mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖

Tactic:
·

Goal state after:
case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖

pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖

Tactic:
·

Goal state after:
case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖

Tactic:
·

Goal state after:
case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
h₂ : ‖((splitMul 𝕜 E) (x⋆ * x)).2‖ = ‖((splitMul 𝕜 E) x).2‖ ^ 2
h₃ : ‖((splitMul 𝕜 E) (x⋆ * x)).1‖ = ‖((splitMul 𝕜 E) x).1‖ ^ 2
h : ‖((splitMul 𝕜 E) x).1‖ ≤ ‖((splitMul 𝕜 E) x).2‖
⊢ max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ * max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ ≤
    max (‖((splitMul 𝕜 E) x).1‖ ^ 2) (‖((splitMul 𝕜 E) x).2‖ ^ 2)

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
h₂ : ‖((splitMul 𝕜 E) (x⋆ * x)).2‖ = ‖((splitMul 𝕜 E) x).2‖ ^ 2
h₃ : ‖((splitMul 𝕜 E) (x⋆ * x)).1‖ = ‖((splitMul 𝕜 E) x).1‖ ^ 2
h : ‖((splitMul 𝕜 E) x).2‖ < ‖((splitMul 𝕜 E) x).1‖
⊢ max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ * max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ ≤
    max (‖((splitMul 𝕜 E) x).1‖ ^ 2) (‖((splitMul 𝕜 E) x).2‖ ^ 2)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
h₂ : ‖((splitMul 𝕜 E) (x⋆ * x)).2‖ = ‖((splitMul 𝕜 E) x).2‖ ^ 2
h₃ : ‖((splitMul 𝕜 E) (x⋆ * x)).1‖ = ‖((splitMul 𝕜 E) x).1‖ ^ 2
h : ‖((splitMul 𝕜 E) x).1‖ ≤ ‖((splitMul 𝕜 E) x).2‖
⊢ max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ * max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ ≤
    max (‖((splitMul 𝕜 E) x).1‖ ^ 2) (‖((splitMul 𝕜 E) x).2‖ ^ 2)
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 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}

pp: 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}

Tactic:
·

Goal state after:
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}
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 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}

Tactic:
·

Goal state after:
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}
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
h₂ : ‖((splitMul 𝕜 E) (x⋆ * x)).2‖ = ‖((splitMul 𝕜 E) x).2‖ ^ 2
h₃ : ‖((splitMul 𝕜 E) (x⋆ * x)).1‖ = ‖((splitMul 𝕜 E) x).1‖ ^ 2
h : ‖((splitMul 𝕜 E) x).2‖ < ‖((splitMul 𝕜 E) x).1‖
⊢ max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ * max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ ≤
    max (‖((splitMul 𝕜 E) x).1‖ ^ 2) (‖((splitMul 𝕜 E) x).2‖ ^ 2)

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
h₂ : ‖((splitMul 𝕜 E) (x⋆ * x)).2‖ = ‖((splitMul 𝕜 E) x).2‖ ^ 2
h₃ : ‖((splitMul 𝕜 E) (x⋆ * x)).1‖ = ‖((splitMul 𝕜 E) x).1‖ ^ 2
h : ‖((splitMul 𝕜 E) x).2‖ < ‖((splitMul 𝕜 E) x).1‖
⊢ max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ * max ‖((splitMul 𝕜 E) x).1‖ ‖((splitMul 𝕜 E) x).2‖ ≤
    max (‖((splitMul 𝕜 E) x).1‖ ^ 2) (‖((splitMul 𝕜 E) x).2‖ ^ 2)
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: case inl
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
φ : ↑(characterSpace 𝕜 A)
h₁ : φ 1 * (1 - φ 1) = 0
h₂ : φ 1 = 0
⊢ φ 1 = 1

pp: case inr
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
φ : ↑(characterSpace 𝕜 A)
h₁ : φ 1 * (1 - φ 1) = 0
h₂ : 1 - φ 1 = 0
⊢ φ 1 = 1

Tactic:
·

Goal state after:
case inl
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
φ : ↑(characterSpace 𝕜 A)
h₁ : φ 1 * (1 - φ 1) = 0
h₂ : φ 1 = 0
⊢ φ 1 = 1
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: case inr
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
φ : ↑(characterSpace 𝕜 A)
h₁ : φ 1 * (1 - φ 1) = 0
h₂ : 1 - φ 1 = 0
⊢ φ 1 = 1

Tactic:
·

Goal state after:
case inr
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
φ : ↑(characterSpace 𝕜 A)
h₁ : φ 1 * (1 - φ 1) = 0
h₂ : 1 - φ 1 = 0
⊢ φ 1 = 1
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
⊢ f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) →
    f ∈ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)})

pp: case mpr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
⊢ f ∈ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) →
    f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))

Tactic:
·

Goal state after:
case mp
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
⊢ f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) →
    f ∈ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)})
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: case 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}

pp: case 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:
·

Goal state after:
case 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}
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: case 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:
·

Goal state after:
case 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
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
⊢ f ∈ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) →
    f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))

Tactic:
·

Goal state after:
case mpr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
⊢ f ∈ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) →
    f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case refine_1
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
⊢ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ)) ⊆ closure[inst✝⁸] ↑s ∩ ↑(RingHom.ker φ)

pp: case refine_2
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
⊢ closure[inst✝⁸] ↑s ∩ ↑(RingHom.ker φ) ⊆ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ))

