/-- 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 _ => (· + ·))
          

Proof state:
case hopi
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 : ι) (x y : ↑↑(K i)),
    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =
      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y
case h
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 : ι) (x y : ↑↑(K i)), (f i) (x + y) = (f i) x + (f i) y
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:
on_goal 3 => rw [coe_iSup_of_directed dir]