ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Subring.opEquiv_symm_apply
@[simps]
def opEquiv : Subring R ≃o Subring Rᵐᵒᵖ 
Arguments: {R : Type u_2} [Ring R] (S : Subring Rᵐᵒᵖ)
Goal: (RelIso.symm Subring.opEquiv) S = S.unop
-- Subring.op_bot
@[simp]
theorem op_bot : (⊥ : Subring R).op = ⊥ 
Arguments: {R : Type u_2} [Ring R]
Goal: ⊥.op = ⊥
-- Subring.unop_bot
@[simp]
theorem unop_bot : (⊥ : Subring Rᵐᵒᵖ).unop = ⊥ 
Arguments: {R : Type u_2} [Ring R]
Goal: ⊥.unop = ⊥
-- Subring.op_top
@[simp]
theorem op_top : (⊤ : Subring R).op = ⊤ 
Arguments: {R : Type u_2} [Ring R]
Goal: ⊤.op = ⊤
-- Subring.unop_top
@[simp]
theorem unop_top : (⊤ : Subring Rᵐᵒᵖ).unop = ⊤ 
Arguments: {R : Type u_2} [Ring R]
Goal: ⊤.unop = ⊤
-- Subring.op_sup
theorem op_sup (S₁ S₂ : Subring R) : (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op 
Arguments: {R : Type u_2} [Ring R] (S₁ S₂ : Subring R)
Goal: (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
-- Subring.unop_sup
theorem unop_sup (S₁ S₂ : Subring Rᵐᵒᵖ) : (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop 
Arguments: {R : Type u_2} [Ring R] (S₁ S₂ : Subring Rᵐᵒᵖ)
Goal: (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
-- Subring.op_inf
theorem op_inf (S₁ S₂ : Subring R) : (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op 
Arguments: {R : Type u_2} [Ring R] (S₁ S₂ : Subring R)
Goal: (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
-- Subring.unop_inf
theorem unop_inf (S₁ S₂ : Subring Rᵐᵒᵖ) : (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop 
Arguments: {R : Type u_2} [Ring R] (S₁ S₂ : Subring Rᵐᵒᵖ)
Goal: (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
-- Subring.op_sSup
theorem op_sSup (S : Set (Subring R)) : (sSup S).op = sSup (.unop ⁻¹' S) 
Arguments: {R : Type u_2} [Ring R] (S : Set (Subring R))
Goal: (sSup S).op = sSup (Subring.unop ⁻¹' S)
-- Subring.unop_sSup
theorem unop_sSup (S : Set (Subring Rᵐᵒᵖ)) : (sSup S).unop = sSup (.op ⁻¹' S) 
Arguments: {R : Type u_2} [Ring R] (S : Set (Subring Rᵐᵒᵖ))
Goal: (sSup S).unop = sSup (Subring.op ⁻¹' S)
-- Subring.op_sInf
theorem op_sInf (S : Set (Subring R)) : (sInf S).op = sInf (.unop ⁻¹' S) 
Arguments: {R : Type u_2} [Ring R] (S : Set (Subring R))
Goal: (sInf S).op = sInf (Subring.unop ⁻¹' S)
-- Subring.unop_sInf
theorem unop_sInf (S : Set (Subring Rᵐᵒᵖ)) : (sInf S).unop = sInf (.op ⁻¹' S) 
Arguments: {R : Type u_2} [Ring R] (S : Set (Subring Rᵐᵒᵖ))
Goal: (sInf S).unop = sInf (Subring.op ⁻¹' S)
-- Subring.op_iSup
theorem op_iSup (S : ι → Subring R) : (iSup S).op = ⨆ i, (S i).op 
Arguments: {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring R)
Goal: (iSup S).op = ⨆ i, (S i).op
-- Subring.unop_iSup
theorem unop_iSup (S : ι → Subring Rᵐᵒᵖ) : (iSup S).unop = ⨆ i, (S i).unop 
Arguments: {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring Rᵐᵒᵖ)
Goal: (iSup S).unop = ⨆ i, (S i).unop
-- Subring.op_iInf
theorem op_iInf (S : ι → Subring R) : (iInf S).op = ⨅ i, (S i).op 
Arguments: {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring R)
Goal: (iInf S).op = ⨅ i, (S i).op
-- Subring.unop_iInf
theorem unop_iInf (S : ι → Subring Rᵐᵒᵖ) : (iInf S).unop = ⨅ i, (S i).unop 
Arguments: {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring Rᵐᵒᵖ)
Goal: (iInf S).unop = ⨅ i, (S i).unop
-- Subring.op_closure
theorem op_closure (s : Set R) : (closure s).op = closure (MulOpposite.unop ⁻¹' s) 
Arguments: {R : Type u_2} [Ring R] (s : Set R)
Goal: (Subring.closure s).op = Subring.closure (MulOpposite.unop ⁻¹' s)
-- Subring.unop_closure
theorem unop_closure (s : Set Rᵐᵒᵖ) : (closure s).unop = closure (MulOpposite.op ⁻¹' s) 
Arguments: {R : Type u_2} [Ring R] (s : Set Rᵐᵒᵖ)
Goal: (Subring.closure s).unop = Subring.closure (MulOpposite.op ⁻¹' s)
-- Subring.addEquivOp_symm_apply_coe
@[simps!]
