ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsSMulRegular.of_injective
lemma of_injective {N F} [SMul R N] [FunLike F M N] [MulActionHomClass F R M N]
    (f : F) {r : R} (h1 : Function.Injective f) (h2 : IsSMulRegular N r) :
    IsSMulRegular M r 
Arguments: {R : Type u_1} {M : Type u_3} [SMul R M] {N : Type u_4} {F : Type u_5} [SMul R N] [FunLike F M N] [MulActionHomClass F R M N] (f : F) {r : R} (h1 : Function.Injective ⇑f) (h2 : IsSMulRegular N r)
Goal: IsSMulRegular M r
-- IsSMulRegular.one
@[simp]
theorem one : IsSMulRegular M (1 : R) 
Arguments: {R : Type u_1} (M : Type u_3) [Monoid R] [MulAction R M]
Goal: IsSMulRegular M 1
-- IsSMulRegular.of_mul_eq_one
theorem of_mul_eq_one (h : a * b = 1) : IsSMulRegular M b 
Arguments: {R : Type u_1} {M : Type u_3} {a b : R} [Monoid R] [MulAction R M] (h : a * b = 1)
Goal: IsSMulRegular M b
-- IsSMulRegular.pow
theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) 
Arguments: {R : Type u_1} {M : Type u_3} {a : R} [Monoid R] [MulAction R M] (n : ℕ) (ra : IsSMulRegular M a)
Goal: IsSMulRegular M (a ^ n)
-- IsSMulRegular.pow_iff
theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a 
Arguments: {R : Type u_1} {M : Type u_3} {a : R} [Monoid R] [MulAction R M] {n : ℕ} (n0 : 0 < n)
Goal: IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a
-- IsSMulRegular.of_smul_eq_one
theorem of_smul_eq_one (h : a • s = 1) : IsSMulRegular M s 
Arguments: {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} {s : S} [Monoid S] [SMul R M] [SMul R S] [MulAction S M] [IsScalarTower R S M] (h : a • s = 1)
Goal: IsSMulRegular M s
-- IsSMulRegular.subsingleton
protected theorem subsingleton (h : IsSMulRegular M (0 : R)) : Subsingleton M 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] (h : IsSMulRegular M 0)
Goal: Subsingleton M
-- IsSMulRegular.zero_iff_subsingleton
theorem zero_iff_subsingleton : IsSMulRegular M (0 : R) ↔ Subsingleton M 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M]
Goal: IsSMulRegular M 0 ↔ Subsingleton M
-- IsSMulRegular.not_zero_iff
theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M]
Goal: ¬IsSMulRegular M 0 ↔ Nontrivial M
-- IsSMulRegular.zero
theorem zero [sM : Subsingleton M] : IsSMulRegular M (0 : R) 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [sM : Subsingleton M]
Goal: IsSMulRegular M 0
-- IsSMulRegular.not_zero
theorem not_zero [nM : Nontrivial M] : ¬IsSMulRegular M (0 : R) 
Arguments: {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [nM : Nontrivial M]
Goal: ¬IsSMulRegular M 0
-- IsSMulRegular.mul_iff
theorem mul_iff : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b 
Arguments: {R : Type u_1} {M : Type u_3} {a b : R} [CommSemigroup R] [SMul R M] [IsScalarTower R R M]
Goal: IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b
-- isSMulRegular_of_group
theorem isSMulRegular_of_group [MulAction G R] (g : G) : IsSMulRegular R g 
Arguments: {R : Type u_1} {G : Type u_4} [Group G] [MulAction G R] (g : G)
Goal: IsSMulRegular R g
-- Units.isSMulRegular
theorem Units.isSMulRegular (a : Rˣ) : IsSMulRegular M (a : R) 
Arguments: {R : Type u_1} (M : Type u_3) [Monoid R] [MulAction R M] (a : Rˣ)
Goal: IsSMulRegular M ↑a
-- IsUnit.isSMulRegular
theorem IsUnit.isSMulRegular (ua : IsUnit a) : IsSMulRegular M a 
Arguments: {R : Type u_1} (M : Type u_3) {a : R} [Monoid R] [MulAction R M] (ua : IsUnit a)
Goal: IsSMulRegular M a
