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, center_le_maxNilpotentIdeal, 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, 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
96
Total106

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
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
RingHom.id
Semiring.toNonAssocSemiring
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
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_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.toPow
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
5 mathmath: lcs_succ, LieSubmodule.lcs_le_lcs_of_is_nilpotent_span_sup_eq_top, 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
40 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.lcs_le_lcs_of_is_nilpotent_span_sup_eq_top, 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€”instReflLe
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.toPow
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
instReflLe
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
LieSubmodule
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.toPow
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.toPow
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

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€”instReflLe
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
LieSubmodule
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
ucs
β€”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
CommSemiring.toSemiring
CommRing.toCommSemiring
AddCommGroup.toAddCommMonoid
Preorder.toLE
PartialOrder.toPreorder
Submodule.instPartialOrder
Submodule.map
RingHom.id
Semiring.toNonAssocSemiring
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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