ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgHom.val_comp_codRestrict
@[simp]
theorem val_comp_codRestrict (f : A →ₐ[R] B) (S : Subalgebra R B) (hf : ∀ x, f x ∈ S) :
    S.val.comp (f.codRestrict S hf) = f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R B) (hf : ∀ (x : A), f x ∈ S)
Goal: S.val.comp (f.codRestrict S hf) = f
-- AlgHom.coe_codRestrict
@[simp]
theorem coe_codRestrict (f : A →ₐ[R] B) (S : Subalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :
    ↑(f.codRestrict S hf x) = f x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R B) (hf : ∀ (x : A), f x ∈ S) (x : A)
Goal: ↑((f.codRestrict S hf) x) = f x
-- AlgHom.injective_codRestrict
theorem injective_codRestrict (f : A →ₐ[R] B) (S : Subalgebra R B) (hf : ∀ x, f x ∈ S) :
    Function.Injective (f.codRestrict S hf) ↔ Function.Injective f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R B) (hf : ∀ (x : A), f x ∈ S)
Goal: Function.Injective ⇑(f.codRestrict S hf) ↔ Function.Injective ⇑f
-- AlgHom.rangeRestrict_surjective
theorem rangeRestrict_surjective (f : A →ₐ[R] B) : Function.Surjective (f.rangeRestrict) 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B)
Goal: Function.Surjective ⇑f.rangeRestrict
-- AlgEquiv.ofLeftInverse_apply
@[simp]
theorem ofLeftInverse_apply {g : B → A} {f : A →ₐ[R] B} (h : Function.LeftInverse g f) (x : A) :
    ↑(ofLeftInverse h x) = f x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {g : B → A} {f : A →ₐ[R] B} (h : Function.LeftInverse g ⇑f) (x : A)
Goal: ↑((AlgEquiv.ofLeftInverse h) x) = f x
-- AlgEquiv.ofLeftInverse_symm_apply
@[simp]
theorem ofLeftInverse_symm_apply {g : B → A} {f : A →ₐ[R] B} (h : Function.LeftInverse g f)
    (x : f.range) : (ofLeftInverse h).symm x = g x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {g : B → A} {f : A →ₐ[R] B} (h : Function.LeftInverse g ⇑f) (x : ↥f.range)
Goal: (AlgEquiv.ofLeftInverse h).symm x = g ↑x
-- AlgEquiv.ofInjective_apply
@[simp]
theorem ofInjective_apply (f : A →ₐ[R] B) (hf : Function.Injective f) (x : A) :
    ↑(ofInjective f hf x) = 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) (hf : Function.Injective ⇑f) (x : A)
Goal: ↑((AlgEquiv.ofInjective f hf) x) = f x
-- AlgEquiv.subalgebraMap_symm_apply_coe
@[simps!]
def subalgebraMap (e : A ≃ₐ[R] B) (S : Subalgebra R A) : S ≃ₐ[R] S.map (e : 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] (e : A ≃ₐ[R] B) (S : Subalgebra R A) (y : ↑(⇑↑e.toRingEquiv.toAddEquiv '' ↑S.toAddSubmonoid))
Goal: ↑((e.subalgebraMap S).symm y) = (↑↑e).symm ↑y
-- AlgEquiv.subalgebraMap_apply_coe
@[simps!]