-- IsSMulRegular.eq_zero_of_smul_eq_zero
protected
lemma IsSMulRegular.eq_zero_of_smul_eq_zero [Zero M] [SMulZeroClass R M]
    {r : R} {x : M} (h1 : IsSMulRegular M r) (h2 : r • x = 0) : x = 0 
Arguments: {R : Type u_1} {M : Type u_3} [Zero M] [SMulZeroClass R M] {r : R} {x : M} (h1 : IsSMulRegular M r) (h2 : r • x = 0)
Goal: x = 0
-- Equiv.isSMulRegular_congr
lemma Equiv.isSMulRegular_congr {R S M M'} [SMul R M] [SMul S M'] {e : M ≃ M'}
    {r : R} {s : S} (h : ∀ x, e (r • x) = s • e x) :
    IsSMulRegular M r ↔ IsSMulRegular M' s 
Arguments: {R : Type u_4} {S : Type u_5} {M : Type u_6} {M' : Type u_7} [SMul R M] [SMul S M'] {e : M ≃ M'} {r : R} {s : S} (h : ∀ (x : M), e (r • x) = s • e x)
Goal: IsSMulRegular M r ↔ IsSMulRegular M' s
-- LieRing.leibniz_lie
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
    protected add_lie : ∀ x y z : L, ⁅x + y, z⁆ = ⁅x, z⁆ + ⁅y, z⁆
    protected lie_add : ∀ x y z : L, ⁅x, y + z⁆ = ⁅x, y⁆ + ⁅x, z⁆
    protected lie_self : ∀ x : L, ⁅x, x⁆ = 0
    protected leibniz_lie : ∀ x y z : L, ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆
Arguments: {L : Type v} [self : LieRing L] (x y z : L)
Goal: ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆
-- LieRing.lie_self
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
    protected add_lie : ∀ x y z : L, ⁅x + y, z⁆ = ⁅x, z⁆ + ⁅y, z⁆
    protected lie_add : ∀ x y z : L, ⁅x, y + z⁆ = ⁅x, y⁆ + ⁅x, z⁆
    protected lie_self : ∀ x : L, ⁅x, x⁆ = 0
    protected leibniz_lie : ∀ x y z : L, ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆
Arguments: {L : Type v} [self : LieRing L] (x : L)
Goal: ⁅x, x⁆ = 0
-- LieRing.add_lie
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
    protected add_lie : ∀ x y z : L, ⁅x + y, z⁆ = ⁅x, z⁆ + ⁅y, z⁆
    protected lie_add : ∀ x y z : L, ⁅x, y + z⁆ = ⁅x, y⁆ + ⁅x, z⁆
    protected lie_self : ∀ x : L, ⁅x, x⁆ = 0
    protected leibniz_lie : ∀ x y z : L, ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆
Arguments: {L : Type v} [self : LieRing L] (x y z : L)
Goal: ⁅x + y, z⁆ = ⁅x, z⁆ + ⁅y, z⁆
-- LieRing.lie_add
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
    protected add_lie : ∀ x y z : L, ⁅x + y, z⁆ = ⁅x, z⁆ + ⁅y, z⁆
    protected lie_add : ∀ x y z : L, ⁅x, y + z⁆ = ⁅x, y⁆ + ⁅x, z⁆
    protected lie_self : ∀ x : L, ⁅x, x⁆ = 0
    protected leibniz_lie : ∀ x y z : L, ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆
Arguments: {L : Type v} [self : LieRing L] (x y z : L)
Goal: ⁅x, y + z⁆ = ⁅x, y⁆ + ⁅x, z⁆
-- LieAlgebra.lie_smul
class LieAlgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] extends Module R L where
    protected lie_smul : ∀ (t : R) (x y : L), ⁅x, t • y⁆ = t • ⁅x, y⁆
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [self : LieAlgebra R L] (t : R) (x y : L)
Goal: ⁅x, t • y⁆ = t • ⁅x, y⁆
-- LieRingModule.add_lie
class LieRingModule (L : Type v) (M : Type w) [LieRing L] [AddCommGroup M] extends Bracket L M where
    protected add_lie : ∀ (x y : L) (m : M), ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆
    protected lie_add : ∀ (x : L) (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆
    protected leibniz_lie : ∀ (x y : L) (m : M), ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [self : LieRingModule L M] (x y : L) (m : M)
Goal: ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆
-- LieRingModule.leibniz_lie
