/-- Define an algebra homomorphism on a directed supremum of subalgebras by defining
it on each subalgebra, and proving that it agrees on the intersection of subalgebras. -/
noncomputable def iSupLift : ↥T →ₐ[R] B :=
  { 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
          dsimp
          rw [hf i k hik, hf j k hjk]
          rfl)
        T (by rw [hT, coe_iSup_of_directed dir])
    map_one' := by apply Set.iUnionLift_const _ (fun _ => 1) <;> simp
    map_zero' := by dsimp; apply Set.iUnionLift_const _ (fun _ => 0) <;> simp
    map_mul' := by
      subst hT; 

Proof state:
R : Type u
A : Type v
B : Type w
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Subalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → Subalgebra R A
dir : Directed (fun x x_1 => x ≤ x_1) K
f : (i : ι) → ↥(K i) →ₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = AlgHom.comp (f j) (inclusion h)
⊢ ∀ (x y : ↥(iSup K)),
    OneHom.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_one' :=
            (_ :
              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)) 1 =
                1) }
        (x * y) =
      OneHom.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_one' :=
              (_ :
                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)) 1 =
                  1) }
          x *
        OneHom.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_one' :=
              (_ :
                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)) 1 =
                  1) }
          y

Next tactic:
dsimp