ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieAlgebra.IsKilling.ker_killingForm_eq_bot
@[simp] lemma ker_killingForm_eq_bot :
    LinearMap.ker (killingForm R L) = ⊥ 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsKilling R L]
Goal: LinearMap.ker (killingForm R L) = ⊥
-- LieAlgebra.IsKilling.killingForm_nondegenerate
lemma killingForm_nondegenerate :
    (killingForm R L).Nondegenerate 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsKilling R L]
Goal: (killingForm R L).Nondegenerate
-- LieAlgebra.IsKilling.ideal_eq_bot_of_isLieAbelian
lemma ideal_eq_bot_of_isLieAbelian [IsDomain R] [IsPrincipalIdealRing R]
    (I : LieIdeal R L) [IsLieAbelian I] : I = ⊥ 
Arguments: {R : Type u_1} {L : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [Module.Free R L] [Module.Finite R L] [LieAlgebra.IsKilling R L] [IsDomain R] [IsPrincipalIdealRing R] (I : LieIdeal R L) [IsLieAbelian ↥↑I]
Goal: I = ⊥
-- LieAlgebra.killingForm_of_equiv_apply
@[simp] lemma killingForm_of_equiv_apply (e : L ≃ₗ⁅R⁆ L') (x y : L) :
    killingForm R L' (e x) (e y) = killingForm R L x y 
Arguments: {R : Type u_1} {L : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] {L' : Type u_5} [LieRing L'] [LieAlgebra R L'] (e : L ≃ₗ⁅R⁆ L') (x y : L)
Goal: ((killingForm R L') (e x)) (e y) = ((killingForm R L) x) y
-- LieAlgebra.isKilling_of_equiv
lemma isKilling_of_equiv [IsKilling R L] (e : L ≃ₗ⁅R⁆ L') : IsKilling R L' 
Arguments: {R : Type u_1} {L : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] {L' : Type u_5} [LieRing L'] [LieAlgebra R L'] [LieAlgebra.IsKilling R L] (e : L ≃ₗ⁅R⁆ L')
Goal: LieAlgebra.IsKilling R L'
-- LieModule.polyCharpoly_coeff_rank_ne_zero
lemma polyCharpoly_coeff_rank_ne_zero :
    (polyCharpoly φ b).coeff (rank R L M) ≠ 0 
Arguments: (R : Type u_1) (L : Type u_3) (M : Type u_4) {ι : Type u_5} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] [Fintype ι] [DecidableEq ι] (b : Basis ι R L)
Goal: ((↑(LieModule.toEnd R L M)).polyCharpoly b).coeff (LieModule.rank R L M) ≠ 0
-- LieModule.rank_eq_natTrailingDegree
lemma rank_eq_natTrailingDegree :
    rank R L M = (polyCharpoly φ b).natTrailingDegree 
Arguments: (R : Type u_1) (L : Type u_3) (M : Type u_4) {ι : Type u_5} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] [Fintype ι] [DecidableEq ι] (b : Basis ι R L)
Goal: LieModule.rank R L M = ((↑(LieModule.toEnd R L M)).polyCharpoly b).natTrailingDegree
-- LieModule.rank_le_card
lemma rank_le_card : rank R L M ≤ Fintype.card ιₘ 
Arguments: (R : Type u_1) (L : Type u_3) (M : Type u_4) {ιₘ : Type u_6} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] [Fintype ιₘ] (bₘ : Basis ιₘ R M)
Goal: LieModule.rank R L M ≤ Fintype.card ιₘ
-- LieModule.rank_le_finrank
lemma rank_le_finrank : rank R L M ≤ finrank R M 
Arguments: (R : Type u_1) (L : Type u_3) (M : Type u_4) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M]
Goal: LieModule.rank R L M ≤ FiniteDimensional.finrank R M
-- LieModule.rank_le_natTrailingDegree_charpoly_ad
lemma rank_le_natTrailingDegree_charpoly_ad :
    rank R L M ≤ (toEnd R L M x).charpoly.natTrailingDegree 
Arguments: (R : Type u_1) {L : Type u_3} (M : Type u_4) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] (x : L)