def addEquivOp (S : Subring R) : S ≃+ S.op 
Arguments: {R : Type u_2} [Ring R] (S : Subring R) (b : { b // (fun x => x ∈ S.op) b })
Goal: ↑(S.addEquivOp.symm b) = MulOpposite.unop ↑b
-- Subring.addEquivOp_apply_coe
@[simps!]
def addEquivOp (S : Subring R) : S ≃+ S.op 
Arguments: {R : Type u_2} [Ring R] (S : Subring R) (a : { a // (fun x => x ∈ S.toSubmonoid) a })
Goal: ↑(S.addEquivOp a) = MulOpposite.op ↑a
-- Subring.ringEquivOpMop_apply
@[simps!]
def ringEquivOpMop (S : Subring R) : S ≃+* (S.op)ᵐᵒᵖ 
Arguments: {R : Type u_2} [Ring R] (S : Subring R)
Goal: ∀ (a : ↥S.toSubsemiring), S.ringEquivOpMop a = MulOpposite.op (S.addEquivOp a)
-- Subring.ringEquivOpMop_symm_apply_coe
@[simps!]
def ringEquivOpMop (S : Subring R) : S ≃+* (S.op)ᵐᵒᵖ 
Arguments: {R : Type u_2} [Ring R] (S : Subring R)
Goal: ∀ (a : (↥S.op)ᵐᵒᵖ), ↑(S.ringEquivOpMop.symm a) = MulOpposite.unop ↑(MulOpposite.unop a)
-- Subring.mopRingEquivOp_apply_coe
@[simps!]
def mopRingEquivOp (S : Subring R) : Sᵐᵒᵖ ≃+* S.op 
Arguments: {R : Type u_2} [Ring R] (S : Subring R)
Goal: ∀ (a : (↥S.toSubsemiring)ᵐᵒᵖ), ↑(S.mopRingEquivOp a) = MulOpposite.op ↑(MulOpposite.unop a)
-- Subring.mopRingEquivOp_symm_apply
@[simps!]
