ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieAlgebra.abelian_radical_iff_solvable_is_abelian
theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
    IsLieAbelian (radical R L) ↔ ∀ I : LieIdeal R L, IsSolvable R I → IsLieAbelian I 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [IsNoetherian R L]
Goal: IsLieAbelian ↥↑(LieAlgebra.radical R L) ↔ ∀ (I : LieIdeal R L), LieAlgebra.IsSolvable R ↥↑I → IsLieAbelian ↥↑I
-- LieAlgebra.ad_ker_eq_bot_of_hasTrivialRadical
theorem ad_ker_eq_bot_of_hasTrivialRadical [HasTrivialRadical R L] : (ad R L).ker = ⊥ 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.HasTrivialRadical R L]
Goal: (LieAlgebra.ad R L).ker = ⊥
-- LieDerivation.IsKilling.killingForm_restrict_range_ad
lemma killingForm_restrict_range_ad : (killingForm R 𝔻).restrict 𝕀 = killingForm R 𝕀 
Arguments: (R : Type u_1) (L : Type u_2) [Field R] [LieRing L] [LieAlgebra R L] [Module.Finite R L]
Goal: (killingForm R (LieDerivation R L L)).restrict (LieDerivation.ad R L).range.toSubmodule =
  killingForm R ↥(LieDerivation.ad R L).range
-- LieDerivation.IsKilling.rangeAdOrthogonal_carrier
@[simps!] noncomputable def rangeAdOrthogonal : LieSubmodule R L (LieDerivation R L L) 
Arguments: (R : Type u_1) (L : Type u_2) [Field R] [LieRing L] [LieAlgebra R L]
Goal: ↑(LieDerivation.IsKilling.rangeAdOrthogonal R L) =
  {m | ∀ (a : L), (killingForm R (LieDerivation R L L)).IsOrtho ((LieDerivation.ad R L) a) m}
-- LieDerivation.IsKilling.ad_mem_orthogonal_of_mem_orthogonal
lemma ad_mem_orthogonal_of_mem_orthogonal {D : LieDerivation R L L} (hD : D ∈ 𝕀ᗮ) (x : L) :
    ad R L (D x) ∈ 𝕀ᗮ 
Arguments: {R : Type u_1} {L : Type u_2} [Field R] [LieRing L] [LieAlgebra R L] {D : LieDerivation R L L} (hD : D ∈ (killingForm R (LieDerivation R L L)).orthogonal (LieDerivation.ad R L).range.toSubmodule) (x : L)
Goal: (LieDerivation.ad R L) (D x) ∈ (killingForm R (LieDerivation R L L)).orthogonal (LieDerivation.ad R L).range.toSubmodule
-- LieDerivation.IsKilling.ad_mem_ker_killingForm_ad_range_of_mem_orthogonal
lemma ad_mem_ker_killingForm_ad_range_of_mem_orthogonal
    {D : LieDerivation R L L} (hD : D ∈ 𝕀ᗮ) (x : L) :
    ad R L (D x) ∈ (LinearMap.ker (killingForm R 𝕀)).map (LieHom.range (ad R L)).subtype 
Arguments: {R : Type u_1} {L : Type u_2} [Field R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] {D : LieDerivation R L L} (hD : D ∈ (killingForm R (LieDerivation R L L)).orthogonal (LieDerivation.ad R L).range.toSubmodule) (x : L)
Goal: (LieDerivation.ad R L) (D x) ∈
  Submodule.map (LieDerivation.ad R L).range.subtype (LinearMap.ker (killingForm R ↥(LieDerivation.ad R L).range))
-- LieDerivation.IsKilling.ad_apply_eq_zero_iff
@[simp] lemma ad_apply_eq_zero_iff (x : L) : ad R L x = 0 ↔ x = 0 
