l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem top_toNonUnitalSubalgebra :
    (⊤ : NonUnitalStarSubalgebra R A).toNonUnitalSubalgebra = ⊤ := by ext; 

Proof state:
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
x✝ : A
⊢ x✝ ∈ ⊤.toNonUnitalSubalgebra ↔ x✝ ∈ ⊤

Next tactic:
simp
theorem mem_sup_left {S T : NonUnitalStarSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ ∀ {x : A}, x ∈ S → x ∈ S ⊔ T

Next tactic:
rw [← SetLike.le_def]
theorem mem_sup_left {S T : NonUnitalStarSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by
  rw [← SetLike.le_def]
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ S ≤ S ⊔ T

Next tactic:
exact le_sup_left
theorem mem_sup_right {S T : NonUnitalStarSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T

Next tactic:
rw [← SetLike.le_def]
theorem mem_sup_right {S T : NonUnitalStarSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by
  rw [← SetLike.le_def]
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ T ≤ S ⊔ T

Next tactic:
exact le_sup_right
theorem mem_sInf {S : Set (NonUnitalStarSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : Set (NonUnitalStarSubalgebra R A)
x : A
⊢ x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

Next tactic:
simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]
@[simp]
theorem sInf_toNonUnitalSubalgebra (S : Set (NonUnitalStarSubalgebra R A)) :
    (sInf S).toNonUnitalSubalgebra = sInf (NonUnitalStarSubalgebra.toNonUnitalSubalgebra '' S) :=
  SetLike.coe_injective <| by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : Set (NonUnitalStarSubalgebra R A)
⊢ ↑(sInf S).toNonUnitalSubalgebra = ↑(sInf (toNonUnitalSubalgebra '' S))

Next tactic:
simp
@[simp, norm_cast]
theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} :
    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
ι : Sort u_1
S : ι → NonUnitalStarSubalgebra R A
⊢ ↑(⨅ i, S i) = ⋂ i, ↑(S i)

Next tactic:
simp [iInf]
theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} {x : A} :
    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
ι : Sort u_1
S : ι → NonUnitalStarSubalgebra R A
x : A
⊢ x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i

Next tactic:
simp only [iInf, mem_sInf, Set.forall_range_iff]
@[simp]
theorem iInf_toNonUnitalSubalgebra {ι : Sort*} (S : ι → NonUnitalStarSubalgebra R A) :
    (⨅ i, S i).toNonUnitalSubalgebra = ⨅ i, (S i).toNonUnitalSubalgebra :=
  SetLike.coe_injective <| by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
ι : Sort u_1
S : ι → NonUnitalStarSubalgebra R A
⊢ ↑(⨅ i, S i).toNonUnitalSubalgebra = ↑(⨅ i, (S i).toNonUnitalSubalgebra)

Next tactic:
simp
theorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalStarSubalgebra R A) ↔ x = 0 :=
  show x ∈ NonUnitalAlgebra.adjoin R (∅ ∪ star ∅ : Set A) ↔ x = 0 by
    

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
x : A
⊢ x ∈ NonUnitalAlgebra.adjoin R (∅ ∪ star ∅) ↔ x = 0

Next tactic:
rw [Set.star_empty, Set.union_empty, NonUnitalAlgebra.adjoin_empty, NonUnitalAlgebra.mem_bot]
theorem toNonUnitalSubalgebra_bot :
    (⊥ : NonUnitalStarSubalgebra R A).toNonUnitalSubalgebra = ⊥ := by
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
⊢ ⊥.toNonUnitalSubalgebra = ⊥

Next tactic:
ext x
theorem toNonUnitalSubalgebra_bot :
    (⊥ : NonUnitalStarSubalgebra R A).toNonUnitalSubalgebra = ⊥ := by
  ext x
  

Proof state:
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
x : A
⊢ x ∈ ⊥.toNonUnitalSubalgebra ↔ x ∈ ⊥