def mopRingEquivOp (S : Subring R) : Sᵐᵒᵖ ≃+* S.op 
Arguments: {R : Type u_2} [Ring R] (S : Subring R)
Goal: ∀ (a : ↥S.op), S.mopRingEquivOp.symm a = MulOpposite.op (S.addEquivOp.symm a)
-- LinearMap.mulLeft_toAddMonoidHom
@[simp]
theorem mulLeft_toAddMonoidHom (a : A) : (mulLeft R a : A →+ A) = AddMonoidHom.mulLeft a 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a : A)
Goal: ↑(LinearMap.mulLeft R a) = AddMonoidHom.mulLeft a
-- LinearMap.mulRight_toAddMonoidHom
@[simp]
theorem mulRight_toAddMonoidHom (a : A) : (mulRight R a : A →+ A) = AddMonoidHom.mulRight a 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a : A)
Goal: ↑(LinearMap.mulRight R a) = AddMonoidHom.mulRight a
-- LinearMap.mul_apply'
@[simp]
theorem mul_apply' (a b : A) : mul R A a b = a * b 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A)
Goal: ((LinearMap.mul R A) a) b = a * b
-- LinearMap.mulLeft_apply
@[simp]
theorem mulLeft_apply (a b : A) : mulLeft R a b = a * b 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A)
Goal: (LinearMap.mulLeft R a) b = a * b
-- LinearMap.mulRight_apply
@[simp]
theorem mulRight_apply (a b : A) : mulRight R a b = b * a 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A)
Goal: (LinearMap.mulRight R a) b = b * a
-- LinearMap.mulLeftRight_apply
@[simp]
theorem mulLeftRight_apply (a b x : A) : mulLeftRight R (a, b) x = a * x * b 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b x : A)
Goal: (LinearMap.mulLeftRight R (a, b)) x = a * x * b
-- LinearMap.mul'_apply
@[simp]
theorem mul'_apply {a b : A} : mul' R A (a ⊗ₜ b) = a * b 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {a b : A}
Goal: (LinearMap.mul' R A) (a ⊗ₜ[R] b) = a * b
-- LinearMap.mulLeft_zero_eq_zero
@[simp]
theorem mulLeft_zero_eq_zero : mulLeft R (0 : A) = 0 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]
Goal: LinearMap.mulLeft R 0 = 0
-- LinearMap.mulRight_zero_eq_zero
@[simp]
theorem mulRight_zero_eq_zero : mulRight R (0 : A) = 0 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]
Goal: LinearMap.mulRight R 0 = 0
-- NonUnitalAlgHom.coe_lmul_eq_mul
@[simp]
theorem _root_.NonUnitalAlgHom.coe_lmul_eq_mul : ⇑(NonUnitalAlgHom.lmul R A) = mul R A 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]
Goal: ⇑(NonUnitalAlgHom.lmul R A) = ⇑(LinearMap.mul R A)
-- LinearMap.commute_mulLeft_right
theorem commute_mulLeft_right (a b : A) : Commute (mulLeft R a) (mulRight R b) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A)
Goal: Commute (LinearMap.mulLeft R a) (LinearMap.mulRight R b)
-- LinearMap.mulLeft_mul
@[simp]
theorem mulLeft_mul (a b : A) : mulLeft R (a * b) = (mulLeft R a).comp (mulLeft R b) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A)
Goal: LinearMap.mulLeft R (a * b) = LinearMap.mulLeft R a ∘ₗ LinearMap.mulLeft R b
-- LinearMap.mulRight_mul
@[simp]
theorem mulRight_mul (a b : A) : mulRight R (a * b) = (mulRight R b).comp (mulRight R a) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A)
Goal: LinearMap.mulRight R (a * b) = LinearMap.mulRight R b ∘ₗ LinearMap.mulRight R a
-- LinearMap.map_mul_iff
theorem map_mul_iff (f : A →ₗ[R] B) :
    (∀ x y, f (x * y) = f x * f y) ↔
      (LinearMap.mul R A).compr₂ f = (LinearMap.mul R B ∘ₗ f).compl₂ f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B)
Goal: (∀ (x y : A), f (x * y) = f x * f y) ↔ (LinearMap.mul R A).compr₂ f = (LinearMap.mul R B ∘ₗ f).compl₂ f
-- Algebra.coe_lmul_eq_mul
@[simp]
