Documentation Verification Report

Nilpotent

πŸ“ Source: Mathlib/Algebra/Lie/Nilpotent.lean

Statistics

MetricCount
DefinitionsmaxNilpotentIdeal, lcs, IsNilpotent, lowerCentralSeries, lowerCentralSeriesLast, maxNilpotentSubmodule, nilpotencyLength, IsNilpotent, lcs, ucs
10
TheoremslieModule_isNilpotent_iff, lieAlgebra_isNilpotent, lieModuleIsNilpotent, lieAlgebra_isNilpotent, lieModuleIsNilpotent, lieModule_lcs_map_eq, isNilpotent_iff_le_maxNilpotentIdeal, ad_nilpotent_of_nilpotent, center_le_maxNilpotentIdeal, isNilpotent_ad_of_isNilpotent, isNilpotent_range_ad_iff, isSolvable_of_isNilpotent, maxNilpotentIdealIsNilpotent, maxNilpotentIdeal_eq_top_of_isNilpotent, maxNilpotentIdeal_le_radical, nilpotent_ad_of_nilpotent_algebra, nilpotent_of_nilpotent_quotient, non_trivial_center_of_isNilpotent, nilpotent_iff_equiv_nilpotent, isNilpotent_range, coe_lcs_eq, instIsNilpotentSubtypeMemLieSubmoduleOfIsNilpotent, lcs_succ, lcs_top, lcs_zero, lowerCentralSeries_map_eq, map_lowerCentralSeries_le, mk, nilpotent, nilpotent_int, antitone_lowerCentralSeries, coe_lcs_range_toEnd_eq, coe_lowerCentralSeries_eq_int, coe_lowerCentralSeries_ideal_le, derivedSeries_le_lowerCentralSeries, disjoint_lowerCentralSeries_maxTrivSubmodule_iff, eventually_iInf_lowerCentralSeries_eq, exists_forall_pow_toEnd_eq_zero, iInf_lcs_le_of_isNilpotent_quot, iInf_lowerCentralSeries_eq_bot_of_isNilpotent, instIsNilpotentSup, instIsNilpotentTensor, instMaxNilpotentSubmoduleIsNilpotent, isNilpotent_iff, isNilpotent_iff_exists_ucs_eq_top, isNilpotent_iff_int, isNilpotent_iff_le_maxNilpotentSubmodule, isNilpotent_of_le, isNilpotent_of_top_iff, isNilpotent_of_top_iff', isNilpotent_quotient_iff, isNilpotent_range_toEnd_iff, isNilpotent_toEnd_of_isNilpotent, isNilpotent_toEnd_of_isNilpotentβ‚‚, isTrivial_of_nilpotencyLength_le_one, iterate_toEnd_mem_lowerCentralSeries, iterate_toEnd_mem_lowerCentralSeriesβ‚‚, lowerCentralSeriesLast_le_max_triv, lowerCentralSeriesLast_le_of_not_isTrivial, lowerCentralSeries_succ, lowerCentralSeries_zero, map_lowerCentralSeries_eq, map_lowerCentralSeries_le, maxGenEigenSpace_toEnd_eq_top, maxNilpotentSubmodule_eq_top_of_isNilpotent, nilpotencyLength_eq_one_iff, nilpotencyLength_eq_succ_iff, nilpotencyLength_eq_zero_iff, nilpotentOfNilpotentQuotient, nontrivial_lowerCentralSeriesLast, nontrivial_max_triv_of_isNilpotent, trivialIsNilpotent, trivial_iff_lower_central_eq_bot, isNilpotent_ad_of_isNilpotent_ad, gc_lcs_ucs, isNilpotent_iff_exists_lcs_eq_bot, isNilpotent_iff_exists_self_le_ucs, lcs_add_le_iff, lcs_le_iff, lcs_le_self, lcs_succ, lcs_sup, lcs_zero, lowerCentralSeries_eq_bot_iff_lcs_eq_bot, lowerCentralSeries_eq_lcs_comap, lowerCentralSeries_map_eq_lcs, lowerCentralSeries_tensor_eq_baseChange, ucs_add, ucs_bot_one, ucs_comap_incl, ucs_eq_self_of_normalizer_eq_self, ucs_eq_top_iff, ucs_le_of_normalizer_eq_self, ucs_mono, ucs_succ, ucs_zero, coe_lowerCentralSeries_ideal_quot_eq, instIsNilpotentSubtypeMemLieSubalgebraTop, lieModule_lcs_map_le
99
Total109

Equiv

Theorems

NameKindAssumesProvesValidatesDepends On
lieModule_isNilpotent_iff πŸ“–mathematicalBracket.bracket
LieRingModule.toBracket
DFunLike.coe
LieEquiv
EquivLike.toFunLike
LieEquiv.instEquivLike
LinearEquiv
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
RingHomInvPair.ids
AddCommGroup.toAddCommMonoid
DFinsupp.instEquivLikeLinearEquiv
LieModule.IsNilpotentβ€”RingHomInvPair.ids
LinearEquiv.surjective
Function.Surjective.lieModuleIsNilpotent
LieEquiv.surjective
LinearEquiv.coe_coe
LieEquiv.coe_toLieHom
LinearEquiv.symm_apply_apply
LieEquiv.apply_symm_apply
LinearEquiv.apply_symm_apply

Function.Injective

Theorems

NameKindAssumesProvesValidatesDepends On
lieAlgebra_isNilpotent πŸ“–mathematicalDFunLike.coe
LieHom
LieHom.instFunLike
LieRing.IsNilpotentβ€”LieRing.IsNilpotent.eq_1
LieModule.isNilpotent_iff
lieAlgebraSelfModule
LieIdeal.bot_of_map_eq_bot
eq_bot_iff
LieIdeal.map_lowerCentralSeries_le
lieModuleIsNilpotent πŸ“–mathematicalBracket.bracket
LieRingModule.toBracket
DFunLike.coe
LieHom
LieHom.instFunLike
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
LieModule.IsNilpotentβ€”LieModule.IsNilpotent.nilpotent
LieModule.isNilpotent_iff
Submodule.map_injective_of_injective
RingHomSurjective.ids
Submodule.map_bot
lieModule_lcs_map_le

