ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NonUnitalSubalgebra.starClosure_eq_adjoin
theorem _root_.NonUnitalSubalgebra.starClosure_eq_adjoin (S : NonUnitalSubalgebra R A) :
    S.starClosure = adjoin R (S : Set A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (S : NonUnitalSubalgebra R A)
Goal: S.starClosure = NonUnitalStarAlgebra.adjoin R ↑S
-- NonUnitalStarAlgebra.coe_top
@[simp]
theorem coe_top : ((⊤ : NonUnitalStarSubalgebra R A) : Set A) = Set.univ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: ↑⊤ = Set.univ
-- NonUnitalStarAlgebra.mem_top
@[simp]
theorem mem_top {x : A} : x ∈ (⊤ : NonUnitalStarSubalgebra R A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {x : A}
Goal: x ∈ ⊤
-- NonUnitalStarAlgebra.top_toNonUnitalSubalgebra
@[simp]
theorem top_toNonUnitalSubalgebra :
    (⊤ : NonUnitalStarSubalgebra R A).toNonUnitalSubalgebra = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: ⊤.toNonUnitalSubalgebra = ⊤
-- NonUnitalStarAlgebra.toNonUnitalSubalgebra_eq_top
@[simp]
theorem toNonUnitalSubalgebra_eq_top {S : NonUnitalStarSubalgebra R A} :
    S.toNonUnitalSubalgebra = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S : NonUnitalStarSubalgebra R A}
Goal: S.toNonUnitalSubalgebra = ⊤ ↔ S = ⊤
-- NonUnitalStarAlgebra.mem_sup_left
theorem mem_sup_left {S T : NonUnitalStarSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} {x : A}
Goal: x ∈ S → x ∈ S ⊔ T
-- NonUnitalStarAlgebra.mem_sup_right
theorem mem_sup_right {S T : NonUnitalStarSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} {x : A}
Goal: x ∈ T → x ∈ S ⊔ T
-- NonUnitalStarAlgebra.mul_mem_sup
theorem mul_mem_sup {S T : NonUnitalStarSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :
    x * y ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T)
Goal: x * y ∈ S ⊔ T
-- NonUnitalStarAlgebra.map_sup
theorem map_sup (f : F) (S T : NonUnitalStarSubalgebra R A) :
    ((S ⊔ T).map f : NonUnitalStarSubalgebra R B) = S.map f ⊔ T.map f 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F) (S T : NonUnitalStarSubalgebra R A)
Goal: NonUnitalStarSubalgebra.map f (S ⊔ T) = NonUnitalStarSubalgebra.map f S ⊔ NonUnitalStarSubalgebra.map f T
-- NonUnitalStarAlgebra.coe_inf
@[simp, norm_cast]
theorem coe_inf (S T : NonUnitalStarSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (S T : NonUnitalStarSubalgebra R A)
Goal: ↑(S ⊓ T) = ↑S ∩ ↑T
-- NonUnitalStarAlgebra.mem_inf
@[simp]
theorem mem_inf {S T : NonUnitalStarSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} {x : A}
Goal: x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T
-- NonUnitalStarAlgebra.inf_toNonUnitalSubalgebra
@[simp]
theorem inf_toNonUnitalSubalgebra (S T : NonUnitalStarSubalgebra R A) :
    (S ⊓ T).toNonUnitalSubalgebra = S.toNonUnitalSubalgebra ⊓ T.toNonUnitalSubalgebra 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (S T : NonUnitalStarSubalgebra R A)
Goal: (S ⊓ T).toNonUnitalSubalgebra = S.toNonUnitalSubalgebra ⊓ T.toNonUnitalSubalgebra
-- NonUnitalStarAlgebra.coe_sInf
@[simp, norm_cast]
theorem coe_sInf (S : Set (NonUnitalStarSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (S : Set (NonUnitalStarSubalgebra R A))
Goal: ↑(sInf S) = ⋂ s ∈ S, ↑s
-- NonUnitalStarAlgebra.mem_sInf
theorem mem_sInf {S : Set (NonUnitalStarSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S : Set (NonUnitalStarSubalgebra R A)} {x : A}
Goal: x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p
-- NonUnitalStarAlgebra.sInf_toNonUnitalSubalgebra
@[simp]
theorem sInf_toNonUnitalSubalgebra (S : Set (NonUnitalStarSubalgebra R A)) :
    (sInf S).toNonUnitalSubalgebra = sInf (NonUnitalStarSubalgebra.toNonUnitalSubalgebra '' S) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (S : Set (NonUnitalStarSubalgebra R A))
Goal: (sInf S).toNonUnitalSubalgebra = sInf (NonUnitalStarSubalgebra.toNonUnitalSubalgebra '' S)