Goal: LieModule.rank R L M ≤ (LinearMap.charpoly ((LieModule.toEnd R L M) x)).natTrailingDegree
-- LieModule.isRegular_def
lemma isRegular_def :
    IsRegular R M x ↔ (toEnd R L M x).charpoly.coeff (rank R L M) ≠ 0 
Arguments: (R : Type u_1) {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] (x : L)
Goal: LieModule.IsRegular R M x ↔ (LinearMap.charpoly ((LieModule.toEnd R L M) x)).coeff (LieModule.rank R L M) ≠ 0
-- LieModule.isRegular_iff_coeff_polyCharpoly_rank_ne_zero
lemma isRegular_iff_coeff_polyCharpoly_rank_ne_zero :
    IsRegular R M x ↔
    MvPolynomial.eval (b.repr x)
      ((polyCharpoly φ b).coeff (rank R L M)) ≠ 0 
Arguments: (R : Type u_1) {L : Type u_3} (M : Type u_4) {ι : Type u_5} [CommRing R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] [Fintype ι] [DecidableEq ι] (b : Basis ι R L) (x : L)
Goal: LieModule.IsRegular R M x ↔
  (MvPolynomial.eval ⇑(b.repr x)) (((↑(LieModule.toEnd R L M)).polyCharpoly b).coeff (LieModule.rank R L M)) ≠ 0
-- LieModule.isRegular_iff_natTrailingDegree_charpoly_eq_rank
lemma isRegular_iff_natTrailingDegree_charpoly_eq_rank :
    IsRegular R M x ↔ (toEnd R L M x).charpoly.natTrailingDegree = rank R L M 
Arguments: (R : Type u_1) {L : Type u_3} (M : Type u_4) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] (x : L)
Goal: LieModule.IsRegular R M x ↔ (LinearMap.charpoly ((LieModule.toEnd R L M) x)).natTrailingDegree = LieModule.rank R L M
-- LieModule.exists_isRegular_of_finrank_le_card
lemma exists_isRegular_of_finrank_le_card (h : finrank R M ≤ #R) :
    ∃ x : L, IsRegular R M x 
Arguments: (R : Type u_1) (L : Type u_3) (M : Type u_4) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] [IsDomain R] (h : ↑(FiniteDimensional.finrank R M) ≤ Cardinal.mk R)
Goal: ∃ x, LieModule.IsRegular R M x
-- LieModule.exists_isRegular
lemma exists_isRegular [Infinite R] : ∃ x : L, IsRegular R M x 
Arguments: (R : Type u_1) (L : Type u_3) (M : Type u_4) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] [IsDomain R] [Infinite R]
Goal: ∃ x, LieModule.IsRegular R M x
-- LieAlgebra.polyCharpoly_coeff_rank_ne_zero
lemma polyCharpoly_coeff_rank_ne_zero :
    (polyCharpoly (ad R L).toLinearMap b).coeff (rank R L) ≠ 0 
Arguments: (R : Type u_1) (L : Type u_3) {ι : Type u_5} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [Fintype ι] [DecidableEq ι] (b : Basis ι R L)
Goal: ((↑(LieAlgebra.ad R L)).polyCharpoly b).coeff (LieAlgebra.rank R L) ≠ 0
-- LieAlgebra.rank_eq_natTrailingDegree
lemma rank_eq_natTrailingDegree :
    rank R L = (polyCharpoly (ad R L).toLinearMap b).natTrailingDegree 
Arguments: (R : Type u_1) (L : Type u_3) {ι : Type u_5} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [Fintype ι] [DecidableEq ι] (b : Basis ι R L)
Goal: LieAlgebra.rank R L = ((↑(LieAlgebra.ad R L)).polyCharpoly b).natTrailingDegree
-- LieAlgebra.rank_le_card
lemma rank_le_card : rank R L ≤ Fintype.card ι 
Arguments: (R : Type u_1) (L : Type u_3) {ι : Type u_5} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [Fintype ι] (b : Basis ι R L)
Goal: LieAlgebra.rank R L ≤ Fintype.card ι
-- LieAlgebra.rank_le_finrank
lemma rank_le_finrank : rank R L ≤ finrank R L 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L]
Goal: LieAlgebra.rank R L ≤ FiniteDimensional.finrank R L