Function.Surjective

Theorems

NameKindAssumesProvesValidatesDepends On
lieAlgebra_isNilpotent πŸ“–mathematicalDFunLike.coe
LieHom
LieHom.instFunLike
LieRing.IsNilpotentβ€”LieRing.IsNilpotent.eq_1
LieModule.isNilpotent_iff
lieAlgebraSelfModule
LieIdeal.lowerCentralSeries_map_eq
lieModuleIsNilpotent πŸ“–mathematicalBracket.bracket
LieRingModule.toBracket
DFunLike.coe
LieHom
LieHom.instFunLike
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
LieModule.IsNilpotentβ€”LieModule.IsNilpotent.nilpotent
LieModule.isNilpotent_iff
RingHomSurjective.ids
lieModule_lcs_map_eq
Submodule.map.congr_simp
Submodule.map_bot
lieModule_lcs_map_eq πŸ“–mathematicalBracket.bracket
LieRingModule.toBracket
DFunLike.coe
LieHom
LieHom.instFunLike
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
Submodule.map
RingHomSurjective.ids
LieSubmodule.toSubmodule
LieModule.lowerCentralSeries
β€”le_antisymm
RingHomSurjective.ids
lieModule_lcs_map_le
Submodule.map_top
LieModule.lowerCentralSeries_succ
Submodule.map.congr_simp
LieSubmodule.lieIdeal_oper_eq_linear_span'
Submodule.map_span
Submodule.span_mono
Set.Subset.trans

LieAlgebra

Definitions

NameCategoryTheorems
maxNilpotentIdeal πŸ“–CompOp
5 mathmath: LieIdeal.isNilpotent_iff_le_maxNilpotentIdeal, maxNilpotentIdeal_eq_top_of_isNilpotent, center_le_maxNilpotentIdeal, maxNilpotentIdeal_le_radical, maxNilpotentIdealIsNilpotent

Theorems

NameKindAssumesProvesValidatesDepends On
ad_nilpotent_of_nilpotent πŸ“–mathematicalIsNilpotent
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieRing.toAddCommGroup
LieRing.ofAssociativeRing
toModule
ofAssociativeAlgebra
LinearMap.instZero
RingHom.id
Semiring.toNonAssocSemiring
Module.End.instMonoid
DFunLike.coe
LieHom
Module.End.instRing
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
LieHom.instFunLike
ad
β€”smulCommClass_self
IsScalarTower.left
Algebra.to_smulCommClass
IsScalarTower.right
ad_eq_lmul_left_sub_lmul_right
LinearMap.isNilpotent_mulLeft_iff
LinearMap.isNilpotent_mulRight_iff
LinearMap.commute_mulLeft_right
Commute.isNilpotent_sub
center_le_maxNilpotentIdeal πŸ“–mathematicalβ€”LieIdeal
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
LieRing.toAddCommGroup
toModule
lieRingSelfModule
center
maxNilpotentIdeal
β€”le_sSup
LieSubmodule.instAddSubgroupClass
LieModule.trivialIsNilpotent
LieModule.instIsTrivialSubtypeMemLieSubmoduleMaxTrivSubmodule
lieAlgebraSelfModule
isNilpotent_ad_of_isNilpotent πŸ“–mathematicalIsNilpotent
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
LieSubalgebra
LieRing.ofAssociativeRing
ofAssociativeAlgebra
SetLike.instMembership
LieSubalgebra.instSetLike
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieRing.toAddCommGroup
LieSubalgebra.lieRing
toModule
LieSubalgebra.lieAlgebra
LinearMap.instZero
RingHom.id
Semiring.toNonAssocSemiring
Module.End.instMonoid
DFunLike.coe
LieHom
Module.End.instRing
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
LieHom.instFunLike
ad
β€”LieSubalgebra.isNilpotent_ad_of_isNilpotent_ad
ad_nilpotent_of_nilpotent
isNilpotent_range_ad_iff πŸ“–mathematicalβ€”LieRing.IsNilpotent
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieRing.toAddCommGroup
toModule
LieSubalgebra
LieRing.ofAssociativeRing
Module.End.instRing
ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
SetLike.instMembership
LieSubalgebra.instSetLike
LieHom.range
ad
LieSubalgebra.lieRing
β€”smulCommClass_self
IsScalarTower.left
self_module_ker_eq_center
nilpotent_of_nilpotent_quotient
le_of_eq
LieEquiv.nilpotent_iff_equiv_nilpotent
LieHom.isNilpotent_range
isSolvable_of_isNilpotent πŸ“–mathematicalβ€”IsSolvableβ€”LieModule.IsNilpotent.nilpotent_int
le_bot_iff
le_trans
LieModule.derivedSeries_le_lowerCentralSeries
maxNilpotentIdealIsNilpotent πŸ“–mathematicalβ€”LieModule.IsNilpotent
LieSubmodule
LieRing.toAddCommGroup
toModule
lieRingSelfModule
SetLike.instMembership
LieSubmodule.instSetLike
maxNilpotentIdeal
LieIdeal.lieRing
LieSubmodule.instLieRingModuleSubtypeMem
β€”LieModule.instMaxNilpotentSubmoduleIsNilpotent
lieAlgebraSelfModule
maxNilpotentIdeal_eq_top_of_isNilpotent πŸ“–mathematicalβ€”maxNilpotentIdeal
Top.top
LieSubmodule
LieRing.toAddCommGroup
toModule
lieRingSelfModule
LieSubmodule.instTop
β€”LieModule.maxNilpotentSubmodule_eq_top_of_isNilpotent
lieAlgebraSelfModule
maxNilpotentIdeal_le_radical πŸ“–mathematicalβ€”LieSubmodule
LieRing.toAddCommGroup
toModule
lieRingSelfModule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
maxNilpotentIdeal
radical
β€”sSup_le_sSup
LieSubmodule.instAddSubgroupClass
isSolvable_of_isNilpotent
LieIdeal.instIsNilpotentSubtypeMemLieSubmoduleOfIsNilpotent
nilpotent_ad_of_nilpotent_algebra πŸ“–mathematicalβ€”Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieRing.toAddCommGroup
toModule
Monoid.toNatPow
Module.End.instMonoid
DFunLike.coe
LieHom
LieRing.ofAssociativeRing
Module.End.instRing
ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
ad
LinearMap.instZero
RingHom.id
Semiring.toNonAssocSemiring
β€”LieModule.exists_forall_pow_toEnd_eq_zero
lieAlgebraSelfModule
nilpotent_of_nilpotent_quotient πŸ“–mathematicalLieIdeal
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
LieRing.toAddCommGroup
toModule
lieRingSelfModule
center
LieRing.IsNilpotentβ€”lieAlgebraSelfModule
LieModule.isNilpotent_iff
LieSubmodule.Quotient.lieQuotientLieModule
coe_lowerCentralSeries_ideal_quot_eq
LieModule.nilpotentOfNilpotentQuotient
non_trivial_center_of_isNilpotent πŸ“–mathematicalβ€”Nontrivial
LieSubmodule
LieRing.toAddCommGroup
toModule
lieRingSelfModule
SetLike.instMembership
LieSubmodule.instSetLike
center
β€”LieModule.nontrivial_max_triv_of_isNilpotent
lieAlgebraSelfModule