Arguments: (R : Type u_1) (L : Type u_2) [Field R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [LieAlgebra.IsKilling R L] (x : L)
Goal: (LieDerivation.ad R L) x = 0 ↔ x = 0
-- LieDerivation.IsKilling.killingForm_restrict_range_ad_nondegenerate
lemma killingForm_restrict_range_ad_nondegenerate : ((killingForm R 𝔻).restrict 𝕀).Nondegenerate 
Arguments: (R : Type u_1) (L : Type u_2) [Field R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [LieAlgebra.IsKilling R L]
Goal: ((killingForm R (LieDerivation R L L)).restrict (LieDerivation.ad R L).range.toSubmodule).Nondegenerate
-- LieDerivation.IsKilling.range_ad_eq_top
@[simp]
lemma range_ad_eq_top : 𝕀 = ⊤ 
Arguments: (R : Type u_1) (L : Type u_2) [Field R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [LieAlgebra.IsKilling R L]
Goal: (LieDerivation.ad R L).range = ⊤
-- LieDerivation.IsKilling.exists_eq_ad
lemma exists_eq_ad (D : 𝔻) : ∃ x, ad R L x = D 
Arguments: {R : Type u_1} {L : Type u_2} [Field R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [LieAlgebra.IsKilling R L] (D : LieDerivation R L L)
Goal: ∃ x, (LieDerivation.ad R L) x = D
-- LieSubmodule.exists_smul_add_of_span_sup_eq_top
theorem exists_smul_add_of_span_sup_eq_top (y : L) : ∃ t : R, ∃ z ∈ I, y = t • x + z 
Arguments: {R : Type u₁} {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {I : LieIdeal R L} {x : L} (hxI : Submodule.span R {x} ⊔ (lieIdealSubalgebra R L I).toSubmodule = ⊤) (y : L)
Goal: ∃ t, ∃ z ∈ I, y = t • x + z
-- LieSubmodule.lie_top_eq_of_span_sup_eq_top
theorem lie_top_eq_of_span_sup_eq_top (N : LieSubmodule R L M) :
    (↑⁅(⊤ : LieIdeal R L), N⁆ : Submodule R M) =
      (N : Submodule R M).map (toEnd R L M x) ⊔ (↑⁅I, N⁆ : Submodule R M) 
Arguments: {R : Type u₁} {L : Type u₂} {M : Type u₄} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {I : LieIdeal R L} {x : L} (hxI : Submodule.span R {x} ⊔ (lieIdealSubalgebra R L I).toSubmodule = ⊤) (N : LieSubmodule R L M)
Goal: ↑⁅⊤, N⁆ = Submodule.map ((LieModule.toEnd R L M) x) ↑N ⊔ ↑⁅I, N⁆
-- LieSubmodule.lcs_le_lcs_of_is_nilpotent_span_sup_eq_top
theorem lcs_le_lcs_of_is_nilpotent_span_sup_eq_top {n i j : ℕ}
    (hxn : toEnd R L M x ^ n = 0) (hIM : lowerCentralSeries R L M i ≤ I.lcs M j) :
    lowerCentralSeries R L M (i + n) ≤ I.lcs M (j + 1) 
Arguments: {R : Type u₁} {L : Type u₂} {M : Type u₄} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {I : LieIdeal R L} {x : L} (hxI : Submodule.span R {x} ⊔ (lieIdealSubalgebra R L I).toSubmodule = ⊤) {n i j : ℕ} (hxn : (LieModule.toEnd R L M) x ^ n = 0) (hIM : LieModule.lowerCentralSeries R L M i ≤ I.lcs M j)
Goal: LieModule.lowerCentralSeries R L M (i + n) ≤ I.lcs M (j + 1)
-- LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop
theorem isNilpotentOfIsNilpotentSpanSupEqTop (hnp : IsNilpotent <| toEnd R L M x)