theorem _root_.Algebra.coe_lmul_eq_mul : ⇑(Algebra.lmul R A) = mul R A 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⇑(Algebra.lmul R A) = ⇑(LinearMap.mul R A)
-- Algebra.lmul_injective
theorem _root_.Algebra.lmul_injective : Function.Injective (Algebra.lmul R A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: Function.Injective ⇑(Algebra.lmul R A)
-- Algebra.lmul_isUnit_iff
theorem _root_.Algebra.lmul_isUnit_iff {x : A} :
    IsUnit (Algebra.lmul R A x) ↔ IsUnit x 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {x : A}
Goal: IsUnit ((Algebra.lmul R A) x) ↔ IsUnit x
-- LinearMap.mulLeft_eq_zero_iff
@[simp]
theorem mulLeft_eq_zero_iff (a : A) : mulLeft R a = 0 ↔ a = 0 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A)
Goal: LinearMap.mulLeft R a = 0 ↔ a = 0
-- LinearMap.mulRight_eq_zero_iff
@[simp]
theorem mulRight_eq_zero_iff (a : A) : mulRight R a = 0 ↔ a = 0 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A)
Goal: LinearMap.mulRight R a = 0 ↔ a = 0
-- LinearMap.mulLeft_one
@[simp]
theorem mulLeft_one : mulLeft R (1 : A) = LinearMap.id 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: LinearMap.mulLeft R 1 = LinearMap.id
-- LinearMap.mulRight_one
@[simp]
theorem mulRight_one : mulRight R (1 : A) = LinearMap.id 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: LinearMap.mulRight R 1 = LinearMap.id
-- LinearMap.pow_mulLeft
@[simp]
theorem pow_mulLeft (a : A) (n : ℕ) : mulLeft R a ^ n = mulLeft R (a ^ n) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) (n : ℕ)
Goal: LinearMap.mulLeft R a ^ n = LinearMap.mulLeft R (a ^ n)
-- LinearMap.pow_mulRight
@[simp]
theorem pow_mulRight (a : A) (n : ℕ) : mulRight R a ^ n = mulRight R (a ^ n) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) (n : ℕ)
Goal: LinearMap.mulRight R a ^ n = LinearMap.mulRight R (a ^ n)
-- LinearMap.mulLeft_injective
theorem mulLeft_injective [NoZeroDivisors A] {x : A} (hx : x ≠ 0) :
    Function.Injective (mulLeft R x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] [NoZeroDivisors A] {x : A} (hx : x ≠ 0)
Goal: Function.Injective ⇑(LinearMap.mulLeft R x)
-- LinearMap.mulRight_injective
theorem mulRight_injective [NoZeroDivisors A] {x : A} (hx : x ≠ 0) :
    Function.Injective (mulRight R x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] [NoZeroDivisors A] {x : A} (hx : x ≠ 0)
Goal: Function.Injective ⇑(LinearMap.mulRight R x)
-- LinearMap.mul_injective
theorem mul_injective [NoZeroDivisors A] {x : A} (hx : x ≠ 0) : Function.Injective (mul R A x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] [NoZeroDivisors A] {x : A} (hx : x ≠ 0)
Goal: Function.Injective ⇑((LinearMap.mul R A) x)
-- Subring.pointwise_smul_def
theorem pointwise_smul_def {a : M} (S : Subring R) :
    a • S = S.map (MulSemiringAction.toRingHom _ _ a) 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] {a : M} (S : Subring R)
Goal: a • S = Subring.map (MulSemiringAction.toRingHom M R a) S
-- Subring.coe_pointwise_smul
@[simp]
theorem coe_pointwise_smul (m : M) (S : Subring R) : ↑(m • S) = m • (S : Set R) 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (m : M) (S : Subring R)
Goal: ↑(m • S) = m • ↑S
-- Subring.pointwise_smul_toAddSubgroup
@[simp]
theorem pointwise_smul_toAddSubgroup (m : M) (S : Subring R) :
    (m • S).toAddSubgroup = m • S.toAddSubgroup 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (m : M) (S : Subring R)
Goal: (m • S).toAddSubgroup = m • S.toAddSubgroup
-- Subring.pointwise_smul_toSubsemiring
@[simp]
theorem pointwise_smul_toSubsemiring (m : M) (S : Subring R) :