LieAlgebra.LieIdeal

Theorems

NameKindAssumesProvesValidatesDepends On
isNilpotent_iff_le_maxNilpotentIdeal πŸ“–mathematicalβ€”LieModule.IsNilpotent
LieSubmodule
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
SetLike.instMembership
LieSubmodule.instSetLike
LieIdeal.lieRing
LieSubmodule.instLieRingModuleSubtypeMem
LieIdeal
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
LieAlgebra.maxNilpotentIdeal
β€”LieModule.isNilpotent_iff_le_maxNilpotentSubmodule
lieAlgebraSelfModule

LieEquiv

Theorems

NameKindAssumesProvesValidatesDepends On
nilpotent_iff_equiv_nilpotent πŸ“–mathematicalβ€”LieRing.IsNilpotentβ€”Function.Injective.lieAlgebra_isNilpotent
injective

LieHom

Theorems

NameKindAssumesProvesValidatesDepends On
isNilpotent_range πŸ“–mathematicalβ€”LieRing.IsNilpotent
LieSubalgebra
SetLike.instMembership
LieSubalgebra.instSetLike
range
LieSubalgebra.lieRing
β€”Function.Surjective.lieAlgebra_isNilpotent
surjective_rangeRestrict

LieIdeal

Definitions

NameCategoryTheorems
lcs πŸ“–CompOp
4 mathmath: lcs_succ, lcs_zero, coe_lcs_eq, lcs_top

Theorems

NameKindAssumesProvesValidatesDepends On
coe_lcs_eq πŸ“–mathematicalβ€”LieSubmodule.toSubmodule
lcs
LieSubmodule
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
SetLike.instMembership
LieSubmodule.instSetLike
lieRing
lieRingModule
LieModule.lowerCentralSeries
lieAlgebra
β€”LieModule.lowerCentralSeries_succ
lcs_succ
LieSubmodule.lieIdeal_oper_eq_linear_span'
lieModule
LieSubmodule.mem_toSubmodule
LieSubalgebra.coe_bracket_of_module
instIsNilpotentSubtypeMemLieSubmoduleOfIsNilpotent πŸ“–mathematicalβ€”LieRing.IsNilpotent
LieSubmodule
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
SetLike.instMembership
LieSubmodule.instSetLike
lieRing
β€”LieSubmodule.instAddSubgroupClass
coe_bracket_of_module
Function.Injective.lieModuleIsNilpotent
lieModule
LieSubmodule.instLieModule
lieAlgebraSelfModule
lcs_succ πŸ“–mathematicalβ€”lcs
Bracket.bracket
LieIdeal
LieSubmodule
LieSubmodule.hasBracket
β€”Function.iterate_succ_apply'
lcs_top πŸ“–mathematicalβ€”lcs
Top.top
LieIdeal
LieSubmodule.instTop
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
LieModule.lowerCentralSeries
β€”β€”
lcs_zero πŸ“–mathematicalβ€”lcs
Top.top
LieSubmodule
LieSubmodule.instTop
β€”β€”
lowerCentralSeries_map_eq πŸ“–mathematicalDFunLike.coe
LieHom
LieHom.instFunLike
map
LieModule.lowerCentralSeries
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
β€”LieHom.idealRange_eq_map
LieHom.idealRange_eq_top_of_surjective
LieModule.lowerCentralSeries_succ
map_bracket_eq
map_lowerCentralSeries_le πŸ“–mathematicalβ€”LieIdeal
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
map
LieModule.lowerCentralSeries
β€”LieModule.lowerCentralSeries_succ
le_trans
map_bracket_le
LieSubmodule.mono_lie
le_top

LieModule

Definitions

NameCategoryTheorems
IsNilpotent πŸ“–CompData
27 mathmath: Equiv.lieModule_isNilpotent_iff, LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot, isNilpotent_of_top_iff, isNilpotent_range_toEnd_iff, IsNilpotent.mk, LieSubmodule.isNilpotent_iff_exists_self_le_ucs, isNilpotent_iff, isNilpotent_iff_int, LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop, instIsNilpotentTensor, instMaxNilpotentSubmoduleIsNilpotent, Function.Surjective.lieModuleIsNilpotent, isNilpotent_quotient_iff, isNilpotent_iff_exists_ucs_eq_top, LieAlgebra.LieIdeal.isNilpotent_iff_le_maxNilpotentIdeal, shiftedGenWeightSpace.instIsNilpotentSubtypeMemLieSubmoduleOfIsNoetherian, isNilpotent_of_top_iff', nilpotentOfNilpotentQuotient, Function.Injective.lieModuleIsNilpotent, isNilpotent_iff_forall, isNilpotent_iff_le_maxNilpotentSubmodule, instIsNilpotentSup, instIsNilpotentSubtypeMemLieSubmoduleGenWeightSpaceOfNatForallOfIsNoetherian, isNilpotent_iff_forall', LieAlgebra.maxNilpotentIdealIsNilpotent, isNilpotent_of_le, trivialIsNilpotent
lowerCentralSeries πŸ“–CompOp
39 mathmath: trivial_iff_lower_central_eq_bot, iInf_lowerCentralSeries_eq_posFittingComp, LieSubmodule.lowerCentralSeries_tensor_eq_baseChange, lowerCentralSeries_zero, posFittingComp_le_iInf_lowerCentralSeries, derivedSeries_le_lowerCentralSeries, lowerCentralSeries_one_inf_center_le_ker_traceForm, nilpotencyLength_eq_succ_iff, LieIdeal.lowerCentralSeries_map_eq, LieSubmodule.lowerCentralSeries_eq_bot_iff_lcs_eq_bot, LieSubmodule.lowerCentralSeries_map_eq_lcs, disjoint_lowerCentralSeries_maxTrivSubmodule_iff, isNilpotent_iff, isNilpotent_iff_int, IsNilpotent.nilpotent, lowerCentralSeries_succ, lowerCentralSeriesLast_le_of_not_isTrivial, antitone_lowerCentralSeries, iInf_lcs_le_of_isNilpotent_quot, coe_lowerCentralSeries_ideal_quot_eq, Function.Surjective.lieModule_lcs_map_eq, LieIdeal.map_lowerCentralSeries_le, iterate_toEnd_mem_lowerCentralSeriesβ‚‚, LieSubmodule.ucs_eq_top_iff, isNilpotent_quotient_iff, LieSubmodule.lowerCentralSeries_eq_lcs_comap, map_lowerCentralSeries_le, coe_lowerCentralSeries_eq_int, IsNilpotent.nilpotent_int, posFittingCompOf_le_lowerCentralSeries, map_lowerCentralSeries_eq, eventually_iInf_lowerCentralSeries_eq, coe_lcs_range_toEnd_eq, iterate_toEnd_mem_lowerCentralSeries, LieIdeal.coe_lcs_eq, iInf_lowerCentralSeries_eq_bot_of_isNilpotent, lieModule_lcs_map_le, LieIdeal.lcs_top, coe_lowerCentralSeries_ideal_le
lowerCentralSeriesLast πŸ“–CompOp
3 mathmath: lowerCentralSeriesLast_le_of_not_isTrivial, lowerCentralSeriesLast_le_max_triv, nontrivial_lowerCentralSeriesLast
maxNilpotentSubmodule πŸ“–CompOp
3 mathmath: instMaxNilpotentSubmoduleIsNilpotent, maxNilpotentSubmodule_eq_top_of_isNilpotent, isNilpotent_iff_le_maxNilpotentSubmodule
nilpotencyLength πŸ“–CompOp
3 mathmath: nilpotencyLength_eq_succ_iff, nilpotencyLength_eq_one_iff, nilpotencyLength_eq_zero_iff

