/-- 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
          

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)
⊢ ∀ (x y : ↥(iSup K)),
    MulActionHom.toFun
        {
          toFun :=
            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)),
          map_smul' :=
            (_ :
              ∀ (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) }
        (x + y) =
      MulActionHom.toFun
          {
            toFun :=
              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)),
            map_smul' :=
              (_ :
                ∀ (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) }
          x +
        MulActionHom.toFun
          {
            toFun :=
              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)),
            map_smul' :=
              (_ :
                ∀ (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) }
          y

Next tactic:
dsimp