    (m • S).toSubsemiring = m • S.toSubsemiring 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (m : M) (S : Subring R)
Goal: (m • S).toSubsemiring = m • S.toSubsemiring
-- Subring.smul_mem_pointwise_smul
theorem smul_mem_pointwise_smul (m : M) (r : R) (S : Subring R) : r ∈ S → m • r ∈ m • S 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (m : M) (r : R) (S : Subring R)
Goal: r ∈ S → m • r ∈ m • S
-- Subring.mem_smul_pointwise_iff_exists
theorem mem_smul_pointwise_iff_exists (m : M) (r : R) (S : Subring R) :
    r ∈ m • S ↔ ∃ s : R, s ∈ S ∧ m • s = r 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (m : M) (r : R) (S : Subring R)
Goal: r ∈ m • S ↔ ∃ s ∈ S, m • s = r
-- Subring.smul_bot
@[simp]
theorem smul_bot (a : M) : a • (⊥ : Subring R) = ⊥ 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (a : M)
Goal: a • ⊥ = ⊥
-- Subring.smul_sup
theorem smul_sup (a : M) (S T : Subring R) : a • (S ⊔ T) = a • S ⊔ a • T 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (a : M) (S T : Subring R)
Goal: a • (S ⊔ T) = a • S ⊔ a • T
-- Subring.smul_closure
theorem smul_closure (a : M) (s : Set R) : a • closure s = closure (a • s) 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (a : M) (s : Set R)
Goal: a • Subring.closure s = Subring.closure (a • s)
-- Subring.smul_mem_pointwise_smul_iff
@[simp]
theorem smul_mem_pointwise_smul_iff {a : M} {S : Subring R} {x : R} : a • x ∈ a • S ↔ x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [Ring R] [MulSemiringAction M R] {a : M} {S : Subring R} {x : R}
Goal: a • x ∈ a • S ↔ x ∈ S
-- Subring.mem_pointwise_smul_iff_inv_smul_mem
theorem mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : Subring R} {x : R} :
    x ∈ a • S ↔ a⁻¹ • x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [Ring R] [MulSemiringAction M R] {a : M} {S : Subring R} {x : R}
Goal: x ∈ a • S ↔ a⁻¹ • x ∈ S
-- Subring.mem_inv_pointwise_smul_iff
theorem mem_inv_pointwise_smul_iff {a : M} {S : Subring R} {x : R} : x ∈ a⁻¹ • S ↔ a • x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [Ring R] [MulSemiringAction M R] {a : M} {S : Subring R} {x : R}
Goal: x ∈ a⁻¹ • S ↔ a • x ∈ S
-- Subring.pointwise_smul_le_pointwise_smul_iff
@[simp]
theorem pointwise_smul_le_pointwise_smul_iff {a : M} {S T : Subring R} : a • S ≤ a • T ↔ S ≤ T 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [Ring R] [MulSemiringAction M R] {a : M} {S T : Subring R}
Goal: a • S ≤ a • T ↔ S ≤ T
-- Subring.pointwise_smul_subset_iff
theorem pointwise_smul_subset_iff {a : M} {S T : Subring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [Ring R] [MulSemiringAction M R] {a : M} {S T : Subring R}
Goal: a • S ≤ T ↔ S ≤ a⁻¹ • T
-- Subring.subset_pointwise_smul_iff
theorem subset_pointwise_smul_iff {a : M} {S T : Subring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [Ring R] [MulSemiringAction M R] {a : M} {S T : Subring R}
Goal: S ≤ a • T ↔ a⁻¹ • S ≤ T
-- Subring.smul_mem_pointwise_smul_iff₀
@[simp]
theorem smul_mem_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : Subring R) (x : R) :
    a • x ∈ a • S ↔ x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Ring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) (S : Subring R) (x : R)
Goal: a • x ∈ a • S ↔ x ∈ S
-- Subring.mem_pointwise_smul_iff_inv_smul_mem₀
theorem mem_pointwise_smul_iff_inv_smul_mem₀ {a : M} (ha : a ≠ 0) (S : Subring R) (x : R) :
    x ∈ a • S ↔ a⁻¹ • x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Ring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) (S : Subring R) (x : R)