Next tactic:
simp only [mem_bot, NonUnitalStarSubalgebra.mem_toNonUnitalSubalgebra, NonUnitalAlgebra.mem_bot]
@[simp]
theorem coe_bot : ((⊥ : NonUnitalStarSubalgebra R A) : Set A) = {0} := by
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
⊢ ↑⊥ = {0}

Next tactic:
simp only [Set.ext_iff, NonUnitalStarAlgebra.mem_bot, SetLike.mem_coe, Set.mem_singleton_iff,
    iff_self_iff, forall_const]
theorem eq_top_iff {S : NonUnitalStarSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=
  ⟨fun h x => by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : S = ⊤
x : A
⊢ x ∈ S

Next tactic:
rw [h]
theorem eq_top_iff {S : NonUnitalStarSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=
  ⟨fun h x => by rw [h]; 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : S = ⊤
x : A
⊢ x ∈ ⊤

Next tactic:
exact mem_top
theorem eq_top_iff {S : NonUnitalStarSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=
  ⟨fun h x => by rw [h]; exact mem_top,
    fun h => by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : ∀ (x : A), x ∈ S
⊢ S = ⊤

Next tactic:
ext x
theorem eq_top_iff {S : NonUnitalStarSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=
  ⟨fun h x => by rw [h]; exact mem_top,
    fun h => by ext x; 

Proof state:
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
h : ∀ (x : A), x ∈ S
x : A
⊢ x ∈ S ↔ x ∈ ⊤

Next tactic:
exact ⟨fun _ => mem_top, fun _ => h x⟩
@[simp]
theorem map_bot (f : F) : (⊥ : NonUnitalStarSubalgebra R A).map f = ⊥ :=
  SetLike.coe_injective <| by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
f : F
⊢ ↑(map f ⊥) = ↑⊥

Next tactic:
simp [NonUnitalAlgebra.coe_bot, NonUnitalStarSubalgebra.coe_map]
instance subsingleton_of_subsingleton [Subsingleton A] :
    Subsingleton (NonUnitalStarSubalgebra R A) :=
  ⟨fun B C => ext fun x => by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B✝ : Type w
C✝ : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B✝
inst✝⁵ : StarRing B✝
inst✝⁴ : Module R B✝
inst✝³ : IsScalarTower R B✝ B✝
inst✝² : SMulCommClass R B✝ B✝
inst✝¹ : StarModule R B✝
hF : NonUnitalStarAlgHomClass F R A B✝
S : NonUnitalStarSubalgebra R A
inst✝ : Subsingleton A
B C : NonUnitalStarSubalgebra R A
x : A
⊢ x ∈ B ↔ x ∈ C

Next tactic:
simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]
@[simp]
theorem prod_top : (prod ⊤ ⊤ : NonUnitalStarSubalgebra R (A × B)) = ⊤ := by 

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
S₁ : NonUnitalStarSubalgebra R B
⊢ prod ⊤ ⊤ = ⊤

Next tactic:
ext
@[simp]
theorem prod_top : (prod ⊤ ⊤ : NonUnitalStarSubalgebra R (A × B)) = ⊤ := by ext; 

Proof state:
case h
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : NonUnitalSemiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R A A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : StarModule R A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
S₁ : NonUnitalStarSubalgebra R B
x✝ : A × B
⊢ x✝ ∈ prod ⊤ ⊤ ↔ x✝ ∈ ⊤

Next tactic:
simp
/-- Define a non-unital star algebra homomorphism on a directed supremum of non-unital star
subalgebras by defining it on each non-unital star subalgebra, and proving that it agrees on the
intersection of non-unital star subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalStarSubalgebra R A)
    (dir : Directed (· ≤ ·) K) (f : ∀ i, K i →⋆ₙₐ[R] B)
    (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))
    (T : NonUnitalStarSubalgebra R A) (hT : T = iSup K) : ↥T →⋆ₙₐ[R] B := by
  

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : 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 = NonUnitalStarAlgHom.comp (f j) (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
⊢ ↥T →⋆ₙₐ[R] B