-- NonUnitalStarAlgebra.coe_iInf
@[simp, norm_cast]
theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} :
    (↑(⨅ i, S i) : Set A) = ⋂ i, S i 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {ι : Sort u_1} {S : ι → NonUnitalStarSubalgebra R A}
Goal: ↑(⨅ i, S i) = ⋂ i, ↑(S i)
-- NonUnitalStarAlgebra.mem_iInf
theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} {x : A} :
    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {ι : Sort u_1} {S : ι → NonUnitalStarSubalgebra R A} {x : A}
Goal: x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i
-- NonUnitalStarAlgebra.iInf_toNonUnitalSubalgebra
@[simp]
theorem iInf_toNonUnitalSubalgebra {ι : Sort*} (S : ι → NonUnitalStarSubalgebra R A) :
    (⨅ i, S i).toNonUnitalSubalgebra = ⨅ i, (S i).toNonUnitalSubalgebra 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {ι : Sort u_1} (S : ι → NonUnitalStarSubalgebra R A)
Goal: (⨅ i, S i).toNonUnitalSubalgebra = ⨅ i, (S i).toNonUnitalSubalgebra
-- NonUnitalStarAlgebra.mem_bot
theorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalStarSubalgebra R A) ↔ x = 0 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {x : A}
Goal: x ∈ ⊥ ↔ x = 0
-- NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot
theorem toNonUnitalSubalgebra_bot :
    (⊥ : NonUnitalStarSubalgebra R A).toNonUnitalSubalgebra = ⊥ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: ⊥.toNonUnitalSubalgebra = ⊥
-- NonUnitalStarAlgebra.coe_bot
@[simp]
theorem coe_bot : ((⊥ : NonUnitalStarSubalgebra R A) : Set A) = {0} 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: ↑⊥ = {0}
-- NonUnitalStarAlgebra.eq_top_iff
theorem eq_top_iff {S : NonUnitalStarSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S : NonUnitalStarSubalgebra R A}
Goal: S = ⊤ ↔ ∀ (x : A), x ∈ S
-- NonUnitalStarAlgebra.range_top_iff_surjective
theorem range_top_iff_surjective (f : F) :
    NonUnitalStarAlgHom.range f = (⊤ : NonUnitalStarSubalgebra R B) ↔ Function.Surjective f 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F)
Goal: NonUnitalStarAlgHom.range f = ⊤ ↔ Function.Surjective ⇑f
-- NonUnitalStarAlgebra.range_id
@[simp]
theorem range_id : NonUnitalStarAlgHom.range (NonUnitalStarAlgHom.id R A) = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: NonUnitalStarAlgHom.range (NonUnitalStarAlgHom.id R A) = ⊤
-- NonUnitalStarAlgebra.map_top
@[simp]
theorem map_top (f : F) : (⊤ : NonUnitalStarSubalgebra R A).map f = NonUnitalStarAlgHom.range f 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F)
Goal: NonUnitalStarSubalgebra.map f ⊤ = NonUnitalStarAlgHom.range f
-- NonUnitalStarAlgebra.map_bot
@[simp]
theorem map_bot (f : F) : (⊥ : NonUnitalStarSubalgebra R A).map f = ⊥ 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F)
Goal: NonUnitalStarSubalgebra.map f ⊥ = ⊥
-- NonUnitalStarAlgebra.comap_top
@[simp]
theorem comap_top (f : F) : (⊤ : NonUnitalStarSubalgebra R B).comap f = ⊤ 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F)
Goal: NonUnitalStarSubalgebra.comap f ⊤ = ⊤
-- NonUnitalStarAlgHom.map_adjoin
lemma _root_.NonUnitalStarAlgHom.map_adjoin (f : F) (s : Set A) :
    map f (adjoin R s) = adjoin R (f '' s) 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F) (s : Set A)
Goal: NonUnitalStarSubalgebra.map f (NonUnitalStarAlgebra.adjoin R s) = NonUnitalStarAlgebra.adjoin R (⇑f '' s)
-- NonUnitalStarAlgHom.map_adjoin_singleton
@[simp]
lemma _root_.NonUnitalStarAlgHom.map_adjoin_singleton (f : F) (x : A) :
    map f (adjoin R {x}) = adjoin R {f x} 
Arguments: {F : Type v'} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B] (f : F) (x : A)
Goal: NonUnitalStarSubalgebra.map f (NonUnitalStarAlgebra.adjoin R {x}) = NonUnitalStarAlgebra.adjoin R {f x}
-- NonUnitalStarSubalgebra.range_val
