ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgHom.coe_comp
@[simp]
theorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B)
Goal: ⇑(φ₁.comp φ₂) = ⇑φ₁ ∘ ⇑φ₂
-- AlgHom.comp_apply
theorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A)
Goal: (φ₁.comp φ₂) p = φ₁ (φ₂ p)
-- AlgHom.comp_toRingHom
theorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :
    (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B)
Goal: ↑(φ₁.comp φ₂) = (↑φ₁).comp ↑φ₂
-- AlgHom.comp_id
@[simp]
theorem comp_id : φ.comp (AlgHom.id R A) = φ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)
Goal: φ.comp (AlgHom.id R A) = φ
-- AlgHom.id_comp
@[simp]
theorem id_comp : (AlgHom.id R B).comp φ = φ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)
Goal: (AlgHom.id R B).comp φ = φ
-- AlgHom.comp_assoc
theorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :
    (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D] [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D] (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B)
Goal: (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃)
-- AlgHom.toLinearMap_apply
@[simp]
theorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (p : A)
Goal: φ.toLinearMap p = φ p
-- AlgHom.toLinearMap_injective
theorem toLinearMap_injective :
    Function.Injective (toLinearMap : _ → A →ₗ[R] B) 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
Goal: Function.Injective AlgHom.toLinearMap
-- AlgHom.comp_toLinearMap
@[simp]
theorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :
    (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (g : B →ₐ[R] C)
Goal: (g.comp f).toLinearMap = g.toLinearMap ∘ₗ f.toLinearMap
-- AlgHom.toLinearMap_id
@[simp]
theorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: (AlgHom.id R A).toLinearMap = LinearMap.id
-- AlgHom.ofLinearMap_apply
@[simps]
def ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :
    A →ₐ[R] B 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y) (a : A)
Goal: (AlgHom.ofLinearMap f map_one map_mul) a = f a
-- AlgHom.ofLinearMap_toLinearMap
@[simp]
theorem ofLinearMap_toLinearMap (map_one) (map_mul) :
    ofLinearMap φ.toLinearMap map_one map_mul = φ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (map_one : φ.toLinearMap 1 = 1) (map_mul : ∀ (x y : A), φ.toLinearMap (x * y) = φ.toLinearMap x * φ.toLinearMap y)
Goal: AlgHom.ofLinearMap φ.toLinearMap map_one map_mul = φ
-- AlgHom.toLinearMap_ofLinearMap
@[simp]
theorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :
    toLinearMap (ofLinearMap f map_one map_mul) = f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y)
Goal: (AlgHom.ofLinearMap f map_one map_mul).toLinearMap = f
-- AlgHom.ofLinearMap_id
@[simp]
theorem ofLinearMap_id (map_one) (map_mul) :
    ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (map_one : LinearMap.id 1 = 1) (map_mul : ∀ (x y : A), LinearMap.id (x * y) = LinearMap.id x * LinearMap.id y)
Goal: AlgHom.ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A
-- AlgHom.map_smul_of_tower
theorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')
    (x : A) : φ (r • x) = r • φ x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {R' : Type u_1} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R') (x : A)
Goal: φ (r • x) = r • φ x
-- AlgHom.map_list_prod
@[deprecated map_list_prod (since := "2024-06-26")]
protected theorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (s : List A)
Goal: φ s.prod = (List.map (⇑φ) s).prod
-- AlgHom.End_toOne_one
@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]
instance End : Monoid (A →ₐ[R] A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: 1 = AlgHom.id R A
-- AlgHom.End_toSemigroup_toMul_mul
@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]
instance End : Monoid (A →ₐ[R] A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ₁ φ₂ : A →ₐ[R] A)
Goal: φ₁ * φ₂ = φ₁.comp φ₂
-- AlgHom.one_apply
