Documentation Verification Report

LinearMap

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

Statistics

MetricCount
DefinitionscodLift, codRestrict, domRestrict, domRestrict', restrict, subtype, inclusion, instAddActionSubtypeMem, subtype
9
TheoremscodLift_apply, codRestrict_apply, coeFn_sum, coe_domRestrict, coe_sum, comp_codLift, comp_codRestrict, domRestrict'_apply, domRestrict_apply, restrict_apply, restrict_coe_apply, restrict_commute, restrict_comp, restrict_eq_codRestrict_domRestrict, restrict_eq_domRestrict_codRestrict, restrict_smul_one, restrict_sub, subtype_comp_codRestrict, subtype_comp_restrict, sum_apply, pow_apply_mem_of_forall_mem, pow_restrict, submodule_pow_eq_zero_of_pow_eq_zero, coe_subtype, subtype_apply, subtype_injective, coe_inclusion, coe_subtype, coe_sum, inclusion_apply, inclusion_injective, injective_subtype, subtype_apply, subtype_comp_inclusion, subtype_injective
35
Total44

LinearMap

Definitions

NameCategoryTheorems
codLift 📖CompOp
2 mathmath: codLift_apply, comp_codLift
codRestrict 📖CompOp
29 mathmath: Rep.resCoindHomEquiv_apply_hom, subtype_comp_codRestrict, setBasisOfLinearIndependentOfCardEqFinrank_repr_apply, Rep.resCoindAdjunction_unit_app_hom_hom, compMultilinearMap_codRestrict, basisOfLinearIndependentOfCardEqFinrank_repr_apply, Rep.invariantsAdjunction_unit_app, ContinuousLinearMap.coe_codRestrict, ker_codRestrict, restrict_eq_codRestrict_domRestrict, comap_codRestrict, Submodule.comap_equiv_self_of_inj_of_le_apply, finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_liftK_hom, Submodule.isIdempotentElemEquiv_apply_coe, groupCohomology.map_id_comp_H0Iso_hom_apply, LinearIndependent.linearCombinationEquiv_symm_apply, LinearEquiv.ofSubmodule'_toLinearMap, kerComplementEquivRange_symm_apply, setBasisOfTopLeSpanOfCardEqFinrank_repr_apply, comp_codRestrict, map_codRestrict, Rep.invariantsAdjunction_homEquiv_apply_hom, compAlternatingMap_codRestrict, Rep.invariantsFunctor_map_hom, codRestrict_apply, range_codRestrict, restrict_eq_domRestrict_codRestrict, finsetBasisOfLinearIndependentOfCardEqFinrank_repr_apply
domRestrict 📖CompOp
18 mathmath: subtype_comp_restrict, range_domRestrict_le_range, RootPairing.range_polarization_domRestrict_le_span_coroot, range_domRestrict_eq_range_iff, restrict_eq_codRestrict_domRestrict, RootPairing.prod_rootForm_smul_coroot_mem_range_domRestrict, ker_domRestrict, coe_domRestrict, Submodule.equivSubtypeMap_apply, LinearEquiv.ofSubmodule'_toLinearMap, range_domRestrict, injective_domRestrict_iff, BilinForm.toLin_restrict_ker_eq_inf_orthogonal, BilinForm.toLin_restrict_range_dualCoannihilator_eq_orthogonal, restrict_eq_domRestrict_codRestrict, domRestrict_apply, BilinForm.restrict_apply, surjective_domRestrict_iff
domRestrict' 📖CompOp
1 mathmath: domRestrict'_apply
restrict 📖CompOp
47 mathmath: Module.End.isSemisimple_restrict_iff, subtype_comp_restrict, Module.End.isNilpotent_restrict_genEigenspace_nat, injective_restrict_iff_disjoint, Module.End.mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo, Module.End.eigenspace_restrict_le_eigenspace, restrict_comp, IsCompactOperator.restrict', Module.End.pow_restrict, restrict_commute, Submodule.inf_genEigenspace, restrict_sub, Module.End.eigenspace_restrict_eq_bot, LieSubmodule.trace_eq_trace_restrict_of_le_idealizer, IsSymmetric.restrict_invariant, Module.End.IsSemisimple.restrict, restrict_eq_codRestrict_domRestrict, LieSubmodule.toEnd_restrict_eq_toEnd, Module.End.IsFinitelySemisimple.restrict, Module.End.isFinitelySemisimple_iff', IsCompactOperator.restrict, Module.End.isNilpotent_restrict_maxGenEigenspace_sub_algebraMap, toMatrix_directSum_collectedBasis_eq_blockDiagonal', Representation.subrepresentation_apply, groupHomology.mapCycles₂_hom, trace_eq_sum_trace_restrict, Module.End.isNilpotent_restrict_sub_algebraMap, Rep.coindMap_hom, restrict_smul_one, AffineMap.restrict.linear, IsSymmetric.orthogonalComplement_iSup_eigenspaces, Representation.coind_apply, Submodule.inf_iInf_maxGenEigenspace_of_forall_mapsTo, trace_restrict_eq_of_forall_mem, groupHomology.mapCycles₁_hom, ker_restrict, Module.End.isNilpotent_restrict_genEigenspace_top, det_eq_det_mul_det, restrict_coe_apply, Module.End.isNilpotent.restrict, Module.End.genEigenspace_restrict_eq_top, trace_eq_sum_trace_restrict', Module.Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix, restrict_eq_domRestrict_codRestrict, Module.End.genEigenspace_restrict, trace_eq_sum_trace_restrict_of_eq_biSup, restrict_apply