Next tactic:
subst hT
/-- Define a non-unital star algebra homomorphism on a directed supremum of non-unital star
subalgebras by defining it on each non-unital star subalgebra, and proving that it agrees on the
intersection of non-unital star subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalStarSubalgebra R A)
    (dir : Directed (· ≤ ·) K) (f : ∀ i, K i →⋆ₙₐ[R] B)
    (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))
    (T : NonUnitalStarSubalgebra R A) (hT : T = iSup K) : ↥T →⋆ₙₐ[R] B := by
  subst hT
  exact
    { toFun :=
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)
          (fun i j x hxi hxj => by
            

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : 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 = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
⊢ (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj }

Next tactic:
let ⟨k, hik, hjk⟩ := dir i j
/-- Define a non-unital star algebra homomorphism on a directed supremum of non-unital star
subalgebras by defining it on each non-unital star subalgebra, and proving that it agrees on the
intersection of non-unital star subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalStarSubalgebra R A)
    (dir : Directed (· ≤ ·) K) (f : ∀ i, K i →⋆ₙₐ[R] B)
    (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))
    (T : NonUnitalStarSubalgebra R A) (hT : T = iSup K) : ↥T →⋆ₙₐ[R] B := by
  subst hT
  exact
    { toFun :=
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)
          (fun i j x hxi hxj => by
            let ⟨k, hik, hjk⟩ := dir i j
            

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : 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 = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
k : ι
hik : (fun x x_1 => x ≤ x_1) (K i) (K k)
hjk : (fun x x_1 => x ≤ x_1) (K j) (K k)
⊢ (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj }

Next tactic:
simp only
/-- Define a non-unital star algebra homomorphism on a directed supremum of non-unital star
subalgebras by defining it on each non-unital star subalgebra, and proving that it agrees on the
intersection of non-unital star subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalStarSubalgebra R A)
    (dir : Directed (· ≤ ·) K) (f : ∀ i, K i →⋆ₙₐ[R] B)
    (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))
    (T : NonUnitalStarSubalgebra R A) (hT : T = iSup K) : ↥T →⋆ₙₐ[R] B := by
  subst hT
  exact
    { toFun :=
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)
          (fun i j x hxi hxj => by
            let ⟨k, hik, hjk⟩ := dir i j
            simp only
            

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : 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 = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
k : ι
hik : (fun x x_1 => x ≤ x_1) (K i) (K k)
hjk : (fun x x_1 => x ≤ x_1) (K j) (K k)
⊢ (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj }

Next tactic:
rw [hf i k hik, hf j k hjk]
/-- Define a non-unital star algebra homomorphism on a directed supremum of non-unital star
subalgebras by defining it on each non-unital star subalgebra, and proving that it agrees on the
intersection of non-unital star subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalStarSubalgebra R A)
    (dir : Directed (· ≤ ·) K) (f : ∀ i, K i →⋆ₙₐ[R] B)
    (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))
    (T : NonUnitalStarSubalgebra R A) (hT : T = iSup K) : ↥T →⋆ₙₐ[R] B := by
  subst hT
  exact
    { toFun :=
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)
          (fun i j x hxi hxj => by
            let ⟨k, hik, hjk⟩ := dir i j
            simp only
            rw [hf i k hik, hf j k hjk]
            

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : 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 = NonUnitalStarAlgHom.comp (f j) (inclusion h)
i j : ι
x : A
hxi : x ∈ (fun i => ↑(K i)) i
hxj : x ∈ (fun i => ↑(K i)) j
k : ι
hik : (fun x x_1 => x ≤ x_1) (K i) (K k)
hjk : (fun x x_1 => x ≤ x_1) (K j) (K k)
⊢ (NonUnitalStarAlgHom.comp (f k) (inclusion hik)) { val := x, property := hxi } =
    (NonUnitalStarAlgHom.comp (f k) (inclusion hjk)) { val := x, property := hxj }

Next tactic:
rfl