theorem range_val : NonUnitalStarAlgHom.range (NonUnitalStarSubalgebraClass.subtype S) = S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] (S : NonUnitalStarSubalgebra R A)
Goal: NonUnitalStarAlgHom.range (NonUnitalStarSubalgebraClass.subtype S) = S
-- NonUnitalStarSubalgebra.inclusion_injective
theorem inclusion_injective {S T : NonUnitalStarSubalgebra R A} (h : S ≤ T) :
    Function.Injective (inclusion h) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} (h : S ≤ T)
Goal: Function.Injective ⇑(NonUnitalStarSubalgebra.inclusion h)
-- NonUnitalStarSubalgebra.inclusion_self
@[simp]
theorem inclusion_self {S : NonUnitalStarSubalgebra R A} :
    inclusion (le_refl S) = NonUnitalAlgHom.id R S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S : NonUnitalStarSubalgebra R A}
Goal: NonUnitalAlgHomClass.toNonUnitalAlgHom (NonUnitalStarSubalgebra.inclusion ⋯) = NonUnitalAlgHom.id R ↥S
-- NonUnitalStarSubalgebra.inclusion_mk
@[simp]
theorem inclusion_mk {S T : NonUnitalStarSubalgebra 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] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S)
Goal: (NonUnitalStarSubalgebra.inclusion h) ⟨x, hx⟩ = ⟨x, ⋯⟩
-- NonUnitalStarSubalgebra.inclusion_right
theorem inclusion_right {S T : NonUnitalStarSubalgebra 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] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} (h : S ≤ T) (x : ↥T) (m : ↑x ∈ S)
Goal: (NonUnitalStarSubalgebra.inclusion h) ⟨↑x, m⟩ = x
-- NonUnitalStarSubalgebra.inclusion_inclusion
@[simp]
theorem inclusion_inclusion {S T U : NonUnitalStarSubalgebra 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] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T U : NonUnitalStarSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : ↥S)
Goal: (NonUnitalStarSubalgebra.inclusion htu) ((NonUnitalStarSubalgebra.inclusion hst) x) =
  (NonUnitalStarSubalgebra.inclusion ⋯) x
-- NonUnitalStarSubalgebra.val_inclusion
@[simp]
theorem val_inclusion {S T : NonUnitalStarSubalgebra R A} (h : S ≤ T) (s : S) :
    (inclusion h s : A) = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {S T : NonUnitalStarSubalgebra R A} (h : S ≤ T) (s : ↥S)
Goal: ↑((NonUnitalStarSubalgebra.inclusion h) s) = ↑s
-- NonUnitalStarSubalgebra.coe_prod
@[simp]
theorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [NonUnitalSemiring B] [StarRing B] [Module R B] (S : NonUnitalStarSubalgebra R A) (S₁ : NonUnitalStarSubalgebra R B)
Goal: ↑(S.prod S₁) = ↑S ×ˢ ↑S₁
-- NonUnitalStarSubalgebra.prod_toNonUnitalSubalgebra
theorem prod_toNonUnitalSubalgebra :
    (S.prod S₁).toNonUnitalSubalgebra = S.toNonUnitalSubalgebra.prod S₁.toNonUnitalSubalgebra 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [NonUnitalSemiring B] [StarRing B] [Module R B] (S : NonUnitalStarSubalgebra R A) (S₁ : NonUnitalStarSubalgebra R B)
Goal: (S.prod S₁).toNonUnitalSubalgebra = S.prod S₁.toNonUnitalSubalgebra
-- NonUnitalStarSubalgebra.mem_prod
@[simp]
theorem mem_prod {S : NonUnitalStarSubalgebra R A} {S₁ : NonUnitalStarSubalgebra R B} {x : A × B} :
    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [NonUnitalSemiring B] [StarRing B] [Module R B] {S : NonUnitalStarSubalgebra R A} {S₁ : NonUnitalStarSubalgebra R B} {x : A × B}
Goal: x ∈ S.prod S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁
-- NonUnitalStarSubalgebra.prod_top
@[simp]
theorem prod_top : (prod ⊤ ⊤ : NonUnitalStarSubalgebra R (A × B)) = ⊤ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B]
Goal: ⊤.prod ⊤ = ⊤
-- NonUnitalStarSubalgebra.prod_mono
theorem prod_mono {S T : NonUnitalStarSubalgebra R A} {S₁ T₁ : NonUnitalStarSubalgebra R B} :
    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] {S T : NonUnitalStarSubalgebra R A} {S₁ T₁ : NonUnitalStarSubalgebra R B}
Goal: S ≤ T → S₁ ≤ T₁ → S.prod S₁ ≤ T.prod T₁
-- NonUnitalStarSubalgebra.prod_inf_prod
@[simp]
theorem prod_inf_prod {S T : NonUnitalStarSubalgebra R A} {S₁ T₁ : NonUnitalStarSubalgebra R B} :
