/-- 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
        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]
        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 only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]
        apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))
        on_goal 3 => rw [coe_iSup_of_directed dir]
        all_goals simp
      map_smul' := fun r => by
        dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]
        apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)
          (fun _ _ => rfl)
        on_goal 2 => rw [coe_iSup_of_directed dir]
        all_goals simp
      map_star' := 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)
⊢ ∀ (a : ↥(iSup K)),
    MulActionHom.toFun
        {
              toDistribMulActionHom :=
                {
                  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) },
              map_mul' :=
                (_ :
                  ∀ (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) }.toDistribMulActionHom.toMulActionHom
        (star a) =
      star
        (MulActionHom.toFun
          {
                toDistribMulActionHom :=
                  {
                    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) },
                map_mul' :=
                  (_ :
                    ∀ (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) }.toDistribMulActionHom.toMulActionHom
          a)

Next tactic:
dsimp only [SetLike.coe_sort_coe, NonUnitalStarAlgHom.comp_apply, inclusion_mk, Eq.ndrec,
          id_eq, eq_mpr_eq_cast, ZeroMemClass.coe_zero, AddSubmonoid.mk_add_mk, Set.inclusion_mk,
          MulMemClass.mk_mul_mk, NonUnitalAlgHom.toDistribMulActionHom_eq_coe,
          DistribMulActionHom.toFun_eq_coe, NonUnitalAlgHom.coe_to_distribMulActionHom,
          NonUnitalAlgHom.coe_mk]