@[simp]
theorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (x : A)
Goal: 1 x = x
-- AlgHom.mul_apply
@[simp]
theorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ ψ : A →ₐ[R] A) (x : A)
Goal: (φ * ψ) x = φ (ψ x)
-- AlgHom.algebraMap_eq_apply
theorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :
    algebraMap R B y = f x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) {y : R} {x : A} (h : (algebraMap R A) y = x)
Goal: (algebraMap R B) y = f x
-- AlgHom.map_multiset_prod
@[deprecated map_multiset_prod (since := "2024-06-26")]
protected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (s : Multiset A)
Goal: φ s.prod = (Multiset.map (⇑φ) s).prod
-- AlgHom.map_prod
@[deprecated map_prod (since := "2024-06-26")]
protected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :
    φ (∏ x ∈ s, f x) = ∏ x ∈ s, φ (f x) 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {ι : Type u_1} (f : ι → A) (s : Finset ι)
Goal: φ (∏ x ∈ s, f x) = ∏ x ∈ s, φ (f x)
-- AlgHom.map_finsupp_prod
@[deprecated map_finsupp_prod (since := "2024-06-26")]
protected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :
    φ (f.prod g) = f.prod fun i a => φ (g i a) 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {α : Type u_1} [Zero α] {ι : Type u_2} (f : ι →₀ α) (g : ι → α → A)
Goal: φ (f.prod g) = f.prod fun i a => φ (g i a)
-- AlgHom.map_neg
@[deprecated map_neg (since := "2024-06-26")]
protected theorem map_neg (x) : φ (-x) = -φ x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x : A)
Goal: φ (-x) = -φ x
-- AlgHom.map_sub
@[deprecated map_sub (since := "2024-06-26")]
protected theorem map_sub (x y) : φ (x - y) = φ x - φ y 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (x y : A)
Goal: φ (x - y) = φ x - φ y
-- RingHom.toIntAlgHom_injective
lemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :
    Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) 
Arguments: {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S]
Goal: Function.Injective RingHom.toIntAlgHom
-- Algebra.ofId_apply
theorem ofId_apply (r) : ofId R A r = algebraMap R A r 
Arguments: {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (r : R)
Goal: (Algebra.ofId R A) r = (algebraMap R A) r
-- Algebra.ext_id
@[ext high]
theorem ext_id (f g : R →ₐ[R] A) : f = g 
Arguments: {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (f g : R →ₐ[R] A)
Goal: f = g
-- Algebra.smul_units_def
@[simp]
theorem smul_units_def (f : A →ₐ[R] A) (x : Aˣ) :
    f • x = Units.map (f : A →* A) x 
Arguments: {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (f : A →ₐ[R] A) (x : Aˣ)
Goal: f • x = (Units.map ↑f) x
-- Algebra.algebraMapSubmonoid_map_eq
lemma algebraMapSubmonoid_map_eq (f : A →ₐ[R] B) :
    (algebraMapSubmonoid A M).map f = algebraMapSubmonoid B M 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (M : Submonoid R) {B : Type w} [CommRing B] [Algebra R B] (f : A →ₐ[R] B)
Goal: Submonoid.map f (Algebra.algebraMapSubmonoid A M) = Algebra.algebraMapSubmonoid B M
-- Algebra.algebraMapSubmonoid_le_comap
lemma algebraMapSubmonoid_le_comap (f : A →ₐ[R] B) :
    algebraMapSubmonoid A M ≤ (algebraMapSubmonoid B M).comap f.toRingHom 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (M : Submonoid R) {B : Type w} [CommRing B] [Algebra R B] (f : A →ₐ[R] B)
Goal: Algebra.algebraMapSubmonoid A M ≤ Submonoid.comap f.toRingHom (Algebra.algebraMapSubmonoid B M)
-- MulSemiringAction.toAlgHom_apply
@[simps]
def toAlgHom (m : M) : A →ₐ[R] A 
Arguments: {M : Type u_1} (R : Type u_3) (A : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] (m : M) (a : A)
Goal: (MulSemiringAction.toAlgHom R A m) a = m • a
-- MulSemiringAction.toAlgHom_injective
theorem toAlgHom_injective [FaithfulSMul M A] :
    Function.Injective (MulSemiringAction.toAlgHom R A : M → A →ₐ[R] A) 
Arguments: {M : Type u_1} (R : Type u_3) (A : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] [FaithfulSMul M A]