def subalgebraMap (e : A ≃ₐ[R] B) (S : Subalgebra R A) : S ≃ₐ[R] S.map (e : 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] (e : A ≃ₐ[R] B) (S : Subalgebra R A) (x : ↑↑S.toAddSubmonoid)
Goal: ↑((e.subalgebraMap S) x) = e ↑x
-- Algebra.gc
protected theorem gc : GaloisConnection (adjoin R : Set A → Subalgebra R A) (↑) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: GaloisConnection (Algebra.adjoin R) SetLike.coe
-- Algebra.coe_top
@[simp]
theorem coe_top : (↑(⊤ : Subalgebra R A) : Set A) = Set.univ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ↑⊤ = Set.univ
-- Algebra.mem_top
@[simp]
theorem mem_top {x : A} : x ∈ (⊤ : Subalgebra R A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {x : A}
Goal: x ∈ ⊤
-- Algebra.top_toSubmodule
@[simp]
theorem top_toSubmodule : Subalgebra.toSubmodule (⊤ : Subalgebra R A) = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: Subalgebra.toSubmodule ⊤ = ⊤
-- Algebra.top_toSubsemiring
@[simp]
theorem top_toSubsemiring : (⊤ : Subalgebra R A).toSubsemiring = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⊤.toSubsemiring = ⊤
-- Algebra.top_toSubring
@[simp]
theorem top_toSubring {R A : Type*} [CommRing R] [Ring A] [Algebra R A] :
    (⊤ : Subalgebra R A).toSubring = ⊤ 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A]
Goal: ⊤.toSubring = ⊤
-- Algebra.toSubmodule_eq_top
@[simp]
theorem toSubmodule_eq_top {S : Subalgebra R A} : Subalgebra.toSubmodule S = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S : Subalgebra R A}
Goal: Subalgebra.toSubmodule S = ⊤ ↔ S = ⊤
-- Algebra.toSubsemiring_eq_top
@[simp]
theorem toSubsemiring_eq_top {S : Subalgebra R A} : S.toSubsemiring = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S : Subalgebra R A}
Goal: S.toSubsemiring = ⊤ ↔ S = ⊤
-- Algebra.toSubring_eq_top
@[simp]
theorem toSubring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A] {S : Subalgebra R A} :
    S.toSubring = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] {S : Subalgebra R A}
Goal: S.toSubring = ⊤ ↔ S = ⊤
-- Algebra.mem_sup_left
theorem mem_sup_left {S T : Subalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} {x : A}
Goal: x ∈ S → x ∈ S ⊔ T
-- Algebra.mem_sup_right
theorem mem_sup_right {S T : Subalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} {x : A}
Goal: x ∈ T → x ∈ S ⊔ T
-- Algebra.mul_mem_sup
theorem mul_mem_sup {S T : Subalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) : x * y ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T)
Goal: x * y ∈ S ⊔ T
-- Algebra.map_sup
theorem map_sup (f : A →ₐ[R] B) (S T : Subalgebra R A) : (S ⊔ T).map f = S.map f ⊔ T.map f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S T : Subalgebra R A)
Goal: Subalgebra.map f (S ⊔ T) = Subalgebra.map f S ⊔ Subalgebra.map f T
-- Algebra.coe_inf
@[simp, norm_cast]
theorem coe_inf (S T : Subalgebra R A) : (↑(S ⊓ T) : Set A) = (S ∩ T : Set A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S T : Subalgebra R A)
Goal: ↑(S ⊓ T) = ↑S ∩ ↑T
-- Algebra.mem_inf
@[simp]
theorem mem_inf {S T : Subalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} {x : A}
Goal: x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T
-- Algebra.inf_toSubmodule
@[simp]
theorem inf_toSubmodule (S T : Subalgebra R A) :
    toSubmodule (S ⊓ T) = toSubmodule S ⊓ toSubmodule T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S T : Subalgebra R A)
Goal: Subalgebra.toSubmodule (S ⊓ T) = Subalgebra.toSubmodule S ⊓ Subalgebra.toSubmodule T
-- Algebra.inf_toSubsemiring
@[simp]
theorem inf_toSubsemiring (S T : Subalgebra R A) :
    (S ⊓ T).toSubsemiring = S.toSubsemiring ⊓ T.toSubsemiring 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S T : Subalgebra R A)
Goal: (S ⊓ T).toSubsemiring = S.toSubsemiring ⊓ T.toSubsemiring
-- Algebra.coe_sInf
@[simp, norm_cast]
theorem coe_sInf (S : Set (Subalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A))
Goal: ↑(sInf S) = ⋂ s ∈ S, ↑s
-- Algebra.mem_sInf
theorem mem_sInf {S : Set (Subalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S : Set (Subalgebra R A)} {x : A}
Goal: x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p
