti_family_span_of_span`, `NumberField.fractionalIdeal_rank`, `GradedAlgHom.coe_mk'`, `LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace`, `toLinearMap_orthogonalProjection_eq_linearProjOfIsCompl`, `Ideal.mul_unit_mem_iff_mem`, `LinearMap.finrank_range_adjoint`, `MvPolynomial.decomposition.decompose'_apply`, `groupCohomology.coboundaries₁.val_eq_coe`, `PeriodPair.derivWeierstrassP_coe`, `mulRightMap_apply`, `MvPolynomial.mem_degreesLE`, `LieHom.quotKerEquivRange_invFun`, `Module.isTorsionBy_iff_mem_ker_lsmul`, `IsLattice.free`, `LinearPMap.map_smul`, `Ideal.pow_mem_of_pow_mem`, `Module.injOn_dualMap_subtype_span_range_range`, `groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_apply`, `mem_affineSpan_iff_exists`, `map₂_le`, `LieSubalgebra.coe_ofLe`, `Real.Convex.dimH_eq_finrank_vectorSpan`, `Ideal.zero_mem`, `nontrivial_span_singleton`, `quotientEquivOfIsCompl_symm_apply`, `Ideal.isCoprime_iff_exists`, `Polynomial.aeval_apply_smul_mem_of_le_comap'`, `Ideal.mem_primeCompl_iff`, `exists_smith_normal_form_of_rank_eq`, `MvPolynomial.monomial_mem_restrictSupport`, `DividedPowers.SubDPIdeal.dpow_mem`, `RootPairing.GeckConstruction.diagonal_elim_mem_span_h_iff`, `GradedTensorProduct.auxEquiv_tmul`, `Ideal.cotangentEquivIdeal_apply`, `ModuleCat.kernelIsoKer_hom_ker_subtype`, `exteriorPower.map_injective_field`, `subset_span_finite_of_subset_span`, `groupHomology.single_ρ_self_add_single_inv_mem_boundaries₁`, `groupHomology.H1ToTensorOfIsTrivial_H1π_single`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_assoc_apply`, `EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangent`, `Ideal.exists_subset_radical_span_sup_of_subset_radical_sup`, `Algebra.Extension.H1Cotangent.val_add`, `Subspace.dualLift_of_mem`, `LinearEquiv.coe_ofEq_apply`, `Polynomial.degreeLT.addLinearEquiv_symm_apply`, `coord_norm'`, `RootPairing.algebraMap_posRootForm_posForm`, `DirectSum.IsInternal.collectedOrthonormalBasis_mem`, `PadicInt.norm_le_pow_iff_mem_span_pow`, `LinearMap.eqOn_sup`, `Ideal.sum_pow_mem_span_pow`, `coe_toIdeal`, `Ideal.apply_mem_of_equiv_iff`, `coe_torsionBy`, `NumberField.mixedEmbedding.volume_fundamentalDomain_fractionalIdealLatticeBasis`, `Rep.FiniteCyclicGroup.groupHomologyπEven_eq_zero_iff`, `Basis.mem_ideal_iff'`, `Ideal.quotientToQuotientRangePowQuotSucc_injective`, `groupCohomology.cocyclesMk₁_eq`, `add_mem_sup`, `Algebra.idealMap_isLocalizedModule`, `IsDedekindDomain.HeightOneSpectrum.intValuation_eq_one_iff`, `PrimeSpectrum.closure_image_comap_zeroLocus`, `range_powSMulQuotInclusion`, `RingHom.one_notMem_ker`, `DirectSum.decomposeLinearEquiv_apply`, `AddMonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_support_val`, `Subalgebra.finiteDimensional_toSubmodule`, `sectionOfRetractionKerToTensor_algebraMap`, `exists_leadingCoeff_pow_smul_mem_radical_conductor`, `exteriorPower.alternatingMapLinearEquiv_comp`, `collinear_iff_rank_le_one`, `Finsupp.linearCombination_restrict`, `Algebra.Extension.H1Cotangent.val_smul`, `PowerSeries.coeff_mul_mem_ideal_of_coeff_left_mem_ideal'`, `Ideal.add_pow_add_pred_mem_of_pow_mem`, `MvPolynomial.mem_homogeneousSubmodule`, `ContinuousLinearMap.HasLeftInverse.isClosed_complement`, `topologicalClosure_coe`, `smul_mem_iff'`, `groupHomology.cyclesOfIsCycle₁_coe`, `LinearEquiv.ofSubmodule'_toLinearMap`, `finrank_top`, `DirectSum.decomposeLinearEquiv_symm_comp_lof`, `exteriorPower.basis_coord`, `Ideal.norm_mem_spanNorm`, `FractionalIdeal.mem_span_mul_finite_of_mem_mul`, `ClosedSubmodule.mem_orthogonal_iff`, `EuclideanGeometry.orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem`, `LinearIndependent.repr_range`, `range_deriv_subset_closure_span_image`, `Algebra.Extension.Cotangent.mk_eq_mk_iff_sub_mem`, `IsDedekindDomain.HeightOneSpectrum.intAdicAbv_eq_one_iff`, `Polynomial.mem_iff_annIdealGenerator_dvd`, `ZLattice.covolume.tendsto_card_div_pow'`, `IsSemisimpleModule.exists_end_ringEquiv`, `Module.End.mem_eigenspace_iff`, `Module.isPrincipal_submodule_iff`, `Ideal.constr_basisSpanSingleton`, `Polynomial.coeff_divModByMonicAux_mem_span_pow_mul_span`, `Function.Exact.iff_linearMap_rangeRestrict`, `Module.isTorsionBySet_annihilator_top`, `Localization.exists_awayMap_injective_of_localRingHom_injective`, `LinearPMap.supSpanSingleton_apply_smul_self`, `Ideal.apply_coe_mem_map`, `RootPairing.range_polarizationIn`, `pow_mem_pow`, `RootPairing.restrictScalars_coe_root`, `mem_span_range_iff_exists_fun`, `exteriorPower.presentation_relation`, `PrimeSpectrum.closure_singleton`, `Ideal.mem_map_C_iff`, `mem_selfAdjointMatricesSubmodule`, `mulMap_one_left_eq`, `groupHomology.d₂₁_apply_mem_cycles₁`, `stereographic_target`, `starProjection_apply_mem`, `coe_set_mk`, `LieModule.Cohomology.smul_apply_apply`, `mem_sSup_of_mem`, `groupCohomology.coboundariesToCocycles₂_apply`, `LinearPMap.inverse_apply_eq`, `EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq`, `AddMonoidAlgebra.gradeBy.gradedMonoid`, `MvPolynomial.instFiniteSubtypeMemSubmoduleRestrictTotalDegreeOfFinite`, `quotDualCoannihilatorToDual_injective`, `Affine.Simplex.centroid_restrict`, `MultilinearMap.codRestrict_apply_coe`, `PrimeSpectrum.coe_primesOverOrderIsoFiber_apply_asIdeal`, `groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_apply`, `groupHomology.cyclesMap_comp_isoCycles₁_hom_apply`, `Module.AEval.mem_mapSubmodule_symm_apply`, `dfinsuppSum_mem`, `GradedAlgebra.coe_algebraMap_apply`, `UniqueFactorizationMonoid.iff_exists_prime_mem_of_isPrime`, `mem_toAddSubmonoid`, `groupHomology.toCycles_comp_isoCycles₂_hom_apply`, `groupCohomology.cocyclesOfIsMulCocycle₁_coe`, `LinearPMap.coprod_apply`, `DirectSum.toBaseChange_injective`, `TensorProduct.exists_finite_submodule_of_setFinite'`, `groupCohomology.H2π_eq_zero_iff`, `MvPolynomial.DirectSum.coeLinearMap_eq_finsum`, `Ideal.mul_add_mem_pow_succ_inj`, `LinearIsometryEquiv.submoduleMap_symm_apply_coe`, `LinearMap.disjoint_ker_iff_injOn`, `LieModule.Cohomology.instLinearMapClassSubtypeLinearMapIdMemSubmoduleTwoCochain`, `Subspace.dualRestrict_surjective`, `Coplanar.finiteDimensional_vectorSpan`, `groupCohomology.mapCocycles₁_comp_i_assoc`, `PointedCone.coe_ofConeComb`, `LinearMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal'`, `groupCohomology.cocycles₁.val_eq_coe`, `Ideal.mem_iInf`, `torsionBy_isTorsion_nonZeroDivisor`, `PointedCone.mem_span_set`, `AddSubgroup.coe_toIntSubmodule`, `ker_subtypeL`, `LinearMap.kerComplementEquivRange_symm_apply`, `LinearPMap.supSpanSingleton_apply_self`, `Polynomial.Chebyshev.T_iterate_derivative_mem_span_T`, `mem_orthogonal`, `groupCohomology.H1π_comp_map_apply`, `torsionBySet_le_torsionBySet_pow`, `StrongDual.mem_polarSubmodule`, `ProjectiveSpectrum.gc_ideal`, `LinearMap.lift_rank_range_add_rank_ker`, `mulMap_comm_of_commute`, `groupCohomology.cocycles₁_map_one`, `TensorAlgebra.GradedAlgebra.ι_apply`, `MeasureTheory.integrable_condExpL2_of_isFiniteMeasure`, `PeriodPair.latticeEquiv_symm_apply`, `HahnEmbedding.ArchimedeanStrata.isInternal_stratum'`, `groupCohomology.cocyclesMap_cocyclesIso₀_hom_f`, `Ideal.not_isPrime_iff`, `EuclideanGeometry.dist_orthogonalProjection_eq_infDist`, `exteriorPower.instFinite`, `EuclideanGeometry.orthogonalProjection_mem`, `LinearEquiv.coord_self`, `Ideal.subset_union_prime_finite`, `isCentralScalar`, `NumberField.canonicalEmbedding.mem_rat_span_latticeBasis`, `PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept`, `exteriorPower.alternatingMapToDual_apply_ιMulti`, `groupCohomology.π_comp_H2Iso_hom_assoc`, `Ideal.IsMaximal.mul_mem_pow`, `Affine.Simplex.orthogonalProjectionSpan_restrict`, `nonempty`, `isArtinian_submodule'`, `Representation.Coinvariants.mk_eq_zero`, `Ideal.smul_mem_pointwise_smul`, `isClosed_nullSubmodule`, `AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mk_mem_carrier`, `KaehlerDifferential.D_tensorProductTo`, `groupCohomology.toCocycles_comp_isoCocycles₂_hom_assoc`, `Ring.KrullDimLE.isNilpotent_iff_mem_maximalIdeal`, `nat_power_gradedMonoid`, `Ideal.mem_span`, `LinearEquiv.ofEq_symm`, `IsSemisimpleModule.exists_end_algEquiv`, `TensorProduct.spanFinrank_top_eq_of_residueField`, `groupHomology.mem_cycles₁_of_mem_boundaries₁`, `Ideal.mem_span_range_iff_exists_fun`, `IsCompl.projection_apply_eq_zero_iff`, `MvPolynomial.monomial_mem_homogeneousSubmodule_pow_degree`, `mulMap_eq_mul'_comp_mapIncl`, `DirectSum.decomposeLinearEquiv_symm_apply`, `existsUnique_from_graph`, `LinearEquiv.ofLeftInverse_apply`, `DirectSum.decompose_bracket`, `setBasisOfTopLeSpanOfCardEqFinrank_repr_apply`, `weightedVSub_mem_vectorSpan`, `liftOfDerivationToSquareZero_apply`, `MonoidAlgebra.supportedEquivFinsupp_apply_support_val`, `AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι`, `LieSubmodule.coe_lieSpan_submodule_eq_iff`, `Algebra.Extension.Cotangent.mk_surjective`, `small_iSup`, `Algebra.exists_formallyUnramified_of_isUnramifiedAt`, `Affine.Simplex.setInterior_restrict`, `isLinearTopology_iff_hasBasis_open_ideal`, `Ideal.exists_mul_add_sub_mem_of_mem_jacobson`, `AffineSubspace.finiteDimensional_sup`, `Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_aeval_zero_of_monic_mem_map`, `IsLocalization.smul_mem_iff`, `PrimeSpectrum.gc_set`, `mem_pNilradical`, `IsLocalization.orderIsoOfPrime_symm_apply_coe`, `LinearIndepOn.mem_span_iff`, `AddMonoidAlgebra.mem_grade_iff`, `rank_add_le_rank_add_rank`, `TwoSidedIdeal.mem_asIdeal`, `RootPairing.RootPositiveForm.isSymm_posForm`, `FractionalIdeal.isNoetherian_iff`, `mem_smul_top_iff`, `ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom`, `ZLattice.covolume.tendsto_card_le_div`, `ProjectiveSpectrum.zeroLocus_anti_mono_ideal`, `DirectSum.decomposeAlgEquiv_apply`, `torsionBySet_span_singleton_eq`, `AffineIndependent.finrank_vectorSpan`, `AddMonoidAlgebra.grade.decompose_single`, `mem_span_image_finset_iff_exists_fun`, `IsCompl.projection_apply`, `mem_annihilator_span`, `LinearDisjoint.not_linearIndependent_pair_of_flat`, `Module.Basis.self_mem_flag_iff`, `add_mem_iff_left`, `LinearMap.comp_codRestrict`, `InnerProductSpace.mem_span_gramSchmidt`, `mulLeftMap_eq_mulMap_comp`, `Representation.subrepresentation_apply`, `groupHomology.mapCycles₂_hom`, `DividedPowers.span_isSubDPIdeal_iff`, `HahnEmbedding.Partial.orderTop_eq_finiteArchimedeanClassMk`, `MeasureTheory.integrable_condExpL2_indicator`, `rank_quotient_add_rank_le`, `LieAlgebra.Extension.d₁₂_oneCochainOfTwoSplitting`, `LinearMap.mem_selfAdjointSubmodule`, `groupHomology.isoCycles₂_inv_comp_iCycles_apply`, `LinearMap.finrank_range_dualMap_eq_finrank_range`, `Subbimodule.coe_toSubmodule`, `Affine.Simplex.excenterWeights_restrict`, `groupHomology.isoCycles₂_inv_comp_iCycles_assoc`, `EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff`, `sub_mem_orthogonal_of_inner_right`, `AffineEquiv.ofEq_symm`, `Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_root_of_monic_mem`, `nontrivial_iff_ne_bot`, `image_span_subset`, `LinearIndepOn.notMem_span_iff_id`, `Algebra.Extension.Hom.subToKer_apply_coe`, `reflection_eq_self_iff`, `stereographic_source`, `ModuleCat.kernelIsoKer_hom_ker_subtype_apply`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorph_self`, `ZLattice.summable_norm_sub_rpow`, `Algebra.Generators.ofComp_kerCompPreimage`, `Rep.indCoindIso_hom_hom_toLinearMap`, `HahnEmbedding.ArchimedeanStrata.iSupIndep_stratum'`, `LinearMap.IsProj.eq_conj_prod_map'`, `IsIntegral.mem_span_pow`, `LinearEquiv.submoduleMap_symm_apply`, `IsHomogeneous.mem_iff`, `IsLocalization.disjoint_comap_iff`, `GradedAlgHom.toGradedRingHom_injective`, `MvPolynomial.mem_restrictDegree_iff_sup`, `LinearMap.map_codRestrict`, `mem_span_pair`, `RootPairing.root_mem_submodule_iff_of_add_mem_invtSubmodule`, `Ideal.mem_inf`, `ZLattice.abs_repr_lt_of_norm_lt`, `EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq`, `LinearMap.mem_range`, `ContinuousLinearMap.isClosed_eigenspace`, `MonoidAlgebra.supportedEquivFinsupp_apply_apply`, `groupHomology.cyclesMk₂_eq`, `LinearMap.BilinForm.finrank_orthogonal`, `mem_nullSubmodule_iff`, `DirectSum.IsInternal.card_filter_subordinateOrthonormalBasisIndex_eq`, `mem_span_of_iInf_ker_le_ker`, `ContinuousMultilinearMap.codRestrict_toMultilinearMap`, `Polynomial.degreeLT.addLinearEquiv_symm_apply'`, `finrank_add_inf_finrank_orthogonal'`, `LinearDisjoint.rank_le_one_of_flat_of_self`, `ContinuousLinearMap.projKerOfRightInverse_comp_inv`, `TensorProduct.eq_of_fg_of_subtype_eq`, `LinearEquiv.toSpanNonzeroSingleton_symm_apply_smul`, `Polynomial.coeff_mem_radical_span_coeff_of_dvd`, `IsZLattice.span_top`, `MonoidAlgebra.mem_ideal_span_of_image`, `Module.Basis.self_mem_span_image`, `Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_is_extension`, `instInvertibleSubtypeMemVal`, `AffineSubspace.linear_equivMapOfInjective`, `groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_assoc`, `EuclideanGeometry.Sphere.IsTangentAt.eq_orthogonalProjection`, `Subbimodule.coe_toSubmodule'`, `groupHomology.H1π_eq_zero_iff`, `DirectSum.IsInternal.toBaseChange_injective`, `Ideal.mem_iInf_smul_pow_eq_bot_iff`, `zero_mem`, `Valuation.Integers.coe_span_singleton_eq_setOf_le_v_algebraMap`, `groupHomology.isoCycles₁_inv_comp_iCycles_assoc`, `MeasureTheory.MemLp.condExpL2_ae_eq_condExp'`, `LinearMap.BilinForm.IsSymm.restrict`, `groupHomology.π_comp_H1Iso_hom_assoc`, `Affine.Simplex.median_restrict`, `isNilpotent_tensor_residueField_iff`, `LinearMap.trace_eq_sum_trace_restrict`, `Affine.Triangle.dist_circumcenter_reflection_orthocenter_finset`, `RootPairing.root'_apply_apply_mem_of_mem_span`, `toIsOrderedCancelAddMonoid`, `Ideal.rank_pow_quot_aux`, `traceDual_eq_span_map_traceDual_of_linearDisjoint`, `isNoetherian_of_fg_of_noetherian`, `LinearMap.range_liftBaseChange`, `groupHomology.H2π_comp_map`, `RootPairing.rootSpanMem_reflectionPerm_self`, `CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_apply`, `groupHomology.mem_cycles₂_of_mem_boundaries₂`, `coe_smul`, `Subspace.dualEquivDual_def`, `PrimeSpectrum.coe_preimageOrderIsoFiber_apply_asIdeal`, `coe_span_eq_span_of_surjective`, `groupCohomology.cocycles₂.val_eq_coe`, `affineSpan_coe_preimage_eq_top`, `mem_one`, `TensorProduct.quotientTensorEquiv_apply_tmul_mk`, `exists_finset_span_eq_linearIndepOn`, `exteriorPower.zeroEquiv_ιMulti`, `NumberField.mixedEmbedding.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIntegerLattice`, `smul_vsub_rev_mem_vectorSpan_pair`, `LinearPMap.mem_graph_iff'`, `map_nonunit`, `reflection_bot`, `LinearMap.surjective_comp_linearProjOfIsCompl`, `LinearMap.exact_subtype_ker_map`, `Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_iff`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorph_fst`, `Algebra.Extension.H1Cotangent.map_apply_coe`, `HahnEmbedding.Seed.truncLT_mem_range_baseEmbedding`, `PeriodPair.analyticOnNhd_derivWeierstrassP`, `Polynomial.eval_eq_sum_degreeLTEquiv`, `Ideal.mul_mem_right`, `mulMap'_surjective`, `MvPowerSeries.coe_hasEvalIdeal`, `subtypeL_apply`, `AffineSubspace.coe_direction_eq_vsub_set_left`, `finrank_span_eq_finrank_span`, `restrictScalars_mem`, `instNonUnitalSubringClassIdeal`, `aeval_derivative_mem_differentIdeal`, `AlgebraicGeometry.Scheme.IdealSheafData.coe_support_eq_eq_iInter_zeroLocus`, `Module.isTorsionBySet_quotient_iff`, `mem_orthogonalBilin_iff`, `mem_smul_span`, `groupHomology.H1π_comp_map_assoc`, `Subspace.dualLift_rightInverse`, `Representation.Coinvariants.mk_eq_iff`, `Polynomial.exists_monic_aeval_eq_zero_forall_mem_pow_of_mem_map`, `Ideal.natAbs_det_equiv`, `LinearMap.map_coe_ker`, `torsion_gc`, `EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq`, `IsPrincipal.generator_mem`, `finrank_map_subtype_eq`, `groupHomology.instEpiModuleCatH1π`, `coe_dualAnnihilator_span`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorph_apply_ker`, `ker_orthogonalProjection`, `NumberField.mem_span_integralBasis`, `Affine.Simplex.closedInterior_restrict`, `ClosedSubmodule.mem_mk`, `GradedAlgHom.map_mem`, `Ideal.isOpen_of_isOpen_subideal`, `mem_carrier`, `groupCohomology.isoCocycles₂_inv_comp_iCocycles`, `MeasureTheory.lpTrimToLpMeas_ae_eq`, `AlgebraicGeometry.Scheme.zeroLocus_biInf_of_nonempty`, `LinearMap.ofIsCompl_add`, `apply_mem_span_image_of_mem_span`, `LinearEquiv.ofSubmodule'_apply`, `starProjection_orthogonalComplement_singleton_eq_zero`, `FG.directedSystem`, `HahnEmbedding.Partial.mem_domain`, `mem_span_of_mem`, `powSMulQuotInclusion_mk`, `EuclideanGeometry.orthogonalProjection_orthogonalProjection`, `exteriorPower.map_injective`, `vectorSpan_add_self`, `Ideal.rank_pow_quot`, `Ideal.subtype_isoBaseOfIsPrincipal_eq_mul`, `LinearDisjoint.rank_le_one_of_commute_of_flat_of_self`, `MvPolynomial.mem_coeffsIn_iff_coeffs_subset`, `mem_iSup_finset_iff_exists_sum`, `groupCohomology.δ₀_apply`, `eq_top_iff_forall_basis_mem`, `EuclideanGeometry.orthogonalProjection_vsub_mem_direction`, `exteriorPower.basis_repr_apply`, `iSup_eq_span'`, `LinearIndependent.repr_eq`, `HasOrthogonalProjection.exists_orthogonal`, `mk_quotientEquivOfIsCompl_apply`, `smul_mem_iff`, `groupHomology.single_one_snd_sub_single_one_snd_mem_boundaries₂`, `MvPolynomial.weightedDecomposition.decompose'_eq`, `prodEquivOfIsCompl_symm_apply_right`, `coe_mapIic_apply`, `le_sigNeg_of_negDef`, `NumberField.mixedEmbedding.fundamentalCone.idealSetEquiv_symm_apply`, `submoduleOf_eq_top`, `Subspace.dualCopairing_nondegenerate`, `HahnEmbedding.Seed.mem_domain_baseEmbedding`, `exteriorPower.presentation_R`, `ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus`, `restrictScalars.isScalarTower`, `mem_span`, `pow_eq_span_pow_set`, `finiteDimensional_direction_affineSpan_image_of_finite`, `groupHomology.instEpiModuleCatH2π`, `ContinuousLinearEquiv.submoduleMap_symm_apply`, `NumberField.mem_span_basisOfFractionalIdeal`, `exists_isInternal_prime_power_torsion`, `LinearIndepOn.image_subset_span_image_extend`, `Algebra.Extension.tensorCotangentInvFun_smul_mk`, `Affine.Simplex.orthogonalProjection_eq_circumcenter_of_dist_eq`, `Subbimodule.coe_baseChange`, `LinearMap.BilinForm.mem_orthogonal_iff`, `Polynomial.mem_ker_modByMonic`, `mem_vectorSpan_pair`, `AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal`, `eq_span_singleton_of_surjective`, `IsArtinianRing.exists_not_mem_forall_mem_of_ne`, `AffineSubspace.topEquiv_apply`, `Ideal.exists_mul_add_mem_pow_succ`, `Basis.mem_ideal_iff`, `LinearEquiv.ofTop_symm_apply`, `Finsupp.mem_span_range_iff_exists_finsupp`, `groupCohomology.cocyclesMk₂_eq`, `LinearIsometry.completeSpace_map`, `GradedTensorProduct.tmul_coe_mul_zero_coe_tmul`, `Affine.Simplex.mongePoint_restrict`, `rank_span_of_finset`, `Rep.indCoindIso_inv_hom_toLinearMap`, `coe_smul_of_tower`, `PeriodPair.instDiscreteTopologySubtypeComplexMemSubmoduleIntLattice`, `MvPolynomial.coeffsIn_eq_span_monomial`, `Module.Dual.exists_forall_mem_ne_zero_of_forall_exists`, `finrank_mono`, `mulLeftMap_apply_single`, `Finsupp.single_mem_span_single`, `AlgebraicGeometry.Scheme.IdealSheafData.supportSet_subset_zeroLocus`, `ContinuousLinearMap.range_eq_map_coprodSubtypeLEquivOfIsCompl`, `support_of_supportDim_eq_zero`, `EuclideanGeometry.hasFDerivAt_inversion`, `mem_coeSubmodule_conductor`, `finite_finset_sup`, `mem_isotypicComponents_iff`, `LinearIndependent.repr_ker`, `factorPow_comp_powSMulQuotInclusion`, `groupHomology.H1π_comp_map`, `Affine.Simplex.eulerPoint_restrict`, `finrank_add_le_finrank_add_finrank`, `groupHomology.d₃₂_apply_mem_cycles₂`, `Pi.dim_spanSubset`, `toLinearPMap_apply_aux`, `OrthonormalBasis.span_apply`, `Ideal.mem_quotient_iff_mem_sup`, `Ideal.mem_torsionOf_iff`, `le_div_iff`, `EuclideanGeometry.vsub_orthogonalProjection_mem_direction`, `span_sSup'`, `mulMap_map_comp_eq`, `mem_span_triple`, `MeasureTheory.Measure.addHaar_submodule`, `GradedTensorProduct.comm_coe_tmul_coe`, `mem_torsionBy_iff`, `AdicCompletion.restrictScalars_range_ofPowSMul_eq_ker_eval`, `Module.Basis.flag_le_iff`, `instCompleteSpaceSubtypeMemSubmoduleOfIsClosedCoe`, `QuadraticMap.restrict_apply`, `Valuation.Integers.isPrincipal_iff_exists_eq_setOf_valuation_le`, `Module.Flat.iff_lift_lsmul_comp_subtype_injective`, `ContinuousMap.adjoin_id_eq_span_one_add`, `ClosedSubmodule.instCanLiftSubmoduleToSubmoduleIsClosedCoe`, `AffineSubspace.coe_inclusion_apply`, `Algebra.exists_etale_of_isEtaleAt`, `Ideal.range_finsuppTotal`, `lt_sup_iff_notMem`, `groupCohomology.toCocycles_comp_isoCocycles₁_hom_apply`, `Algebra.exists_aeval_invOf_eq_zero_of_idealMap_adjoin_sup_span_eq_top`, `Affine.Simplex.ExcenterExists.touchpoint_restrict`, `AffineSubspace.signedInfDist_singleton`, `groupHomology.boundariesOfIsBoundary₂_coe`, `Affine.Simplex.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection`, `Module.Finite.top`, `LinearPMap.mem_domain_of_mem_graph`, `biUnion_associatedPrimes_eq_compl_nonZeroDivisors`, `IsSemisimpleRing.exists_ringEquiv_pi_matrix_end_mulOpposite`, `finrank_le_one_iff_isPrincipal`, `groupHomology.cyclesMk₁_eq`, `ne_bot_iff`, `Module.End.mem_invtSubmodule_iff_forall_mem_of_mem`, `Module.End.isNilpotent_restrict_sub_algebraMap`, `Ideal.primaryComponent_mem`, `FG.lTensor.directLimit_apply`, `IsCoprime.exists`, `LieSubmodule.coe_toSubmodule`, `finrank_span_eq_finrank`, `sup_torsionBySet_ideal_eq_torsionBySet_inf`, `LinearMap.IsProj.codRestrict_ker`, `Collinear.finrank_le_one`, `AffineSubspace.vadd_mem_iff_mem_direction`, `Module.fgSystem.equiv_comp_of`, `Ideal.Quotient.isUnit_mk_pow_iff_notMem`, `AddMonoidAlgebra.grade.isInternal`, `NumberField.FinitePlace.norm_eq_one_iff_notMem`, `groupHomology.mapCycles₂_comp_i_assoc`, `Ideal.unit_mul_mem_iff_mem`, `RootPairing.posRootForm_posForm_nondegenerate`, `isLinearTopology_iff_hasBasis_open_submodule`, `ArchimedeanClass.mem_closedBall_iff`, `Affine.Simplex.excenterWeightsUnnorm_restrict`, `groupCohomology.isoCocycles₁_hom_comp_i`, `Ideal.Quotient.mk_eq_mk_iff_sub_mem`, `LinearMap.singularValues_finrank_range_self`, `AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_supportSet`, `range_inclusion`, `Module.Flat.instSubtypeMemSubmoduleSubmoduleAlgebra`, `MvPolynomial.HomogeneousSubmodule.gradedMonoid`, `ContinuousLinearMap.projKerOfRightInverse_apply_idem`, `mem_traceDual`, `spanFinrank_top`, `coe_iSup_of_directed`, `LinearMap.domRestrict₂_apply`, `AffineSubspace.subtypeₐᵢ_linearIsometry`, `Ideal.coe_closure`, `LinearMap.IsIdempotentElem.mem_range_iff`, `Algebra.Extension.Hom.sub_aux`, `mul_subset_mul`, `mem_span_image_finset_iff_exists_fun'`, `PointedCone.mem_closure`, `Module.End.mem_genEigenspace_one`, `Polynomial.IsEisensteinAt.mem`, `PeriodPair.hasSumLocallyUniformly_aux`, `Ideal.powQuotSuccInclusion_injective`, `PointedCone.subset_dual_flip_iff