Theorems

NameKindAssumesProvesValidatesDepends On
codLift_apply 📖mathematicalDFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
Set
Set.instMembership
Set.range
codLift
codRestrict_apply 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
instFunLike
Submodule.addCommMonoid
Submodule.module
codRestrict
coeFn_sum 📖mathematicalDFunLike.coe
LinearMap
instFunLike
Finset.sum
addCommMonoid
Pi.addCommMonoid
coe_sum
coe_domRestrict 📖mathematicalDFunLike.coe
LinearMap
Submodule
SetLike.instMembership
Submodule.setLike
Submodule.addCommMonoid
Submodule.module
instFunLike
domRestrict
Set.restrict
SetLike.coe
coe_sum 📖mathematicalDFunLike.coe
LinearMap
instFunLike
Finset.sum
addCommMonoid
Pi.addCommMonoid
map_sum
AddMonoidHom.instAddMonoidHomClass
comp_codLift 📖mathematicalDFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
Set
Set.instMembership
Set.range
comp
RingHomCompTriple.right_ids
codLift
ext
RingHomCompTriple.right_ids
comp_apply
codLift_apply
comp_codRestrict 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
instFunLike
comp
Submodule.addCommMonoid
Submodule.module
codRestrict
ext
domRestrict'_apply 📖mathematicalDFunLike.coe
LinearMap
CommSemiring.toSemiring
RingHom.id
Semiring.toNonAssocSemiring
Submodule
SetLike.instMembership
Submodule.setLike
Submodule.addCommMonoid
Submodule.module
instFunLike
addCommMonoid
module
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
domRestrict'
smulCommClass_self
domRestrict_apply 📖mathematicalDFunLike.coe
LinearMap
Submodule
SetLike.instMembership
Submodule.setLike
Submodule.addCommMonoid
Submodule.module
instFunLike
domRestrict
restrict_apply 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
Submodule.addCommMonoid
Submodule.module
restrict
restrict_coe_apply 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
Submodule.addCommMonoid
Submodule.module
restrict
restrict_commute 📖mathematicalCommute
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
Module.End.instMul
Set.MapsTo
DFunLike.coe
instFunLike
SetLike.coe
Submodule
Submodule.setLike
SetLike.instMembership
Submodule.addCommMonoid
Submodule.module
restrict
RingHomCompTriple.ids
Set.MapsTo.comp
restrict_comp 📖mathematicalSet.MapsTo
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
SetLike.coe
Submodule
Submodule.setLike
comp
RingHomCompTriple.ids
Set.MapsTo.comp
restrict
SetLike.instMembership
Submodule.addCommMonoid
Submodule.module
RingHomCompTriple.ids
restrict_eq_codRestrict_domRestrict 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
restrict
codRestrict
Submodule.addCommMonoid
Submodule.module
domRestrict
restrict_eq_domRestrict_codRestrict 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
restrict
domRestrict
Submodule.addCommMonoid
Submodule.module
codRestrict
restrict_smul_one 📖mathematicalSubmodule.smul_mem
CommSemiring.toSemiring
restrict
Module.End
instSMul
RingHom.id
Semiring.toNonAssocSemiring
DistribMulAction.toDistribSMul
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
smulCommClass_self
CommSemiring.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
Module.End.instOne
Submodule
SetLike.instMembership
Submodule.setLike
Submodule.addCommMonoid
Submodule.module
smulCommClass_self
restrict_sub 📖mathematicalSet.MapsTo
DFunLike.coe
LinearMap
Ring.toSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
instFunLike
SetLike.coe
Submodule
Submodule.setLike
instSub
Submodule.sub_mem
SetLike.instMembership
Submodule.addCommMonoid
Submodule.module
Submodule.addCommGroup
restrict
ext
subtype_comp_codRestrict 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
instFunLike
comp
Submodule.addCommMonoid
Submodule.module
RingHom.id
Semiring.toNonAssocSemiring
RingHomCompTriple.right_ids
Submodule.subtype
codRestrict
ext
RingHomCompTriple.right_ids
subtype_comp_restrict 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
instFunLike
comp
Submodule.addCommMonoid
Submodule.module
RingHomCompTriple.ids
Submodule.subtype
restrict
domRestrict
RingHomCompTriple.ids
sum_apply 📖mathematicalDFunLike.coe
LinearMap
instFunLike
Finset.sum
addCommMonoid
map_sum
AddMonoidHom.instAddMonoidHomClass

