l3lab/ntp-mathlib huggingface dataset:
theorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by
  

Proof state:
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
⊢ NonUnitalSubalgebra.toSubmodule ⊥ = ⊥

Next tactic:
ext
theorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by
  ext
  

Proof state:
case h
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
x✝ : A
⊢ x✝ ∈ NonUnitalSubalgebra.toSubmodule ⊥ ↔ x✝ ∈ ⊥

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

Proof state:
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
⊢ ↑⊥ = {0}

Next tactic:
simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]
theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=
  ⟨fun h x => by 

Proof state:
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
S : NonUnitalSubalgebra R A
h : S = ⊤
x : A
⊢ x ∈ S

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

Proof state:
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
S : NonUnitalSubalgebra R A
h : S = ⊤
x : A
⊢ x ∈ ⊤

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

Proof state:
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
S : NonUnitalSubalgebra R A
h : ∀ (x : A), x ∈ S
⊢ S = ⊤

Next tactic:
ext x
theorem eq_top_iff {S : NonUnitalSubalgebra 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 u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
S : NonUnitalSubalgebra 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 : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=
  SetLike.coe_injective <| by 

Proof state:
F : Type u_1
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
inst✝ : NonUnitalAlgHomClass F R A B
f : A →ₙₐ[R] B
⊢ ↑(NonUnitalSubalgebra.map f ⊥) = ↑⊥

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

Proof state:
R : Type u
A : Type v
B✝ : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B✝
inst✝³ : Module R B✝
inst✝² : IsScalarTower R B✝ B✝
inst✝¹ : SMulCommClass R B✝ B✝
S : NonUnitalSubalgebra R A
inst✝ : Subsingleton A
B C : NonUnitalSubalgebra 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 ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by 

Proof state:
R : Type u
A : Type v
B : Type w
inst✝⁸ : CommSemiring R
inst✝⁷ : NonUnitalNonAssocSemiring A
inst✝⁶ : Module R A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : NonUnitalNonAssocSemiring B
inst✝² : Module R B
inst✝¹ : IsScalarTower R B B
inst✝ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
S₁ : NonUnitalSubalgebra R B
⊢ prod ⊤ ⊤ = ⊤

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

Proof state:
case h
R : Type u
A : Type v
B : Type w
inst✝⁸ : CommSemiring R
inst✝⁷ : NonUnitalNonAssocSemiring A
inst✝⁶ : Module R A
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : NonUnitalNonAssocSemiring B
inst✝² : Module R B
inst✝¹ : IsScalarTower R B B
inst✝ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
S₁ : NonUnitalSubalgebra R B
x✝ : A × B
⊢ x✝ ∈ prod ⊤ ⊤ ↔ x✝ ∈ ⊤

Next tactic:
simp
/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by
  

Proof state:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.comp (f j) (inclusion h)
T : NonUnitalSubalgebra R A
hT : T = iSup K
⊢ ↥T →ₙₐ[R] B

Next tactic:
subst hT
/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by
  subst hT
  

Proof state:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.comp (f j) (inclusion h)
⊢ ↥(iSup K) →ₙₐ[R] B

Next tactic:
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]
              rfl)
            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])
        map_zero' := by
          dsimp
          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)
        map_mul' := by
          dsimp
          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))
          on_goal 3 => rw [coe_iSup_of_directed dir]
          all_goals simp
        map_add' := by
          dsimp
          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))
          on_goal 3 => rw [coe_iSup_of_directed dir]
          all_goals simp
        map_smul' := fun r => by
          dsimp
          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)
            (fun _ _ => rfl)
          on_goal 2 => rw [coe_iSup_of_directed dir]
          all_goals simp }
/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra 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:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.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 an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra 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:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.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 an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra 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:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.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 an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra 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:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.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)
⊢ (NonUnitalAlgHom.comp (f k) (inclusion hik)) { val := x, property := hxi } =
    (NonUnitalAlgHom.comp (f k) (inclusion hjk)) { val := x, property := hxj }

Next tactic:
rfl
/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra 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]
              rfl)
            (↑(iSup K)) (by 

Proof state:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.comp (f j) (inclusion h)
⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)

Next tactic:
rw [coe_iSup_of_directed dir]
/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra 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]
              rfl)
            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])
        map_zero' := by
          dsimp
          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)
        map_mul' := by
          dsimp
          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))
          on_goal 3 => rw [coe_iSup_of_directed dir]
          all_goals simp
        map_add' := by
          dsimp
          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))
          on_goal 3 => rw [coe_iSup_of_directed dir]
          all_goals simp
        map_smul' := fun r => by
          

Proof state:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (x : ↥(iSup K)),
    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)
        (_ :
          ∀ (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 })
        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =
      r •
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)
          (_ :
            ∀ (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 })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x

Next tactic:
dsimp
/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining
it on each non-unital subalgebra, and proving that it agrees on the intersection of
non-unital subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra 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 : NonUnitalSubalgebra 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]
              rfl)
            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])
        map_zero' := by
          dsimp
          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)
        map_mul' := by
          dsimp
          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))
          on_goal 3 => rw [coe_iSup_of_directed dir]
          all_goals simp
        map_add' := by
          dsimp
          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))
          on_goal 3 => rw [coe_iSup_of_directed dir]
          all_goals simp
        map_smul' := fun r => by
          dsimp
          

Proof state:
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : NonUnitalNonAssocSemiring B
inst✝³ : Module R B
inst✝² : IsScalarTower R B B
inst✝¹ : SMulCommClass R B B
S : NonUnitalSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalSubalgebra 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 = NonUnitalAlgHom.comp (f j) (inclusion h)
r : R
⊢ ∀ (x : ↥(iSup K)),
    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)
        (_ :
          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
            (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })
        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =
      r •
        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)
          (_ :
            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),
              (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })
          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x

Next tactic:
apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)
            (fun _ _ => rfl)