-- LieAlgebra.rank_le_natTrailingDegree_charpoly_ad
lemma rank_le_natTrailingDegree_charpoly_ad :
    rank R L ≤ (ad R L x).charpoly.natTrailingDegree 
Arguments: (R : Type u_1) {L : Type u_3} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] (x : L)
Goal: LieAlgebra.rank R L ≤ (LinearMap.charpoly ((LieAlgebra.ad R L) x)).natTrailingDegree
-- LieAlgebra.isRegular_def
lemma isRegular_def :
    IsRegular R x ↔ (Polynomial.coeff (ad R L x).charpoly (rank R L) ≠ 0) 
Arguments: (R : Type u_1) {L : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] (x : L)
Goal: LieAlgebra.IsRegular R x ↔ (LinearMap.charpoly ((LieAlgebra.ad R L) x)).coeff (LieAlgebra.rank R L) ≠ 0
-- LieAlgebra.isRegular_iff_coeff_polyCharpoly_rank_ne_zero
lemma isRegular_iff_coeff_polyCharpoly_rank_ne_zero :
    IsRegular R x ↔
    MvPolynomial.eval (b.repr x)
      ((polyCharpoly (ad R L).toLinearMap b).coeff (rank R L)) ≠ 0 
Arguments: (R : Type u_1) {L : Type u_3} {ι : Type u_5} [CommRing R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [Fintype ι] [DecidableEq ι] (b : Basis ι R L) (x : L)
Goal: LieAlgebra.IsRegular R x ↔
  (MvPolynomial.eval ⇑(b.repr x)) (((↑(LieAlgebra.ad R L)).polyCharpoly b).coeff (LieAlgebra.rank R L)) ≠ 0
-- LieAlgebra.isRegular_iff_natTrailingDegree_charpoly_eq_rank
lemma isRegular_iff_natTrailingDegree_charpoly_eq_rank :
    IsRegular R x ↔ (ad R L x).charpoly.natTrailingDegree = rank R L 
Arguments: (R : Type u_1) {L : Type u_3} [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] (x : L)
Goal: LieAlgebra.IsRegular R x ↔ (LinearMap.charpoly ((LieAlgebra.ad R L) x)).natTrailingDegree = LieAlgebra.rank R L
-- LieAlgebra.exists_isRegular_of_finrank_le_card
lemma exists_isRegular_of_finrank_le_card (h : finrank R L ≤ #R) :
    ∃ x : L, IsRegular R x 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [IsDomain R] (h : ↑(FiniteDimensional.finrank R L) ≤ Cardinal.mk R)
Goal: ∃ x, LieAlgebra.IsRegular R x
-- LieAlgebra.exists_isRegular
lemma exists_isRegular [Infinite R] : ∃ x : L, IsRegular R x 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [Nontrivial R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [IsDomain R] [Infinite R]
Goal: ∃ x, LieAlgebra.IsRegular R x
-- LieAlgebra.finrank_engel
lemma finrank_engel (x : L) :
    finrank K (engel K x) = (ad K L x).charpoly.natTrailingDegree 
Arguments: (K : Type u_7) {L : Type u_8} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (x : L)
Goal: FiniteDimensional.finrank K ↥(LieSubalgebra.engel K x) = (LinearMap.charpoly ((LieAlgebra.ad K L) x)).natTrailingDegree
-- LieAlgebra.rank_le_finrank_engel
lemma rank_le_finrank_engel (x : L) :
    rank K L ≤ finrank K (engel K x) 
Arguments: (K : Type u_7) {L : Type u_8} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (x : L)
Goal: LieAlgebra.rank K L ≤ FiniteDimensional.finrank K ↥(LieSubalgebra.engel K x)
-- LieAlgebra.isRegular_iff_finrank_engel_eq_rank
lemma isRegular_iff_finrank_engel_eq_rank (x : L) :
    IsRegular K x ↔ finrank K (engel K x) = rank K L 
Arguments: (K : Type u_7) {L : Type u_8} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (x : L)
Goal: LieAlgebra.IsRegular K x ↔ FiniteDimensional.finrank K ↥(LieSubalgebra.engel K x) = LieAlgebra.rank K L
-- LieSubalgebra.engel_carrier
@[simps!]