    (hIM : IsNilpotent R I M) : IsNilpotent R L M 
Arguments: {R : Type u₁} {L : Type u₂} {M : Type u₄} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {I : LieIdeal R L} {x : L} (hxI : Submodule.span R {x} ⊔ (lieIdealSubalgebra R L I).toSubmodule = ⊤) (hnp : IsNilpotent ((LieModule.toEnd R L M) x)) (hIM : LieModule.IsNilpotent R (↥↑I) M)
Goal: LieModule.IsNilpotent R L M
-- LieAlgebra.isEngelian_of_subsingleton
theorem LieAlgebra.isEngelian_of_subsingleton [Subsingleton L] : LieAlgebra.IsEngelian R L 
Arguments: {R : Type u₁} {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] [Subsingleton L]
Goal: LieAlgebra.IsEngelian R L
-- Function.Surjective.isEngelian
theorem Function.Surjective.isEngelian {f : L →ₗ⁅R⁆ L₂} (hf : Function.Surjective f)
    (h : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R L) : LieAlgebra.IsEngelian.{u₁, u₃, u₄} R L₂ 
Arguments: {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L₂] [LieAlgebra R L₂] {f : L →ₗ⁅R⁆ L₂} (hf : Function.Surjective ⇑f) (h : LieAlgebra.IsEngelian R L)
Goal: LieAlgebra.IsEngelian R L₂
-- LieEquiv.isEngelian_iff
theorem LieEquiv.isEngelian_iff (e : L ≃ₗ⁅R⁆ L₂) :
    LieAlgebra.IsEngelian.{u₁, u₂, u₄} R L ↔ LieAlgebra.IsEngelian.{u₁, u₃, u₄} R L₂ 
Arguments: {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L₂] [LieAlgebra R L₂] (e : L ≃ₗ⁅R⁆ L₂)
Goal: LieAlgebra.IsEngelian R L ↔ LieAlgebra.IsEngelian R L₂
-- LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer
theorem LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer {K : LieSubalgebra R L}
    (hK₁ : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R K) (hK₂ : K < K.normalizer) :
    ∃ (K' : LieSubalgebra R L), LieAlgebra.IsEngelian.{u₁, u₂, u₄} R K' ∧ K < K' 
Arguments: {R : Type u₁} {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} (hK₁ : LieAlgebra.IsEngelian R ↥K) (hK₂ : K < K.normalizer)
Goal: ∃ K', LieAlgebra.IsEngelian R ↥K' ∧ K < K'
-- LieAlgebra.isEngelian_of_isNoetherian
theorem LieAlgebra.isEngelian_of_isNoetherian [IsNoetherian R L] : LieAlgebra.IsEngelian R L 
Arguments: {R : Type u₁} {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] [IsNoetherian R L]
Goal: LieAlgebra.IsEngelian R L
-- LieModule.isNilpotent_iff_forall
theorem LieModule.isNilpotent_iff_forall [IsNoetherian R L] :
    LieModule.IsNilpotent R L M ↔ ∀ x, _root_.IsNilpotent <| toEnd R L M x 
Arguments: {R : Type u₁} {L : Type u₂} {M : Type u₄} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R L]
Goal: LieModule.IsNilpotent R L M ↔ ∀ (x : L), IsNilpotent ((LieModule.toEnd R L M) x)
-- LieModule.isNilpotent_iff_forall'
theorem LieModule.isNilpotent_iff_forall' [IsNoetherian R M] :
    LieModule.IsNilpotent R L M ↔ ∀ x, _root_.IsNilpotent <| toEnd R L M x 