Goal: Function.Injective (MulSemiringAction.toAlgHom R A)
-- AlgHom.ker_rangeRestrict
theorem ker_rangeRestrict (f : A →ₐ[R] B) : RingHom.ker f.rangeRestrict = RingHom.ker f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B)
Goal: RingHom.ker f.rangeRestrict = RingHom.ker f
-- Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem
theorem mem_of_finset_sum_eq_one_of_pow_smul_mem
    {ι : Type*} (ι' : Finset ι) (s : ι → S) (l : ι → S)
    (e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
    (H : ∀ i, ∃ n : ℕ, (s i ^ n : S) • x ∈ S') : x ∈ S' 
Arguments: {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommRing S] [Algebra R S] (S' : Subalgebra R S) {ι : Type u_3} (ι' : Finset ι) (s l : ι → S) (e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ (i : ι), s i ∈ S') (hl : ∀ (i : ι), l i ∈ S') (x : S) (H : ∀ (i : ι), ∃ n, s i ^ n • x ∈ S')
Goal: x ∈ S'
-- Subalgebra.mem_of_span_eq_top_of_smul_pow_mem
theorem mem_of_span_eq_top_of_smul_pow_mem
    (s : Set S) (l : s →₀ S) (hs : Finsupp.total s S S (↑) l = 1)
    (hs' : s ⊆ S') (hl : ∀ i, l i ∈ S') (x : S) (H : ∀ r : s, ∃ n : ℕ, (r : S) ^ n • x ∈ S') :
    x ∈ S' 
Arguments: {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommRing S] [Algebra R S] (S' : Subalgebra R S) (s : Set S) (l : ↑s →₀ S) (hs : (Finsupp.total (↑s) S S Subtype.val) l = 1) (hs' : s ⊆ ↑S') (hl : ∀ (i : ↑s), l i ∈ S') (x : S) (H : ∀ (r : ↑s), ∃ n, ↑r ^ n • x ∈ S')
Goal: x ∈ S'
-- Subalgebra.op_toSubsemiring
@[simps toSubsemiring]
protected def op (S : Subalgebra R A) : Subalgebra R Aᵐᵒᵖ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: S.op.toSubsemiring = S.op
-- Subalgebra.op_coe
@[simp, norm_cast]
theorem op_coe (S : Subalgebra R A) : S.op = MulOpposite.unop ⁻¹' (S : Set A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: ↑S.op = MulOpposite.unop ⁻¹' ↑S
-- Subalgebra.mem_op
@[simp]
theorem mem_op {x : Aᵐᵒᵖ} {S : Subalgebra R A} : x ∈ S.op ↔ x.unop ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {x : Aᵐᵒᵖ} {S : Subalgebra R A}
Goal: x ∈ S.op ↔ MulOpposite.unop x ∈ S
-- Subalgebra.unop_toSubsemiring
@[simps toSubsemiring]
protected def unop (S : Subalgebra R Aᵐᵒᵖ) : Subalgebra R A 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ)
Goal: S.unop.toSubsemiring = S.unop
-- Subalgebra.unop_coe
@[simp, norm_cast]
theorem unop_coe (S : Subalgebra R Aᵐᵒᵖ) : S.unop = MulOpposite.op ⁻¹' (S : Set Aᵐᵒᵖ) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ)
Goal: ↑S.unop = MulOpposite.op ⁻¹' ↑S
-- Subalgebra.mem_unop
@[simp]
theorem mem_unop {x : A} {S : Subalgebra R Aᵐᵒᵖ} : x ∈ S.unop ↔ MulOpposite.op x ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {x : A} {S : Subalgebra R Aᵐᵒᵖ}
Goal: x ∈ S.unop ↔ MulOpposite.op x ∈ S
-- Subalgebra.unop_op
@[simp]
theorem unop_op (S : Subalgebra R A) : S.op.unop = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: S.op.unop = S
-- Subalgebra.op_unop
@[simp]
theorem op_unop (S : Subalgebra R Aᵐᵒᵖ) : S.unop.op = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ)
Goal: S.unop.op = S
-- Subalgebra.op_le_iff
theorem op_le_iff {S₁ : Subalgebra R A} {S₂ : Subalgebra R Aᵐᵒᵖ} : S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ : Subalgebra R A} {S₂ : Subalgebra R Aᵐᵒᵖ}
Goal: S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop
-- Subalgebra.le_op_iff
theorem le_op_iff {S₁ : Subalgebra R Aᵐᵒᵖ} {S₂ : Subalgebra R A} : S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ : Subalgebra R Aᵐᵒᵖ} {S₂ : Subalgebra R A}
Goal: S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂
-- Subalgebra.op_le_op_iff
@[simp]
theorem op_le_op_iff {S₁ S₂ : Subalgebra R A} : S₁.op ≤ S₂.op ↔ S₁ ≤ S₂ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ S₂ : Subalgebra R A}
Goal: S₁.op ≤ S₂.op ↔ S₁ ≤ S₂
-- Subalgebra.unop_le_unop_iff
@[simp]
theorem unop_le_unop_iff {S₁ S₂ : Subalgebra R Aᵐᵒᵖ} : S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {S₁ S₂ : Subalgebra R Aᵐᵒᵖ}