-- Algebra.sInf_toSubmodule
@[simp]
theorem sInf_toSubmodule (S : Set (Subalgebra R A)) :
    Subalgebra.toSubmodule (sInf S) = sInf (Subalgebra.toSubmodule '' S) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A))
Goal: Subalgebra.toSubmodule (sInf S) = sInf (⇑Subalgebra.toSubmodule '' S)
-- Algebra.sInf_toSubsemiring
@[simp]
theorem sInf_toSubsemiring (S : Set (Subalgebra R A)) :
    (sInf S).toSubsemiring = sInf (Subalgebra.toSubsemiring '' S) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A))
Goal: (sInf S).toSubsemiring = sInf (Subalgebra.toSubsemiring '' S)
-- Algebra.coe_iInf
@[simp, norm_cast]
theorem coe_iInf {ι : Sort*} {S : ι → Subalgebra R A} : (↑(⨅ i, S i) : Set A) = ⋂ i, S i 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {ι : Sort u_1} {S : ι → Subalgebra R A}
Goal: ↑(⨅ i, S i) = ⋂ i, ↑(S i)
-- Algebra.mem_iInf
theorem mem_iInf {ι : Sort*} {S : ι → Subalgebra R A} {x : A} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {ι : Sort u_1} {S : ι → Subalgebra R A} {x : A}
Goal: x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i
-- Algebra.iInf_toSubmodule
@[simp]
theorem iInf_toSubmodule {ι : Sort*} (S : ι → Subalgebra R A) :
    toSubmodule (⨅ i, S i) = ⨅ i, toSubmodule (S i) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {ι : Sort u_1} (S : ι → Subalgebra R A)
Goal: Subalgebra.toSubmodule (⨅ i, S i) = ⨅ i, Subalgebra.toSubmodule (S i)
-- Algebra.mem_bot
theorem mem_bot {x : A} : x ∈ (⊥ : Subalgebra R A) ↔ x ∈ Set.range (algebraMap R A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {x : A}
Goal: x ∈ ⊥ ↔ x ∈ Set.range ⇑(algebraMap R A)
-- Algebra.toSubmodule_bot
theorem toSubmodule_bot : Subalgebra.toSubmodule (⊥ : Subalgebra R A) = 1 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: Subalgebra.toSubmodule ⊥ = 1
-- Algebra.coe_bot
@[simp]
theorem coe_bot : ((⊥ : Subalgebra R A) : Set A) = Set.range (algebraMap R A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ↑⊥ = Set.range ⇑(algebraMap R A)
-- Algebra.eq_top_iff
theorem eq_top_iff {S : Subalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S : Subalgebra R A}
Goal: S = ⊤ ↔ ∀ (x : A), x ∈ S
-- Algebra.range_top_iff_surjective
theorem range_top_iff_surjective (f : A →ₐ[R] B) :
    f.range = (⊤ : Subalgebra R B) ↔ Function.Surjective f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B)
Goal: f.range = ⊤ ↔ Function.Surjective ⇑f
-- Algebra.range_id
@[simp]
theorem range_id : (AlgHom.id R A).range = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: (AlgHom.id R A).range = ⊤
-- Algebra.map_top
@[simp]
theorem map_top (f : A →ₐ[R] B) : (⊤ : Subalgebra R A).map f = f.range 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B)
Goal: Subalgebra.map f ⊤ = f.range
-- Algebra.map_bot
@[simp]
theorem map_bot (f : A →ₐ[R] B) : (⊥ : Subalgebra R A).map f = ⊥ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B)
Goal: Subalgebra.map f ⊥ = ⊥
-- Algebra.comap_top
@[simp]
theorem comap_top (f : A →ₐ[R] B) : (⊤ : Subalgebra R B).comap f = ⊤ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B)
Goal: Subalgebra.comap f ⊤ = ⊤
-- Algebra.surjective_algebraMap_iff
theorem surjective_algebraMap_iff :
    Function.Surjective (algebraMap R A) ↔ (⊤ : Subalgebra R A) = ⊥ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: Function.Surjective ⇑(algebraMap R A) ↔ ⊤ = ⊥
-- Algebra.bijective_algebraMap_iff
theorem bijective_algebraMap_iff {R A : Type*} [Field R] [Semiring A] [Nontrivial A]
    [Algebra R A] : Function.Bijective (algebraMap R A) ↔ (⊤ : Subalgebra R A) = ⊥ 
Arguments: {R : Type u_1} {A : Type u_2} [Field R] [Semiring A] [Nontrivial A] [Algebra R A]
Goal: Function.Bijective ⇑(algebraMap R A) ↔ ⊤ = ⊥
-- Algebra.botEquiv_symm_apply
@[simps! symm_apply]
noncomputable def botEquiv (F R : Type*) [Field F] [Semiring R] [Nontrivial R] [Algebra F R] :
    (⊥ : Subalgebra F R) ≃ₐ[F] F 
Arguments: (F : Type u_1) (R : Type u_2) [Field F] [Semiring R] [Nontrivial R] [Algebra F R]
Goal: ∀ (a : F), (Algebra.botEquiv F R).symm a = (Algebra.ofId F ↥⊥) a
-- Subalgebra.topEquiv_symm_apply_coe
@[simps!]
