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

tactic: ext φ
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
⊢ characterSpace 𝕜 A = {φ | ↑φ 1 = 1 ∧ ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}

tactic: refine' ⟨_, _⟩
state_before: case h
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
⊢ φ ∈ characterSpace 𝕜 A ↔ φ ∈ {φ | ↑φ 1 = 1 ∧ ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}

tactic: rintro hφ
state_before: case h.refine'_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
⊢ φ ∈ characterSpace 𝕜 A → φ ∈ {φ | ↑φ 1 = 1 ∧ ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}

tactic: lift φ to characterSpace 𝕜 A using hφ
state_before: case h.refine'_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ φ ∈ {φ | ↑φ 1 = 1 ∧ ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}

tactic: exact ⟨map_one φ, map_mul φ⟩
state_before: case h.refine'_1.intro
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : { x // x ∈ characterSpace 𝕜 A }
⊢ ↑φ ∈ {φ | ↑φ 1 = 1 ∧ ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}

tactic: rintro ⟨hφ₁, hφ₂⟩
state_before: case h.refine'_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
⊢ φ ∈ {φ | ↑φ 1 = 1 ∧ ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y} → φ ∈ characterSpace 𝕜 A

tactic: refine' ⟨_, hφ₂⟩
state_before: case h.refine'_2.intro
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
hφ₁ : ↑φ 1 = 1
hφ₂ : ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y
⊢ φ ∈ characterSpace 𝕜 A

tactic: rintro rfl
state_before: case h.refine'_2.intro
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
hφ₁ : ↑φ 1 = 1
hφ₂ : ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y
⊢ φ ≠ 0

tactic: exact zero_ne_one hφ₁
state_before: case h.refine'_2.intro
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
hφ₁ : ↑0 1 = 1
hφ₂ : ∀ (x y : A), ↑0 (x * y) = ↑0 x * ↑0 y
⊢ False