Tactic:
·

Goal state after:
case refine_1
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
⊢ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ)) ⊆ closure[inst✝⁸] ↑s ∩ ↑(RingHom.ker φ)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁹ : CommSemiring R
inst✝²⁸ : Nontrivial R
inst✝²⁷ : StarRing R
inst✝²⁶ : MetricSpace R
inst✝²⁵ : IsTopologicalSemiring R
inst✝²⁴ : ContinuousStar R
inst✝²³ : CommRing S
inst✝²² : Algebra R S
inst✝²¹ : NonUnitalRing A
inst✝²⁰ : StarRing A
inst✝¹⁹ : TopologicalSpace A
inst✝¹⁸ : Module R A
inst✝¹⁷ : IsScalarTower R A A
inst✝¹⁶ : SMulCommClass R A A
inst✝¹⁵ : NonUnitalRing B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Module R B
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : SMulCommClass R B B
inst✝⁹ : Module S A
inst✝⁸ : Module S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : NonUnitalContinuousFunctionalCalculus R A p
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMapZero.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (quasispectrum R a)
hf₀ : f 0 = 0
hφ : Continuous[inst✝¹⁹, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₙₐ[R] B := ⋯
this : Continuous[inst✝¹⁹, inst✝¹³] ⇑ψ
h_spec : quasispectrum R (φ a) ⊆ quasispectrum R a
hψa : q (ψ a)
ι : C(↑(quasispectrum R (ψ a)), ↑(quasispectrum R a))₀ := ⋯
⊢ (cfcₙHom hψa) (ContinuousMapZero.id (quasispectrum R (ψ a))) =
    ((cfcₙHom hψa).comp (ContinuousMapZero.nonUnitalStarAlgHom_precomp R ι)) (ContinuousMapZero.id (quasispectrum R a))

Tactic:
·

Goal state after:
F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁹ : CommSemiring R
inst✝²⁸ : Nontrivial R
inst✝²⁷ : StarRing R
inst✝²⁶ : MetricSpace R
inst✝²⁵ : IsTopologicalSemiring R
inst✝²⁴ : ContinuousStar R
inst✝²³ : CommRing S
inst✝²² : Algebra R S
inst✝²¹ : NonUnitalRing A
inst✝²⁰ : StarRing A
inst✝¹⁹ : TopologicalSpace A
inst✝¹⁸ : Module R A
inst✝¹⁷ : IsScalarTower R A A
inst✝¹⁶ : SMulCommClass R A A
inst✝¹⁵ : NonUnitalRing B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Module R B
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : SMulCommClass R B B
inst✝⁹ : Module S A
inst✝⁸ : Module S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : NonUnitalContinuousFunctionalCalculus R A p
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMapZero.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (quasispectrum R a)
hf₀ : f 0 = 0
hφ : Continuous[inst✝¹⁹, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₙₐ[R] B := ⋯
this : Continuous[inst✝¹⁹, inst✝¹³] ⇑ψ
h_spec : quasispectrum R (φ a) ⊆ quasispectrum R a
hψa : q (ψ a)
ι : C(↑(quasispectrum R (ψ a)), ↑(quasispectrum R a))₀ := ⋯
⊢ (cfcₙHom hψa) (ContinuousMapZero.id (quasispectrum R (ψ a))) =
    ((cfcₙHom hψa).comp (ContinuousMapZero.nonUnitalStarAlgHom_precomp R ι)) (ContinuousMapZero.id (quasispectrum R a))
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case refine_2
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
⊢ closure[inst✝⁸] ↑s ∩ ↑(RingHom.ker φ) ⊆ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ))

Tactic:
·

Goal state after:
case refine_2
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
⊢ closure[inst✝⁸] ↑s ∩ ↑(RingHom.ker φ) ⊆ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := ⋯
this : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := ⋯
⊢ (cfcHom hψa) (ContinuousMap.restrict (spectrum R (ψ a)) (ContinuousMap.id R)) =
    ((cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι))
      (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R))

Tactic:
·

Goal state after:
F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := ⋯
this : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := ⋯
⊢ (cfcHom hψa) (ContinuousMap.restrict (spectrum R (ψ a)) (ContinuousMap.id R)) =
    ((cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι))
      (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R))
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
⊢ ∀ ⦃x : Set (Set α)⦄,
    x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U} →
      IsOpen[generateFrom
            (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]
        {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty}

pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
⊢ ∀ ⦃x : Set (Set (Set α))⦄,
    x ∈
        {f |
          f.Finite ∧
            f ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}} →
      ∀ x_1 ∈ ⋂₀ x,
        ∃ v ∈ (fun u ↦ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}) '' {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
          x_1 ∈ v ∧ v ⊆ ⋂₀ x

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
⊢ ∀ ⦃x : Set (Set α)⦄,
    x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U} →
      IsOpen[generateFrom
            (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]
        {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
⊢ ∀ ⦃x : Set (Set (Set α))⦄,
    x ∈
        {f |
          f.Finite ∧
            f ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}} →
      ∀ x_1 ∈ ⋂₀ x,
        ∃ v ∈ (fun u ↦ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}) '' {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
          x_1 ∈ v ∧ v ⊆ ⋂₀ x

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
⊢ ∀ ⦃x : Set (Set (Set α))⦄,
    x ∈
        {f |
          f.Finite ∧
            f ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}} →
      ∀ x_1 ∈ ⋂₀ x,
        ∃ v ∈ (fun u ↦ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}) '' {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
          x_1 ∈ v ∧ v ⊆ ⋂₀ x