Arguments: {R : Type u₁} {L : Type u₂} {M : Type u₄} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M]
Goal: LieModule.IsNilpotent R L M ↔ ∀ (x : L), IsNilpotent ((LieModule.toEnd R L M) x)
-- LieAlgebra.isNilpotent_iff_forall
theorem LieAlgebra.isNilpotent_iff_forall [IsNoetherian R L] :
    LieAlgebra.IsNilpotent R L ↔ ∀ x, _root_.IsNilpotent <| LieAlgebra.ad R L x 
Arguments: {R : Type u₁} {L : Type u₂} [CommRing R] [LieRing L] [LieAlgebra R L] [IsNoetherian R L]
Goal: LieAlgebra.IsNilpotent R L ↔ ∀ (x : L), IsNilpotent ((LieAlgebra.ad R L) x)
-- DirectSum.lie_module_bracket_apply
@[simp]
theorem lie_module_bracket_apply (x : L) (m : ⨁ i, M i) (i : ι) : ⁅x, m⁆ i = ⁅x, m i⁆ 
Arguments: {ι : Type v} {L : Type w₁} {M : ι → Type w} [LieRing L] [(i : ι) → AddCommGroup (M i)] [(i : ι) → LieRingModule L (M i)] (x : L) (m : DirectSum ι fun i => M i) (i : ι)
Goal: ⁅x, m⁆ i = ⁅x, m i⁆
-- DirectSum.bracket_apply
@[simp]
theorem bracket_apply (x y : ⨁ i, L i) (i : ι) : ⁅x, y⁆ i = ⁅x i, y i⁆ 
Arguments: {ι : Type v} (L : ι → Type w) [(i : ι) → LieRing (L i)] (x y : DirectSum ι fun i => L i) (i : ι)
Goal: ⁅x, y⁆ i = ⁅x i, y i⁆
-- DirectSum.lie_of_same
theorem lie_of_same [DecidableEq ι] {i : ι} (x y : L i) :
    ⁅of L i x, of L i y⁆ = of L i ⁅x, y⁆ 
Arguments: {ι : Type v} (L : ι → Type w) [(i : ι) → LieRing (L i)] [DecidableEq ι] {i : ι} (x y : L i)
Goal: ⁅(DirectSum.of L i) x, (DirectSum.of L i) y⁆ = (DirectSum.of L i) ⁅x, y⁆
-- DirectSum.lie_of_of_ne
theorem lie_of_of_ne [DecidableEq ι] {i j : ι} (hij : i ≠ j) (x : L i) (y : L j) :
    ⁅of L i x, of L j y⁆ = 0 
Arguments: {ι : Type v} (L : ι → Type w) [(i : ι) → LieRing (L i)] [DecidableEq ι] {i j : ι} (hij : i ≠ j) (x : L i) (y : L j)
Goal: ⁅(DirectSum.of L i) x, (DirectSum.of L j) y⁆ = 0
-- DirectSum.lie_of
@[simp]
theorem lie_of [DecidableEq ι] {i j : ι} (x : L i) (y : L j) :
    ⁅of L i x, of L j y⁆ = if hij : i = j then of L i ⁅x, hij.symm.recOn y⁆ else 0 
Arguments: {ι : Type v} (L : ι → Type w) [(i : ι) → LieRing (L i)] [DecidableEq ι] {i j : ι} (x : L i) (y : L j)
Goal: ⁅(DirectSum.of L i) x, (DirectSum.of L j) y⁆ = if hij : i = j then (DirectSum.of L i) ⁅x, Eq.recOn ⋯ y⁆ else 0
-- DirectSum.lieAlgebraOf_apply
@[simps]
def lieAlgebraOf [DecidableEq ι] (j : ι) : L j →ₗ⁅R⁆ ⨁ i, L i 
Arguments: (R : Type u) (ι : Type v) [CommRing R] (L : ι → Type w) [(i : ι) → LieRing (L i)] [(i : ι) → LieAlgebra R (L i)] [DecidableEq ι] (j : ι) (a : L j)
Goal: (DirectSum.lieAlgebraOf R ι L j) a = (DirectSum.of L j) a
-- DirectSum.lieAlgebraComponent_apply
@[simps]
def lieAlgebraComponent (j : ι) : (⨁ i, L i) →ₗ⁅R⁆ L j 
Arguments: (R : Type u) (ι : Type v) [CommRing R] (L : ι → Type w) [(i : ι) → LieRing (L i)] [(i : ι) → LieAlgebra R (L i)] (j : ι) (a : DirectSum ι fun i => L i)
