/-- Define a non-unital star algebra homomorphism on a directed supremum of non-unital star
subalgebras by defining it on each non-unital star subalgebra, and proving that it agrees on the
intersection of non-unital star subalgebras. -/
noncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalStarSubalgebra 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 : NonUnitalStarSubalgebra 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 only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]
        exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl)  _ (by simp)
      map_mul' := by
        

Proof state:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁴ : CommSemiring R
inst✝¹³ : StarRing R
inst✝¹² : NonUnitalSemiring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R A A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : StarModule R A
inst✝⁶ : NonUnitalSemiring B
inst✝⁵ : StarRing B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
hF : NonUnitalStarAlgHomClass F R A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra 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 = NonUnitalStarAlgHom.comp (f j) (inclusion h)
⊢ ∀ (x y : ↥(iSup K)),
    MulActionHom.toFun
        {
            toMulActionHom :=
              {
                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) },
            map_zero' :=
              (_ :
                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) }
                    0 =
                  0),
            map_add' :=
              (_ :
                ∀ (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) }.toMulActionHom
        (x * y) =
      MulActionHom.toFun
          {
              toMulActionHom :=
                {
                  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) },
              map_zero' :=
                (_ :
                  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) }
                      0 =
                    0),
              map_add' :=
                (_ :
                  ∀ (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) }.toMulActionHom
          x *
        MulActionHom.toFun
          {
              toMulActionHom :=
                {
                  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) },
              map_zero' :=
                (_ :
                  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) }
                      0 =
                    0),
              map_add' :=
                (_ :
                  ∀ (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) }.toMulActionHom
          y

Next tactic:
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast, ZeroMemClass.coe_zero,
          AddSubmonoid.mk_add_mk, Set.inclusion_mk]