def engel (x : L) : LieSubalgebra R L 
Arguments: (R : Type u_1) {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L)
Goal: ↑(LieSubalgebra.engel R x) = ⋂ s ∈ fun x_1 => ∀ (a : ℕ), LinearMap.ker ((LieAlgebra.ad R L) x ^ a) ≤ x_1, ↑s
-- LieSubalgebra.mem_engel_iff
lemma mem_engel_iff (x y : L) :
    y ∈ engel R x ↔ ∃ n : ℕ, ((ad R L x) ^ n) y = 0 
Arguments: (R : Type u_1) {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (x y : L)
Goal: y ∈ LieSubalgebra.engel R x ↔ ∃ n, ((LieAlgebra.ad R L) x ^ n) y = 0
-- LieSubalgebra.self_mem_engel
lemma self_mem_engel (x : L) : x ∈ engel R x 
Arguments: (R : Type u_1) {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L)
Goal: x ∈ LieSubalgebra.engel R x
-- LieSubalgebra.engel_zero
@[simp]
lemma engel_zero : engel R (0 : L) = ⊤ 
Arguments: (R : Type u_1) {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: LieSubalgebra.engel R 0 = ⊤
-- LieSubalgebra.normalizer_engel
@[simp]
lemma normalizer_engel (x : L) : normalizer (engel R x) = engel R x 
Arguments: (R : Type u_1) {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L)
Goal: (LieSubalgebra.engel R x).normalizer = LieSubalgebra.engel R x
-- LieSubalgebra.normalizer_eq_self_of_engel_le
lemma normalizer_eq_self_of_engel_le [IsArtinian R L]
    (H : LieSubalgebra R L) (x : L) (h : engel R x ≤ H) :
    normalizer H = H 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [IsArtinian R L] (H : LieSubalgebra R L) (x : L) (h : LieSubalgebra.engel R x ≤ H)
Goal: H.normalizer = H
-- LieSubalgebra.isNilpotent_of_forall_le_engel
lemma isNilpotent_of_forall_le_engel [IsNoetherian R L]
    (H : LieSubalgebra R L) (h : ∀ x ∈ H, H ≤ engel R x) :
    LieAlgebra.IsNilpotent R H 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [IsNoetherian R L] (H : LieSubalgebra R L) (h : ∀ x ∈ H, H ≤ LieSubalgebra.engel R x)
Goal: LieAlgebra.IsNilpotent R ↥H
-- LieHom.toNonUnitalAlgHom_apply
@[simps]
def toNonUnitalAlgHom (f : L →ₗ⁅R⁆ L₂) : CommutatorRing L →ₙₐ[R] CommutatorRing L₂ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (a : L)
Goal: f.toNonUnitalAlgHom a = f a
-- LieHom.toNonUnitalAlgHom_toFun
@[simps]
def toNonUnitalAlgHom (f : L →ₗ⁅R⁆ L₂) : CommutatorRing L →ₙₐ[R] CommutatorRing L₂ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (a : L)
Goal: f.toNonUnitalAlgHom a = f a
-- LieHom.toNonUnitalAlgHom_injective
theorem toNonUnitalAlgHom_injective :
    Function.Injective (toNonUnitalAlgHom : _ → CommutatorRing L →ₙₐ[R] CommutatorRing L₂) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂]
Goal: Function.Injective LieHom.toNonUnitalAlgHom
-- LieSubmodule.Quotient.is_quotient_mk
theorem is_quotient_mk (m : M) : Quotient.mk'' m = (mk m : M ⧸ N) 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} (m : M)
Goal: Quotient.mk'' m = LieSubmodule.Quotient.mk m
-- LieSubmodule.Quotient.mk_bracket
@[simp]
theorem mk_bracket (x y : L) : mk ⁅x, y⁆ = ⁅(mk x : L ⧸ I), (mk y : L ⧸ I)⁆ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {I : LieIdeal R L} (x y : L)
Goal: LieSubmodule.Quotient.mk ⁅x, y⁆ = ⁅LieSubmodule.Quotient.mk x, LieSubmodule.Quotient.mk y⁆
-- LieSubmodule.Quotient.mk'_apply
@[simps]
def mk' : M →ₗ⁅R,L⁆ M ⧸ 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)