def topEquiv : (⊤ : Subalgebra R A) ≃ₐ[R] A 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (a : A)
Goal: ↑(Subalgebra.topEquiv.symm a) = a
-- Subalgebra.topEquiv_apply
@[simps!]
def topEquiv : (⊤ : Subalgebra R A) ≃ₐ[R] A 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (a : ↥⊤)
Goal: Subalgebra.topEquiv a = ↑a
-- Subalgebra.range_val
theorem range_val : S.val.range = S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: S.val.range = S
-- Subalgebra.inclusion_injective
theorem inclusion_injective {S T : Subalgebra R A} (h : S ≤ T) : Function.Injective (inclusion h) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} (h : S ≤ T)
Goal: Function.Injective ⇑(Subalgebra.inclusion h)
-- Subalgebra.inclusion_self
@[simp]
theorem inclusion_self {S : Subalgebra R A} : inclusion (le_refl S) = AlgHom.id R S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S : Subalgebra R A}
Goal: Subalgebra.inclusion ⋯ = AlgHom.id R ↥S
-- Subalgebra.inclusion_mk
@[simp]
theorem inclusion_mk {S T : Subalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :
    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S)
Goal: (Subalgebra.inclusion h) ⟨x, hx⟩ = ⟨x, ⋯⟩
-- Subalgebra.inclusion_right
theorem inclusion_right {S T : Subalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :
    inclusion h ⟨x, m⟩ = x 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} (h : S ≤ T) (x : ↥T) (m : ↑x ∈ S)
Goal: (Subalgebra.inclusion h) ⟨↑x, m⟩ = x
-- Subalgebra.inclusion_inclusion
@[simp]
theorem inclusion_inclusion {S T U : Subalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :
    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T U : Subalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : ↥S)
Goal: (Subalgebra.inclusion htu) ((Subalgebra.inclusion hst) x) = (Subalgebra.inclusion ⋯) x
-- Subalgebra.coe_inclusion
@[simp]
theorem coe_inclusion {S T : Subalgebra R A} (h : S ≤ T) (s : S) : (inclusion h s : A) = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {S T : Subalgebra R A} (h : S ≤ T) (s : ↥S)
Goal: ↑((Subalgebra.inclusion h) s) = ↑s
-- Subalgebra.equivOfEq_apply
@[simps apply]
def equivOfEq (S T : Subalgebra R A) (h : S = T) : S ≃ₐ[R] T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S T : Subalgebra R A) (h : S = T) (x : ↥S)
Goal: (S.equivOfEq T h) x = ⟨↑x, ⋯⟩
-- Subalgebra.equivOfEq_symm
@[simp]
theorem equivOfEq_symm (S T : Subalgebra R A) (h : S = T) :
    (equivOfEq S T h).symm = equivOfEq T S h.symm 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S T : Subalgebra R A) (h : S = T)
Goal: (S.equivOfEq T h).symm = T.equivOfEq S ⋯
-- Subalgebra.equivOfEq_rfl
@[simp]
theorem equivOfEq_rfl (S : Subalgebra R A) : equivOfEq S S rfl = AlgEquiv.refl 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: S.equivOfEq S ⋯ = AlgEquiv.refl
-- Subalgebra.equivOfEq_trans
@[simp]
theorem equivOfEq_trans (S T U : Subalgebra R A) (hST : S = T) (hTU : T = U) :
    (equivOfEq S T hST).trans (equivOfEq T U hTU) = equivOfEq S U (hST.trans hTU) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S T U : Subalgebra R A) (hST : S = T) (hTU : T = U)
Goal: (S.equivOfEq T hST).trans (T.equivOfEq U hTU) = S.equivOfEq U ⋯
-- Subalgebra.range_comp_val
theorem range_comp_val : (f.comp S.val).range = S.map f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B)