Goal: S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂
-- Subalgebra.opEquiv_symm_apply
@[simps]
def opEquiv : Subalgebra R A ≃o Subalgebra R Aᵐᵒᵖ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ)
Goal: (RelIso.symm Subalgebra.opEquiv) S = S.unop
-- Subalgebra.opEquiv_apply
@[simps]
def opEquiv : Subalgebra R A ≃o Subalgebra R Aᵐᵒᵖ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: Subalgebra.opEquiv S = S.op
-- Subalgebra.op_bot
@[simp]
theorem op_bot : (⊥ : Subalgebra R A).op = ⊥ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⊥.op = ⊥
-- Subalgebra.unop_bot
@[simp]
theorem unop_bot : (⊥ : Subalgebra R Aᵐᵒᵖ).unop = ⊥ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⊥.unop = ⊥
-- Subalgebra.op_top
@[simp]
theorem op_top : (⊤ : Subalgebra R A).op = ⊤ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⊤.op = ⊤
-- Subalgebra.unop_top
@[simp]
theorem unop_top : (⊤ : Subalgebra R Aᵐᵒᵖ).unop = ⊤ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⊤.unop = ⊤
-- Subalgebra.op_sup
theorem op_sup (S₁ S₂ : Subalgebra R A) : (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R A)
Goal: (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
-- Subalgebra.unop_sup
theorem unop_sup (S₁ S₂ : Subalgebra R Aᵐᵒᵖ) : (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R Aᵐᵒᵖ)
Goal: (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
-- Subalgebra.op_inf
theorem op_inf (S₁ S₂ : Subalgebra R A) : (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R A)
Goal: (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
-- Subalgebra.unop_inf
theorem unop_inf (S₁ S₂ : Subalgebra R Aᵐᵒᵖ) : (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S₁ S₂ : Subalgebra R Aᵐᵒᵖ)
Goal: (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
-- Subalgebra.op_sSup
theorem op_sSup (S : Set (Subalgebra R A)) : (sSup S).op = sSup (.unop ⁻¹' S) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A))
Goal: (sSup S).op = sSup (Subalgebra.unop ⁻¹' S)
-- Subalgebra.unop_sSup
theorem unop_sSup (S : Set (Subalgebra R Aᵐᵒᵖ)) : (sSup S).unop = sSup (.op ⁻¹' S) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R Aᵐᵒᵖ))
Goal: (sSup S).unop = sSup (Subalgebra.op ⁻¹' S)
-- Subalgebra.op_sInf
theorem op_sInf (S : Set (Subalgebra R A)) : (sInf S).op = sInf (.unop ⁻¹' S) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A))
Goal: (sInf S).op = sInf (Subalgebra.unop ⁻¹' S)
-- Subalgebra.unop_sInf
theorem unop_sInf (S : Set (Subalgebra R Aᵐᵒᵖ)) : (sInf S).unop = sInf (.op ⁻¹' S) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R Aᵐᵒᵖ))
Goal: (sInf S).unop = sInf (Subalgebra.op ⁻¹' S)
-- Subalgebra.op_iSup
theorem op_iSup (S : ι → Subalgebra R A) : (iSup S).op = ⨆ i, (S i).op 
Arguments: {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R A)
Goal: (iSup S).op = ⨆ i, (S i).op
-- Subalgebra.unop_iSup
theorem unop_iSup (S : ι → Subalgebra R Aᵐᵒᵖ) : (iSup S).unop = ⨆ i, (S i).unop 
Arguments: {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R Aᵐᵒᵖ)
Goal: (iSup S).unop = ⨆ i, (S i).unop
-- Subalgebra.op_iInf
theorem op_iInf (S : ι → Subalgebra R A) : (iInf S).op = ⨅ i, (S i).op 
Arguments: {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R A)
Goal: (iInf S).op = ⨅ i, (S i).op
-- Subalgebra.unop_iInf
theorem unop_iInf (S : ι → Subalgebra R Aᵐᵒᵖ) : (iInf S).unop = ⨅ i, (S i).unop 
Arguments: {ι : Sort u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : ι → Subalgebra R Aᵐᵒᵖ)
Goal: (iInf S).unop = ⨅ i, (S i).unop
-- Subalgebra.op_adjoin
theorem op_adjoin (s : Set A) :
    (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (s : Set A)
Goal: (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s)
-- Subalgebra.unop_adjoin
theorem unop_adjoin (s : Set Aᵐᵒᵖ) :
    (Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (s : Set Aᵐᵒᵖ)
Goal: (Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s)
-- Subalgebra.linearEquivOp_apply_coe
@[simps!]