_subset_dual`, `LinearMap.IsProj.trace`, `IsLocalRing.CotangentSpace.map_eq_top_iff`, `LinearPMap.mem_graph`, `rank_range_of_injective`, `HahnEmbedding.ArchimedeanStrata.archimedean_stratum`, `Module.isTorsionBy_quotient_iff`, `basisOfPid_bot`, `IsLocalRing.isOpen_maximalIdeal_pow`, `exists_maximal_linearIndepOn`, `EuclideanGeometry.Sphere.IsTangent.isTangentAt`, `groupCohomology.mapCocycles₁_comp_i_apply`, `ContinuousLinearMap.coe_codRestrict_apply`, `ZLattice.isAddFundamentalDomain`, `LinearMap.isClosed_or_dense_ker`, `isInternal_prime_power_torsion`, `LinearMap.restrict_smul_one`, `Module.End.mapsTo_maxGenEigenspace_of_comm`, `groupHomology.mapCycles₂_id_comp_apply`, `image2_subset_map₂`, `coe_eq_zero`, `coe_closure`, `Ideal.sub_mem`, `LinearMap.range_domRestrict`, `Polynomial.det_taylorLinearEquiv`, `IsSemisimpleModule.exists_end_algEquiv_pi_matrix_end`, `Ideal.mem_map_iff_of_surjective`, `instIsTorsionFree`, `groupCohomology.cocycles₂_ext_iff`, `LinearMap.continuous_domRestrict`, `Algebra.Extension.h1Cotangentι_apply`, `coe_subtype`, `mulMap_comp_lTensor`, `LinearEquiv.rank_map_eq`, `isNoetherian_span_of_finite`, `exists_leadingCoeff_pow_smul_mem_conductor`, `mem_sInf`, `RootPairing.algebraMap_coroot'In_apply`, `smul_torsionBy`, `coe_neg`, `ContinuousLinearMap.IsPositive.orthogonalProjection_comp`, `LinearMap.linearEquiv_of_ne_zero`, `LinearMap.IsSymm.nondegenerate_restrict_of_isCompl_ker`, `smul_le`, `mem_localized₀`, `LinearPMap.adjointDomainMkCLMExtend_apply`, `sup_eq_top_iff`, `dimH_orthogonalProjection_le`, `LinearMap.range_ofIsCompl`, `GradedLieAlgebra.toGradedBracket`, `ClassGroup.exists_mk0_eq_mk0`, `RootPairing.rootForm_restrict_nondegenerate_of_isAnisotropic`, `instDiscreteTopologySubtypeMemSubmoduleIntComap`, `Module.End.mem_maxGenEigenspace`, `derivationOfSectionOfKerSqZero_apply_coe`, `torsionBy_isTorsionBy`, `EuclideanGeometry.Sphere.mem_inter_orthRadius_iff_vsub_mem_and_norm_sq`, `add_mem_iff_right`, `span_span`, `Module.Finite.iff_fg`, `groupHomology.H2π_comp_map_apply`, `mem_iSup_of_mem`, `adjoint_orthogonalProjection`, `lTensorOne_symm_apply`, `LinearMap.tensorEqLocus_tmul`, `exteriorPower.map_comp_ιMulti_family`, `MvPolynomial.mul_monomial_mem_coeffsIn`, `EuclideanGeometry.reflection_apply'`, `mul_le`, `fg_top`, `sectionOfRetractionKerToTensorAux_algebraMap`, `AffineIsometryEquiv.ofEq_rfl`, `RootPairing.rootFormIn_self_smul_coroot`, `MvPolynomial.C_mem_pow_idealOfVars_iff`, `tensorToSpan_apply_tmul`, `ZLattice.le_norm_of_le_abs_repr`, `Valuation.integer.coe_span_singleton_eq_setOf_le_v_coe`, `exteriorPower.map_id`, `LinearMap.ofIsCompl_left_apply`, `EuclideanGeometry.oangle_self_orthogonalProjection`, `nilpotent_iff_mem_prime`, `isCompl_comap_subtype_of_isCompl_of_le`, `Ideal.zeroLocus_inj_of_pure`, `le_isotypicComponent_iff`, `TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_apply_tmul`, `AddMonoidAlgebra.supportedEquivFinsupp_symm_apply_coe_apply`, `Projectivization.instFiniteDimensionalSubtypeMemSubmoduleSubmodule`, `LinearMap.ker_le_iff`, `LieModule.Cohomology.twoCochain_val_apply`, `Algebra.mem_ideal_map_adjoin`, `Module.Finite.of_fg`, `image_specComap_zeroLocus_eq_zeroLocus_comap`, `HahnEmbedding.Seed.hahnCoeff_apply`, `AlgebraicGeometry.Scheme.zeroLocus_span`, `Ideal.nonempty_inter_nonZeroDivisors_of_faithfulSMul`, `Ideal.exists_coeff_ne_zero_mem_comap_of_root_mem`, `LinearIsometry.strictConvexSpace_range_iff`, `linearIndependent_iff_notMem_span`, `DividedPowers.SubDPIdeal.memCarrier`, `mem_orthogonal_singleton_iff_inner_left`, `groupCohomology.coboundaries₁_ext_iff`, `fg_iff_finiteDimensional`, `PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal`, `iSupIndep_iff_dfinsupp_lsum_injective`, `mem_adjoint_iff`, `Ideal.add_pow_mem_of_pow_mem_of_le_of_commute`, `MvPolynomial.weightedHomogeneousComponent_mem`, `span_biUnion`, `Finsupp.supportedEquivFinsupp_symm_apply_coe`, `span_singleton_eq_range`, `Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_iff`, `Ideal.disjoint_primeCompl_of_liesOver`, `exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul`, `LinearEquiv.toSpanNonzeroSingleton_one`, `lTensorOne'_one_tmul`, `LinearMap.compAlternatingMap_codRestrict`, `Module.Basis.self_mem_flag`, `Ideal.IsPrime.exists_mem_prime_of_ne_bot`, `Ideal.colon_span`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_K`, `LinearMap.mem_graph_iff`, `LinearMap.coe_quotientInfToSupQuotient`, `Subspace.dualLift_injective`, `Algebra.Generators.cotangentRestrict_mk`, `MatrixModCat.toModuleCat_map`, `Polynomial.toMatrix_sylvesterMap`, `TensorProduct.exists_finite_submodule_right_of_finite`, `LinearMap.subtype_compMultilinearMap_codRestrict`, `Finsupp.supportedEquivFinsupp_apply_support_val`, `LinearIsometryEquiv.reflections_generate_dim`, `GradedAlgebra.proj_apply`, `Ideal.coe_piOrderIso_symm_apply`, `Polynomial.coeff_divByMonic_mem_pow_natDegree_mul`, `FractionalIdeal.mem_coe`, `exists_mem_ne_zero_of_rank_pos`, `Irreducible.maximalIdeal_eq_closedBall`, `AffineEquiv.coe_ofEq_apply`, `PrimeSpectrum.coe_vanishingIdeal`, `finiteDimensional_bot`, `ContinuousLinearMap.exist_extension_of_finiteDimensional_range`, `span_inter`, `groupCohomology.π_comp_H2Iso_hom_apply`, `LinearPMap.mkSpanSingleton_apply`, `PeriodPair.hasSum_weierstrassPExcept`, `Ideal.mul_mem_mul_rev`, `MeasureTheory.hausdorffMeasure_orthogonalProjection_le`, `small_span`, `LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_invariant`, `coe_span_eq_self`, `Algebra.Generators.H1Cotangent.δ_eq_δAux`, `CliffordAlgebra.even.lift.aux_apply`, `isPathConnected_compl_of_isPathConnected_compl_zero`, `AffineSubspace.mem_direction_iff_eq_vsub_right`, `norm_coe`, `exists_lieSubalgebra_coe_eq_iff`, `finrank_add_finrank_orthogonal`, `instIsSemisimpleModuleSubtypeMemSubmoduleValSetIsotypicComponents`, `AffineMap.restrict.linear`, `ZLattice.summable_norm_sub_zpow`, `map_subtype_top`, `AddSubmonoid.coe_toNatSubmodule`, `mem_toConvexCone`, `exteriorPower.pairingDual_apply_apply_eq_one_zero`, `lipschitzWith_orthogonalProjection`, `Affine.Simplex.abs_signedInfDist_eq_dist_of_mem_affineSpan_range`, `DividedPowers.IsSubDPIdeal.dpow_eq`, `Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson`, `Ideal.mem_span_pair`, `TopModuleCat.kerι_apply`, `Ideal.isRadical_iff_pow_one_lt`, `LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces`, `NumberField.mixedEmbedding.fractionalIdealLatticeBasis_apply`, `coe_copy`, `exists_fg_le_subset_range_rTensor_inclusion`, `Ideal.exists_mem_pow_notMem_pow_succ`, `derivationToSquareZeroEquivLift_apply_coe_apply`, `orthogonalFamily_iff_pairwise`, `span_span_coe_preimage`, `Representation.mem_coindV`, `AffineSubspace.coe_subtype`, `Module.support_eq_zeroLocus`, `Representation.coind_apply`, `mem_reesAlgebra_iff`, `groupHomology.isoCycles₁_hom_comp_i_assoc`, `ContinuousLinearEquiv.submoduleMap_apply`, `FractionalIdeal.abs_det_basis_change`, `ClosedComplemented.exists_submodule_equiv_prod`, `LinearMap.map_span_le`, `instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallHSMul`, `Rep.invariantsFunctor_map_hom`, `Module.End.iInf_maxGenEigenspace_restrict_map_subtype_eq`, `finrank_span_singleton`, `LieAlgebra.mem_corootSpace'`, `MonoidAlgebra.mem_supported`, `Ideal.add_mem_iff_left`, `Affine.Simplex.orthogonalProjectionSpan_reindex`, `Subalgebra.coe_toSubmodule`, `riesz_lemma_of_lt_one`, `LinearPMap.vadd_apply`, `LinearMap.le_eqLocus`, `span_sup`, `liftOfDerivationToSquareZero_mk_apply'`, `ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection`, `Module.Finite.map`, `rank_le_one_iff_isPrincipal`, `subalgebra_top_finrank_eq_submodule_top_finrank`, `Ideal.univ_eq_iUnion_image_add`, `PrimeSpectrum.basicOpen_le_basicOpen_iff`, `stereographic_apply`, `MeasureTheory.inner_condExpL2_eq_inner_fun`, `Ideal.mem_mul_span_singleton`, `ContinuousMap.mem_setOfIdeal`, `finite_dualAnnihilator_iff`, `linearProjOfIsCompl_comp_bijective_of_exact`, `RootPairing.RootPositiveForm.algebraMap_apply_eq_form_iff`, `MeasureTheory.mem_lpMeas_self`, `groupCohomology.π_comp_H1Iso_hom_apply`, `LinearMap.continuous_iff_isClosed_ker`, `MeasureTheory.integrableOn_condExpL2_of_measure_ne_top`, `ClassGroup.exists_min`, `exists_of_finrank_lt`, `LinearMap.rTensor_range`, `Module.Basis.SmithNormalForm.repr_comp_embedding_eq_smul`, `Representation.Coinvariants.mem_ker_of_eq`, `LinearIsometry.strictConvexSpace_range`, `PrimeSpectrum.mem_basicOpen`, `TensorProduct.exists_finite_submodule_right_of_finite'`, `Algebra.IsUnramifiedAt.exists_notMem_forall_ne_mem_and_adjoin_eq_top`, `IsLocalRing.CotangentSpace.span_image_eq_top_iff`, `AlternatingMap.codRestrict_apply_coe`, `LinearEquiv.coord_apply_smul`, `LinearPMap.ext_iff`, `topologicalAddGroup`, `inf_iInf_maxGenEigenspace_of_forall_mapsTo`, `IsLocalRing.notMem_maximalIdeal`, `Ideal.isPrime_iff`, `RootPairing.restrictScalars_toLinearMap_apply_apply`, `LinearMap.finrank_maxGenEigenspace_zero_eq`, `fstL_comp_coe_orthogonalDecomposition`, `Polynomial.isWeaklyEisensteinAt_iff`, `RootPairing.finrank_corootSpan_eq`, `isPathConnected_compl_of_one_lt_codim`, `LinearMap.ofIsCompl_right_apply`, `NumberField.mixedEmbedding.fundamentalCone.preimage_of_IdealSetMap`, `Ideal.coe_prod`, `mem_span_set'`, `Polynomial.sylvesterMap_apply_coe`, `vaddCommClass`, `Polynomial.degreeLT.addLinearEquiv_natAdd`, `linearDepOn_iffₛ`, `riesz_extension`, `starProjection_minimal`, `PeriodPair.differentiableOn_weierstrassPExcept`, `EuclideanGeometry.orthogonalProjection_linear`, `PrimeSpectrum.subset_zeroLocus_vanishingIdeal`, `PrimeSpectrum.exist_mem_one_of_mem_maximal_ideal`, `Valuation.mem_ltSubmodule_iff`, `ContinuousLinearEquiv.ofSubmodule'_symm_apply`, `pinGroup.involute_act_ι_mem_range_ι`, `Module.End.mem_genEigenspace_nat`, `inclusion_injective`, `Ideal.neg_mem_iff`, `mulMap_comp_map_inclusion`, `ContinuousLinearMap.completeSpace_ker`, `orthogonalProjection_orthogonal_apply_eq_zero`, `AffineSubspace.signedInfDist_def`, `PeriodPair.derivWeierstrassPExcept_add_coe`, `LinearMap.submoduleComap_surjective_of_surjective`, `IsSemisimpleModule.exists_sSupIndep_sSup_simples_eq_top`, `starProjection_mem_subspace_eq_self`, `Ideal.Quotient.mk_eq_one_iff_sub_mem`, `exteriorPower.basis_repr`, `AlgebraicGeometry.Scheme.IdealSheafData.coe_support_inter`, `angle_coe`, `exists_extension_norm_eq`, `MvPolynomial.mem_ideal_span_monomial_image_iff_dvd`, `NumberField.Units.unitLattice_rank`, `LinearMap.mem_fixedSubmodule_iff`, `ProjectiveSpectrum.zeroLocus_sup_ideal`, `CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_apply`, `IsSMulRegular.subsingleton_linearMap_iff`, `finrank_le`, `instNontrivialSubtypeMemSubmoduleValSetIsotypicComponents`, `groupHomology.