/-- 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 

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)
T : Subalgebra R A
hT : T = iSup K
⊢ OneHom.toFun
      (↑{
          toOneHom :=
            {
              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 })
                  ↑T (_ : ↑T ⊆ ⋃ 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 })
                      ↑T (_ : ↑T ⊆ ⋃ i, ↑(K i)) 1 =
                    1) },
          map_mul' :=
            (_ :
              ∀ (x y : ↥T),
                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 })
                          ↑T (_ : ↑T ⊆ ⋃ 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 })
                              ↑T (_ : ↑T ⊆ ⋃ 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 })
                            ↑T (_ : ↑T ⊆ ⋃ 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 })
                                ↑T (_ : ↑T ⊆ ⋃ 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 })
                            ↑T (_ : ↑T ⊆ ⋃ 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 })
                                ↑T (_ : ↑T ⊆ ⋃ i, ↑(K i)) 1 =
                              1) }
                      y) })
      0 =
    0

Next tactic:
dsimp