def linearEquivOp (S : Subalgebra R A) : S ≃ₗ[R] S.op 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: ∀ (a : ↥S.toSubsemiring), ↑(S.linearEquivOp a) = MulOpposite.op ↑a
-- Subalgebra.linearEquivOp_symm_apply_coe
@[simps!]
def linearEquivOp (S : Subalgebra R A) : S ≃ₗ[R] S.op 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: ∀ (a : ↥S.op), ↑(S.linearEquivOp.symm a) = MulOpposite.unop ↑a
-- Subalgebra.algEquivOpMop_symm_apply_coe
@[simps!]
def algEquivOpMop (S : Subalgebra R A) : S ≃ₐ[R] (S.op)ᵐᵒᵖ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: ∀ (a : (↥S.op)ᵐᵒᵖ), ↑(S.algEquivOpMop.symm a) = MulOpposite.unop ↑(MulOpposite.unop a)
-- Subalgebra.algEquivOpMop_apply
@[simps!]
def algEquivOpMop (S : Subalgebra R A) : S ≃ₐ[R] (S.op)ᵐᵒᵖ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: ∀ (a : ↥S.toSubsemiring), S.algEquivOpMop a = MulOpposite.op (S.addEquivOp a)
-- Subalgebra.mopAlgEquivOp_symm_apply
@[simps!]
def mopAlgEquivOp (S : Subalgebra R A) : Sᵐᵒᵖ ≃ₐ[R] S.op 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: ∀ (a : ↥S.op), S.mopAlgEquivOp.symm a = MulOpposite.op (S.addEquivOp.symm a)
-- Subalgebra.mopAlgEquivOp_apply_coe
@[simps!]
def mopAlgEquivOp (S : Subalgebra R A) : Sᵐᵒᵖ ≃ₐ[R] S.op 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: ∀ (a : (↥S.toSubsemiring)ᵐᵒᵖ), ↑(S.mopAlgEquivOp a) = MulOpposite.op ↑(MulOpposite.unop a)
-- Subalgebra.op_toSubring
@[simp]
theorem op_toSubring (S : Subalgebra R A) : S.op.toSubring = S.toSubring.op 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (S : Subalgebra R A)
Goal: S.op.toSubring = S.toSubring.op
-- Subalgebra.unop_toSubring
@[simp]
theorem unop_toSubring (S : Subalgebra R Aᵐᵒᵖ) : S.unop.toSubring = S.toSubring.unop 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (S : Subalgebra R Aᵐᵒᵖ)
Goal: S.unop.toSubring = S.toSubring.unop
-- Algebra.smul_def'
-- Porting note(#5171): unsupported @[nolint has_nonempty_instance]
class Algebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends SMul R A,
  R →+* A where
  commutes' : ∀ r x, toRingHom r * x = x * toRingHom r
  smul_def' : ∀ r x, r • x = toRingHom r * x
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : Algebra R A] (r : R) (x : A)
Goal: r • x = Algebra.toRingHom r * x
-- Algebra.commutes'
-- Porting note(#5171): unsupported @[nolint has_nonempty_instance]
class Algebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends SMul R A,
  R →+* A where
  commutes' : ∀ r x, toRingHom r * x = x * toRingHom r
  smul_def' : ∀ r x, r • x = toRingHom r * x
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : Algebra R A] (r : R) (x : A)
Goal: Algebra.toRingHom r * x = x * Algebra.toRingHom r
-- algebraMap.coe_zero
@[norm_cast]
theorem coe_zero : (↑(0 : R) : A) = 0 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ↑0 = 0
-- algebraMap.coe_one
@[norm_cast]
theorem coe_one : (↑(1 : R) : A) = 1 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ↑1 = 1
-- algebraMap.coe_natCast
@[norm_cast]
theorem coe_natCast (a : ℕ) : (↑(a : R) : A) = a 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : ℕ)
Goal: ↑↑a = ↑a