Goal: (DirectSum.lieAlgebraComponent R ι L j) a = (DirectSum.component R ι L j) a
-- DirectSum.lieAlgebra_ext
@[ext]
theorem lieAlgebra_ext {x y : ⨁ i, L i}
    (h : ∀ i, lieAlgebraComponent R ι L i x = lieAlgebraComponent R ι L i y) : x = y 
Arguments: (R : Type u) (ι : Type v) [CommRing R] (L : ι → Type w) [(i : ι) → LieRing (L i)] [(i : ι) → LieAlgebra R (L i)] {x y : DirectSum ι fun i => L i} (h : ∀ (i : ι), (DirectSum.lieAlgebraComponent R ι L i) x = (DirectSum.lieAlgebraComponent R ι L i) y)
Goal: x = y
-- DirectSum.toLieAlgebra_apply
@[simps]
def toLieAlgebra [DecidableEq ι] (L' : Type w₁) [LieRing L'] [LieAlgebra R L']
    (f : ∀ i, L i →ₗ⁅R⁆ L') (hf : Pairwise fun i j => ∀ (x : L i) (y : L j), ⁅f i x, f j y⁆ = 0) :
    (⨁ i, L i) →ₗ⁅R⁆ L' 
Arguments: {R : Type u} {ι : Type v} [CommRing R] {L : ι → Type w} [(i : ι) → LieRing (L i)] [(i : ι) → LieAlgebra R (L i)] [DecidableEq ι] (L' : Type w₁) [LieRing L'] [LieAlgebra R L'] (f : (i : ι) → L i →ₗ⁅R⁆ L') (hf : Pairwise fun i j => ∀ (x : L i) (y : L j), ⁅(f i) x, (f j) y⁆ = 0) (a : DirectSum ι fun i => L i)
Goal: (DirectSum.toLieAlgebra L' f hf) a = (DirectSum.toModule R ι L' fun i => ↑(f i)) a
-- LieRing.of_associative_ring_bracket
theorem of_associative_ring_bracket (x y : A) : ⁅x, y⁆ = x * y - y * x 
Arguments: {A : Type v} [Ring A] (x y : A)
Goal: ⁅x, y⁆ = x * y - y * x
-- LieRing.lie_apply
@[simp]
theorem lie_apply {α : Type*} (f g : α → A) (a : α) : ⁅f, g⁆ a = ⁅f a, g a⁆ 
Arguments: {A : Type v} [Ring A] {α : Type u_1} (f g : α → A) (a : α)
Goal: ⁅f, g⁆ a = ⁅f a, g a⁆
-- lie_eq_smul
theorem lie_eq_smul (a : A) (m : M) : ⁅a, m⁆ = a • m 
Arguments: {A : Type v} [Ring A] {M : Type w} [AddCommGroup M] [Module A M] (a : A) (m : M)
Goal: ⁅a, m⁆ = a • m
-- LieModule.ofAssociativeModule
theorem LieModule.ofAssociativeModule : LieModule R A M 
Arguments: {A : Type v} [Ring A] {R : Type u} [CommRing R] [Algebra R A] {M : Type w} [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M]
Goal: LieModule R A M
-- Module.End.lie_apply
@[simp] lemma Module.End.lie_apply (f : Module.End R M) (m : M) : ⁅f, m⁆ = f m 
Arguments: {R : Type u} [CommRing R] {M : Type w} [AddCommGroup M] [Module R M] (f : Module.End R M) (m : M)
Goal: ⁅f, m⁆ = f m
-- AlgHom.coe_toLieHom
@[simp]
theorem coe_toLieHom : ((f : A →ₗ⁅R⁆ B) : A → B) = f 
Arguments: {A : Type v} [Ring A] {R : Type u} [CommRing R] [Algebra R A] {B : Type w} [Ring B] [Algebra R B] (f : A →ₐ[R] B)
Goal: ⇑f.toLieHom = ⇑f
-- AlgHom.toLieHom_apply
theorem toLieHom_apply (x : A) : f.toLieHom x = f x 
Arguments: {A : Type v} [Ring A] {R : Type u} [CommRing R] [Algebra R A] {B : Type w} [Ring B] [Algebra R B] (f : A →ₐ[R] B) (x : A)
Goal: f.toLieHom x = f x
-- AlgHom.toLieHom_id
@[simp]
