/-- 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; 

Proof state:
case h
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
a b : A
a✝ : ↑(characterSpace 𝕜 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))
      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 +
        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)
      a✝

Next tactic:
simp only [map_add, coe_mk, coe_add, Pi.add_apply]