π_comp_H2Iso_hom_apply`, `isArtinian_sup`, `lTensorOne_tmul`, `instIsSimpleModuleSubtypeMemSubmoduleValSetSetOfNonemptyLinearEquivId`, `coe_pi`, `LinearIndepOn.notMem_span_iff`, `PeriodPair.differentiableOn_weierstrassP`, `mem_colon_singleton`, `Module.Flat.iff_rTensor_injectiveₛ`, `LinearDisjoint.val_mulMap_tmul`, `FG.span`, `pow_subset_pow`, `LinearMap.trace_restrict_eq_of_forall_mem`, `Affine.Simplex.orthogonalProjectionSpan_eulerPoint_mem_ninePointCircle`, `Module.Dual.finrank_ker_add_one_of_ne_zero`, `Module.Finite.bot`, `Algebra.Extension.Hom.sub_tmul`, `exteriorPower.alternatingMapLinearEquiv_symm_apply`, `LinearIndependent.notMem_span_image`, `isLinearTopology_iff_hasBasis_submodule`, `Ideal.primaryComponent.map_ker_eq`, `ZLattice.normBound_spec`, `LieAlgebra.IsKilling.coe_invtSubmoduleToLieIdeal_eq_iSup`, `groupHomology.mapCycles₁_hom`, `Module.isTorsionBySet_iff_subset_annihilator`, `instFreeSubtypeMemIdealOfFiniteOfIsTorsionFree`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_π_hom`, `AddValuation.mem_supp_iff`, `mem_iSup_of_chain`, `mem_map_of_mem`, `LinearMap.exists_extend`, `isNoetherian_submodule_left`, `AffineSubspace.mem_mk'`, `groupHomology.isoCycles₁_inv_comp_iCycles`, `Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisors`, `Ideal.mem_span_insert`, `Ideal.isPrimary_iff`, `smul_mem_span_smul'`, `affineIndependent_iff_finrank_vectorSpan_eq`, `GradedAlgHom.comp_toGradedRingHom`, `PrimeSpectrum.coe_preimageHomeomorphFiber_symm_apply_coe_asIdeal`, `PointedCone.mem_comap`, `orthogonalProjectionFn_eq`, `mem_iSup_iff_exists_finset`, `ZSpan.discreteTopology_pi_basisFun`, `mul_eq_span_mul_set`, `RootPairing.finrank_rootSpanIn`, `Affine.Simplex.signedInfDist_affineCombination`, `Ideal.cotangentEquivIdeal_symm_apply`, `Module.flat_iff`, `mem_colon`, `LinearMap.ker_id_sub_eq_of_proj`, `Polynomial.taylorLinearEquiv_symm`, `groupHomology.cyclesMap_comp_isoCycles₁_hom_assoc`, `valFromGraph_mem`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne`, `Ideal.IsPrime.mul_mem_left_iff`, `coe_toNonUnitalSubalgebra`, `EuclideanGeometry.angle_self_orthogonalProjection`, `exists_fg_le_eq_rTensor_subtype`, `groupHomology.single_mem_cycles₁_of_mem_invariants`, `LieSubalgebra.mem_mk_iff'`, `FractionalIdeal.isNoetherian`, `Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_aux`, `groupHomology.coe_mapCycles₁`, `NumberField.Units.logEmbeddingEquiv_apply`, `ZLattice.covolume.tendsto_card_le_div''`, `Ideal.mem_radical_iff`, `groupHomology.toCycles_comp_isoCycles₁_hom`, `Module.End.isNilpotent_restrict_of_le`, `Ideal.IsPrime.mem_or_mem'`, `Polynomial.degreeLT.addLinearEquiv_symm_apply_inl_basis`, `iSup_eq_toSubmodule_range`, `TensorProduct.exists_finite_submodule_of_setFinite`, `TensorProduct.exists_finite_submodule_left_of_setFinite'`, `RootPairing.RootPositiveForm.zero_lt_apply_root_root_iff`, `exteriorPower.map_apply_ιMulti`, `MvPolynomial.one_coeffsIn`, `NumberField.basisOfFractionalIdeal_apply`, `PeriodPair.hasSum_sumInvPow`, `ContinuousAlternatingMap.codRestrict_apply_coe`, `AddMonoidAlgebra.mem_supported'`, `Module.Basis.ofZLatticeComap_repr_apply`, `signedDist_eq_dist_iff_vsub_mem_span`, `iSup_torsionBySet_ideal_eq_torsionBySet_iInf`, `LieSubalgebra.coe_lieSpan_submodule_eq_iff`, `HahnEmbedding.Partial.archimedeanClassMk_le_of_eval_eq`, `TensorProduct.rTensor_injective_of_forall_fg_rTensor_injective`, `AffineSubspace.angle_coe`, `LinearMap.finrank_eq_of_isPerfPair`, `LinearMap.ker_restrict`, `MonomialOrder.sPolynomial_mem_ideal`, `MvPolynomial.mem_ideal_of_coeff_mem_ideal`, `RootPairing.posRootForm_posForm_anisotropic`, `mem_sSup_iff_exists_finset`, `PrimeSpectrum.zeroLocus_inf`, `isScalarTower_of_section_of_ker_sqZero`, `LinearMap.IsSymmetric.LinearMap.IsSymmetric.directSum_isInternal_of_pairwise_commute`, `Finsupp.mem_ideal_span_range_iff_exists_finsupp`, `AddMonoidAlgebra.mem_span_support'`, `QuadraticMap.IsometryEquiv.map_negDef_iff`, `LieAlgebra.IsKilling.mem_rootSet_invtSubmoduleToLieIdeal`, `PrimeSpectrum.mem_vanishingIdeal`, `mem_finsetInf`, `groupHomology.mapCycles₂_comp_i_apply`, `riesz_lemma`, `Valuation.mem_leIdeal_iff`, `rank_quotient_add_rank_of_divisionRing`, `PowerSeries.mem_hasEvalIdeal_iff`, `Module.End.isNilpotent_restrict_genEigenspace_top`, `torsionBySet_eq_torsionBySet_span`, `EuclideanGeometry.orthogonalProjection_eq_self_iff`, `linearIndepOn_insert`, `groupCohomology.cocyclesIso₀_hom_comp_f_apply`, `Ideal.subset_union_prime`, `groupHomology.boundariesToCycles₂_apply`, `Module.Finite.span_finset`, `Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule_1`, `ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id`, `groupCohomology.subtype_comp_d₀₁`, `AddMonoidAlgebra.decomposeAux_single`, `EuclideanGeometry.reflection_subtype`, `subset_span`, `PeriodPair.periodic_weierstrassP`, `LinearPMap.mem_domain_iff_of_eq_graph`, `map_range_rTensor_subtype_lid`, `span_singleton_le_iff_mem`, `isOpen_mono`, `groupHomology.cyclesOfIsCycle₂_coe`, `LinearMap.det_eq_det_mul_det`, `isNoetherian_sup`, `FiniteDimensional.instSubtypeMemSubmoduleMap`, `eq_zero_of_bot_submodule`, `IsLinearTopology.hasBasis_ideal`, `ContinuousLinearEquiv.coord_norm'`, `LinearEquiv.coe_ofTop_symm_apply`, `Ideal.coe_torsionOf`, `IsLattice.finite`, `mem_sup`, `groupHomology.isoCycles₂_hom_comp_i`, `LinearMap.injective_domRestrict_iff`, `groupHomology.π_comp_H1Iso_hom`, `PointedCone.tmul_mem_minTensorProduct`, `CategoryTheory.ShortComplex.moduleCatLeftHomologyData_H`, `LinearMap.support_singularValues`, `LinearMap.mem_span_iff_bound`, `fstEquiv_symm_apply_coe`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_assoc_apply`, `coe_set_smul`, `snd_orthogonalDecomposition_apply`, `groupHomology.isoCycles₂_inv_comp_iCycles`, `LinearMap.BilinForm.toLin_restrict_ker_eq_inf_orthogonal`, `asymptoticCone_submodule`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_apply`, `Module.Basis.ofZLatticeBasis_repr_apply`, `groupHomology.mkH1OfIsTrivial_apply`, `tendsto_card_div_pow_atTop_volume'`, `CliffordAlgebra.ι_mem_evenOdd_one`, `fromModuleCatToModuleCatLinearEquiv_apply`, `Module.Finite.span_of_finite`, `OrzechProperty.injective_of_surjective_of_submodule'`, `PointedCone.DualFG.dual_of_fg`, `notMem_of_ortho`, `PrimeSpectrum.coe_primesOverOrderIsoFiber_symm_apply_coe`, `top_coe`, `IsIsotypic.linearEquiv_finsupp`, `groupCohomology.isoCocycles₁_inv_comp_iCocycles_assoc`, `isCompact_of_fg`, `fromModuleCatToModuleCatLinearEquiv_symm_apply_coe`, `groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom`, `PointedCone.mem_toConvexCone`, `LinearMap.surjective_rangeRestrict`, `AffineSubspace.isometryEquivMap.apply_symm_apply`, `coe_torsion_eq_annihilator_ne_bot`, `AffineSubspace.inclusion_rfl`, `AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff_of_mem`, `IsSemisimpleModule.eq_bot_or_exists_simple_le`, `linearProjOfIsCompl_range`, `Irreducible.maximalIdeal_pow_eq_closedBall_pow`, `LieAlgebra.IsKilling.restr_invtSubmoduleToLieIdeal_eq_iSup`, `NumberField.instFiniteIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule`, `toBaseChange_surjective'`, `instOrthogonalCompleteSpace`, `CharacterModule.intSpanEquivQuotAddOrderOf_apply_self`, `LinearMap.restrict_coe_apply`, `linearIndependent_option'`, `LieAlgebra.IsKilling.ker_restrict_eq_bot_of_isCartanSubalgebra`, `AlgebraicGeometry.Scheme.IdealSheafData.supportSet_inter`, `linearIndepOn_id_insert`, `MvPolynomial.mem_vanishingIdeal_iff`, `MvPolynomial.mem_vanishingIdeal_singleton_iff`, `Ideal.pow_mem_pow`, `coe_mapAlgHom_apply`, `AlgHom.toKerIsLocalization_isLocalizedModule`, `finite_iSup`, `CategoryTheory.ShortComplex.moduleCat_pOpcycles_eq_iff`, `FractionalIdeal.equivNum_apply`, `Irreducible.maximalIdeal_pow_eq_setOf_le_v_coe_pow`, `IsTopologicallyNilpotent.mem_topologicalNilradical_iff`, `Ideal.iUnion_minimalPrimes`, `exists_mem_sup`, `DividedPowers.isSubDPIdeal_inf_iff`, `finsuppSum_mem`, `ContinuousMap.mem_idealOfSet_compl_singleton`, `AddSubgroup.relIndex_eq_natAbs_det`, `isNoetherian_of_submodule_of_noetherian`, `coe_norm`, `orthogonalDecomposition_apply`, `Ideal.Quotient.quotient_ring_saturate`, `smul_mem_smul`, `groupHomology.single_one_mem_boundaries₁`, `Ideal.relNorm_apply`, `map_comap_subtype`, `span_range_subtype_eq_top_iff`, `spanRank_top`, `AntilipschitzWith.completeSpace_range_clm`, `IsLocalizedModule.smul_mem_finsetIntegerMultiple_span`, `LinearMap.isCompl_of_proj`, `LinearIsometry.isComplete_map_iff'`, `Finsupp.mem_span_image_iff_linearCombination`, `IsDedekindDomain.isOpen_of_ne_bot`, `ProjectiveSpectrum.zeroLocus_span`, `exteriorPower.map_comp`, `PeriodPair.mul_ω₁_add_mul_ω₂_mem_lattice`, `Module.End.isNilpotent.restrict`, `groupHomology.π_comp_H1Iso_inv_apply`, `NumberField.mixedEmbedding.fundamentalCone.mem_idealSet`, `Ideal.mem_pi`, `lift_rank_range_le`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_left`, `ZSpan.vadd_mem_fundamentalDomain`, `exteriorPower.ιMultiDual_apply_nondiag`, `LinearMap.codRestrict_apply`, `MvPolynomial.mem_ideal_span_monomial_image`, `sSup_simples_eq_top_iff_isSemisimpleModule`, `Polynomial.taylor_mem_degreeLT`, `Ideal.mem_span_singleton_mul`, `Subrepresentation.mem_ofSubmodule_iff`, `AddMonoidAlgebra.decomposeAux_eq_decompose`, `ValuationSubring.coe_primeSpectrumOrderEquiv_symm_apply_asIdeal`, `exteriorPower.presentation_var`, `PowerSeries.coeff_mul_mem_ideal_of_coeff_left_mem_ideal`, `GradedTensorProduct.tmul_algebraMap_mul_coe_tmul`, `ZSpan.fract_apply`, `groupCohomology.isoCocycles₂_hom_comp_i_apply`, `LinearIsometryEquiv.ofEq_rfl`, `LieAlgebra.mem_corootSpace`, `mulMap_op`, `LinearEquiv.ofEq_rfl`, `torsionBySet.mk_smul`, `EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal`, `Module.End.genEigenspace_restrict_eq_top`, `AffineSubspace.vsub_right_mem_direction_iff_mem`, `isNoetherian_top_iff`, `Ideal.IsPrime.prod_mem_iff_exists_mem`, `LieModule.Cohomology.mem_twoCocycle_iff_of_trivial`, `HahnEmbedding.Partial.sSupFun_strictMono`, `MeasureTheory.lpMeasToLpTrim_ae_eq`, `orthogonal_eq_inter`, `NumberField.mixedEmbedding.fundamentalCone.idealSetEquiv_apply`, `FG.small`, `MeasureTheory.eLpNorm_condExpL2_le`, `mem_sSup_of_directed`, `smul_vsub_mem_vectorSpan_pair`, `isPathConnected`, `finrank_vectorSpan_range_add_one_le`, `mem_annihilator_span_singleton`, `PointedCone.isClosed_dual`, `CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_assoc_apply`, `Module.Flat.tensorProduct_mapIncl_injective_of_right`, `ContinuousLinearEquiv.toSpanNonzeroSingleton_apply_coe`, `mem_span_set_iff_exists_finsupp_le_finrank`, `ContinuousLinearEquiv.coord_self`, `UniformOnFun.continuousSMul_submodule_of_image_bounded`, `isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule`, `Algebra.idealMap_mul`, `LinearMap.linearProjOfIsCompl_of_proj`, `RingHom.sub_mem_ker_iff`, `LinearMap.ofIsCompl_subtype_zero_eq`, `Subbimodule.smul_mem`, `Derivation.liftKaehlerDifferential_apply`, `one_le`, `LinearMap.trace_eq_sum_trace_restrict'`, `LinearEquiv.finrank_map_eq`, `LinearEquiv.toLinearMap_ofTop`, `groupHomology.cycles₁IsoOfIsTrivial_inv_apply`, `map_subtype_range_inclusion`, `finrank_bot`, `ContinuousLinearEquiv.coord_norm`, `existsUnique_add_of_isCompl`, `FiniteDimensional.subalgebra_toSubmodule`, `MonomialOrder.sPolynomial_mem_sup_ideal`, `exteriorPower.map_surjective`, `KaehlerDifferential.DLinearMap_apply`, `Ideal.mem_span_singleton`, `MeasureTheory.lpMeasToLpTrim_smul`, `groupCohomology.coboundaries₂_ext_iff`, `NonUnitalSubalgebra.mem_toSubmodule`, `PointedCone.mem_ofSubmodule_iff`, `MvPolynomial.C_mem_coeffsIn`, `torsionBySet_ideal_isTorsion_of_noZeroDivisors`, `LinearPMap.mem_adjoint_domain_iff`, `Module.Basis.coe_mkFinCons`, `IsLocalization.mem_coeSubmodule`, `vsub_mem_vectorSpan_of_mem_affineSpan_of_mem_affineSpan`, `CliffordAlgebra.GradedAlgebra.lift_ι_eq`, `AddSubgroup.mem_toZModSubmodule`, `linearProjOfIsCompl_ker`, `EuclideanGeometry.eq_orthogonalProjection