Goal: (f.comp S.val).range = Subalgebra.map f S
-- Subalgebra.coe_equivMapOfInjective_apply
@[simp]
theorem coe_equivMapOfInjective_apply (x : S) : ↑(equivMapOfInjective S f hf x) = f x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) (x : ↥S)
Goal: ↑((S.equivMapOfInjective f hf) x) = f ↑x
-- Subalgebra.smul_def
theorem smul_def [SMul A α] {S : Subalgebra R A} (g : S) (m : α) : g • m = (g : A) • m 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {α : Type u_1} [SMul A α] {S : Subalgebra R A} (g : ↥S) (m : α)
Goal: g • m = ↑g • m
-- Subalgebra.algebraMap_eq
theorem algebraMap_eq {R A : Type*} [CommSemiring R] [CommSemiring A] [Semiring α] [Algebra R A]
    [Algebra A α] (S : Subalgebra R A) : algebraMap S α = (algebraMap A α).comp S.val 
Arguments: {α : Type u_1} {R : Type u_3} {A : Type u_4} [CommSemiring R] [CommSemiring A] [Semiring α] [Algebra R A] [Algebra A α] (S : Subalgebra R A)
Goal: algebraMap (↥S) α = (algebraMap A α).comp ↑S.val
-- Subalgebra.rangeS_algebraMap
@[simp]
theorem rangeS_algebraMap {R A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S : Subalgebra R A) : (algebraMap S A).rangeS = S.toSubsemiring 
Arguments: {R : Type u_3} {A : Type u_4} [CommSemiring R] [CommSemiring A] [Algebra R A] (S : Subalgebra R A)
Goal: (algebraMap (↥S) A).rangeS = S.toSubsemiring
-- Subalgebra.range_algebraMap
@[simp]
theorem range_algebraMap {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]
    (S : Subalgebra R A) : (algebraMap S A).range = S.toSubring 
Arguments: {R : Type u_3} {A : Type u_4} [CommRing R] [CommRing A] [Algebra R A] (S : Subalgebra R A)
Goal: (algebraMap (↥S) A).range = S.toSubring
-- Set.algebraMap_mem_center
theorem _root_.Set.algebraMap_mem_center (r : R) : algebraMap R A r ∈ Set.center A 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (r : R)
Goal: (algebraMap R A) r ∈ Set.center A
-- Subalgebra.coe_center
theorem coe_center : (center R A : Set A) = Set.center A 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ↑(Subalgebra.center R A) = Set.center A
-- Subalgebra.center_toSubsemiring
@[simp]
theorem center_toSubsemiring : (center R A).toSubsemiring = Subsemiring.center A 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A]
Goal: (Subalgebra.center R A).toSubsemiring = Subsemiring.center A
-- Subalgebra.center_toSubring
@[simp]
theorem center_toSubring (R A : Type*) [CommRing R] [Ring A] [Algebra R A] :
    (center R A).toSubring = Subring.center A 
Arguments: (R : Type u_1) (A : Type u_2) [CommRing R] [Ring A] [Algebra R A]
Goal: (Subalgebra.center R A).toSubring = Subring.center A
-- Subalgebra.center_eq_top
@[simp]
theorem center_eq_top (A : Type*) [CommSemiring A] [Algebra R A] : center R A = ⊤ 
Arguments: (R : Type u) [CommSemiring R] (A : Type u_1) [CommSemiring A] [Algebra R A]
Goal: Subalgebra.center R A = ⊤
-- Subalgebra.mem_center_iff
theorem mem_center_iff {a : A} : a ∈ center R A ↔ ∀ b : A, b * a = a * b 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {a : A}
Goal: a ∈ Subalgebra.center R A ↔ ∀ (b : A), b * a = a * b
-- Set.algebraMap_mem_centralizer
@[simp]
theorem _root_.Set.algebraMap_mem_centralizer {s : Set A} (r : R) :
    algebraMap R A r ∈ s.centralizer 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {s : Set A} (r : R)