Module.End

Theorems

NameKindAssumesProvesValidatesDepends On
pow_apply_mem_of_forall_mem 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
LinearMap.instFunLike
Monoid.toNatPow
instMonoid
pow_zero
pow_restrict 📖mathematicalSubmodule
SetLike.instMembership
Submodule.setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
LinearMap.instFunLike
pow_apply_mem_of_forall_mem
Submodule.addCommMonoid
Submodule.module
Monoid.toNatPow
instMonoid
LinearMap.restrict
LinearMap.ext
LinearMap.restrict_coe_apply
coe_pow
Function.Semiconj.iterate_right
submodule_pow_eq_zero_of_pow_eq_zero 📖LinearMap.comp
Submodule
SetLike.instMembership
Submodule.setLike
Submodule.addCommMonoid
Submodule.module
RingHom.id
Semiring.toNonAssocSemiring
RingHomCompTriple.ids
Submodule.subtype
Module.End
Monoid.toNatPow
instMonoid
LinearMap.instZero
RingHomCompTriple.ids
LinearMap.ext
RingHomCompTriple.right_ids
commute_pow_left_of_commute
LinearMap.zero_comp
LinearMap.zero_apply
AddSubmonoidClass.toZeroMemClass
Submodule.addSubmonoidClass

SMulMemClass

Definitions

NameCategoryTheorems
subtype 📖CompOp
3 mathmath: subtype_injective, subtype_apply, coe_subtype

Theorems

NameKindAssumesProvesValidatesDepends On
coe_subtype 📖mathematicalDFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
SetLike.instMembership
AddSubmonoidClass.toAddCommMonoid
SubmoduleClass.module
LinearMap.instFunLike
subtype
subtype_apply 📖mathematicalDFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
SetLike.instMembership
AddSubmonoidClass.toAddCommMonoid
SubmoduleClass.module
LinearMap.instFunLike
subtype
subtype_injective 📖mathematicalSetLike.instMembership
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
AddSubmonoidClass.toAddCommMonoid
SubmoduleClass.module
LinearMap.instFunLike
subtype
Subtype.coe_injective