_of_eq_subspace`, `groupCohomology.cocyclesOfIsCocycle₂_coe`, `LinearEquiv.lift_rank_map_eq`, `EuclideanGeometry.orthogonalProjection_apply`, `Module.Basis.extendOfIsLattice_apply`, `groupHomology.isoCycles₁_hom_comp_i`, `Ideal.exists_disjoint_powers_of_span_eq_top`, `mem_selfAdjointMatricesSubmodule'`, `groupCohomology.cocyclesIso₀_inv_comp_iCocycles_apply`, `Ideal.norm_mem_relNorm`, `Module.End.hasGenEigenvector_iff`, `torsion_isTorsion`, `Convex.semilinear_range`, `PeriodPair.analyticOnNhd_weierstrassPExcept`, `rank_span`, `norm_orthogonalProjection`, `LinearMap.range_eq_of_proj`, `LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat`, `AffineSubspace.signedInfDist_apply_self`, `EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection`, `PrimeSpectrum.zeroLocus_sup`, `EuclideanGeometry.orthogonalProjection_mem_orthogonal`, `ProjectiveSpectrum.zeroLocus_iSup_ideal`, `Polynomial.degreeLT.basisProd_natAdd`, `torsionBy_isInternal`, `Algebra.FormallySmooth.iff_injective_cotangentComplexBaseChange`, `exteriorPower.presentation_G`, `mem_set_smul`, `goursat_surjective`, `GradedTensorProduct.tmul_coe_mul_algebraMap_tmul`, `groupCohomology.H1π_comp_map`, `lTensorOne_one_tmul`, `LinearIndependent.sumElim_of_quotient`, `PointedCone.mem_positive`, `groupHomology.single_inv_ρ_self_add_single_mem_boundaries₁`, `Module.End.mem_genEigenspace_zero`, `LinearMap.finrank_maxGenEigenspace_eq`, `LinearMap.IsProj.codRestrict_apply`, `Ideal.mul_mem_left`, `DirectSum.decomposeLinearEquiv_symm_lof`, `lipschitzGroup.involute_act_ι_mem_range_ι`, `Representation.coindMap_coe_apply`, `map_coe`, `ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id`, `mem_toSubalgebra`, `LinearMap.BilinForm.toLin_restrict_range_dualCoannihilator_eq_orthogonal`, `LinearMap.nondegenerate_restrict_of_disjoint_orthogonal`, `rank_le_of_isSMulRegular`, `LinearPMap.coe_vadd`, `AffineSubspace.mem_direction_iff_eq_vsub_left`, `Module.Dual.eq_of_preReflection_mapsTo'`, `span_range_eq_top_of_injective_of_rank_le`, `finrank_lt`, `rank_sup_add_rank_inf_eq`, `ZSpan.repr_floor_apply`, `PointedCone.mem_maxTensorProduct`, `MeasureTheory.condExpIndSMul_ae_eq_smul`, `rank_top`, `Module.fgSystem.instDirectedSystemSubtypeSubmoduleFGMemValCoeLinearMapId`, `isNoetherian_range`, `linearProjOfIsCompl_apply_eq_zero_iff`, `coplanar_iff_finrank_le_two`, `notMem_span_of_apply_notMem_span_image`, `LinearMap.tensorKerEquiv_apply`, `Localization.exists_awayMap_bijective_of_localRingHom_bijective`, `WittVector.mem_span_p_iff_coeff_zero_eq_zero`, `Polynomial.mem_degreeLE`, `Ideal.exists_notMem_dvd_algebraMap_of_primesOver_eq_singleton`, `map_mem_span_algebraMap_image`, `Module.Basis.span_neg`, `finite_sup`, `Ideal.coeff_zero_mem_comap_of_root_mem`, `IsLocalization.mem_map_algebraMap_iff`, `LinearMap.range_codRestrict`, `mem_biSup_iff_exists_dfinsupp`, `TensorProduct.AlgebraTensorModule.range_lTensor_idealMap`, `exteriorPower.ιMulti_family_linearIndependent_ofBasis`, `MeasureTheory.lpMeas.ae_fin_strongly_measurable'`, `Ideal.mem_span_insert'`, `PrimeSpectrum.localization_comap_range`, `RingHom.ker_eq`, `Ideal.IsMaximal.exists_inv_pow`, `IsLocalization.orderIsoOfPrime_apply_coe`, `LieAlgebra.LoopAlgebra.twoCocycleOfBilinear_coe`, `PointedCone.dual_flip_dual_dual_flip`, `groupCohomology.cocyclesMk₀_eq`, `mem_sup_left`, `isSemisimpleModule_biSup_of_isSemisimpleModule_submodule`, `EuclideanGeometry.two_zsmul_oangle_orthogonalProjection_self`, `range_subtype`, `coe_toSubMulAction`, `range_derivWithin_subset_closure_span_image`, `NumberField.instIsLocalizedModuleIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmoduleValFractionalIdealNonZeroDivisorsRestrictScalarsSubtype`, `isArtinian_iSup`, `vsub_set_subset_vectorSpan`, `ContinuousLinearMap.finite_dimensional_eigenspace`, `Ideal.ext_iff`, `ProperCone.mem_toPointedCone`, `subtype_injective`, `Set.MapsTo.submoduleSpan`, `LinearPMap.mem_adjoint_domain_of_exists`, `mem_div_iff_smul_subset`, `dualRestrict_apply`, `LinearMap.BilinForm.finrank_add_finrank_orthogonal`, `LinearMap.mapsTo_biSup_of_mapsTo`, `smulMemClass`, `Ideal.mem_comap`, `Module.End.mem_genEigenspace`, `IsSemisimpleModule.exists_linearEquiv_dfinsupp`, `neg_coe`, `groupHomology.single_mem_cycles₂_iff`, `HasStrictFDerivAt.implicitFunctionOfComplemented_apply_image`, `Module.Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix`, `IsPrincipal.mem_iff_generator_dvd`, `GradedAlgebra.mem_support_iff`, `finrank_quotient`, `exists_C_coeff_notMem`, `Int.absNorm_under_eq_sInf`, `comapSubtypeEquivOfLe_apply_coe`, `Ideal.IsLocal.mem_jacobson_or_exists_inv`, `Ideal.algebraMap_residueField_eq_zero`, `PeriodPair.summable_weierstrassPExceptSummand`, `RootPairing.RootPositiveForm.algebraMap_posForm`, `LinearDisjoint.rank_inf_le_one_of_commute_of_flat_left`, `PrimeSpectrum.range_comap_fst`, `PointedCone.mem_hull_set`, `AmpleSet.of_one_lt_codim`, `QuadraticForm.sigPos_add_finrank_le_of_nonpos`, `ModuleCat.mono_as_hom'_subtype`, `Ideal.dvd_span_singleton`, `DirectSum.isInternal_biSup_submodule_of_iSupIndep`, `QuadraticMap.Nondegenerate.rank_rad_polar_le`, `dfinsuppSumAddHom_mem`, `mem_sup_right`, `LinearMap.ker_le_range_iff`, `exists_mem_and_smul_eq_self_of_fg_of_le_smul`, `Ideal.map_toCotangent_ker`, `PrimeSpectrum.zeroLocus_empty_iff_eq_top`, `ProjectiveSpectrum.zeroLocus_inf`, `LinearMap.ofIsCompl_zero`, `LinearPMap.exists_of_le`, `Module.Baer.ExtensionOfMaxAdjoin.eqn`, `Pi.mem_span_range_single_inl_iff`, `span_biInter`, `sup_span`, `Ideal.smul_mem_pointwise_smul_iff`, `AffineSubspace.equivMapOfInjective_toFun`, `exteriorPower.ιMulti_family_span`, `LinearPMap.supSpanSingleton_apply_mk`, `ZSpan.fract_eq_fract`, `Ideal.mem_sup_right`, `Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero`, `toLocalized'_apply_coe`, `mul_def`, `orthogonalProjection_comp_subtypeL_eq_zero_iff`, `EuclideanGeometry.orthogonalProjection_sup_of_orthogonalProjection_eq`, `PrimeSpectrum.zeroLocus_anti_mono_ideal`, `Affine.Simplex.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter`, `powSMulQuotInclusion_injective`, `dualQuotEquivDualAnnihilator_symm_apply_mk`, `LinearMap.BilinForm.finrank_add_finrank_orthogonal'`, `LieDerivation.IsKilling.killingForm_restrict_range_ad_nondegenerate`, `Ideal.quotTorsionOfEquivSpanSingleton_apply_mk`, `Ideal.quotientToQuotientRangePowQuotSuccAux_mk`, `Module.Basis.sumQuot_inl`, `ZLattice.sum_piFinset_Icc_rpow_le`, `LinearEquiv.ofInjective_symm_apply`, `LieSubalgebra.lie_mem_sup_of_mem_normalizer`, `Int.cast_mem_ideal_iff`, `smul_of_tower_mem`, `LieAlgebra.Extension.twoCocycleOf_coe_coe`, `RingHom.surjectiveOnStalks_iff_forall_ideal`, `mem_map_equiv`, `closedComplemented_iff_isClosed_exists_isClosed_isCompl`, `EuclideanGeometry.coe_orthogonalProjection_eq_iff_mem`, `ExteriorAlgebra.GradedAlgebra.ι_apply`, `Algebra.SubmersivePresentation.basisCotangent_apply`, `linearProjOfIsCompl_surjective`, `LinearMap.exists_extend_of_notMem`, `RootPairing.posRootForm_rootFormIn_posDef`, `Ideal.mem_span_pow_iff_exists_isHomogeneous`, `HahnEmbedding.Partial.truncLT_eval_mem_range_extendFun`, `LinearMap.comp_ker_subtype`, `mulMap_range`, `AddMonoidAlgebra.single_mem_gradeBy`, `weightedVSub_mem_vectorSpan_pair`, `isFullyInvariant_iff_le_imp_isotypicComponent_le`, `TensorProduct.tensorQuotientEquiv_symm_apply_tmul_mk`, `groupHomology.coinvariantsKerOfIsBoundary₀_coe`, `Affine.Simplex.map_subtype_restrict`, `Algebra.kerTensorProductMapIdToAlgHomEquiv_symm_apply`, `exteriorPower.oneEquiv_ιMulti`, `ConvexCone.mem_toPointedCone`, `Ideal.toIdeal_homogeneousHull_eq_iSup`, `Affine.Simplex.restrict_map_subtype`, `continuous_stereoInvFun`, `Ideal.exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem`, `Algebra.Extension.Cotangent.map_mk`, `RootPairing.CoPolarizationIn_apply`, `Module.isTorsionBy_iff_mem_annihilator`, `mem_iSup_iff_exists_dfinsupp'`, `LinearDisjoint.linearIndependent_left_of_flat`, `orthogonalProjection_eq_zero_iff`, `linearDepOn_iff'`, `groupCohomology.cocycles₁_ext_iff`, `CliffordAlgebra.GradedAlgebra.ι_sq_scalar`, `ZLattice.covolume.tendsto_card_div_pow''`, `AdicCompletion.ofPowSMul_val_apply`, `Polynomial.sylveserMap_comp_adjSylvester`, `smul_mem_iff''`, `isAdic_iff`, `SkewMonoidAlgebra.mem_span_support`, `groupCohomology.map_H0Iso_hom_f_assoc`, `AffineSubspace.subtypeₐᵢ_toAffineMap`, `exteriorPower.ιMulti_span`, `map_equivMapOfInjective_symm_apply`, `Polynomial.Chebyshev.U_mem_span_T`, `Finsupp.supportedEquivFinsupp_symm_apply_coe_apply`, `PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal'`, `NonUnitalSubalgebra.coe_toSubmodule`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.num_mem_carrier_iff`, `orthogonalProjection_starProjection_of_le`, `toNonUnitalSubalgebra_mk`, `Real.exists_extension_norm_eq`, `LinearMap.finrank_range_add_finrank_ker`, `IsCompl.projection_apply_mem`, `exteriorPower.toTensorPower_apply_ιMulti`, `coe_orthogonalProjection_apply`, `Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOver`, `ModuleCat.kernelIsoKer_inv_kernel_ι`, `FractionalIdeal.mem_extended_iff`, `IsIsotypicOfType.isotypicComponent`, `Quotient.eq'`, `coe_matrix`, `Ideal.IsMaximal.exists_inv`, `carrier_eq_coe`, `torsion'_torsion'_eq_top`, `groupCohomology.cocycles₁.coe_mk`, `instModuleFree_of_discrete_submodule`, `PeriodPair.ω₁_mem_lattice`, `starProjection_apply_eq_zero_iff`, `Ideal.natAbs_det_basis_change`, `ContinuousLinearMap.apply_val_ker`, `PadicInt.coe_dpow_eq`, `Module.Flat.iff_rTensor_injective'`, `PeriodPair.analyticOnNhd_weierstrassP`, `AddMonoidAlgebra.grade.gradedMonoid`, `ModuleCat.imageIsoRange_hom_subtype_apply`, `skewAdjointPart_comp_subtype_skewAdjoint`, `Module.Basis.mem_span_repr_support`, `mem_top`, `mem_iSup_of_directed`, `KaehlerDifferential.fromIdeal_surjective`, `Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op`, `HasStrictFDerivAt.implicitFunction_apply_image`, `connectedComponentIn_eq_self_of_one_lt_codim`, `SubmodulesRingBasis.mul`, `Polynomial.degreeLT.basisProd_castAdd`, `Polynomial.IsEisensteinAt.leading`, `MvPolynomial.coeffsIn_le`, `IsLocalization.map_algebraMap_ne_top_iff_disjoint`, `QuotSMulTop.mem_annihilator`, `Ideal.mem_of_localization_maximal`, `finrank_add_eq_of_isCompl`, `FG.generators_mem`, `Finsupp.supportedEquivFinsupp_apply_apply`, `FG.rTensor.directLimit_apply'`, `LinearMap.restrict_eq_domRestrict_codRestrict`, `MeasureTheory.lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero`, `finiteDimensional_of_le`, `groupHomology.H1AddEquivOfIsTrivial_symm_tmul`, `LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure`, `IsLocalRing.local_hom_TFAE`, `Ideal.one_notMem`, `LieModule.Cohomology.d₂₃_comp_d₁₂`, `Subspace.quotDualCoannihilatorToDual_bijective`, `groupHomology.single_mem_cycles₁_iff`, `Affine.Triangle.dist_orthocenter_reflection_circumcenter_finset`, `Affine.Simplex.affineSpan_pair_eq_altitude_iff`, `Ideal.mem_iSup_of_mem`, `Ideal.toCotangent_surjective`, `lTensor_mkQ`, `norm_subtypeL_le`, `Finsupp.single_mem_supported`, `Ideal.coe_piOrderIso_apply`, `coe_scott_continuous`, `IsSemisimpleModule.range`, `Ideal.mul_sub_mul_mem`, `ContinuousLinearEquiv.fst_equivOfRightInverse`, `LinearMap.quotientInfEquivSupQuotient_symm_apply_right`, `coe_eq_univ`, `instFaithfulVAddSubtypeMem`, `exteriorPower.zeroEquiv_symm_apply`, `isAtom_iff_