Goal: ∀ (a : M), (LieSubmodule.Quotient.mk' N) a = LieSubmodule.Quotient.mk a
-- LieSubmodule.Quotient.surjective_mk'
@[simp]
theorem surjective_mk' : Function.Surjective (mk' 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)
Goal: Function.Surjective ⇑(LieSubmodule.Quotient.mk' N)
-- LieSubmodule.Quotient.range_mk'
@[simp]
theorem range_mk' : LieModuleHom.range (mk' 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)
Goal: (LieSubmodule.Quotient.mk' N).range = ⊤
-- LieSubmodule.Quotient.mk_eq_zero
theorem mk_eq_zero {m : M} : mk' N m = 0 ↔ m ∈ 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) {m : M}
Goal: (LieSubmodule.Quotient.mk' N) m = 0 ↔ m ∈ N
-- LieSubmodule.Quotient.mk_eq_zero'
@[simp]
theorem mk_eq_zero' {m : M} : mk (N := N) m = 0 ↔ m ∈ N 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) {m : M}
Goal: LieSubmodule.Quotient.mk m = 0 ↔ m ∈ N
-- LieSubmodule.Quotient.mk'_ker
@[simp]
theorem mk'_ker : (mk' N).ker = 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)
Goal: (LieSubmodule.Quotient.mk' N).ker = N
-- LieSubmodule.Quotient.map_mk'_eq_bot_le
@[simp]
theorem map_mk'_eq_bot_le : map (mk' N) N' = ⊥ ↔ 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 N' : LieSubmodule R L M)
Goal: LieSubmodule.map (LieSubmodule.Quotient.mk' N) N' = ⊥ ↔ N' ≤ N
-- LieSubmodule.Quotient.lieModuleHom_ext
@[ext]
theorem lieModuleHom_ext ⦃f g : M ⧸ N →ₗ⁅R,L⁆ M⦄ (h : f.comp (mk' N) = g.comp (mk' N)) : f = g 
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) ⦃f g : M ⧸ N →ₗ⁅R,L⁆ M⦄ (h : f.comp (LieSubmodule.Quotient.mk' N) = g.comp (LieSubmodule.Quotient.mk' N))
Goal: f = g
-- LieSubmodule.Quotient.toEnd_comp_mk'
lemma toEnd_comp_mk' (x : L) :
    LieModule.toEnd R L (M ⧸ N) x ∘ₗ mk' N = mk' N ∘ₗ 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] (N : LieSubmodule R L M) (x : L)
Goal: (LieModule.toEnd R L (M ⧸ N)) x ∘ₗ ↑(LieSubmodule.Quotient.mk' N) =
  ↑(LieSubmodule.Quotient.mk' N) ∘ₗ (LieModule.toEnd R L M) x
-- LieHom.quotKerEquivRange_toFun
@[simps]
noncomputable def quotKerEquivRange : (L ⧸ f.ker) ≃ₗ⁅R⁆ f.range 
Arguments: {R : Type u_1} {L : Type u_2} {L' : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R⁆ L') (a : L ⧸ LinearMap.ker ↑f)
Goal: f.quotKerEquivRange a = (↑f).quotKerEquivRange a
-- LieHom.quotKerEquivRange_invFun
@[simps]
noncomputable def quotKerEquivRange : (L ⧸ f.ker) ≃ₗ⁅R⁆ f.range 
Arguments: {R : Type u_1} {L : Type u_2} {L' : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R⁆ L')
Goal: ∀ (a : ↥(LinearMap.range ↑f)), f.quotKerEquivRange.invFun a = (↑f).quotKerEquivRange.invFun a
-- LieModule.IsTrivial.trivial
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
  trivial : ∀ (x : L) (m : M), ⁅x, m⁆ = 0
Arguments: {L : Type v} {M : Type w} [Bracket L M] [Zero M] [self : LieModule.IsTrivial L M] (x : L) (m : M)
Goal: ⁅x, m⁆ = 0
-- trivial_lie_zero
@[simp]
theorem trivial_lie_zero (L : Type v) (M : Type w) [Bracket L M] [Zero M] [LieModule.IsTrivial L M]
    (x : L) (m : M) : ⁅x, m⁆ = 0 
Arguments: (L : Type v) (M : Type w) [Bracket L M] [Zero M] [LieModule.IsTrivial L M] (x : L) (m : M)
Goal: ⁅x, m⁆ = 0
-- Function.Injective.isLieAbelian
theorem Function.Injective.isLieAbelian {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R]