theorem toLieHom_id : (AlgHom.id R A : A →ₗ⁅R⁆ A) = LieHom.id 
Arguments: {A : Type v} [Ring A] {R : Type u} [CommRing R] [Algebra R A]
Goal: (AlgHom.id R A).toLieHom = LieHom.id
-- AlgHom.toLieHom_comp
@[simp]
theorem toLieHom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C).comp (f : A →ₗ⁅R⁆ B) 
Arguments: {A : Type v} [Ring A] {R : Type u} [CommRing R] [Algebra R A] {B : Type w} {C : Type w₁} [Ring B] [Ring C] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (g : B →ₐ[R] C)
Goal: (g.comp f).toLieHom = g.toLieHom.comp f.toLieHom
-- AlgHom.toLieHom_injective
theorem toLieHom_injective {f g : A →ₐ[R] B} (h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g 
Arguments: {A : Type v} [Ring A] {R : Type u} [CommRing R] [Algebra R A] {B : Type w} [Ring B] [Algebra R B] {f g : A →ₐ[R] B} (h : f.toLieHom = g.toLieHom)
Goal: f = g
-- LieModule.toEnd_apply_apply
@[simps]
def LieModule.toEnd : L →ₗ⁅R⁆ Module.End R M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (m : M)
Goal: ((LieModule.toEnd R L M) x) m = ⁅x, m⁆
-- LieAlgebra.ad_apply
@[simp]
theorem LieAlgebra.ad_apply (x y : L) : LieAlgebra.ad R L x y = ⁅x, y⁆ 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (x y : L)
Goal: ((LieAlgebra.ad R L) x) y = ⁅x, y⁆
-- LieModule.toEnd_module_end
@[simp]
theorem LieModule.toEnd_module_end :
    LieModule.toEnd R (Module.End R M) M = LieHom.id 
Arguments: (R : Type u) (M : Type w) [CommRing R] [AddCommGroup M] [Module R M]
Goal: LieModule.toEnd R (Module.End R M) M = LieHom.id
-- LieSubalgebra.toEnd_eq
theorem LieSubalgebra.toEnd_eq (K : LieSubalgebra R L) {x : K} :
    LieModule.toEnd R K M x = LieModule.toEnd R L M x 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (K : LieSubalgebra R L) {x : ↥K}
Goal: (LieModule.toEnd R (↥K) M) x = (LieModule.toEnd R L M) ↑x
-- LieSubalgebra.toEnd_mk
@[simp]
theorem LieSubalgebra.toEnd_mk (K : LieSubalgebra R L) {x : L} (hx : x ∈ K) :
    LieModule.toEnd R K M ⟨x, hx⟩ = LieModule.toEnd R L M x 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (K : LieSubalgebra R L) {x : L} (hx : x ∈ K)
Goal: (LieModule.toEnd R (↥K) M) ⟨x, hx⟩ = (LieModule.toEnd R L M) x
-- LieSubmodule.coe_toEnd
lemma LieSubmodule.coe_toEnd (N : LieSubmodule R L M) (x : L) (y : N) :
    (toEnd R L N x y : M) = toEnd R L M x y 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (x : L) (y : ↥N)
Goal: ↑(((LieModule.toEnd R L ↥↑N) x) y) = ((LieModule.toEnd R L M) x) ↑y
-- LieSubmodule.coe_toEnd_pow
lemma LieSubmodule.coe_toEnd_pow (N : LieSubmodule R L M) (x : L) (y : N) (n : ℕ) :
    ((toEnd R L N x ^ n) y : M) = (toEnd R L M x ^ n) y 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (x : L) (y : ↥N) (n : ℕ)
Goal: ↑(((LieModule.toEnd R L ↥↑N) x ^ n) y) = ((LieModule.toEnd R L M) x ^ n) ↑y
-- LieSubalgebra.coe_ad