finrank_eq_one`, `finsetBasisOfLinearIndependentOfCardEqFinrank_repr_apply`, `Module.AEval.mem_mapSubmodule_apply`, `Polynomial.IsWeaklyEisensteinAt.mem`, `Affine.Simplex.restrict_map_inclusion`, `basis_of_pid_aux`, `torsion'_isTorsion'`, `DirectSum.isInternal_submodule_iff_isCompl`, `Representation.asAlgebraHom_mem_of_forall_mem`, `HahnEmbedding.Partial.apply_of_mem_stratum`, `exists_norm_eq_iInf_of_complete_subspace`, `Polynomial.coeff_mem_contentIdeal`, `Ideal.mem_span_singleton_self`, `finiteDimensional_direction_affineSpan_singleton`, `EuclideanGeometry.dist_orthogonalProjection_line_eq_iff_two_zsmul_oangle_eq`, `sndEquiv_apply`, `instSMulCommClassSubtypeMemTorsion'`, `RingHom.toKerIsLocalization_isLocalizedModule`, `LinearPMap.adjointAux_inner`, `sndEquiv_symm_apply_coe`, `Algebra.idealMap_apply_coe`, `comapSubtypeEquivOfLe_symm_apply`, `groupCohomology.map_id_comp_H0Iso_hom_assoc`, `PrimeSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal`, `Module.End.genEigenspace_restrict`, `groupCohomology.isoCocycles₂_hom_comp_i_assoc`, `Matrix.rank_eq_finrank_span_cols`, `EuclideanGeometry.orthogonalProjection_eq_iff_mem`, `LinearMap.trace_eq_sum_trace_restrict_of_eq_biSup`, `IsLocalRing.isOpen_iff_finite_quotient`, `Nat.one_mem_span_iff`, `EuclideanGeometry.dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq`, `Absorbent.subset_range_iff_surjective`, `Module.Basis.mem_span_iff_repr_mem`, `Module.Basis.SmithNormalForm.repr_eq_zero_of_notMem_range`, `injective_lTensor_quotient_iff_inf_eq_mul`, `coe_subtypeL`, `mem_sSup`, `PrimeSpectrum.isCompact_isOpen_iff_ideal`, `orthogonalProjection_coe_eq_linearProjOfIsCompl`, `AffineMap.linear_eqOn_vectorSpan`, `IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_eq_one_iff`, `bot_colon`, `quotient_prod_linearEquiv`, `LinearMap.domRestrict₁₂_apply`, `PeriodPair.ω₂_mem_lattice`, `disjoint_torsionBySet_ideal`, `Ideal.primaryComponent_torsionBySet_of_isCoprime`, `Ideal.map_includeLeft_eq`, `mem_graph_toLinearPMap`, `Algebra.Generators.H1Cotangent.equiv_apply`, `MeasureTheory.lpMeasToLpTrimLie_symm_toLp`, `groupCohomology.coe_mapCocycles₂`, `mk_eq_zero`, `orthogonalProjection_norm_le`, `ContinuousLinearEquiv.equivOfRightInverse_symm_apply`, `sub_mem_iff_right`, `span_le`, `LinearRecurrence.is_sol_iff_mem_solSpace`, `LinearMap.injective_rangeRestrict_iff`, `span_sInf`, `LinearDisjoint.rank_inf_le_one_of_flat_right`, `LinearMap.quotientInfEquivSupQuotient_surjective`, `mul_mem_mul_rev`, `colon_bot`, `LinearPMap.graph_map_snd_eq_range`, `TrivSqZeroExt.mem_kerIdeal_iff_inr`, `ZLattice.comap_span_top`, `PrimeSpectrum.zeroLocus_pow`, `ModuleCat.toKernelSubobject_arrow`, `orthogonalProjection_orthogonal`, `LinearMap.finrank_eigenspace_le`, `Module.Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord`, `LinearMap.domRestrict_apply`, `LinearMap.surjective_comp_subtype_of_isComplemented`, `affineSpan_insert_zero`, `Polynomial.exists_monic_aeval_eq_zero_forall_mem_of_mem_map`, `Polynomial.taylorLinearEquiv_apply_coe`, `MeasureTheory.norm_condExpL2_le_one`, `mem_mk`, `LinearMap.mem_span_iff_continuous`, `Module.finrank_quotient_add_finrank_le`, `groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_assoc`, `Module.Flat.iff_lTensor_injective'`, `groupCohomology.H1π_eq_iff`, `Ideal.Fiber.exists_smul_eq_one_tmul`, `mem_span_insert`, `DirectSum.decompose_smul`, `SchauderBasis.RankOneDecomposition.e_mem_range`, `IsFractionRing.ideal_span_singleton_map_subset`, `Module.Basis.addSubgroupOfClosure_apply`, `AffineSubspace.vsub_left_mem_direction_iff_mem`, `MeasureTheory.condExpL2_comp_continuousLinearMap`, `Ideal.hasBasis_nhds_adic`, `mapsTo_span`, `Polynomial.mem_degreeLT`, `ZSpan.setFinite_inter`, `LinearEquiv.ofTop_apply`, `PeriodPair.derivWeierstrassPExcept_def`, `groupCohomology.isoCocycles₁_hom_comp_i_assoc`, `mulMap_comp_rTensor`, `coe_set_neg`, `Algebra.Extension.Cotangent.ker_mk`, `torsionBySet_isTorsionBySet`, `IsPrimary.mem_or_mem`, `orthogonalProjection_orthogonalComplement_singleton_eq_zero`, `PointedCone.closure_eq`, `ClosedSubmodule.coe_toSubmodule`, `Polynomial.degreeLT.basis_repr`, `IsSemisimpleModule.exists_end_ringEquiv_pi_matrix_end`, `ZLattice.coe_comap`, `Module.End.pos_finrank_genEigenspace_of_hasEigenvalue`, `LinearIsometry.submoduleMap_apply_coe`, `LinearDisjoint.rank_inf_le_one_of_flat_left`, `IsLocalization.coe_primeSpectrumOrderIso_apply_coe_asIdeal`, `EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangentAt`, `AddMonoidAlgebra.mem_gradeBy_iff`, `Ideal.primaryComponent_torsionBySet_eq_inf`, `Polynomial.IsEisensteinAt.notMem`, `orthogonalProjection_eq_linearProjOfIsCompl`, `LinearPMap.adjoint_apply_of_dense`, `groupHomology.mapCycles₁_quotientGroupMk'_epi`, `Quotient.mk_eq_zero`, `PointedCone.smul_mem`, `RootPairing.range_polarizationIn_le_span_coroot`, `HomogeneousIdeal.mem_iff`, `groupHomology.mapCycles₁_comp_i_assoc`, `exists_smith_normal_form_of_le`, `PowerSeries.IsWeierstrassDivisorAt.coeff_div_sub_seq_mem`, `linearDepOn_iff'ₛ`, `MeasureTheory.condExpL2_const_inner`, `LinearPMap.toFun_eq_coe`, `exteriorPower.ιMulti_family_apply_coe`, `PrimeSpectrum.le_vanishingIdeal_zeroLocus`, `map_subtype_le`, `IsCoercive.isClosed_range`, `mem_traceDual_iff_isIntegral`, `ZSpan.ceil_eq_self_of_mem`, `AffineEquiv.ofEq_rfl`, `CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_apply`, `IsLocalization.mem_span_iff`, `PadicInt.existsUnique_mem_range`, `LinearMap.finiteDimensional_range`, `coe_prodEquivOfClosedCompl`, `Module.Basis.restrictScalars_toMatrix`, `sub_mem`, `Nat.coe_maximalIdeal`, `IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_lt_one_iff`, `TensorProduct.exists_of_fg`, `torsionBy.mk_smul`, `AddMonoidAlgebra.mem_span_support`, `Subalgebra.rank_toSubmodule`, `groupHomology.π_comp_H2Iso_inv_apply`, `mem_torsion'_iff`, `Valuation.Integers.maximalIdeal_pow_eq_setOf_le_v_algebraMap_pow`, `Polynomial.IsEisensteinAt.coeff_mem`, `LinearMap.BilinForm.restrict_apply`, `groupCohomology.coboundariesOfIsCoboundary₂_coe`, `topologicalClosure.completeSpace`, `DirectSum.range_coeLinearMap`, `PadicInt.toZMod_spec`, `PointedCone.DualFG.iff_exists_fg_dual`, `submoduleOf_self`, `LinearPMap.mem_range_iff`, `EuclideanGeometry.dist_set_eq_iff_dist_orthogonalProjection_eq`, `MonoidAlgebra.mem_supported'`, `HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorph_source`, `PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure`, `HasStrictFDerivAt.eq_implicitFunction`, `IsLocalRing.spanFinrank_eq_finrank_quotient`, `Finsupp.range_restrictDom`, `starProjection_eq_self_iff`, `QuadraticMap.mem_radical_iff'`, `Projectivization.Subspace.mem_submodule_iff`, `Ideal.mem_inv_pointwise_smul_iff`, `LinearMap.BilinForm.nondegenerate_restrict_of_disjoint_orthogonal`, `linearIndepOn_iff_linearCombinationOnₛ`, `finiteDimensional_inf_left`, `EuclideanGeometry.Sphere.finrank_orthRadius`, `ContinuousLinearEquiv.ofSubmodule'_toContinuousLinearMap`, `TwoSidedIdeal.coe_asIdeal`, `LinearMap.rank_range_add_rank_ker`, `finrank_vectorSpan_insert_le_set`, `smul_mem`, `coe_equivMapOfInjective_apply`, `mem_neg`, `mem_iSup`, `Ideal.IsPrime.mul_mem_iff_mem_or_mem`, `MvPolynomial.mem_ideal_span_X_image`, `LinearMap.exists_map_addHaar_eq_smul_addHaar'`, `InnerProductSpace.gramSchmidt_mem_span`, `MvPolynomial.mem_restrictTotalDegree`, `iSupIndep_iff_forall_dfinsupp`, `groupCohomology.coboundaries₁.coe_mk`, `IsLinearTopology.hasBasis_right_ideal`, `disjoint_iff_comap_eq_bot`, `IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_mem`, `groupHomology.mem_cycles₁_iff`, `instProperSpaceSubtypeMemSubmoduleOfCompleteSpaceOfLocallyCompactSpace`, `LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm`, `LinearMap.IsSymmetric.orthogonalFamily_iInf_eigenspaces`, `Subspace.finrank_add_finrank_dualAnnihilator_eq`, `IsDedekindDomain.isOpen_iff`, `TensorProduct.toLinearMap_mapInclIsometry`, `Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_zero_iff`, `small_sup`, `FiniteArchimedeanClass.mem_closedBall_iff`, `existsUnique_add_of_isCompl_prod`, `SimpleGraph.mem_ker_toLin'_lapMatrix_of_connectedComponent`, `lsum_comp_mapRange_toSpanSingleton`, `MvPolynomial.coe_coeffsIn`, `RootPairing.algebraMap_root'In_apply`, `groupCohomology.mapCocycles₁_comp_i`, `groupHomology.boundariesToCycles₁_apply`, `orthonormal_span`, `groupHomology.single_mem_cycles₂_iff_inv`, `OrthogonalFamily.projection_directSum_coeAddHom`, `MvPolynomial.mem_image_comap_C_basicOpen`, `TensorProduct.range_mapIncl_mono`, `surjective_stereographic`, `Module.Finite.of_isComplemented_codomain`, `MatrixModCat.fromMatrixLinear_apply_coe`, `mem_closure_iff`, `TensorProduct.spanRank_top_le`, `coe_ofLinearComb`, `coe_toAddSubgroup`, `PeriodPair.finrank_lattice`, `PeriodPair.derivWeierstrassP_add_coe`, `IsLocalRing.isOpen_maximalIdeal`, `Algebra.Presentation.mem_ker_naive`, `linearIndepOn_iff_linearCombinationOn`, `PointedCone.dual_eq_iInter_dual_singleton`, `Module.Flat.instInvertibleSubtypeMemSubmoduleSubmoduleAlgebra`, `LinearMap.toKerIsLocalized_apply_coe`, `coe_sInf`, `LinearMap.submoduleMap_surjective`, `hasFDerivAt_stereoInvFunAux_comp_coe`, `LinearMap.finrank_le_of_isSMulRegular`, `LinearIsometryEquiv.ofEq_symm`, `AffineSubspace.isClosed_direction_iff`, `finiteDimensional_vectorSpan_insert`, `LinearPMap.inverse_domain`, `exteriorPower.basis_repr_self`, `PointedCone.dual_dual_flip_dual`, `Ideal.mem_prod`, `MeasureTheory.condExpL2_indicator_nonneg`, `flip_quotDualCoannihilatorToDual_injective`, `PointedCone.coe_comap`, `coe_prodEquivOfClosedCompl_symm`, `isClosed_or_dense_of_isCoatom`, `groupCohomology.cocycles₁.d₁₂_apply`, `Module.Finite.instLinearMapIdSubtypeMemSubmoduleOfIsSemisimpleModule`, `Algebra.Extension.cotangentComplex_mk`, `mulRightMap_apply_single`, `MvPolynomial.WeightedHomogeneousSubmodule.gradedMonoid`, `RingHom.surjective_localRingHom_iff`, `Subspace.dualPairing_eq`, `groupHomology.isoCycles₂_hom_comp_i_assoc`, `PointedCone.dual_eq_dual_id_map`, `IdealFilter.isUniform_iff_exists_ringFilterBasis`, `NumberField.mixedEmbedding.mem_rat_span_latticeBasis`, `subalgebra_top_rank_eq_submodule_top_rank`, `Int.submodule_toAddSubgroup_index_ne_zero_iff`, `fstEquiv_apply`, `groupCohomology.π_comp_H2Iso_hom`, `FiniteDimensional.span_singleton`, `Ideal.to_quotient_square_comp_toCotangent`, `retractionOfSectionOfKerSqZero_tmul_D`, `Ideal.IsHomogeneous.mem_iff`, `DirectSum.IsInternal.subordinateOrthonormalBasisIndex_def`, `groupHomology.H2π_eq_zero_iff`, `finrank_vectorSpan_range_le`, `GeneralSchauderBasis.finrank_range_proj`, `ContinuousLinearEquiv.ofSubmodules_apply`, `AffineSubspace.coe_direction_eq_vsub_set_right`, `add_mem`, `Ideal.mem_jacobson_bot`, `LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul`, `BoxIntegral.unitPartition.tag_mem_smul_span`, `LinearMap.ofIsCompl_eq_add`, `AlgebraicGeometry.Scheme.IdealSheafData.supportSet_eq_iInter_zeroLocus`, `PointedCone.mem_lineal`, `span_range_inclusionSpan`, `PeriodPair.weierstrassPExcept_add`, `LinearMap.surjective_domRestrict_iff`, `coe_iSup_of_chain`, `span_coe_eq_restrictScalars`, `surjOn_iff_le_map`, `ZLattice.exists_finsetSum_norm_rpow_le_tsum`, `mem_nilradical`, `Subspace.dualEquivDual_apply`, `isHilbertSumOrthogonal`, `instIsLocalizedModuleSubtypeMemLocalized₀ToLocalized₀`, `Affine.Simplex.restrict_points_coe`, `Ideal.isoBaseOfIsPrincipal_apply`, `smul_coe`, `Polynomial.Chebyshev.T_derivative_mem_span_T`, `groupCohomology.isoCocycles₂_inv_comp_iCocycles_assoc`, `ZLattice.abs_repr_le`, `LinearMap.singularValues_eq_zero_iff_le_finrank_range`, `DirectSum.IsInternal.collectedBasis_repr_of_mem`, `Affine.Simplex.excenterExists_restrict`, `MvPolynomial.decomposition.decompose'_eq`, `restrictScalarsEquiv_apply`, `ZLattice.covolume_eq_det`, `Algebra.Generators.C_mul_X_sub_one_mem_ker`, `RingFilterBasis.SubmodulesBasis.smul`, `Finsupp.mem_supported'`, `rank_range_of_surjective`, `PointedCone.salient_iff_inter_neg_eq_singleton`, `card_eq_card_quotient_mul_card`, `groupHomology.