Goal: x ∈ a • S ↔ a⁻¹ • x ∈ S
-- Subring.mem_inv_pointwise_smul_iff₀
theorem mem_inv_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : Subring R) (x : R) :
    x ∈ a⁻¹ • S ↔ a • x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Ring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) (S : Subring R) (x : R)
Goal: x ∈ a⁻¹ • S ↔ a • x ∈ S
-- Subring.pointwise_smul_le_pointwise_smul_iff₀
@[simp]
theorem pointwise_smul_le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : Subring R} :
    a • S ≤ a • T ↔ S ≤ T 
Arguments: {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Ring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) {S T : Subring R}
Goal: a • S ≤ a • T ↔ S ≤ T
-- Subring.pointwise_smul_le_iff₀
theorem pointwise_smul_le_iff₀ {a : M} (ha : a ≠ 0) {S T : Subring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T 
Arguments: {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Ring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) {S T : Subring R}
Goal: a • S ≤ T ↔ S ≤ a⁻¹ • T
-- Subring.le_pointwise_smul_iff₀
theorem le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : Subring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T 
Arguments: {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Ring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) {S T : Subring R}
Goal: S ≤ a • T ↔ a⁻¹ • S ≤ T
-- IsIdempotentElem.of_isIdempotent
theorem of_isIdempotent [Std.IdempotentOp (α := M) (· * ·)] (a : M) : IsIdempotentElem a 
Arguments: {M : Type u_1} [Mul M] [Std.IdempotentOp fun x x_1 => x * x_1] (a : M)
Goal: IsIdempotentElem a
-- IsIdempotentElem.eq
theorem eq {p : M} (h : IsIdempotentElem p) : p * p = p 
Arguments: {M : Type u_1} [Mul M] {p : M} (h : IsIdempotentElem p)
Goal: p * p = p
-- IsIdempotentElem.mul_of_commute
theorem mul_of_commute {p q : S} (h : Commute p q) (h₁ : IsIdempotentElem p)
    (h₂ : IsIdempotentElem q) : IsIdempotentElem (p * q) 
Arguments: {S : Type u_3} [Semigroup S] {p q : S} (h : Commute p q) (h₁ : IsIdempotentElem p) (h₂ : IsIdempotentElem q)
Goal: IsIdempotentElem (p * q)
-- IsIdempotentElem.zero
theorem zero : IsIdempotentElem (0 : M₀) 
Arguments: {M₀ : Type u_4} [MulZeroClass M₀]
Goal: IsIdempotentElem 0
-- IsIdempotentElem.one
theorem one : IsIdempotentElem (1 : M₁) 
Arguments: {M₁ : Type u_5} [MulOneClass M₁]
Goal: IsIdempotentElem 1
-- IsIdempotentElem.one_sub
theorem one_sub {p : R} (h : IsIdempotentElem p) : IsIdempotentElem (1 - p) 
Arguments: {R : Type u_6} [NonAssocRing R] {p : R} (h : IsIdempotentElem p)
Goal: IsIdempotentElem (1 - p)
-- IsIdempotentElem.one_sub_iff
@[simp]
theorem one_sub_iff {p : R} : IsIdempotentElem (1 - p) ↔ IsIdempotentElem p 
Arguments: {R : Type u_6} [NonAssocRing R] {p : R}
Goal: IsIdempotentElem (1 - p) ↔ IsIdempotentElem p
-- IsIdempotentElem.pow
theorem pow {p : N} (n : ℕ) (h : IsIdempotentElem p) : IsIdempotentElem (p ^ n) 
Arguments: {N : Type u_2} [Monoid N] {p : N} (n : ℕ) (h : IsIdempotentElem p)
Goal: IsIdempotentElem (p ^ n)
-- IsIdempotentElem.pow_succ_eq
theorem pow_succ_eq {p : N} (n : ℕ) (h : IsIdempotentElem p) : p ^ (n + 1) = p 
Arguments: {N : Type u_2} [Monoid N] {p : N} (n : ℕ) (h : IsIdempotentElem p)
Goal: p ^ (n + 1) = p
-- IsIdempotentElem.iff_eq_one
@[simp]
theorem iff_eq_one {p : G} : IsIdempotentElem p ↔ p = 1 
Arguments: {G : Type u_7} [Group G] {p : G}
Goal: IsIdempotentElem p ↔ p = 1
-- IsIdempotentElem.iff_eq_zero_or_one
@[simp]
theorem iff_eq_zero_or_one {p : G₀} : IsIdempotentElem p ↔ p = 0 ∨ p = 1 