lemma LieSubalgebra.coe_ad (H : LieSubalgebra R L) (x y : H) :
    (ad R H x y : L) = ad R L x y 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) (x y : ↥H)
Goal: ↑(((LieAlgebra.ad R ↥H) x) y) = ((LieAlgebra.ad R L) ↑x) ↑y
-- LieSubalgebra.coe_ad_pow
lemma LieSubalgebra.coe_ad_pow (H : LieSubalgebra R L) (x y : H) (n : ℕ) :
    ((ad R H x ^ n) y : L) = (ad R L x ^ n) y 
Arguments: (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) (x y : ↥H) (n : ℕ)
Goal: ↑(((LieAlgebra.ad R ↥H) x ^ n) y) = ((LieAlgebra.ad R L) ↑x ^ n) ↑y
-- LieModule.toEnd_lie
lemma LieModule.toEnd_lie (x y : L) (z : M) :
    (φ x) ⁅y, z⁆ = ⁅ad R L x y, z⁆ + ⁅y, φ x z⁆ 
Arguments: (R : Type u) {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x y : L) (z : M)
Goal: ((LieModule.toEnd R L M) x) ⁅y, z⁆ = ⁅((LieAlgebra.ad R L) x) y, z⁆ + ⁅y, ((LieModule.toEnd R L M) x) z⁆
-- LieAlgebra.ad_lie
lemma LieAlgebra.ad_lie (x y z : L) :
    (ad R L x) ⁅y, z⁆ = ⁅ad R L x y, z⁆ + ⁅y, ad R L x z⁆ 
Arguments: (R : Type u) {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (x y z : L)
Goal: ((LieAlgebra.ad R L) x) ⁅y, z⁆ = ⁅((LieAlgebra.ad R L) x) y, z⁆ + ⁅y, ((LieAlgebra.ad R L) x) z⁆
-- LieModule.toEnd_pow_lie
lemma LieModule.toEnd_pow_lie (x y : L) (z : M) (n : ℕ) :
    ((φ x) ^ n) ⁅y, z⁆ =
      ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L x) ^ ij.1) y, ((φ x) ^ ij.2) z⁆ 
Arguments: (R : Type u) {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x y : L) (z : M) (n : ℕ)
Goal: ((LieModule.toEnd R L M) x ^ n) ⁅y, z⁆ =
  ∑ ij ∈ Finset.antidiagonal n, n.choose ij.1 • ⁅((LieAlgebra.ad R L) x ^ ij.1) y, ((LieModule.toEnd R L M) x ^ ij.2) z⁆
-- LieAlgebra.ad_pow_lie
lemma LieAlgebra.ad_pow_lie (x y z : L) (n : ℕ) :
    ((ad R L x) ^ n) ⁅y, z⁆ =
      ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L x) ^ ij.1) y, ((ad R L x) ^ ij.2) z⁆ 
Arguments: (R : Type u) {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (x y z : L) (n : ℕ)
Goal: ((LieAlgebra.ad R L) x ^ n) ⁅y, z⁆ =
  ∑ ij ∈ Finset.antidiagonal n, n.choose ij.1 • ⁅((LieAlgebra.ad R L) x ^ ij.1) y, ((LieAlgebra.ad R L) x ^ ij.2) z⁆
-- LieModule.toEnd_pow_comp_lieHom
lemma toEnd_pow_comp_lieHom :
    (toEnd R L M₂ x ^ k) ∘ₗ f = f ∘ₗ toEnd R L M x ^ k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] (f : M →ₗ⁅R,L⁆ M₂) (k : ℕ) (x : L)
Goal: ((LieModule.toEnd R L M₂) x ^ k) ∘ₗ ↑f = ↑f ∘ₗ (LieModule.toEnd R L M) x ^ k
-- LieModule.toEnd_pow_apply_map
lemma toEnd_pow_apply_map (m : M) :
    (toEnd R L M₂ x ^ k) (f m) = f ((toEnd R L M x ^ k) m) 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] (f : M →ₗ⁅R,L⁆ M₂) (k : ℕ) (x : L) (m : M)
Goal: ((LieModule.toEnd R L M₂) x ^ k) (f m) = f (((LieModule.toEnd R L M) x ^ k) m)
-- LieSubmodule.coe_map_toEnd_le
theorem coe_map_toEnd_le :