Theorems

NameKindAssumesProvesValidatesDepends On
antitone_lowerCentralSeries πŸ“–mathematicalβ€”Antitone
LieSubmodule
Nat.instPreorder
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
lowerCentralSeries
β€”le_rfl
Nat.of_le_succ
lowerCentralSeries_succ
LE.le.trans
LieSubmodule.mono_lie_right
LieSubmodule.lie_le_right
coe_lcs_range_toEnd_eq πŸ“–mathematicalβ€”LieSubmodule.toSubmodule
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieSubalgebra
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
SetLike.instMembership
LieSubalgebra.instSetLike
LieHom.range
toEnd
LieSubalgebra.lieRing
LieSubalgebra.lieRingModule
Module.End.instLieRingModule
lowerCentralSeries
LieSubalgebra.lieAlgebra
β€”smulCommClass_self
IsScalarTower.left
lowerCentralSeries_succ
LieSubmodule.lieIdeal_oper_eq_linear_span'
LieSubalgebra.lieModule
Module.End.instLieModule
LieSubmodule.mem_toSubmodule
coe_lowerCentralSeries_eq_int πŸ“–mathematicalβ€”SetLike.coe
LieSubmodule
LieSubmodule.instSetLike
lowerCentralSeries
Int.instCommRing
AddCommGroup.toIntModule
LieRing.instLieAlgebra
β€”LieSubmodule.coe_toSubmodule
lowerCentralSeries_succ
LieSubmodule.lieIdeal_oper_eq_linear_span'
instLieModuleInt
Set.ext_iff
le_antisymm
coe_lowerCentralSeries_ideal_le πŸ“–mathematicalβ€”Submodule
LieSubmodule
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
SetLike.instMembership
LieSubmodule.instSetLike
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieIdeal.lieRing
LieIdeal.lieAlgebra
Preorder.toLE
PartialOrder.toPreorder
Submodule.instPartialOrder
LieSubmodule.toSubmodule
LieIdeal.lieRingModule
LieSubmodule.instLieRingModuleSubtypeMem
lowerCentralSeries
β€”lowerCentralSeries_succ
LieSubmodule.lieIdeal_oper_eq_linear_span
LieIdeal.lieModule
LieSubmodule.instLieModule
lieAlgebraSelfModule
Submodule.span_mono
LieSubmodule.mem_top
derivedSeries_le_lowerCentralSeries πŸ“–mathematicalβ€”LieIdeal
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
LieAlgebra.derivedSeries
lowerCentralSeries
β€”LieAlgebra.derivedSeries_def
LieAlgebra.derivedSeriesOfIdeal_zero
lowerCentralSeries_zero
le_refl
LieAlgebra.derivedSeriesOfIdeal_succ
lowerCentralSeries_succ
LieSubmodule.mono_lie
disjoint_lowerCentralSeries_maxTrivSubmodule_iff πŸ“–mathematicalβ€”Disjoint
LieSubmodule
LieSubmodule.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
LieSubmodule.instCompleteLattice
lowerCentralSeries
maxTrivSubmodule
IsTrivial
LieRingModule.toBracket
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
instIsTrivialOfSubsingleton'
le_inf_iff
lowerCentralSeriesLast_le_of_not_isTrivial
lowerCentralSeriesLast_le_max_triv
not_nontrivial
Unique.instSubsingleton
le_bot_iff
Disjoint.eq_bot
nontrivial_lowerCentralSeriesLast
lowerCentralSeries_succ
LieSubmodule.trivial_lie_oper_zero
eventually_iInf_lowerCentralSeries_eq πŸ“–mathematicalβ€”Filter.Eventually
LieSubmodule
iInf
LieSubmodule.instInfSet
lowerCentralSeries
Filter.atTop
Nat.instPreorder
β€”LieSubmodule.wellFoundedLT_of_isArtinian
antitone_lowerCentralSeries
WellFoundedGT.monotone_chain_condition
Filter.eventually_atTop
instIsDirectedOrder
IsStrictOrderedRing.toIsOrderedRing
Nat.instIsStrictOrderedRing
instArchimedeanNat
le_antisymm
iInf_le
le_iInf
le_or_gt
LE.le.trans
le_refl
le_of_lt
exists_forall_pow_toEnd_eq_zero πŸ“–mathematicalβ€”Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
Monoid.toNatPow
Module.End.instMonoid
DFunLike.coe
LieHom
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
toEnd
LinearMap.instZero
RingHom.id
Semiring.toNonAssocSemiring
β€”smulCommClass_self
IsScalarTower.left
IsNilpotent.nilpotent
LinearMap.ext
Module.End.pow_apply
LinearMap.zero_apply
LieSubmodule.mem_bot
iterate_toEnd_mem_lowerCentralSeries
iInf_lcs_le_of_isNilpotent_quot πŸ“–mathematicalβ€”LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
iInf
LieSubmodule.instInfSet
lowerCentralSeries
β€”isNilpotent_quotient_iff
iInf_le_of_le
iInf_lowerCentralSeries_eq_bot_of_isNilpotent πŸ“–mathematicalβ€”iInf
LieSubmodule
LieSubmodule.instInfSet
lowerCentralSeries
Bot.bot
LieSubmodule.instBot
β€”IsNilpotent.nilpotent
eq_bot_iff
iInf_le
instIsNilpotentSup πŸ“–mathematicalβ€”IsNilpotent
LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
LieSubmodule.instMax
AddSubgroupClass.toAddCommGroup
LieSubmodule.instAddSubgroupClass
LieSubmodule.instLieRingModuleSubtypeMem
β€”LieSubmodule.instAddSubgroupClass
LieSubmodule.instSMulMemClass
IsNilpotent.nilpotent
LieSubmodule.instLieModule
antitone_lowerCentralSeries
isNilpotent_iff
LieSubmodule.lowerCentralSeries_eq_lcs_comap
LieSubmodule.lcs_sup
LieSubmodule.lowerCentralSeries_eq_bot_iff_lcs_eq_bot
sup_of_le_left
inf_of_le_right
instIsNilpotentTensor πŸ“–mathematicalβ€”IsNilpotent
TensorProduct
CommRing.toCommSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
AddCommGroup.toAddCommMonoid
LieRing.toAddCommGroup
LieAlgebra.toModule
LieRing.ofAssociativeRing
CommRing.toRing