Arguments: {G₀ : Type u_8} [CancelMonoidWithZero G₀] {p : G₀}
Goal: IsIdempotentElem p ↔ p = 0 ∨ p = 1
-- IsIdempotentElem.coe_zero
@[simp]
theorem coe_zero : ↑(0 : { p : M₀ // IsIdempotentElem p }) = (0 : M₀) 
Arguments: {M₀ : Type u_4} [MulZeroClass M₀]
Goal: ↑0 = 0
-- IsIdempotentElem.coe_one
@[simp]
theorem coe_one : ↑(1 : { p : M₁ // IsIdempotentElem p }) = (1 : M₁) 
Arguments: {M₁ : Type u_5} [MulOneClass M₁]
Goal: ↑1 = 1
-- IsIdempotentElem.coe_compl
@[simp]
theorem coe_compl (p : { p : R // IsIdempotentElem p }) : ↑pᶜ = (1 : R) - ↑p 
Arguments: {R : Type u_6} [NonAssocRing R] (p : { p // IsIdempotentElem p })
Goal: ↑pᶜ = 1 - ↑p
-- IsIdempotentElem.compl_compl
@[simp]
theorem compl_compl (p : { p : R // IsIdempotentElem p }) : pᶜᶜ = p 
Arguments: {R : Type u_6} [NonAssocRing R] (p : { p // IsIdempotentElem p })
Goal: pᶜᶜ = p
-- IsIdempotentElem.zero_compl
@[simp]
theorem zero_compl : (0 : { p : R // IsIdempotentElem p })ᶜ = 1 
Arguments: {R : Type u_6} [NonAssocRing R]
Goal: 0ᶜ = 1
-- IsIdempotentElem.one_compl
@[simp]
theorem one_compl : (1 : { p : R // IsIdempotentElem p })ᶜ = 0 
Arguments: {R : Type u_6} [NonAssocRing R]
Goal: 1ᶜ = 0
-- SemiconjBy.add_right
@[simp]
theorem add_right [Distrib R] {a x y x' y' : R} (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
    SemiconjBy a (x + x') (y + y') 
Arguments: {R : Type x} [Distrib R] {a x y x' y' : R} (h : SemiconjBy a x y) (h' : SemiconjBy a x' y')
Goal: SemiconjBy a (x + x') (y + y')
-- SemiconjBy.add_left
@[simp]
theorem add_left [Distrib R] {a b x y : R} (ha : SemiconjBy a x y) (hb : SemiconjBy b x y) :
    SemiconjBy (a + b) x y 
Arguments: {R : Type x} [Distrib R] {a b x y : R} (ha : SemiconjBy a x y) (hb : SemiconjBy b x y)
Goal: SemiconjBy (a + b) x y
-- SemiconjBy.neg_right
theorem neg_right (h : SemiconjBy a x y) : SemiconjBy a (-x) (-y) 
Arguments: {R : Type x} [Mul R] [HasDistribNeg R] {a x y : R} (h : SemiconjBy a x y)
Goal: SemiconjBy a (-x) (-y)
-- SemiconjBy.neg_right_iff
@[simp]
theorem neg_right_iff : SemiconjBy a (-x) (-y) ↔ SemiconjBy a x y 
Arguments: {R : Type x} [Mul R] [HasDistribNeg R] {a x y : R}
Goal: SemiconjBy a (-x) (-y) ↔ SemiconjBy a x y
-- SemiconjBy.neg_left
theorem neg_left (h : SemiconjBy a x y) : SemiconjBy (-a) x y 
Arguments: {R : Type x} [Mul R] [HasDistribNeg R] {a x y : R} (h : SemiconjBy a x y)
Goal: SemiconjBy (-a) x y
-- SemiconjBy.neg_left_iff
@[simp]
theorem neg_left_iff : SemiconjBy (-a) x y ↔ SemiconjBy a x y 
Arguments: {R : Type x} [Mul R] [HasDistribNeg R] {a x y : R}
Goal: SemiconjBy (-a) x y ↔ SemiconjBy a x y
-- SemiconjBy.neg_one_right
theorem neg_one_right (a : R) : SemiconjBy a (-1) (-1) 
Arguments: {R : Type x} [MulOneClass R] [HasDistribNeg R] (a : R)
Goal: SemiconjBy a (-1) (-1)
-- SemiconjBy.neg_one_left
theorem neg_one_left (x : R) : SemiconjBy (-1) x x 
Arguments: {R : Type x} [MulOneClass R] [HasDistribNeg R] (x : R)
Goal: SemiconjBy (-1) x x
-- SemiconjBy.sub_right
@[simp]
theorem sub_right (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
    SemiconjBy a (x - x') (y - y') 
Arguments: {R : Type x} [NonUnitalNonAssocRing R] {a x y x' y' : R} (h : SemiconjBy a x y) (h' : SemiconjBy a x' y')
Goal: SemiconjBy a (x - x') (y - y')
-- SemiconjBy.sub_left
@[simp]
theorem sub_left (ha : SemiconjBy a x y) (hb : SemiconjBy b x y) :
    SemiconjBy (a - b) x y 
Arguments: {R : Type x} [NonUnitalNonAssocRing R] {a b x y : R} (ha : SemiconjBy a x y) (hb : SemiconjBy b x y)
Goal: SemiconjBy (a - b) x y