class LieRingModule (L : Type v) (M : Type w) [LieRing L] [AddCommGroup M] extends Bracket L M where
    protected add_lie : ∀ (x y : L) (m : M), ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆
    protected lie_add : ∀ (x : L) (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆
    protected leibniz_lie : ∀ (x y : L) (m : M), ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [self : LieRingModule L M] (x y : L) (m : M)
Goal: ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆
-- LieRingModule.lie_add
class LieRingModule (L : Type v) (M : Type w) [LieRing L] [AddCommGroup M] extends Bracket L M where
    protected add_lie : ∀ (x y : L) (m : M), ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆
    protected lie_add : ∀ (x : L) (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆
    protected leibniz_lie : ∀ (x y : L) (m : M), ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [self : LieRingModule L M] (x : L) (m n : M)
Goal: ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆
-- LieModule.lie_smul
class LieModule (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L]
  [AddCommGroup M] [Module R M] [LieRingModule L M] : Prop where
    protected smul_lie : ∀ (t : R) (x : L) (m : M), ⁅t • x, m⁆ = t • ⁅x, m⁆
    protected lie_smul : ∀ (t : R) (x : L) (m : M), ⁅x, t • m⁆ = t • ⁅x, 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] [self : LieModule R L M] (t : R) (x : L) (m : M)
Goal: ⁅x, t • m⁆ = t • ⁅x, m⁆
-- LieModule.smul_lie
class LieModule (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L]
  [AddCommGroup M] [Module R M] [LieRingModule L M] : Prop where
    protected smul_lie : ∀ (t : R) (x : L) (m : M), ⁅t • x, m⁆ = t • ⁅x, m⁆
    protected lie_smul : ∀ (t : R) (x : L) (m : M), ⁅x, t • m⁆ = t • ⁅x, 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] [self : LieModule R L M] (t : R) (x : L) (m : M)
Goal: ⁅t • x, m⁆ = t • ⁅x, m⁆
-- add_lie
@[simp]
theorem add_lie : ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x y : L) (m : M)
Goal: ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆
-- lie_add
@[simp]
theorem lie_add : ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m n : M)
Goal: ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆
-- smul_lie
@[simp]
theorem smul_lie : ⁅t • x, m⁆ = t • ⁅x, 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] (t : R) (x : L) (m : M)
Goal: ⁅t • x, m⁆ = t • ⁅x, m⁆
-- lie_smul
@[simp]
theorem lie_smul : ⁅x, t • m⁆ = t • ⁅x, 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] (t : R) (x : L) (m : M)
Goal: ⁅x, t • m⁆ = t • ⁅x, m⁆
-- leibniz_lie
theorem leibniz_lie : ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x y : L) (m : M)
Goal: ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆
-- lie_zero
@[simp]
theorem lie_zero : ⁅x, 0⁆ = (0 : M) 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L)
Goal: ⁅x, 0⁆ = 0
-- zero_lie
@[simp]
theorem zero_lie : ⁅(0 : L), m⁆ = 0 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (m : M)
Goal: ⁅0, m⁆ = 0
-- lie_self
@[simp]
theorem lie_self : ⁅x, x⁆ = 0 
Arguments: {L : Type v} [LieRing L] (x : L)
Goal: ⁅x, x⁆ = 0
-- lie_skew
@[simp]
theorem lie_skew : -⁅y, x⁆ = ⁅x, y⁆ 
Arguments: {L : Type v} [LieRing L] (x y : L)
Goal: -⁅y, x⁆ = ⁅x, y⁆
-- neg_lie
@[simp]
theorem neg_lie : ⁅-x, m⁆ = -⁅x, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M)
Goal: ⁅-x, m⁆ = -⁅x, m⁆
-- lie_neg