Submodule

Definitions

NameCategoryTheorems
inclusion 📖CompOp
37 mathmath: FG.rTensor.directedSystem, subtype_comp_inclusion, TensorProduct.eq_of_fg_of_subtype_eq', exists_fg_le_eq_rTensor_inclusion, FG.lTensor.directedSystem, Module.Basis.coe_mkFinConsOfLE, FG.exists_rTensor_fg_inclusion_eq, TensorProduct.exists_finite_submodule_of_finite', span_range_inclusion_restrictScalars_eq_top, AffineSubspace.inclusion_linear, inclusion_apply, span_range_inclusion_eq_top, LinearMap.submoduleImage_apply_of_le, LinearMap.quotientInfEquivSupQuotient_apply_mk, coe_inclusion, LinearPMap.apply_comp_inclusion, FG.lTensor.directLimit_apply', FG.rTensor.directLimit_apply, TensorProduct.exists_finite_submodule_right_of_setFinite', ker_inclusion, TensorProduct.exists_finite_submodule_left_of_finite', TensorProduct.eq_zero_of_fg_of_subtype_eq_zero, LieSubalgebra.coe_ofLe, TensorProduct.exists_finite_submodule_of_setFinite', TensorProduct.eq_of_fg_of_subtype_eq, FG.directedSystem, FG.lTensor.directLimit_apply, range_inclusion, mulMap_comp_lTensor, exists_fg_le_subset_range_rTensor_inclusion, TensorProduct.exists_finite_submodule_right_of_finite', inclusion_injective, mulMap_comp_map_inclusion, TensorProduct.exists_finite_submodule_left_of_setFinite', map_subtype_range_inclusion, FG.rTensor.directLimit_apply', mulMap_comp_rTensor
instAddActionSubtypeMem 📖CompOp
subtype 📖CompOp
195 mathmath: LinearMap.lTensor_ker_subtype_tensorKerEquiv_symm, Rep.resCoindHomEquiv_symm_apply_hom, StarModule.decomposeProdAdjointL_symm_apply, 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, iSupIndep.dfinsupp_lsum_injective, subtype_comp_inclusion, groupCohomology.isoCocycles₁_hom_comp_i_apply, LinearMap.subtype_comp_codRestrict, LinearMap.IsProj.subtype_comp_codRestrict, LinearMap.subtype_comp_restrict, LinearMap.BilinForm.ker_restrict_eq_of_codisjoint, covBy_iff_quot_is_simple, Rep.resCoindAdjunction_counit_app_hom_hom, LinearMap.le_ker_iff_comp_subtype_eq_zero, mulMap_one_right_eq, Module.Flat.ker_lTensor_eq, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_i_hom, LinearMap.tensorKer_coe, coe_subtypeL', selfAdjointPart_comp_subtype_skewAdjoint, TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul, Module.End.eigenspace_restrict_le_eigenspace, groupHomology.isoCycles₁_inv_comp_iCycles_apply, biSup_comap_subtype_eq_top, RootPairing.injOn_dualMap_subtype_span_root_coroot, biSup_eq_range_dfinsupp_lsum, selfAdjointPart_comp_subtype_selfAdjoint, Module.Flat.out, equivSubtypeMap_symm_apply, groupCohomology.mapCocycles₂_comp_i_apply, ker_subtype, LinearMap.exact_subtype_mkQ, sup_eq_range, goursat, subtypeₗᵢ_toLinearMap, LinearMap.lTensor_range, subtype_apply, LinearMap.tensorEqLocus_coe, ModuleCat.imageIsoRange_hom_subtype, coe_prodEquivOfIsCompl, groupCohomology.shortComplexH0_f, Module.Flat.iff_rTensor_injective, Rep.subtype_hom, Ideal.map_includeRight_eq, inf_genEigenspace, TensorProduct.exists_finite_submodule_left_of_setFinite, TensorProduct.quotientTensorEquiv_symm_apply_mk_tmul, strictMono_comap_prod_map, LieIdeal.incl_coe, LinearMap.comap_leq_ker_subToSupQuotient, LinearMap.rTensor_injective_iff_subtype, LinearMap.subtype_compAlternatingMap_codRestrict, LinearMap.quotientInfEquivSupQuotient_apply_mk, mem_iSup_iff_exists_dfinsupp, TensorProduct.exists_finite_submodule_right_of_setFinite, iSup_eq_range_dfinsupp_lsum, LinearIndependent.linearCombination_comp_repr, StarModule.decomposeProdAdjoint_symm_apply, LieSubmodule.toEnd_comp_subtype_mem, FG.lTensor.directLimit_apply', dualRestrict_def, groupHomology.mapCycles₁_comp_i_apply, FG.rTensor.directLimit_apply, LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict, comap_subtype_self, groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply, Module.Flat.eqLocus_lTensor_eq, TensorProduct.exists_finite_submodule_left_of_finite, LinearMap.comap_codRestrict, comap_equiv_self_of_inj_of_le_apply, exists_fg_le_subset_range_rTensor_subtype, TensorProduct.rTensor_injective_of_forall_vanishesTrivially, set_smul_eq_map, LinearMap.quotientInfEquivSupQuotient_injective, LinearMap.ker_domRestrict, realPart_comp_subtype_selfAdjoint, map_subtype_embedding_eq, ModuleCat.imageIsoRange_hom_subtype_assoc, comap_subtype_eq_top, natAbs_det_equiv, map_subtype_span_singleton, groupHomology.isoCycles₂_hom_comp_i_apply, groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply, skewAdjointPart_comp_subtype_selfAdjoint, ModuleCat.imageIsoRange_inv_image_ι, Module.Flat.iff_lTensor_injective, AffineSubspace.subtype_linear, TensorProduct.quotientTensorQuotientEquiv_apply_tmul_mk_tmul_mk, TensorProduct.tensorQuotientEquiv_apply_mk_tmul, injective_subtype, linearProjOfIsCompl_comp_subtype, coe_subtypeₗᵢ, Finsupp.restrictDom_comp_subtype, rTensor_mkQ, groupCohomology.map_id_comp_H0Iso_hom_apply, groupCohomology.subtype_comp_d₀₁_assoc, imaginaryPart_comp_subtype_selfAdjoint, iSupIndep.linearEquiv_apply, Module.Flat.iff_lTensor_injectiveₛ, Function.Exact.linearMap_rangeRestrict, Module.End.invtSubmodule.map_subtype_mem_of_mem_invtSubmodule, TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective, equivSubtypeMap_apply, isIdempotentElemEquiv_symm_apply_coe, groupHomology.isoCycles₁_hom_comp_i_apply, AffineSubspace.subtypeₐᵢ_linear, ModuleCat.imageIsoRange_inv_image_ι_assoc, Rep.invariantsAdjunction_counit_app_hom, Module.injOn_dualMap_subtype_span_range_range, ModuleCat.kernelIsoKer_hom_ker_subtype, Finsupp.linearCombination_restrict, DirectSum.decomposeLinearEquiv_symm_comp_lof, Function.Exact.iff_linearMap_rangeRestrict, mulMap_one_left_eq, LinearMap.kerComplementEquivRange_symm_apply, groupHomology.isoCycles₂_inv_comp_iCycles_apply, LinearMap.map_codRestrict, TensorProduct.quotientTensorEquiv_apply_tmul_mk, LinearMap.exact_subtype_ker_map, Ideal.natAbs_det_equiv, finrank_map_subtype_eq, Ideal.subtype_isoBaseOfIsPrincipal_eq_mul, coe_mapIic_apply, Module.Flat.iff_lift_lsmul_comp_subtype_injective, FG.lTensor.directLimit_apply, Module.fgSystem.equiv_comp_of, range_inclusion, groupCohomology.mapCocycles₁_comp_i_apply, coe_subtype, isCompl_comap_subtype_of_isCompl_of_le, iSupIndep_iff_dfinsupp_lsum_injective, LinearMap.coe_quotientInfToSupQuotient, TensorProduct.exists_finite_submodule_right_of_finite, LinearMap.subtype_compMultilinearMap_codRestrict, map_subtype_top, Rep.invariantsFunctor_map_hom, Module.End.iInf_maxGenEigenspace_restrict_map_subtype_eq, LinearMap.rTensor_range, inf_iInf_maxGenEigenspace_of_forall_mapsTo, Module.Flat.iff_rTensor_injectiveₛ, LinearMap.exists_extend, Module.flat_iff, LinearMap.ker_id_sub_eq_of_proj, exists_fg_le_eq_rTensor_subtype, LinearMap.ker_restrict, groupHomology.mapCycles₂_comp_i_apply, groupCohomology.subtype_comp_d₀₁, map_range_rTensor_subtype_lid, LinearMap.BilinForm.toLin_restrict_ker_eq_inf_orthogonal, LieSubmodule.incl_coe, map_comap_subtype, LinearMap.quotientInfEquivSupQuotient_symm_apply_left, groupCohomology.isoCocycles₂_hom_comp_i_apply, LinearMap.ofIsCompl_subtype_zero_eq, map_subtype_range_inclusion, LinearPMap.coe_vadd, Module.Dual.eq_of_preReflection_mapsTo', LinearMap.range_codRestrict, mem_biSup_iff_exists_dfinsupp, range_subtype, NumberField.instIsLocalizedModuleIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmoduleValFractionalIdealNonZeroDivisorsRestrictScalarsSubtype, subtype_injective, comapSubtypeEquivOfLe_apply_coe, ModuleCat.mono_as_hom'_subtype, DirectSum.isInternal_biSup_submodule_of_iSupIndep, LinearMap.ker_le_range_iff, Ideal.map_toCotangent_ker, exteriorPower.ιMulti_family_span, LinearMap.exists_extend_of_notMem, LinearMap.comp_ker_subtype, TensorProduct.tensorQuotientEquiv_symm_apply_tmul_mk, ModuleCat.kernelIsoKer_inv_kernel_ι, Module.Flat.iff_rTensor_injective', skewAdjointPart_comp_subtype_skewAdjoint, FG.rTensor.directLimit_apply', lTensor_mkQ, LinearMap.quotientInfEquivSupQuotient_symm_apply_right, comapSubtypeEquivOfLe_symm_apply, Module.End.genEigenspace_restrict, coe_subtypeL, Ideal.map_includeLeft_eq, LinearMap.quotientInfEquivSupQuotient_surjective, Module.Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord, LinearMap.surjective_comp_subtype_of_isComplemented, Module.Flat.iff_lTensor_injective', map_subtype_le, TensorProduct.exists_of_fg, disjoint_iff_comap_eq_bot, LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm, lsum_comp_mapRange_toSpanSingleton, Ideal.to_quotient_square_comp_toCotangent, comap_subtype_le_iff