LieAlgebra.ofAssociativeAlgebra
LieAlgebra.ExtendScalars.instLieRing
TensorProduct.addCommGroup
LieAlgebra.ExtendScalars.instLieRingModule
β€”IsNilpotent.nilpotent
Algebra.to_smulCommClass
isNilpotent_iff
LieAlgebra.ExtendScalars.instLieModule
LieSubmodule.lowerCentralSeries_tensor_eq_baseChange
LieSubmodule.baseChange_bot
instMaxNilpotentSubmoduleIsNilpotent πŸ“–mathematicalβ€”IsNilpotent
LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
maxNilpotentSubmodule
AddSubgroupClass.toAddCommGroup
LieSubmodule.instAddSubgroupClass
LieSubmodule.instLieRingModuleSubtypeMem
β€”CompleteLattice.WellFoundedGT.isSupClosedCompact
LieSubmodule.wellFoundedGT_of_noetherian
LieSubmodule.instAddSubgroupClass
trivialIsNilpotent
instIsTrivialOfSubsingleton'
Unique.instSubsingleton
instIsNilpotentSup
isNilpotent_iff πŸ“–mathematicalβ€”IsNilpotent
lowerCentralSeries
Bot.bot
LieSubmodule
LieSubmodule.instBot
β€”coe_lowerCentralSeries_eq_int
isNilpotent_iff_exists_ucs_eq_top πŸ“–mathematicalβ€”IsNilpotent
LieSubmodule.ucs
Bot.bot
LieSubmodule
LieSubmodule.instBot
Top.top
LieSubmodule.instTop
β€”isNilpotent_iff
isNilpotent_iff_int πŸ“–mathematicalβ€”IsNilpotent
lowerCentralSeries
Int.instCommRing
LieRing.instLieAlgebra
AddCommGroup.toIntModule
Bot.bot
LieSubmodule
LieSubmodule.instBot
β€”β€”
isNilpotent_iff_le_maxNilpotentSubmodule πŸ“–mathematicalβ€”IsNilpotent
LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
AddSubgroupClass.toAddCommGroup
LieSubmodule.instAddSubgroupClass
LieSubmodule.instLieRingModuleSubtypeMem
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
maxNilpotentSubmodule
β€”LieSubmodule.instAddSubgroupClass
le_sSup
isNilpotent_of_le
instMaxNilpotentSubmoduleIsNilpotent
isNilpotent_of_le πŸ“–mathematicalLieSubmodule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
IsNilpotent
SetLike.instMembership
LieSubmodule.instSetLike
AddSubgroupClass.toAddCommGroup
LieSubmodule.instAddSubgroupClass
LieSubmodule.instLieRingModuleSubtypeMem
β€”LieSubmodule.instAddSubgroupClass
LieSubmodule.instSMulMemClass
Function.Injective.lieModuleIsNilpotent
LieSubmodule.instLieModule
Submodule.inclusion_injective
isNilpotent_of_top_iff πŸ“–mathematicalβ€”IsNilpotent
LieSubalgebra
SetLike.instMembership
LieSubalgebra.instSetLike
Top.top
LieSubalgebra.instTop
LieSubalgebra.lieRing
LieSubalgebra.lieRingModule
β€”Equiv.lieModule_isNilpotent_iff
LieSubalgebra.lieModule
RingHomInvPair.ids
isNilpotent_of_top_iff' πŸ“–mathematicalβ€”IsNilpotent
LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
Top.top
LieSubmodule.instTop
AddSubgroupClass.toAddCommGroup
LieSubmodule.instAddSubgroupClass
LieSubmodule.instLieRingModuleSubtypeMem
β€”Equiv.lieModule_isNilpotent_iff
LieSubmodule.instAddSubgroupClass
LieSubmodule.instLieModule
RingHomInvPair.ids
isNilpotent_quotient_iff πŸ“–mathematicalβ€”IsNilpotent
HasQuotient.Quotient
LieSubmodule
LieSubmodule.instHasQuotient
LieSubmodule.Quotient.addCommGroup
LieSubmodule.Quotient.lieQuotientLieRingModule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
lowerCentralSeries
β€”isNilpotent_iff
LieSubmodule.Quotient.lieQuotientLieModule
LieSubmodule.Quotient.map_mk'_eq_bot_le
map_lowerCentralSeries_eq
LieSubmodule.Quotient.surjective_mk'
isNilpotent_range_toEnd_iff πŸ“–mathematicalβ€”IsNilpotent
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieSubalgebra
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
SetLike.instMembership
LieSubalgebra.instSetLike
LieHom.range
toEnd
LieSubalgebra.lieRing
LieSubalgebra.lieRingModule
Module.End.instLieRingModule
β€”smulCommClass_self
IsScalarTower.left
isNilpotent_iff
LieSubalgebra.lieModule
Module.End.instLieModule
coe_lcs_range_toEnd_eq
isNilpotent_toEnd_of_isNilpotent πŸ“–mathematicalβ€”IsNilpotent
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instZero
RingHom.id
Semiring.toNonAssocSemiring
Monoid.toNatPow
Module.End.instMonoid
DFunLike.coe
LieHom
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
toEnd
β€”smulCommClass_self
IsScalarTower.left
exists_forall_pow_toEnd_eq_zero
isNilpotent_toEnd_of_isNilpotentβ‚‚ πŸ“–mathematicalβ€”IsNilpotent
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instZero
Monoid.toNatPow
Module.End.instMonoid
LinearMap.comp
RingHomCompTriple.ids
DFunLike.coe
LieHom
Module.End
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
toEnd
β€”RingHomCompTriple.ids
smulCommClass_self
IsScalarTower.left
IsNilpotent.nilpotent
eq_bot_iff
antitone_lowerCentralSeries
LinearMap.ext
Module.End.pow_apply
LinearMap.zero_apply
LieSubmodule.mem_bot
iterate_toEnd_mem_lowerCentralSeriesβ‚‚
isTrivial_of_nilpotencyLength_le_one πŸ“–mathematicalnilpotencyLengthIsTrivial
LieRingModule.toBracket
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim
instIsTrivialOfSubsingleton'
nilpotencyLength_eq_zero_iff
nilpotencyLength_eq_one_iff
iterate_toEnd_mem_lowerCentralSeries πŸ“–mathematicalβ€”LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