@[simp]
theorem lie_neg : ⁅x, -m⁆ = -⁅x, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M)
Goal: ⁅x, -m⁆ = -⁅x, m⁆
-- sub_lie
@[simp]
theorem sub_lie : ⁅x - y, m⁆ = ⁅x, m⁆ - ⁅y, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x y : L) (m : M)
Goal: ⁅x - y, m⁆ = ⁅x, m⁆ - ⁅y, m⁆
-- lie_sub
@[simp]
theorem lie_sub : ⁅x, m - n⁆ = ⁅x, m⁆ - ⁅x, n⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m n : M)
Goal: ⁅x, m - n⁆ = ⁅x, m⁆ - ⁅x, n⁆
-- nsmul_lie
@[simp]
theorem nsmul_lie (n : ℕ) : ⁅n • x, m⁆ = n • ⁅x, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (n : ℕ)
Goal: ⁅n • x, m⁆ = n • ⁅x, m⁆
-- lie_nsmul
@[simp]
theorem lie_nsmul (n : ℕ) : ⁅x, n • m⁆ = n • ⁅x, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (n : ℕ)
Goal: ⁅x, n • m⁆ = n • ⁅x, m⁆
-- zsmul_lie
@[simp]
theorem zsmul_lie (a : ℤ) : ⁅a • x, m⁆ = a • ⁅x, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (a : ℤ)
Goal: ⁅a • x, m⁆ = a • ⁅x, m⁆
-- lie_zsmul
@[simp]
theorem lie_zsmul (a : ℤ) : ⁅x, a • m⁆ = a • ⁅x, m⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x : L) (m : M) (a : ℤ)
Goal: ⁅x, a • m⁆ = a • ⁅x, m⁆
-- lie_lie
@[simp]
lemma lie_lie : ⁅⁅x, y⁆, m⁆ = ⁅x, ⁅y, m⁆⁆ - ⁅y, ⁅x, m⁆⁆ 
Arguments: {L : Type v} {M : Type w} [LieRing L] [AddCommGroup M] [LieRingModule L M] (x y : L) (m : M)
Goal: ⁅⁅x, y⁆, m⁆ = ⁅x, ⁅y, m⁆⁆ - ⁅y, ⁅x, m⁆⁆
-- lie_jacobi
theorem lie_jacobi : ⁅x, ⁅y, z⁆⁆ + ⁅y, ⁅z, x⁆⁆ + ⁅z, ⁅x, y⁆⁆ = 0 
Arguments: {L : Type v} [LieRing L] (x y z : L)
Goal: ⁅x, ⁅y, z⁆⁆ + ⁅y, ⁅z, x⁆⁆ + ⁅z, ⁅x, y⁆⁆ = 0
-- LieHom.lie_apply
@[simp]
theorem LieHom.lie_apply (f : M →ₗ[R] N) (x : L) (m : M) : ⁅x, f⁆ m = ⁅x, f m⁆ - f ⁅x, 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] N) (x : L) (m : M)
Goal: ⁅x, f⁆ m = ⁅x, f m⁆ - f ⁅x, m⁆
-- Module.Dual.lie_apply
@[simp] lemma Module.Dual.lie_apply (f : M →ₗ[R] R) : ⁅x, f⁆ m = - f ⁅x, 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) (f : M →ₗ[R] R)
Goal: ⁅x, f⁆ m = -f ⁅x, m⁆
-- LieHom.map_lie'
structure LieHom (R L L' : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
  [LieRing L'] [LieAlgebra R L'] extends L →ₗ[R] L' where
    map_lie' : ∀ {x y : L}, toFun ⁅x, y⁆ = ⁅toFun x, toFun y⁆
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'] (self : L →ₗ⁅R⁆ L') {x y : L}
Goal: (↑self).toFun ⁅x, y⁆ = ⁅(↑self).toFun x, (↑self).toFun y⁆
-- LieHom.coe_toLinearMap
@[simp, norm_cast]
theorem coe_toLinearMap (f : L₁ →ₗ⁅R⁆ L₂) : ⇑(f : L₁ →ₗ[R] L₂) = f 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂)
Goal: ⇑↑f = ⇑f
-- LieHom.toFun_eq_coe
@[simp]
theorem toFun_eq_coe (f : L₁ →ₗ⁅R⁆ L₂) : f.toFun = ⇑f 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂)
Goal: (↑f).toFun = ⇑f
-- LieHom.map_smul
@[simp]
theorem map_smul (f : L₁ →ₗ⁅R⁆ L₂) (c : R) (x : L₁) : f (c • x) = c • f x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (c : R) (x : L₁)
Goal: f (c • x) = c • f x
-- LieHom.map_add
@[simp]
theorem map_add (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f (x + y) = f x + f y 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁)
Goal: f (x + y) = f x + f y
-- LieHom.map_sub
@[simp]
theorem map_sub (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f (x - y) = f x - f y 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁)
Goal: f (x - y) = f x - f y
-- LieHom.map_neg
@[simp]
theorem map_neg (f : L₁ →ₗ⁅R⁆ L₂) (x : L₁) : f (-x) = -f x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x : L₁)
