mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
⊢ φ = ψ

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝⁴ : Semiring A
inst✝³ : Algebra ℝ A
inst✝² : TopologicalSpace A
inst✝¹ : T2Space A
s : Set ℝ
inst✝ : CompactSpace ↑s
φ ψ : C(↑s, ℝ) →ₐ[ℝ] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ((toContinuousMapOnAlgHom s) X) = ψ ((toContinuousMapOnAlgHom s) X)
⊢ φ = ψ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
⊢ φ = ψ

Tactic:
by
    let s' : Set ℝ := (↑) '' s
    let e : s ≃ₜ s' := NNReal.isEmbedding_coe.homeomorphImage s
    have : Fact (0 ∈ s') := ⟨0, Fact.out, coe_zero⟩
    have e0 : e 0 = 0 := rfl
    have e0' : e.symm 0 = 0 := e.symm_apply_eq.mpr e0
    have (ξ : C(s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A) (hξ : Continuous ξ) :
        (let ξ' := ξ.realContinuousMapZeroOfNNReal.comp <|
          ContinuousMapZero.nonUnitalStarAlgHom_precomp ℝ ⟨e, e0⟩;
          Continuous ξ' ∧ ξ' (.id s') = ξ (.id s)) := by
      intro ξ'
      refine ⟨ξ.continuous_realContinuousMapZeroOfNNReal hξ |>.comp <| ?_, ?_⟩
      · rw [continuous_induced_rng]
        exact ContinuousMap.continuous_precomp _ |>.comp continuous_induced_dom
      · exact ξ.realContinuousMapZeroOfNNReal_apply_comp_toReal (.id _)
    obtain ⟨hφ', hφ_id⟩ := this φ hφ
    obtain ⟨hψ', hψ_id⟩ := this ψ hψ
    have hs' : CompactSpace s' := e.compactSpace
    have : ContinuousMapZero.UniqueHom ℝ A := inferInstance
    have h' := ContinuousMapZero.UniqueHom.eq_of_continuous_of_map_id
      s' _ _ hφ' hψ' (hφ_id ▸ hψ_id ▸ h)
    have h'' := congr($(h').comp <|
      ContinuousMapZero.nonUnitalStarAlgHom_precomp ℝ ⟨(e.symm : C(s', s)), e0'⟩)
    have : (ContinuousMapZero.nonUnitalStarAlgHom_precomp ℝ ⟨(e : C(s, s')), e0⟩).comp
        (ContinuousMapZero.nonUnitalStarAlgHom_precomp ℝ ⟨(e.symm : C(s', s)), e0'⟩) =
        NonUnitalStarAlgHom.id _ _ := by
      ext; simp
    simp only [NonUnitalStarAlgHom.comp_assoc, this, NonUnitalStarAlgHom.comp_id] at h''
    exact NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_injective h''

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakDual

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
⊢ Continuous[inst✝, _] (↑(toWeakSpace 𝕜 E)).toFun

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakDual

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_4
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
x : E
⊢ (toWeakSpaceCLM 𝕜 E) x = (toWeakSpace 𝕜 E) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakDual

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_4
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
V : Set E
hV : IsOpen (⇑(toWeakSpaceCLM 𝕜 E) '' V)
⊢ IsOpen[inst✝] V

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : T2Space A
s : Set 𝕜
inst✝ : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ((toContinuousMapOnAlgHom s) X) = ψ ((toContinuousMapOnAlgHom s) X)
⊢ φ = ψ

Tactic:
by

Goal state after:
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‖
⊢ ‖((splitMul 𝕜 E) x).2‖ * ‖((splitMul 𝕜 E) x).2‖ ≤
    max (‖((splitMul 𝕜 E) x).1‖ * ‖((splitMul 𝕜 E) x).1‖) (‖((splitMul 𝕜 E) x).2‖ * ‖((splitMul 𝕜 E) x).2‖)

Tactic:
sup_eq_right.mpr (mul_self_le_mul_self (norm_nonneg _) h)

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‖
⊢ ‖((splitMul 𝕜 E) x).2‖ * ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x).2‖ * ‖((splitMul 𝕜 E) x).2‖
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 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 ψ : ↑(characterSpace 𝕜 A)
h : (fun φ ↦ ⇑↑φ) φ = (fun φ ↦ ⇑↑φ) ψ
⊢ φ = ψ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : TopologicalSpace A
inst✝¹ : Module 𝕜 A
inst✝ : Subsingleton A
φ : ↑(characterSpace 𝕜 A)
x : A
⊢ ↑φ x = 0 x

Tactic:
by

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

Goal state before:
pp: 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✝

Tactic:
by

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

Goal state before:
pp: 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}

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring 𝕜
inst✝⁷ : TopologicalSpace 𝕜
inst✝⁶ : ContinuousAdd 𝕜
inst✝⁵ : ContinuousConstSMul 𝕜 𝕜
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : TopologicalSpace A
inst✝² : Module 𝕜 A
inst✝¹ : T2Space 𝕜
inst✝ : ContinuousMul 𝕜
⊢ IsClosed (characterSpace 𝕜 A ∪ {0})

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
⊢ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) =
    ↑(span 𝕜 ({1} ∪ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)})))

Tactic:
by

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

Goal state before:
pp: 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)
⊢ φ 1 * (1 - φ 1) = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
⊢ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) = ↑(𝕜 ∙ 1) + ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)})

Tactic:
by

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

Goal state before:
pp: 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
a : A
⊢ φ (a * 1) = 0

Tactic:
by

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

Goal state before:
pp: 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
this : ∀ (a : A), φ (a * 1) = 0
⊢ ∀ (x : A), ↑φ x = 0 x

Tactic:
by

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

Goal state before:
pp: 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)
⊢ φ 1 = 1

Tactic:
by

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

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

Tactic:
by

Goal state after:
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 := NonUnitalStarAlgHom.restrictScalars R ↑φ
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 :=
  { toFun := