Theorem: WeakDual.gelfandTransform_apply_apply
File: Mathlib/Topology/Algebra/Module/CharacterSpace.lean

Proof steps:
state_before: 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
⊢ (fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }) 1 = 1
state_after: 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 : ↑(characterSpace 𝕜 A)
⊢ ((fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }) 1) a = 1 a
tactic: ext a
state_before: 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 : ↑(characterSpace 𝕜 A)
⊢ ((fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }) 1) a = 1 a
state_after: no goals
tactic: simp only [coe_mk, coe_one, Pi.one_apply, map_one a]
state_before: 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
⊢ { toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun (a * b) =
    { toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun a *
      { toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun b
state_after: 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)
⊢ ({ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun (a * b)) a✝ =
    ({ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun a *
        { toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun b)
      a✝
tactic: ext
state_before: 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)
⊢ ({ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun (a * b)) a✝ =
    ({ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun a *
        { toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯ }.toFun b)
      a✝
state_after: no goals
tactic: simp only [map_mul, coe_mk, coe_mul, Pi.mul_apply]
state_before: 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
⊢ (↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun 0 = 0
state_after: 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✝ : ↑(characterSpace 𝕜 A)
⊢ ((↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun 0) a✝ =
    0 a✝
tactic: ext
state_before: 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✝ : ↑(characterSpace 𝕜 A)
⊢ ((↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun 0) a✝ =
    0 a✝
state_after: no goals
tactic: simp only [map_zero, coe_mk, coe_mul, coe_zero, Pi.zero_apply]
state_before: 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
⊢ (↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun
      (a + b) =
    (↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun a +
      (↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun b
state_after: 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)
⊢ ((↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun
        (a + b))
      a✝ =
    ((↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun a +
        (↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun b)
      a✝
tactic: ext
state_before: 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)
⊢ ((↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun
        (a + b))
      a✝ =
    ((↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun a +
        (↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }).toFun b)
      a✝
state_after: no goals
tactic: simp only [map_add, coe_mk, coe_add, Pi.add_apply]
state_before: 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
⊢ (↑↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯,
              map_add' := ⋯ }).toFun
      ((algebraMap 𝕜 A) k) =
    (algebraMap 𝕜 C(↑(characterSpace 𝕜 A), 𝕜)) k
state_after: 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
k : 𝕜
a✝ : ↑(characterSpace 𝕜 A)
⊢ ((↑↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯,
                map_zero' := ⋯, map_add' := ⋯ }).toFun
        ((algebraMap 𝕜 A) k))
      a✝ =
    ((algebraMap 𝕜 C(↑(characterSpace 𝕜 A), 𝕜)) k) a✝
tactic: ext
state_before: 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
k : 𝕜
a✝ : ↑(characterSpace 𝕜 A)
⊢ ((↑↑{ toFun := fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }, map_one' := ⋯, map_mul' := ⋯,
                map_zero' := ⋯, map_add' := ⋯ }).toFun
        ((algebraMap 𝕜 A) k))
      a✝ =
    ((algebraMap 𝕜 C(↑(characterSpace 𝕜 A), 𝕜)) k) a✝
state_after: no goals
tactic: simp [AlgHomClass.commutes]