Theorems

NameKindAssumesProvesValidatesDepends On
coe_inclusion 📖mathematicalSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.instMembership
setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
addCommMonoid
module
LinearMap.instFunLike
inclusion
coe_subtype 📖mathematicalDFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
Submodule
SetLike.instMembership
setLike
addCommMonoid
module
LinearMap.instFunLike
subtype
coe_sum 📖mathematicalSubmodule
SetLike.instMembership
setLike
Finset.sum
addCommMonoid
map_sum
DistribMulActionSemiHomClass.toAddMonoidHomClass
SemilinearMapClass.distribMulActionSemiHomClass
LinearMap.semilinearMapClass
inclusion_apply 📖mathematicalSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
SetLike.instMembership
setLike
addCommMonoid
module
LinearMap.instFunLike
inclusion
inclusion_injective 📖mathematicalSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SetLike.instMembership
setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
addCommMonoid
module
LinearMap.instFunLike
inclusion
Subtype.val_injective
injective_subtype 📖mathematicalSubmodule
SetLike.instMembership
setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
addCommMonoid
module
LinearMap.instFunLike
subtype
Subtype.coe_injective
subtype_apply 📖mathematicalDFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
Submodule
SetLike.instMembership
setLike
addCommMonoid
module
LinearMap.instFunLike
subtype
subtype_comp_inclusion 📖mathematicalSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
LinearMap.comp
SetLike.instMembership
setLike
addCommMonoid
module
RingHom.id
Semiring.toNonAssocSemiring
RingHomCompTriple.ids
subtype
inclusion
RingHomCompTriple.ids
subtype_injective 📖mathematicalSubmodule
SetLike.instMembership
setLike
DFunLike.coe
LinearMap
RingHom.id
Semiring.toNonAssocSemiring
addCommMonoid
module
LinearMap.instFunLike
subtype
Subtype.coe_injective

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