    [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] {f : L₁ →ₗ⁅R⁆ L₂}
    (h₁ : Function.Injective f) (_ : IsLieAbelian L₂) : IsLieAbelian L₁ 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] {f : L₁ →ₗ⁅R⁆ L₂} (h₁ : Function.Injective ⇑f)
Goal: IsLieAbelian L₂ → IsLieAbelian L₁
-- Function.Surjective.isLieAbelian
theorem Function.Surjective.isLieAbelian {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R]
    [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] {f : L₁ →ₗ⁅R⁆ L₂}
    (h₁ : Function.Surjective f) (h₂ : IsLieAbelian L₁) : IsLieAbelian L₂ 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] {f : L₁ →ₗ⁅R⁆ L₂} (h₁ : Function.Surjective ⇑f) (h₂ : IsLieAbelian L₁)
Goal: IsLieAbelian L₂
-- lie_abelian_iff_equiv_lie_abelian
theorem lie_abelian_iff_equiv_lie_abelian {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R]
    [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) :
    IsLieAbelian L₁ ↔ IsLieAbelian L₂ 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂)
Goal: IsLieAbelian L₁ ↔ IsLieAbelian L₂
-- commutative_ring_iff_abelian_lie_ring
theorem commutative_ring_iff_abelian_lie_ring {A : Type v} [Ring A] :
    Std.Commutative (α := A) (· * ·) ↔ IsLieAbelian A 
Arguments: {A : Type v} [Ring A]
Goal: (Std.Commutative fun x x_1 => x * x_1) ↔ IsLieAbelian A
-- LieModule.mem_ker
@[simp]
protected theorem mem_ker (x : L) : x ∈ LieModule.ker R L M ↔ ∀ m : M, ⁅x, m⁆ = 0 
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)
Goal: x ∈ LieModule.ker R L M ↔ ∀ (m : M), ⁅x, m⁆ = 0
-- LieModule.mem_maxTrivSubmodule
@[simp]
theorem mem_maxTrivSubmodule (m : M) : m ∈ maxTrivSubmodule R L M ↔ ∀ x : L, ⁅x, m⁆ = 0 
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] (m : M)
Goal: m ∈ LieModule.maxTrivSubmodule R L M ↔ ∀ (x : L), ⁅x, m⁆ = 0
-- LieModule.ideal_oper_maxTrivSubmodule_eq_bot
@[simp]
theorem ideal_oper_maxTrivSubmodule_eq_bot (I : LieIdeal R L) :
    ⁅I, maxTrivSubmodule R L 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] (I : LieIdeal R L)
Goal: ⁅I, LieModule.maxTrivSubmodule R L M⁆ = ⊥
-- LieModule.le_max_triv_iff_bracket_eq_bot
theorem le_max_triv_iff_bracket_eq_bot {N : LieSubmodule R L M} :
    N ≤ maxTrivSubmodule R L M ↔ ⁅(⊤ : LieIdeal R L), 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}
Goal: N ≤ LieModule.maxTrivSubmodule R L M ↔ ⁅⊤, N⁆ = ⊥
-- LieModule.trivial_iff_le_maximal_trivial
theorem trivial_iff_le_maximal_trivial (N : LieSubmodule R L M) :
    IsTrivial L N ↔ N ≤ maxTrivSubmodule R L 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)
Goal: LieModule.IsTrivial L ↥↑N ↔ N ≤ LieModule.maxTrivSubmodule R L M
-- LieModule.isTrivial_iff_max_triv_eq_top
theorem isTrivial_iff_max_triv_eq_top : IsTrivial L M ↔ maxTrivSubmodule R L 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]
Goal: LieModule.IsTrivial L M ↔ LieModule.maxTrivSubmodule R L M = ⊤
-- LieModule.coe_maxTrivHom_apply
@[norm_cast, simp]
theorem coe_maxTrivHom_apply (f : M →ₗ⁅R,L⁆ N) (m : maxTrivSubmodule R L M) :
    (maxTrivHom f m : N) = f m 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [AddCommGroup N] [Module R N] [LieRingModule L N] [LieModule R L N] (f : M →ₗ⁅R,L⁆ N) (m : ↥(LieModule.maxTrivSubmodule R L M))
Goal: ↑((LieModule.maxTrivHom f) m) = f ↑m