lowerCentralSeries
Nat.iterate
DFunLike.coe
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
RingHom.id
Semiring.toNonAssocSemiring
LieHom
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
toEnd
β€”smulCommClass_self
IsScalarTower.left
lowerCentralSeries_succ
Function.iterate_succ'
LieSubmodule.lie_mem_lie
LieSubmodule.mem_top
iterate_toEnd_mem_lowerCentralSeriesβ‚‚ πŸ“–mathematicalβ€”LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
lowerCentralSeries
Nat.iterate
DFunLike.coe
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
LinearMap.comp
RingHomCompTriple.ids
LieHom
Module.End
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
toEnd
β€”RingHomCompTriple.ids
smulCommClass_self
IsScalarTower.left
MulZeroClass.mul_zero
lowerCentralSeries_succ
Function.iterate_succ'
LieSubmodule.lie_mem_lie
LieSubmodule.mem_top
lowerCentralSeriesLast_le_max_triv πŸ“–mathematicalβ€”LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
lowerCentralSeriesLast
maxTrivSubmodule
β€”lowerCentralSeriesLast.eq_1
bot_le
le_max_triv_iff_bracket_eq_bot
lowerCentralSeries_succ
nilpotencyLength_eq_succ_iff
lowerCentralSeriesLast_le_of_not_isTrivial πŸ“–mathematicalIsTrivial
LieRingModule.toBracket
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
lowerCentralSeriesLast
lowerCentralSeries
β€”lowerCentralSeriesLast.eq_1
isTrivial_of_nilpotencyLength_le_one
not_lt
antitone_lowerCentralSeries
lowerCentralSeries_succ πŸ“–mathematicalβ€”lowerCentralSeries
Bracket.bracket
LieIdeal
LieSubmodule
LieSubmodule.hasBracket
Top.top
LieSubmodule.instTop
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
β€”LieSubmodule.lcs_succ
lowerCentralSeries_zero πŸ“–mathematicalβ€”lowerCentralSeries
Top.top
LieSubmodule
LieSubmodule.instTop
β€”β€”
map_lowerCentralSeries_eq πŸ“–mathematicalDFunLike.coe
LieModuleHom
LieModuleHom.instFunLike
LieSubmodule.map
lowerCentralSeries
β€”le_antisymm
map_lowerCentralSeries_le
lowerCentralSeries_zero
top_le_iff
LieModuleHom.map_top
LieModuleHom.range_eq_top
lowerCentralSeries_succ
LieSubmodule.map_bracket_eq
LieSubmodule.mono_lie_right
map_lowerCentralSeries_le πŸ“–mathematicalβ€”LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
LieSubmodule.map
lowerCentralSeries
β€”lowerCentralSeries_succ
LieSubmodule.map_bracket_eq
LieSubmodule.mono_lie_right
maxGenEigenSpace_toEnd_eq_top πŸ“–mathematicalβ€”Module.End.maxGenEigenspace
DFunLike.coe
LieHom
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
toEnd
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Top.top
Submodule
Submodule.instTop
β€”Submodule.ext
smulCommClass_self
IsScalarTower.left
zero_smul
sub_zero
exists_forall_pow_toEnd_eq_zero
maxNilpotentSubmodule_eq_top_of_isNilpotent πŸ“–mathematicalβ€”maxNilpotentSubmodule
Top.top
LieSubmodule
LieSubmodule.instTop
β€”eq_top_iff
le_sSup
LieSubmodule.instAddSubgroupClass
nilpotencyLength_eq_one_iff πŸ“–mathematicalβ€”nilpotencyLength
IsTrivial
LieRingModule.toBracket
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
β€”nilpotencyLength_eq_succ_iff
instLieModuleInt
trivial_iff_lower_central_eq_bot
LieSubmodule.instNontrivial
nilpotencyLength_eq_succ_iff πŸ“–mathematicalβ€”nilpotencyLength
lowerCentralSeries
Bot.bot
LieSubmodule
LieSubmodule.instBot
β€”coe_lowerCentralSeries_eq_int
eq_bot_iff
antitone_lowerCentralSeries
Nat.sInf_upward_closed_eq_succ_iff
nilpotencyLength_eq_zero_iff πŸ“–mathematicalβ€”nilpotencyLengthβ€”IsNilpotent.nilpotent
instLieModuleInt
LieSubmodule.subsingleton_iff
subsingleton_iff_bot_eq_top
lowerCentralSeries_zero
Nat.sInf_mem
Nat.sInf_eq_zero
nilpotentOfNilpotentQuotient πŸ“–mathematicalLieSubmodule
Preorder.toLE
PartialOrder.toPreorder
LieSubmodule.instPartialOrder
maxTrivSubmodule
IsNilpotentβ€”isNilpotent_iff
LieSubmodule.Quotient.lieQuotientLieModule
lowerCentralSeries_succ
LieSubmodule.Quotient.map_mk'_eq_bot_le
le_bot_iff
map_lowerCentralSeries_le
LieSubmodule.mono_lie_right
le_trans
ideal_oper_maxTrivSubmodule_eq_bot
nontrivial_lowerCentralSeriesLast πŸ“–mathematicalβ€”Nontrivial
LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
lowerCentralSeriesLast
β€”LieSubmodule.nontrivial_iff_ne_bot
lowerCentralSeriesLast.eq_1
not_nontrivial_iff_subsingleton
nilpotencyLength_eq_zero_iff
nilpotencyLength_eq_succ_iff
nontrivial_max_triv_of_isNilpotent πŸ“–mathematicalβ€”Nontrivial
LieSubmodule
SetLike.instMembership
LieSubmodule.instSetLike
maxTrivSubmodule
β€”Set.nontrivial_mono
lowerCentralSeriesLast_le_max_triv
nontrivial_lowerCentralSeriesLast
trivialIsNilpotent πŸ“–mathematicalβ€”IsNilpotentβ€”lowerCentralSeries_succ
LieSubmodule.trivial_lie_oper_zero
trivial_iff_lower_central_eq_bot πŸ“–mathematicalβ€”IsTrivial
LieRingModule.toBracket
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
lowerCentralSeries
Bot.bot
LieSubmodule
LieSubmodule.instBot
β€”lowerCentralSeries_succ
LieSubmodule.trivial_lie_oper_zero
LieSubmodule.eq_bot_iff
LieSubmodule.subset_lieSpan

