case inl
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : NonUnitalNonAssocSemiring A
inst✝¹ : TopologicalSpace A
inst✝ : Module 𝕜 A
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ φ ∈ {φ | ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}

case inr
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : NonUnitalNonAssocSemiring A
inst✝¹ : TopologicalSpace A
inst✝ : Module 𝕜 A
⊢ 0 ∈ {φ | ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : NonUnitalNonAssocSemiring A
inst✝¹ : TopologicalSpace A
inst✝ : Module 𝕜 A
⊢ characterSpace 𝕜 A ∪ {0} ≤ {φ | ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}
rintro φ (hφ | rfl)
no goals
case inl
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : NonUnitalNonAssocSemiring A
inst✝¹ : TopologicalSpace A
inst✝ : Module 𝕜 A
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ φ ∈ {φ | ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}
exact hφ.2
no goals
case inr
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : NonUnitalNonAssocSemiring A
inst✝¹ : TopologicalSpace A
inst✝ : Module 𝕜 A
⊢ 0 ∈ {φ | ∀ (x y : A), ↑φ (x * y) = ↑φ x * ↑φ y}
exact fun _ _ => by exact (zero_mul (0 : 𝕜)).symm
no goals
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : NonUnitalNonAssocSemiring A
inst✝¹ : TopologicalSpace A
inst✝ : Module 𝕜 A
x✝¹ x✝ : A
⊢ ↑0 (x✝¹ * x✝) = ↑0 x✝¹ * ↑0 x✝
exact (zero_mul (0 : 𝕜)).symm