    (N : Submodule R M).map (LieModule.toEnd R L M x) ≤ N 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {N : LieSubmodule R L M} {x : L}
Goal: Submodule.map ((LieModule.toEnd R L M) x) ↑N ≤ ↑N
-- LieSubmodule.toEnd_comp_subtype_mem
theorem toEnd_comp_subtype_mem (m : M) (hm : m ∈ (N : Submodule R M)) :
    (toEnd R L M x).comp (N : Submodule R M).subtype ⟨m, hm⟩ ∈ (N : Submodule R M) 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (x : L) (m : M) (hm : m ∈ ↑N)
Goal: ((LieModule.toEnd R L M) x ∘ₗ (↑N).subtype) ⟨m, hm⟩ ∈ ↑N
-- LieSubmodule.toEnd_restrict_eq_toEnd
@[simp]
theorem toEnd_restrict_eq_toEnd (h := N.toEnd_comp_subtype_mem x) :
    (toEnd R L M x).restrict h = toEnd R L N x 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (x : L) (h : ∀ (m : M) (hm : m ∈ ↑N), ((LieModule.toEnd R L M) x ∘ₗ (↑N).subtype) ⟨m, hm⟩ ∈ ↑N := ⋯)
Goal: LinearMap.restrict ((LieModule.toEnd R L M) x) h = (LieModule.toEnd R L ↥↑N) x
-- LieSubmodule.mapsTo_pow_toEnd_sub_algebraMap
lemma mapsTo_pow_toEnd_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
    MapsTo ((toEnd R L M x - algebraMap R (Module.End R M) φ) ^ k) N N 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) {φ : R} {k : ℕ} {x : L}
Goal: Set.MapsTo ⇑(((LieModule.toEnd R L M) x - (algebraMap R (Module.End R M)) φ) ^ k) ↑N ↑N
-- LieAlgebra.ad_eq_lmul_left_sub_lmul_right
theorem LieAlgebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [Ring A] [Algebra R A] :
    (ad R A : A → Module.End R A) = LinearMap.mulLeft R - LinearMap.mulRight R 
Arguments: {R : Type u} [CommRing R] (A : Type v) [Ring A] [Algebra R A]
Goal: ⇑(LieAlgebra.ad R A) = LinearMap.mulLeft R - LinearMap.mulRight R
-- LieSubalgebra.ad_comp_incl_eq
theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
    (ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) (x : ↥K)
Goal: (LieAlgebra.ad R L) ↑x ∘ₗ ↑K.incl = ↑K.incl ∘ₗ (LieAlgebra.ad R ↥K) x
-- LinearEquiv.lieConj_apply
@[simp]
theorem lieConj_apply (f : Module.End R M₁) : e.lieConj f = e.conj f 
Arguments: {R : Type u} {M₁ : Type v} {M₂ : Type w} [CommRing R] [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂] (e : M₁ ≃ₗ[R] M₂) (f : Module.End R M₁)
Goal: e.lieConj f = e.conj f
-- LinearEquiv.lieConj_symm
@[simp]
theorem lieConj_symm : e.lieConj.symm = e.symm.lieConj 
Arguments: {R : Type u} {M₁ : Type v} {M₂ : Type w} [CommRing R] [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂] (e : M₁ ≃ₗ[R] M₂)
Goal: e.lieConj.symm = e.symm.lieConj
-- AlgEquiv.toLieEquiv_apply
@[simp]
theorem toLieEquiv_apply (x : A₁) : e.toLieEquiv x = e x 
Arguments: {R : Type u} {A₁ : Type v} {A₂ : Type w} [CommRing R] [Ring A₁] [Ring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ₐ[R] A₂) (x : A₁)
Goal: e.toLieEquiv x = e x