LieModule.IsNilpotent

Theorems

NameKindAssumesProvesValidatesDepends On
mk πŸ“–mathematicalLieModule.lowerCentralSeries
Bot.bot
LieSubmodule
LieSubmodule.instBot
LieModule.IsNilpotentβ€”LieModule.isNilpotent_iff
nilpotent πŸ“–mathematicalβ€”LieModule.lowerCentralSeries
Bot.bot
LieSubmodule
LieSubmodule.instBot
β€”LieModule.isNilpotent_iff
nilpotent_int πŸ“–mathematicalβ€”LieModule.lowerCentralSeries
Int.instCommRing
LieRing.instLieAlgebra
AddCommGroup.toIntModule
Bot.bot
LieSubmodule
LieSubmodule.instBot
β€”β€”

LieRing

Definitions

NameCategoryTheorems
IsNilpotent πŸ“–MathDef
12 mathmath: Function.Injective.lieAlgebra_isNilpotent, LieIdeal.instIsNilpotentSubtypeMemLieSubmoduleOfIsNilpotent, LieHom.isNilpotent_range, LieSubalgebra.IsCartanSubalgebra.nilpotent, Function.Surjective.lieAlgebra_isNilpotent, LieEquiv.nilpotent_iff_equiv_nilpotent, instIsNilpotentSubtypeMemLieSubalgebraTop, LieAlgebra.nilpotent_of_nilpotent_quotient, LieSubalgebra.isNilpotent_of_forall_le_engel, LieSubalgebra.instIsNilpotentSubtypeMemOfIsCartanSubalgebra, LieAlgebra.isNilpotent_range_ad_iff, LieAlgebra.isNilpotent_iff_forall

LieSubalgebra

Theorems

NameKindAssumesProvesValidatesDepends On
isNilpotent_ad_of_isNilpotent_ad πŸ“–mathematicalIsNilpotent
Module.End
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
LieRing.toAddCommGroup
LieAlgebra.toModule
LinearMap.instZero
RingHom.id
Semiring.toNonAssocSemiring
Monoid.toNatPow
Module.End.instMonoid
DFunLike.coe
LieHom
LieRing.ofAssociativeRing
Module.End.instRing
LieAlgebra.ofAssociativeAlgebra
Module.End.instAlgebra
smulCommClass_self
CommRing.toCommMonoid
DistribMulAction.toMulAction
CommMonoid.toMonoid
AddCommMonoid.toAddMonoid
Module.toDistribMulAction
Ring.toSemiring
CommRing.toRing
Algebra.toSMul
Algebra.id
IsScalarTower.left
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
LieHom.instFunLike
LieAlgebra.ad
LieSubalgebra
SetLike.instMembership
instSetLike
lieRing
lieAlgebra
β€”smulCommClass_self
IsScalarTower.left
Module.End.submodule_pow_eq_zero_of_pow_eq_zero
ad_comp_incl_eq

LieSubmodule

Definitions

NameCategoryTheorems
lcs πŸ“–CompOp
11 mathmath: lcs_sup, lcs_add_le_iff, isNilpotent_iff_exists_lcs_eq_bot, lowerCentralSeries_eq_bot_iff_lcs_eq_bot, lowerCentralSeries_map_eq_lcs, lowerCentralSeries_eq_lcs_comap, lcs_le_iff, lcs_zero, lcs_succ, gc_lcs_ucs, lcs_le_self
ucs πŸ“–CompOp
17 mathmath: ucs_mono, lcs_add_le_iff, LieModule.iSup_ucs_le_genWeightSpace_zero, ucs_zero, isNilpotent_iff_exists_self_le_ucs, ucs_comap_incl, ucs_succ, ucs_eq_self_of_normalizer_eq_self, LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra, ucs_eq_top_iff, LieModule.isNilpotent_iff_exists_ucs_eq_top, lcs_le_iff, ucs_le_of_normalizer_eq_self, LieModule.iSup_ucs_eq_genWeightSpace_zero, ucs_add, gc_lcs_ucs, ucs_bot_one