Goal: f (-x) = -f x
-- LieHom.map_lie
@[simp]
theorem map_lie (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f ⁅x, y⁆ = ⁅f x, f y⁆ 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁)
Goal: f ⁅x, y⁆ = ⁅f x, f y⁆
-- LieHom.map_zero
@[simp]
theorem map_zero (f : L₁ →ₗ⁅R⁆ L₂) : f 0 = 0 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂)
Goal: f 0 = 0
-- LieHom.coe_id
@[simp]
theorem coe_id : ⇑(id : L₁ →ₗ⁅R⁆ L₁) = _root_.id 
Arguments: {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁]
Goal: ⇑LieHom.id = id
-- LieHom.id_apply
theorem id_apply (x : L₁) : (id : L₁ →ₗ⁅R⁆ L₁) x = x 
Arguments: {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (x : L₁)
Goal: LieHom.id x = x
-- LieHom.coe_zero
@[norm_cast, simp]
theorem coe_zero : ((0 : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) = 0 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂]
Goal: ⇑0 = 0
-- LieHom.zero_apply
theorem zero_apply (x : L₁) : (0 : L₁ →ₗ⁅R⁆ L₂) x = 0 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (x : L₁)
Goal: 0 x = 0
-- LieHom.coe_one
@[simp]
theorem coe_one : ((1 : L₁ →ₗ⁅R⁆ L₁) : L₁ → L₁) = _root_.id 
Arguments: {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁]
Goal: ⇑1 = id
-- LieHom.one_apply
theorem one_apply (x : L₁) : (1 : L₁ →ₗ⁅R⁆ L₁) x = x 
Arguments: {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (x : L₁)
Goal: 1 x = x
-- LieHom.coe_injective
theorem coe_injective : @Function.Injective (L₁ →ₗ⁅R⁆ L₂) (L₁ → L₂) (↑) 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂]
Goal: Function.Injective DFunLike.coe
-- LieHom.ext
@[ext]
theorem ext {f g : L₁ →ₗ⁅R⁆ L₂} (h : ∀ x, f x = g x) : f = g 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] {f g : L₁ →ₗ⁅R⁆ L₂} (h : ∀ (x : L₁), f x = g x)
Goal: f = g
-- LieHom.ext_iff
theorem ext_iff {f g : L₁ →ₗ⁅R⁆ L₂} : f = g ↔ ∀ x, f x = g x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] {f g : L₁ →ₗ⁅R⁆ L₂}
Goal: f = g ↔ ∀ (x : L₁), f x = g x
-- LieHom.congr_fun
theorem congr_fun {f g : L₁ →ₗ⁅R⁆ L₂} (h : f = g) (x : L₁) : f x = g x 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] {f g : L₁ →ₗ⁅R⁆ L₂} (h : f = g) (x : L₁)
Goal: f x = g x
-- LieHom.mk_coe
@[simp]
theorem mk_coe (f : L₁ →ₗ⁅R⁆ L₂) (h₁ h₂ h₃) : (⟨⟨⟨f, h₁⟩, h₂⟩, h₃⟩ : L₁ →ₗ⁅R⁆ L₂) = f 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h₁ : ∀ (x y : L₁), f (x + y) = f x + f y) (h₂ :
  ∀ (m : R) (x : L₁),
    { toFun := ⇑f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₁ }.toFun x) (h₃ :
  ∀ {x y : L₁},
    { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun ⁅x, y⁆ =
      ⁅{ toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun x,
        { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }.toFun y⁆)
Goal: { toFun := ⇑f, map_add' := h₁, map_smul' := h₂, map_lie' := h₃ } = f
-- LieHom.coe_mk
@[simp]
theorem coe_mk (f : L₁ → L₂) (h₁ h₂ h₃) : ((⟨⟨⟨f, h₁⟩, h₂⟩, h₃⟩ : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) = f 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ → L₂) (h₁ : ∀ (x y : L₁), f (x + y) = f x + f y) (h₂ :
  ∀ (m : R) (x : L₁),
    { toFun := f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h₁ }.toFun x) (h₃ :
  ∀ {x y : L₁},
    { toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun ⁅x, y⁆ =
      ⁅{ toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun x,
        { toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun y⁆)