δ₁_apply`, `LinearDisjoint.linearIndependent_mul_of_flat`, `Ideal.isOpen_of_isMaximal`, `Ideal.span_singleton_mul_le_iff`, `LinearMap.restrict_apply`, `Valuation.HasExtension.algebraMap_mem_maximalIdeal_iff`, `AddMonoidAlgebra.mem_supported`, `EuclideanGeometry.orthogonalProjection_mem_subspace_eq_self`, `mem_toNonUnitalSubalgebra`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply`, `natAbs_det_basis_change`, `exists_sub_one_mem_and_smul_le_of_fg_of_le_sup`, `NumberField.mixedEmbedding.fundamentalCone.intNorm_idealSetEquiv_apply`, `orthogonalProjectionFn_mem`, `groupHomology.H1π_eq_iff`, `nonempty_basis_of_pid`, `LinearIsometryEquiv.submoduleMap_apply_coe`, `LinearPMap.image_iff`, `PointedCone.dual_span`, `comap_subtype_le_iff`, `finiteDimensional_iSup`, `ZModModule.exists_submodule_subset_card_le`, `MeasureTheory.lpMeas.aestronglyMeasurable`, `Ideal.Cotangent.lift_surjective_iff`, `Affine.Triangle.dist_orthogonalProjectionSpan_faceOpposite_eq_iff_two_zsmul_oangle_eq`, `mem_iff_norm_starProjection`, `isSemisimpleModule_iff_exists_linearEquiv_dfinsupp`, `groupHomology.cyclesMap_comp_isoCycles₁_hom`, `PeriodPair.differentiableOn_derivWeierstrassPExcept`, `PrimeSpectrum.zeroLocus_eq_singleton`, `ContinuousLinearEquiv.ofSubmodules_symm_apply`, `LinearMap.card_support_singularValues`, `KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk_symm_apply_eq_span`, `Module.notMem_support_iff'`, `Algebra.ZariskisMainProperty.exists_fg_and_exists_notMem_and_awayMap_bijective`, `PowerSeries.IsWeierstrassDivisionAt.coeff_f_sub_r_mem`, `exists_linearIndepOn_id_extension`, `Rep.coindFunctorIso_hom_app_hom_toFun_hom_toFun`, `AlgebraicGeometry.IsAffineOpen.mem_ideal_iff`, `HahnEmbedding.Partial.toOrderAddMonoidHom_injective`
`smul` 📖	CompOp	35 math math: `instIsScalarTowerQuotientIdealSubtypeMemTorsionBySetCoe`, `OrthonormalBasis.orthogonalProjection_eq_sum`, `isScalarTower`, `LinearMap.IsSymmetric.diagonalization_apply_self_apply`, `OrthonormalBasis.orthogonalProjection_apply_eq_sum`, `Module.Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul`, `isSMulRegular_submodule_iff_right_eq_zero_of_smul`, `HilbertBasis.hasSum_orthogonalProjection`, `stereoToFun_apply`, `IsSMulRegular.submodule`, `lTensorOne'_tmul`, `rTensorOne'_tmul`, `isScalarTower'`, `rTensorOne_tmul`, `instIsScalarTowerQuotientSpanSingletonSetSubtypeMemTorsionBy`, `torsionBy.mk_ideal_smul`, `LinearPMap.map_smul`, `LieModule.Cohomology.smul_apply_apply`, `isCentralScalar`, `coe_smul`, `restrictScalars.isScalarTower`, `coe_smul_of_tower`, `smul_torsionBy`, `stereographic_apply`, `lTensorOne_tmul`, `isScalarTower_of_section_of_ker_sqZero`, `torsionBySet.mk_smul`, `UniformOnFun.continuousSMul_submodule_of_image_bounded`, `isSMulRegular_on_submodule_iff_disjoint_ker_lsmul_submodule`, `MeasureTheory.lpMeasToLpTrim_smul`, `mem_set_smul`, `Ideal.quotTorsionOfEquivSpanSingleton_apply_mk`, `exteriorPower.zeroEquiv_symm_apply`, `instSMulCommClassSubtypeMemTorsion'`, `torsionBy.mk_smul`
`toAddSubgroup` 📖	CompOp	61 math math: `toAddSubgroup_toZModSubmodule`, `LinearMap.range_toAddSubgroup`, `FiniteArchimedeanClass.toAddSubgroup_ball`, `ContinuousMap.range_toLp`, `fg_toAddSubgroup`, `SubmodulesRingBasis.toRing_subgroups_basis`, `ZLattice.comap_toAddSubgroup`, `pointwise_smul_toAddSubgroup`, `AdjoinRoot.evalEval_apply`, `torsion_int`, `NumberField.Units.span_basisOfIsMaxRank`, `Quotient.quotientAddGroupMk_eq_mk`, `AddSubgroup.toZModSubmodule_toAddSubgroup`, `exists_finiteIndex_of_cover`, `span_int_eq`, `ZLattice.covolume_div_covolume_eq_relIndex`, `toAddSubgroup_toAddSubmonoid`, `Ideal.index_pow_le`, `AddSubgroup.toIntSubmodule_symm`, `top_toAddSubgroup`, `Ideal.Quotient.index_eq_zero`, `RingPreordering.supportAddSubgroup_eq`, `toAddSubgroup_mono`, `span_singleton_toAddSubgroup_eq_zmultiples`, `toIntSubmodule_toAddSubgroup`, `toAddSubgroup_injective`, `isFiniteRelIndex_of_map_linearMapMulLeft_le`, `toAddSubgroup_toIntSubmodule`, `ZSpan.isAddFundamentalDomain'`, `fg_iff_addSubgroup_fg`, `AddSubgroup.toZModSubmodule_symm`, `mem_toAddSubgroup`, `lp.coe_lpSubmodule`, `span_int_eq_addSubgroupClosure`, `goursatSnd_toAddSubgroup`, `toAddSubgroup_strictMono`, `Ideal.span_singleton_toAddSubgroup_eq_zmultiples`, `ArchimedeanClass.toAddSubgroup_ball`, `goursatFst_toAddSubgroup`, `ZSpan.instDiscreteTopologySubtypeMemAddSubgroupToAddSubgroupIntSpanRangeCoeBasisRealOfFinite`, `Module.Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod`, `toAddSubgroup_eq_top`, `PointedCone.support_eq`, `Ideal.pointwise_smul_toAddSubgroup`, `MeasureTheory.Lp.coe_LpSubmodule`, `AddSubgroup.toIntSubmodule_toAddSubgroup`, `map_toAddSubgroup`, `index_smul_le`, `FiniteArchimedeanClass.toAddSubgroup_closedBall`, `toAddSubgroup_le`, `Module.Basis.SmithNormalForm.toAddSubgroup_index_eq_ite`, `BoundedContinuousFunction.range_toLp`, `LinearMap.ker_toAddSubgroup`, `KaehlerDifferential.quotKerTotalEquiv_apply`, `toAddSubgroup_inj`, `sup_toAddSubgroup`, `bot_toAddSubgroup`, `ZLattice.covolume_div_covolume_eq_relIndex'`, `coe_toAddSubgroup`, `ArchimedeanClass.toAddSubgroup_closedBall`, `Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply`
`toAddSubmonoid` 📖	CompOp	66 math math: `toAddSubmonoid_toNatSubmodule`, `Ideal.Filtration.submodule_closure_single`, `ClosedSubmodule.mem_iff`, `top_toAddSubmonoid`, `coe_toAddSubmonoid`, `toAddSubmonoid_injective`, `toSubalgebra_toSubsemiring`, `ClosedSubmodule.coe_sSup`, `coe_toSubalgebra`, `smul_mem'`, `closure_le_toAddSubmonoid_span`, `LinearMap.range_toAddSubmonoid`, `HomogeneousIdeal.toAddSubmonoid_irrelevant_le`, `span_nat_eq`, `LieSubalgebra.mem_carrier`, `AddSubmonoid.toNatSubmodule_toAddSubmonoid`, `le_one_toAddSubmonoid`, `Ideal.disjoint_powers_iff_notMem`, `toAddSubgroup_toAddSubmonoid`, `toAddSubmonoid_strictMono`, `pointwise_smul_toAddSubmonoid`, `ClosedSubmodule.ext_iff`, `LinearMap.ker_toAddSubmonoid`, `bot_toAddSubmonoid`, `HomogeneousIdeal.irrelevant_le`, `toAddSubmonoid_mono`, `smul_toAddSubmonoid`, `toAddSubmonoid_sSup`, `LieSubmodule.lie_mem`, `eta`, `span_eq_closure`, `AddSubmonoid.toNatSubmodule_symm`, `mul_toAddSubmonoid`, `ClosedSubmodule.mem_sup`, `ClosedSubmodule.mem_iSup`, `fg_iff_addSubmonoid_fg`, `toAddSubmonoid_le`, `mem_toAddSubmonoid`, `Representation.invariants_eq_inter`, `Subalgebra.pi_toSubsemiring`, `span_nat_eq_addSubmonoidClosure`, `toAddSubmonoid_inj`, `ClosedSubmodule.mem_sSup`, `mem_carrier`, `Subalgebra.coe_pi`, `LinearMap.eqLocus_toAddSubmonoid`, `toAddSubmonoid_restrictScalars`, `PointedCone.support_eq`, `HomogeneousIdeal.irrelevant_eq_iSup`, `LieSubmodule.mem_carrier`, `sup_toAddSubmonoid`, `HomogeneousIdeal.irrelevant_eq_closure`, `ClosedSubmodule.coe_iSup`, `HomogeneousIdeal.toIdeal_irrelevant_le`, `map_toAddSubmonoid'`, `ClosedSubmodule.coe_sup`, `neg_toAddSubmonoid`, `carrier_eq_coe`, `pow_toAddSubmonoid`, `LieSubalgebra.lie_mem'`, `iSup_toAddSubmonoid`, `le_pow_toAddSubmonoid`, `ClosedSubmodule.carrier_eq_coe`, `map_toAddSubmonoid`, `Ideal.pointwise_smul_toAddSubmonoid`, `ClosedSubmodule.isClosed'`
`toSubMulAction` 📖	CompOp	6 math math: `toSubMulAction_inj`, `toSubMulAction_strictMono`, `coe_toSubMulAction`, `toSubMulAction_one`, `toSubMulAction_injective`, `toSubMulAction_mono`
`zero` 📖	CompOp	76 math math: `HasStrictFDerivAt.map_implicitFunctionOfComplemented_eq`, `NumberField.mixedEmbedding.fundamentalDomain_integerLattice`, `iSupIndep.linearEquiv_symm_apply`, `linearProjOfIsCompl_apply_right`, `prodEquivOfIsCompl_symm_apply_snd_eq_zero`, `Ideal.finsuppTotal_apply_eq_of_fintype`, `eq_zero_of_coe_mem_of_disjoint`, `Ideal.finsuppTotal_apply`, `linearProjOfIsCompl_apply_right'`, `Polynomial.degreeLT.addLinearEquiv_castAdd`, `IsCompactOperator.restrict'`, `mulLeftMap_apply`, `stereographic_apply_neg`, `HasStrictFDerivAt.to_implicitFunctionOfComplemented`, `HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorphOfComplemented_target`, `NumberField.mixedEmbedding.fundamentalDomain_idealLattice`, `mem_left_iff_eq_zero_of_disjoint`, `orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero`, `botEquivPUnit_symm_apply`, `LinearDisjoint.linearIndependent_right_of_flat`, `IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp`, `HasStrictFDerivAt.map_implicitFunction_eq`, `mem_iSup_iff_exists_dfinsupp`, `prodEquivOfIsCompl_symm_apply_fst_eq_zero`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem_ne`, `HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorph_target`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_self`, `LinearPMap.map_zero`, `HasStrictFDerivAt.to_implicitFunction`, `Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent`, `HahnEmbedding.Seed.coeff_baseEmbedding`, `coe_zero`, `IsCompactOperator.restrict`, `prodEquivOfIsCompl_symm_apply_left`, `stereographic_neg_apply`, `iSupIndep.linearEquiv_apply`, `Function.Exact.linearMap_rangeRestrict`, `ZSpan.isAddFundamentalDomain`, `mem_right_iff_eq_zero_of_disjoint`, `mulRightMap_apply`, `Function.Exact.iff_linearMap_rangeRestrict`, `HasStrictFDerivAt.implicitToOpenPartialHomeomorph_self`, `ContinuousLinearMap.projKerOfRightInverse_comp_inv`, `prodEquivOfIsCompl_symm_apply_right`, `mulLeftMap_apply_single`, `Ideal.range_finsuppTotal`, `ZLattice.isAddFundamentalDomain`, `coe_eq_zero`, `smul_torsionBy`, `HahnEmbedding.Seed.hahnCoeff_apply`, `CliffordAlgebra.even.lift.aux_apply`, `ClosedComplemented.exists_submodule_equiv_prod`, `Polynomial.degreeLT.addLinearEquiv_natAdd`, `orthogonalProjection_orthogonal_apply_eq_zero`, `DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne`, `eq_zero_of_bot_submodule`, `IsIsotypic.linearEquiv_finsupp`, `EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection`, `Polynomial.degreeLT.basisProd_natAdd`, `mem_set_smul`, `linearProjOfIsCompl_apply_eq_zero_iff`, `mem_biSup_iff_exists_dfinsupp`, `IsSemisimpleModule.exists_linearEquiv_dfinsupp`, `HasStrictFDerivAt.implicitFunctionOfComplemented_apply_image`, `mem_iSup_iff_exists_dfinsupp'`, `LinearDisjoint.linearIndependent_left_of_flat`, `orthogonalProjection_eq_zero_iff`, `Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op`, `HasStrictFDerivAt.implicitFunction_apply_image`, `Polynomial.degreeLT.basisProd_castAdd`, `mk_eq_zero`, `orthogonalProjection_orthogonalComplement_singleton_eq_zero`, `iSupIndep_iff_forall_dfinsupp`, `hasFDerivAt_stereoInvFunAux_comp_coe`, `mulRightMap_apply_single`, `isSemisimpleModule_iff_exists_linearEquiv_dfinsupp`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`addSubgroupClass` 📖	mathematical	—	`AddSubgroupClass` `Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `AddGroup.toSubNegMonoid` `AddCommGroup.toAddGroup` `setLike`	—	`addSubmonoidClass` `SubMulAction.neg_mem`