Goal: (algebraMap R A) r ∈ s.centralizer
-- Subalgebra.coe_centralizer
@[simp, norm_cast]
theorem coe_centralizer (s : Set A) : (centralizer R s : Set A) = s.centralizer 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (s : Set A)
Goal: ↑(Subalgebra.centralizer R s) = s.centralizer
-- Subalgebra.mem_centralizer_iff
theorem mem_centralizer_iff {s : Set A} {z : A} : z ∈ centralizer R s ↔ ∀ g ∈ s, g * z = z * g 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {s : Set A} {z : A}
Goal: z ∈ Subalgebra.centralizer R s ↔ ∀ g ∈ s, g * z = z * g
-- Subalgebra.center_le_centralizer
theorem center_le_centralizer (s) : center R A ≤ centralizer R s 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (s : Set A)
Goal: Subalgebra.center R A ≤ Subalgebra.centralizer R s
-- Subalgebra.centralizer_le
theorem centralizer_le (s t : Set A) (h : s ⊆ t) : centralizer R t ≤ centralizer R s 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (s t : Set A) (h : s ⊆ t)
Goal: Subalgebra.centralizer R t ≤ Subalgebra.centralizer R s
-- Subalgebra.centralizer_eq_top_iff_subset
@[simp]
theorem centralizer_eq_top_iff_subset {s : Set A} : centralizer R s = ⊤ ↔ s ⊆ center R A 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {s : Set A}
Goal: Subalgebra.centralizer R s = ⊤ ↔ s ⊆ ↑(Subalgebra.center R A)
-- Subalgebra.centralizer_univ
@[simp]
theorem centralizer_univ : centralizer R Set.univ = center R A 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: Subalgebra.centralizer R Set.univ = Subalgebra.center R A
-- Subalgebra.le_centralizer_centralizer
lemma le_centralizer_centralizer {s : Subalgebra R A} :
    s ≤ centralizer R (centralizer R (s : Set A)) 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {s : Subalgebra R A}
Goal: s ≤ Subalgebra.centralizer R ↑(Subalgebra.centralizer R ↑s)
-- Subalgebra.centralizer_centralizer_centralizer
@[simp]
lemma centralizer_centralizer_centralizer {s : Set A} :
    centralizer R s.centralizer.centralizer = centralizer R s 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {s : Set A}
Goal: Subalgebra.centralizer R s.centralizer.centralizer = Subalgebra.centralizer R s
-- mem_subalgebraOfSubsemiring
@[simp]
theorem mem_subalgebraOfSubsemiring {x : R} {S : Subsemiring R} :
    x ∈ subalgebraOfSubsemiring S ↔ x ∈ S 
Arguments: {R : Type u_1} [Semiring R] {x : R} {S : Subsemiring R}
Goal: x ∈ subalgebraOfSubsemiring S ↔ x ∈ S
-- mem_subalgebraOfSubring
@[simp]
theorem mem_subalgebraOfSubring {x : R} {S : Subring R} : x ∈ subalgebraOfSubring S ↔ x ∈ S 
Arguments: {R : Type u_1} [Ring R] {x : R} {S : Subring R}
Goal: x ∈ subalgebraOfSubring S ↔ x ∈ S
-- AlgHom.mem_equalizer
@[simp]
theorem mem_equalizer (φ ψ : F) (x : A) : x ∈ equalizer φ ψ ↔ φ x = ψ x 
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 : Type u_4} [FunLike F A B] [AlgHomClass F R A B] (φ ψ : F) (x : A)
Goal: x ∈ AlgHom.equalizer φ ψ ↔ φ x = ψ x
-- AlgHom.equalizer_toSubmodule
theorem equalizer_toSubmodule {φ ψ : F} :
    Subalgebra.toSubmodule (equalizer φ ψ) = LinearMap.eqLocus φ ψ 
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 : Type u_4} [FunLike F A B] [AlgHomClass F R A B] {φ ψ : F}
Goal: Subalgebra.toSubmodule (AlgHom.equalizer φ ψ) = LinearMap.eqLocus φ ψ