    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] {S T : NonUnitalStarSubalgebra R A} {S₁ T₁ : NonUnitalStarSubalgebra R B}
Goal: S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁)
-- NonUnitalStarSubalgebra.coe_iSup_of_directed
theorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalStarSubalgebra R A}
    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] {ι : Type u_1} [Nonempty ι] {S : ι → NonUnitalStarSubalgebra R A} (dir : Directed (fun x x_1 => x ≤ x_1) S)
Goal: ↑(iSup S) = ⋃ i, ↑(S i)
-- NonUnitalStarSubalgebra.iSupLift_inclusion
@[simp]
theorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :
    iSupLift K dir f hf T hT (inclusion h x) = f i x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] {ι : Type u_1} [Nonempty ι] {K : ι → NonUnitalStarSubalgebra R A} {dir : Directed (fun x x_1 => x ≤ x_1) K} {f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (NonUnitalStarSubalgebra.inclusion h)} {T : NonUnitalStarSubalgebra R A} {hT : T = iSup K} {i : ι} (x : ↥(K i)) (h : K i ≤ T)
Goal: (NonUnitalStarSubalgebra.iSupLift K dir f hf T hT) ((NonUnitalStarSubalgebra.inclusion h) x) = (f i) x
-- NonUnitalStarSubalgebra.iSupLift_comp_inclusion
@[simp]
theorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :
    (iSupLift K dir f hf T hT).comp (inclusion h) = f i 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] {ι : Type u_1} [Nonempty ι] {K : ι → NonUnitalStarSubalgebra R A} {dir : Directed (fun x x_1 => x ≤ x_1) K} {f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (NonUnitalStarSubalgebra.inclusion h)} {T : NonUnitalStarSubalgebra R A} {hT : T = iSup K} {i : ι} (h : K i ≤ T)
Goal: (NonUnitalStarSubalgebra.iSupLift K dir f hf T hT).comp (NonUnitalStarSubalgebra.inclusion h) = f i
-- NonUnitalStarSubalgebra.iSupLift_mk
@[simp]
theorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :
    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] {ι : Type u_1} [Nonempty ι] {K : ι → NonUnitalStarSubalgebra R A} {dir : Directed (fun x x_1 => x ≤ x_1) K} {f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (NonUnitalStarSubalgebra.inclusion h)} {T : NonUnitalStarSubalgebra R A} {hT : T = iSup K} {i : ι} (x : ↥(K i)) (hx : ↑x ∈ T)
Goal: (NonUnitalStarSubalgebra.iSupLift K dir f hf T hT) ⟨↑x, hx⟩ = (f i) x
-- NonUnitalStarSubalgebra.iSupLift_of_mem
theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :
    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] [NonUnitalSemiring B] [StarRing B] [Module R B] {ι : Type u_1} [Nonempty ι] {K : ι → NonUnitalStarSubalgebra R A} {dir : Directed (fun x x_1 => x ≤ x_1) K} {f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (NonUnitalStarSubalgebra.inclusion h)} {T : NonUnitalStarSubalgebra R A} {hT : T = iSup K} {i : ι} (x : ↥T) (hx : ↑x ∈ K i)
Goal: (NonUnitalStarSubalgebra.iSupLift K dir f hf T hT) x = (f i) ⟨↑x, hx⟩
-- NonUnitalStarSubalgebra.coe_center
theorem coe_center : (center R A : Set A) = Set.center A 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ↑(NonUnitalStarSubalgebra.center R A) = Set.center A
-- NonUnitalStarSubalgebra.center_toNonUnitalSubalgebra
@[simp]
theorem center_toNonUnitalSubalgebra :
    (center R A).toNonUnitalSubalgebra = NonUnitalSubalgebra.center R A 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: (NonUnitalStarSubalgebra.center R A).toNonUnitalSubalgebra = NonUnitalSubalgebra.center R A
-- NonUnitalStarSubalgebra.center_eq_top
@[simp]
theorem center_eq_top (A : Type*) [NonUnitalCommSemiring A] [StarRing A] [Module R A]
    [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] : center R A = ⊤ 
Arguments: (R : Type u) [CommSemiring R] [StarRing R] (A : Type u_1) [NonUnitalCommSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A]
Goal: NonUnitalStarSubalgebra.center R A = ⊤