`addSubmonoidClass` 📖	mathematical	—	`AddSubmonoidClass` `Submodule` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `setLike`	—	`AddSubsemigroup.add_mem'` `AddSubmonoid.zero_mem'`
`add_mem` 📖	mathematical	`Submodule` `SetLike.instMembership` `setLike`	`Submodule` `SetLike.instMembership` `setLike` `AddCommMagma.toAdd` `AddCommSemigroup.toAddCommMagma` `AddCommMonoid.toAddCommSemigroup`	—	`AddMemClass.add_mem` `AddSubmonoidClass.toAddMemClass` `addSubmonoidClass`
`add_mem_iff_left` 📖	mathematical	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike`	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `AddCommMagma.toAdd` `AddCommSemigroup.toAddCommMagma` `AddCommMonoid.toAddCommSemigroup`	—	`add_mem_cancel_right` `addSubgroupClass`
`add_mem_iff_right` 📖	mathematical	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike`	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `AddCommMagma.toAdd` `AddCommSemigroup.toAddCommMagma` `AddCommMonoid.toAddCommSemigroup`	—	`add_mem_cancel_left` `addSubgroupClass`
`carrier_eq_coe` 📖	mathematical	—	`AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `toAddSubmonoid` `SetLike.coe` `Submodule` `setLike`	—	—
`coe_add` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `add` `AddCommMagma.toAdd` `AddCommSemigroup.toAddCommMagma` `AddCommMonoid.toAddCommSemigroup`	—	—
`coe_copy` 📖	mathematical	`SetLike.coe` `Submodule` `setLike`	`SetLike.coe` `Submodule` `setLike` `copy`	—	—
`coe_eq_zero` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `zero`	—	`SetLike.coe_eq_coe`
`coe_mem` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike`	—	—
`coe_mk` 📖	mathematical	`Submodule` `SetLike.instMembership` `setLike`	`Submodule` `SetLike.instMembership` `setLike`	—	—
`coe_neg` 📖	mathematical	—	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `NegMemClass.neg` `NegZeroClass.toNeg` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `AddSubgroupClass.toNegMemClass` `AddGroup.toSubNegMonoid` `AddCommGroup.toAddGroup` `addSubgroupClass`	—	`NegMemClass.coe_neg` `addSubgroupClass`
`coe_ofClass` 📖	mathematical	—	`SetLike.coe` `Submodule` `setLike` `ofClass`	—	—
`coe_ofLinearComb` 📖	mathematical	`Set.Nonempty` `Set` `Set.instMembership` `AddCommMagma.toAdd` `AddCommSemigroup.toAddCommMagma` `AddCommMonoid.toAddCommSemigroup` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	`SetLike.coe` `Submodule` `setLike` `ofLinearComb`	—	—
`coe_set_mk` 📖	mathematical	`Set` `Set.instMembership` `AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `SMulZeroClass.toSMul` `AddZero.toZero` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	`SetLike.coe` `Submodule` `setLike` `AddSubmonoid` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.instSetLike`	—	—
`coe_smul` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `smul` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddMonoidWithOne.toAddMonoid` `AddCommMonoidWithOne.toAddMonoidWithOne` `NonAssocSemiring.toAddCommMonoidWithOne` `Semiring.toNonAssocSemiring` `DistribSMul.toSMulZeroClass` `instDistribSMul` `NonAssocSemiring.toNonUnitalNonAssocSemiring` `AddCommMonoid.toAddMonoid` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction` `IsScalarTower.left` `DistribMulAction.toMulAction`	—	`IsScalarTower.left`
`coe_smul_of_tower` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `smul`	—	—
`coe_sub` 📖	mathematical	—	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `AddSubgroupClass.sub` `AddGroup.toSubNegMonoid` `AddCommGroup.toAddGroup` `addSubgroupClass` `SubNegMonoid.toSub`	—	`AddSubgroupClass.coe_sub` `addSubgroupClass`
`coe_toAddSubgroup` 📖	mathematical	—	`SetLike.coe` `AddSubgroup` `AddCommGroup.toAddGroup` `AddSubgroup.instSetLike` `toAddSubgroup` `Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `setLike`	—	—
`coe_toAddSubmonoid` 📖	mathematical	—	`SetLike.coe` `AddSubmonoid` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.instSetLike` `toAddSubmonoid` `Submodule` `setLike`	—	—
`coe_toSubMulAction` 📖	mathematical	—	`SetLike.coe` `SubMulAction` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction` `SubMulAction.instSetLike` `toSubMulAction` `Submodule` `setLike`	—	—
`coe_zero` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `zero` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid`	—	—
`copy_eq` 📖	mathematical	`SetLike.coe` `Submodule` `setLike`	`copy`	—	`SetLike.coe_injective`
`eta` 📖	mathematical	`Set` `Set.instMembership` `AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `toAddSubmonoid` `SMulZeroClass.toSMul` `AddZero.toZero` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	`toAddSubmonoid`	—	—
`ext` 📖	—	`Submodule` `SetLike.instMembership` `setLike`	—	—	`SetLike.ext`
`ext_iff` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike`	—	`ext`
`instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallHSMul` 📖	mathematical	—	`CanLift` `Set` `Submodule` `SetLike.coe` `setLike` `Set.instMembership` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid`	—	—
`isScalarTower` 📖	mathematical	—	`IsScalarTower` `Submodule` `SetLike.instMembership` `setLike` `smul` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddMonoidWithOne.toAddMonoid` `AddCommMonoidWithOne.toAddMonoidWithOne` `NonAssocSemiring.toAddCommMonoidWithOne` `Semiring.toNonAssocSemiring` `DistribSMul.toSMulZeroClass` `instDistribSMul` `NonAssocSemiring.toNonUnitalNonAssocSemiring` `AddCommMonoid.toAddMonoid` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction` `IsScalarTower.left` `DistribMulAction.toMulAction`	—	`SubMulAction.isScalarTower`
`isScalarTower'` 📖	mathematical	—	`IsScalarTower` `Submodule` `SetLike.instMembership` `setLike` `smul`	—	`SubMulAction.isScalarTower'`
`mem_carrier` 📖	mathematical	—	`Set` `Set.instMembership` `AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `toAddSubmonoid` `SetLike.coe` `Submodule` `setLike`	—	—
`mem_mk` 📖	mathematical	`Set` `Set.instMembership` `AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `SMulZeroClass.toSMul` `AddZero.toZero` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	`Submodule` `SetLike.instMembership` `setLike` `AddSubmonoid` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.instSetLike`	—	—
`mem_toAddSubgroup` 📖	mathematical	—	`AddSubgroup` `AddCommGroup.toAddGroup` `SetLike.instMembership` `AddSubgroup.instSetLike` `toAddSubgroup` `Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `setLike`	—	—
`mem_toAddSubmonoid` 📖	mathematical	—	`AddSubmonoid` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `SetLike.instMembership` `AddSubmonoid.instSetLike` `toAddSubmonoid` `Submodule` `setLike`	—	—
`mk_eq_zero` 📖	mathematical	`Submodule` `SetLike.instMembership` `setLike`	`Submodule` `SetLike.instMembership` `setLike` `zero` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid`	—	—
`mk_le_mk` 📖	mathematical	`Set` `Set.instMembership` `AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `SMulZeroClass.toSMul` `AddZero.toZero` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	`Submodule` `Preorder.toLE` `PartialOrder.toPreorder` `instPartialOrder` `AddSubmonoid` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.instPartialOrder`	—	—
`neg_mem` 📖	mathematical	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike`	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `NegZeroClass.toNeg` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid`	—	`NegMemClass.neg_mem` `AddSubgroupClass.toNegMemClass` `addSubgroupClass`
`neg_mem_iff` 📖	mathematical	—	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `NegZeroClass.toNeg` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid`	—	`neg_mem_iff` `AddSubgroupClass.toNegMemClass` `addSubgroupClass`
`nonempty` 📖	mathematical	—	`Set.Nonempty` `SetLike.coe` `Submodule` `setLike`	—	`zero_mem`
`smulMemClass` 📖	mathematical	—	`SMulMemClass` `Submodule` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction` `setLike`	—	`SubMulAction.smul_mem'`
`smul_mem` 📖	mathematical	`Submodule` `SetLike.instMembership` `setLike`	`Submodule` `SetLike.instMembership` `setLike` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	—	`smul_mem'`
`smul_mem'` 📖	mathematical	`Set` `Set.instMembership` `AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `toAddSubmonoid`	`Set` `Set.instMembership` `AddSubsemigroup.carrier` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `AddSubmonoid.toAddSubsemigroup` `toAddSubmonoid` `SMulZeroClass.toSMul` `AddZero.toZero` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	—	—
`smul_mem_iff'` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `SemigroupAction.toSMul` `Monoid.toSemigroup` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid` `MulAction.toSemigroupAction`	—	`SubMulAction.smul_mem_iff'`
`smul_mem_iff''` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	—	`invOf_smul_smul` `smul_mem`
`smul_mem_iff_of_isUnit` 📖	mathematical	`IsUnit` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero`	`Submodule` `SetLike.instMembership` `setLike` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction`	—	`smul_mem_iff''`
`smul_of_tower_mem` 📖	mathematical	`Submodule` `SetLike.instMembership` `setLike`	`Submodule` `SetLike.instMembership` `setLike`	—	`SubMulAction.smul_of_tower_mem`
`sub_mem` 📖	mathematical	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike`	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddCommGroup.toAddGroup`	—	`sub_mem` `addSubgroupClass`
`sub_mem_iff_left` 📖	mathematical	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike`	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddCommGroup.toAddGroup`	—	`sub_eq_add_neg` `add_mem_iff_left` `neg_mem`
`sub_mem_iff_right` 📖	mathematical	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike`	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `SetLike.instMembership` `setLike` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `AddCommGroup.toAddGroup`	—	`sub_eq_add_neg` `add_mem_iff_right` `neg_mem_iff`
`toAddSubgroup_inj` 📖	mathematical	—	`toAddSubgroup`	—	`toAddSubgroup_injective`
`toAddSubgroup_injective` 📖	mathematical	—	`Submodule` `Ring.toSemiring` `AddCommGroup.toAddCommMonoid` `AddSubgroup` `AddCommGroup.toAddGroup` `toAddSubgroup`	—	`SetLike.ext` `SetLike.ext_iff`
`toAddSubmonoid_inj` 📖	mathematical	—	`toAddSubmonoid`	—	`toAddSubmonoid_injective`
`toAddSubmonoid_injective` 📖	mathematical	—	`Submodule` `AddSubmonoid` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `toAddSubmonoid`	—	`SetLike.ext'_iff`
`toSubMulAction_inj` 📖	mathematical	—	`toSubMulAction`	—	`toSubMulAction_injective`
`toSubMulAction_injective` 📖	mathematical	—	`Submodule` `SubMulAction` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `MonoidWithZero.toMonoid` `Semiring.toMonoidWithZero` `Module.toDistribMulAction` `toSubMulAction`	—	`SetLike.ext'_iff`
`zero_mem` 📖	mathematical	—	`Submodule` `SetLike.instMembership` `setLike` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddCommMonoid.toAddMonoid`	—	`ZeroMemClass.zero_mem` `AddSubmonoidClass.toZeroMemClass` `addSubmonoidClass`

SubmoduleClass

Definitions

Name	Category	Theorems
`module'` 📖	CompOp	—

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