/-- The **Gelfand transform** is an algebra homomorphism (over `𝕜`) from a topological `𝕜`-algebra
`A` into the `𝕜`-algebra of continuous `𝕜`-valued functions on the `characterSpace 𝕜 A`.
The character space itself consists of all algebra homomorphisms from `A` to `𝕜`.  -/
@[simps]
def gelfandTransform : A →ₐ[𝕜] C(characterSpace 𝕜 A, 𝕜) where
  toFun a :=
    { toFun := fun φ => φ a
      continuous_toFun := (eval_continuous a).comp continuous_induced_dom }
  map_one' := by ext a; simp only [coe_mk, coe_one, Pi.one_apply, map_one a]
  map_mul' a b := by ext; simp only [map_mul, coe_mk, coe_mul, Pi.mul_apply]
  map_zero' := by ext; simp only [map_zero, coe_mk, coe_mul, coe_zero, Pi.zero_apply]
  map_add' a b := by ext; simp only [map_add, coe_mk, coe_add, Pi.add_apply]
  commutes' k := by 

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : TopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
k : 𝕜
⊢ OneHom.toFun
      (↑↑{
            toMonoidHom :=
              { toOneHom := { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) },
                map_mul' :=
                  (_ :
                    ∀ (a b : A),
                      OneHom.toFun
                          { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                          (a * b) =
                        OneHom.toFun
                            { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                            a *
                          OneHom.toFun
                            { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                            b) },
            map_zero' :=
              (_ :
                OneHom.toFun
                    (↑{
                        toOneHom :=
                          { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) },
                        map_mul' :=
                          (_ :
                            ∀ (a b : A),
                              OneHom.toFun
                                  { toFun := fun a => mk fun φ => φ a,
                                    map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                  (a * b) =
                                OneHom.toFun
                                    { toFun := fun a => mk fun φ => φ a,
                                      map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                    a *
                                  OneHom.toFun
                                    { toFun := fun a => mk fun φ => φ a,
                                      map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                    b) })
                    0 =
                  0),
            map_add' :=
              (_ :
                ∀ (a b : A),
                  OneHom.toFun
                      (↑{
                          toOneHom :=
                            { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) },
                          map_mul' :=
                            (_ :
                              ∀ (a b : A),
                                OneHom.toFun
                                    { toFun := fun a => mk fun φ => φ a,
                                      map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                    (a * b) =
                                  OneHom.toFun
                                      { toFun := fun a => mk fun φ => φ a,
                                        map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                      a *
                                    OneHom.toFun
                                      { toFun := fun a => mk fun φ => φ a,
                                        map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                      b) })
                      (a + b) =
                    OneHom.toFun
                        (↑{
                            toOneHom :=
                              { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) },
                            map_mul' :=
                              (_ :
                                ∀ (a b : A),
                                  OneHom.toFun
                                      { toFun := fun a => mk fun φ => φ a,
                                        map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                      (a * b) =
                                    OneHom.toFun
                                        { toFun := fun a => mk fun φ => φ a,
                                          map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                        a *
                                      OneHom.toFun
                                        { toFun := fun a => mk fun φ => φ a,
                                          map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                        b) })
                        a +
                      OneHom.toFun
                        (↑{
                            toOneHom :=
                              { toFun := fun a => mk fun φ => φ a, map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) },
                            map_mul' :=
                              (_ :
                                ∀ (a b : A),
                                  OneHom.toFun
                                      { toFun := fun a => mk fun φ => φ a,
                                        map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                      (a * b) =
                                    OneHom.toFun
                                        { toFun := fun a => mk fun φ => φ a,
                                          map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                        a *
                                      OneHom.toFun
                                        { toFun := fun a => mk fun φ => φ a,
                                          map_one' := (_ : (fun a => mk fun φ => φ a) 1 = 1) }
                                        b) })
                        b) })
      ((algebraMap 𝕜 A) k) =
    (algebraMap 𝕜 C(↑(characterSpace 𝕜 A), 𝕜)) k

Next tactic:
ext