Theorems

NameKindAssumesProvesValidatesDepends On
gc_lcs_ucs πŸ“–mathematicalβ€”GaloisConnection
LieSubmodule
PartialOrder.toPreorder
instPartialOrder
lcs
ucs
β€”lcs_le_iff
isNilpotent_iff_exists_lcs_eq_bot πŸ“–mathematicalβ€”LieModule.IsNilpotent
LieSubmodule
SetLike.instMembership
instSetLike
AddSubgroupClass.toAddCommGroup
instAddSubgroupClass
instLieRingModuleSubtypeMem
lcs
Bot.bot
instBot
β€”instAddSubgroupClass
instSMulMemClass
LieModule.isNilpotent_iff
instLieModule
lowerCentralSeries_eq_lcs_comap
comap_incl_eq_bot
inf_eq_right
lcs_le_self
isNilpotent_iff_exists_self_le_ucs πŸ“–mathematicalβ€”LieModule.IsNilpotent
LieSubmodule
SetLike.instMembership
instSetLike
AddSubgroupClass.toAddCommGroup
instAddSubgroupClass
instLieRingModuleSubtypeMem
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
ucs
Bot.bot
instBot
β€”instAddSubgroupClass
instSMulMemClass
instLieModule
LieModule.isNilpotent_iff_exists_ucs_eq_top
lcs_add_le_iff πŸ“–mathematicalβ€”LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
lcs
ucs
β€”add_zero
Mathlib.Tactic.Abel.subst_into_add
Mathlib.Tactic.Abel.term_atom
Mathlib.Tactic.Abel.term_add_const
zero_add
Mathlib.Tactic.Abel.const_add_term
ucs_succ
lcs_succ
top_lie_le_iff_le_normalizer
lcs_le_iff πŸ“–mathematicalβ€”LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
lcs
ucs
β€”zero_add
lcs_add_le_iff
lcs_le_self πŸ“–mathematicalβ€”LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
lcs
β€”lcs_succ
LE.le.trans
mono_lie_right
lie_le_right
lcs_succ πŸ“–mathematicalβ€”lcs
Bracket.bracket
LieIdeal
LieSubmodule
hasBracket
Top.top
instTop
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
β€”Function.iterate_succ_apply'
lcs_sup πŸ“–mathematicalβ€”lcs
LieSubmodule
instMax
β€”lcs_succ
lie_sup
lcs_zero πŸ“–mathematicalβ€”lcsβ€”β€”
lowerCentralSeries_eq_bot_iff_lcs_eq_bot πŸ“–mathematicalβ€”LieModule.lowerCentralSeries
LieSubmodule
SetLike.instMembership
instSetLike
AddSubgroupClass.toAddCommGroup
instAddSubgroupClass
SubmoduleClass.module
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
AddCommGroup.toAddGroup
instSMulMemClass
instLieRingModuleSubtypeMem
Bot.bot
instBot
lcs
β€”instAddSubgroupClass
instSMulMemClass
lowerCentralSeries_map_eq_lcs
LieModuleHom.le_ker_iff_map
ker_incl
lowerCentralSeries_eq_lcs_comap
comap_incl_eq_bot
inf_of_le_right
lowerCentralSeries_eq_lcs_comap πŸ“–mathematicalβ€”LieModule.lowerCentralSeries
LieSubmodule
SetLike.instMembership
instSetLike
AddSubgroupClass.toAddCommGroup
instAddSubgroupClass
SubmoduleClass.module
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
AddCommGroup.toAddGroup
instSMulMemClass
instLieRingModuleSubtypeMem
comap
incl
lcs
β€”instAddSubgroupClass
instSMulMemClass
comap_incl_self
LieModule.lowerCentralSeries_succ
lcs_succ
range_incl
lcs_le_self
comap_bracket_eq
instLieModule
ker_incl
lowerCentralSeries_map_eq_lcs πŸ“–mathematicalβ€”map
LieSubmodule
SetLike.instMembership
instSetLike
AddSubgroupClass.toAddCommGroup
instAddSubgroupClass
SubmoduleClass.module
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
AddCommGroup.toAddGroup
instSMulMemClass
instLieRingModuleSubtypeMem
incl
LieModule.lowerCentralSeries
lcs
β€”instAddSubgroupClass
instSMulMemClass
lowerCentralSeries_eq_lcs_comap
map_comap_incl
inf_eq_right
lcs_le_self
lowerCentralSeries_tensor_eq_baseChange πŸ“–mathematicalβ€”LieModule.lowerCentralSeries
TensorProduct
CommRing.toCommSemiring
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
AddCommGroup.toAddCommMonoid
LieRing.toAddCommGroup
LieAlgebra.toModule
LieRing.ofAssociativeRing
CommRing.toRing
LieAlgebra.ofAssociativeAlgebra
LieAlgebra.ExtendScalars.instLieRing
LieAlgebra.ExtendScalars.instLieAlgebra
TensorProduct.addCommGroup
TensorProduct.leftModule
CommSemiring.toSemiring
Algebra.id
Algebra.to_smulCommClass
LieAlgebra.ExtendScalars.instLieRingModule
baseChange
β€”Algebra.to_smulCommClass
baseChange_top
lieAlgebraSelfModule
LieModule.lowerCentralSeries_succ
lie_baseChange
ucs_add πŸ“–mathematicalβ€”ucsβ€”Function.iterate_add_apply
ucs_bot_one πŸ“–mathematicalβ€”ucs
Bot.bot
LieSubmodule
instBot
LieModule.maxTrivSubmodule
β€”ucs_succ
ucs_comap_incl πŸ“–mathematicalβ€”comap
LieSubmodule
SetLike.instMembership
instSetLike
AddSubgroupClass.toAddCommGroup
instAddSubgroupClass
SubmoduleClass.module
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
AddCommGroup.toAddGroup
instSMulMemClass
instLieRingModuleSubtypeMem
incl
ucs
Bot.bot
instBot
instLieModule
β€”instAddSubgroupClass
instSMulMemClass
instLieModule
ker_incl
ucs_succ
comap_normalizer
normalizer.congr_simp
ucs_eq_self_of_normalizer_eq_self πŸ“–mathematicalnormalizerucsβ€”ucs_succ
ucs_eq_top_iff πŸ“–mathematicalβ€”ucs
Top.top
LieSubmodule
instTop
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
LieModule.lowerCentralSeries
β€”eq_top_iff
lcs_le_iff
ucs_le_of_normalizer_eq_self πŸ“–mathematicalnormalizerLieSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
ucs
Bot.bot
instBot
β€”ucs_eq_self_of_normalizer_eq_self
ucs_mono
ucs_mono πŸ“–mathematicalLieSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
ucsβ€”ucs_succ
normalizer_mono
ucs_succ πŸ“–mathematicalβ€”ucs
normalizer
β€”Function.iterate_succ_apply'
ucs_zero πŸ“–mathematicalβ€”ucsβ€”β€”

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
coe_lowerCentralSeries_ideal_quot_eq πŸ“–mathematicalβ€”LieSubmodule.toSubmodule
HasQuotient.Quotient
LieIdeal
LieSubmodule.instHasQuotient
LieRing.toAddCommGroup
LieAlgebra.toModule
lieRingSelfModule
LieSubmodule.Quotient.addCommGroup
LieSubmodule.Quotient.module
LieSubmodule.Quotient.lieQuotientLieRingModule
lieAlgebraSelfModule
LieModule.lowerCentralSeries
LieSubmodule.Quotient.lieQuotientLieRing
LieSubmodule.Quotient.lieQuotientLieAlgebra
β€”lieAlgebraSelfModule
LieModule.lowerCentralSeries_succ
LieSubmodule.lieIdeal_oper_eq_linear_span
LieSubmodule.Quotient.lieQuotientLieModule
LieSubmodule.mem_top
LieSubmodule.mem_toSubmodule
instIsNilpotentSubtypeMemLieSubalgebraTop πŸ“–mathematicalβ€”LieRing.IsNilpotent
LieSubalgebra
SetLike.instMembership
LieSubalgebra.instSetLike
Top.top
LieSubalgebra.instTop
LieSubalgebra.lieRing
β€”LieEquiv.nilpotent_iff_equiv_nilpotent
lieModule_lcs_map_le πŸ“–mathematicalBracket.bracket
LieRingModule.toBracket
DFunLike.coe
LieHom
LieHom.instFunLike
LinearMap
CommSemiring.toSemiring
CommRing.toCommSemiring
RingHom.id
Semiring.toNonAssocSemiring
AddCommGroup.toAddCommMonoid
LinearMap.instFunLike
Submodule
Preorder.toLE
PartialOrder.toPreorder
Submodule.instPartialOrder
Submodule.map
RingHomSurjective.ids
LieSubmodule.toSubmodule
LieModule.lowerCentralSeries
β€”RingHomSurjective.ids
Submodule.map_top
LieModule.lowerCentralSeries_succ
LieSubmodule.lieIdeal_oper_eq_linear_span'
Submodule.map_span
Submodule.span_le
LieSubmodule.instAddSubgroupClass
Submodule.mem_map_of_mem
LieSubmodule.lie_mem_lie
SetLike.coe_mem

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