Goal: ⇑{ toFun := f, map_add' := h₁, map_smul' := h₂, map_lie' := h₃ } = f
-- LieHom.comp_apply
theorem comp_apply (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) (x : L₁) : f.comp g x = f (g x) 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) (x : L₁)
Goal: (f.comp g) x = f (g x)
-- LieHom.coe_comp
@[norm_cast, simp]
theorem coe_comp (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) : (f.comp g : L₁ → L₃) = f ∘ g 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- LieHom.coe_linearMap_comp
@[norm_cast, simp]
theorem coe_linearMap_comp (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) :
    (f.comp g : L₁ →ₗ[R] L₃) = (f : L₂ →ₗ[R] L₃).comp (g : L₁ →ₗ[R] L₂) 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [LieRing L₃] [LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂)
Goal: ↑(f.comp g) = ↑f ∘ₗ ↑g
-- LieHom.comp_id
@[simp]
theorem comp_id (f : L₁ →ₗ⁅R⁆ L₂) : f.comp (id : L₁ →ₗ⁅R⁆ L₁) = f 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂)
Goal: f.comp LieHom.id = f
-- LieHom.id_comp
@[simp]
theorem id_comp (f : L₁ →ₗ⁅R⁆ L₂) : (id : L₂ →ₗ⁅R⁆ L₂).comp f = f 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂)
Goal: LieHom.id.comp f = f
-- LieRingModule.compLieHom_apply
theorem LieRingModule.compLieHom_apply (x : L₁) (m : M) :
    haveI := LieRingModule.compLieHom M f
    ⁅x, m⁆ = ⁅f x, m⁆ 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} (M : Type w₁) [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M] [LieRingModule L₂ M] (f : L₁ →ₗ⁅R⁆ L₂) (x : L₁) (m : M)
Goal: ⁅x, m⁆ = ⁅f x, m⁆
-- LieModule.compLieHom
theorem LieModule.compLieHom [Module R M] [LieModule R L₂ M] :
    @LieModule R L₁ M _ _ _ _ _ (LieRingModule.compLieHom M f) 
Arguments: {R : Type u} {L₁ : Type v} {L₂ : Type w} (M : Type w₁) [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] [AddCommGroup M] [LieRingModule L₂ M] (f : L₁ →ₗ⁅R⁆ L₂) [Module R M] [LieModule R L₂ M]
Goal: LieModule R L₁ M
-- LieEquiv.left_inv
structure LieEquiv (R : Type u) (L : Type v) (L' : Type w) [CommRing R] [LieRing L] [LieAlgebra R L]
  [LieRing L'] [LieAlgebra R L'] extends L →ₗ⁅R⁆ L' where
    invFun : L' → L
    left_inv : Function.LeftInverse invFun toLieHom.toFun
    right_inv : Function.RightInverse invFun toLieHom.toFun
Arguments: {R : Type u} {L : Type v} {L' : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (self : L ≃ₗ⁅R⁆ L')
Goal: Function.LeftInverse self.invFun (↑self.toLieHom).toFun
-- LieEquiv.right_inv
structure LieEquiv (R : Type u) (L : Type v) (L' : Type w) [CommRing R] [LieRing L] [LieAlgebra R L]
  [LieRing L'] [LieAlgebra R L'] extends L →ₗ⁅R⁆ L' where
    invFun : L' → L
    left_inv : Function.LeftInverse invFun toLieHom.toFun
    right_inv : Function.RightInverse invFun toLieHom.toFun
Arguments: {R : Type u} {L : Type v} {L' : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (self : L ≃ₗ⁅R⁆ L')
Goal: Function.RightInverse self.invFun (↑self.toLieHom).toFun
-- LieEquiv.coe_to_lieHom
theorem coe_to_lieHom (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑(e : L₁ →ₗ⁅R⁆ L₂) = e 
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: ⇑e.toLieHom = ⇑e
-- LieEquiv.coe_to_linearEquiv
@[simp]
theorem coe_to_linearEquiv (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑(e : L₁ ≃ₗ[R] L₂) = e 
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: ⇑e.toLinearEquiv = ⇑e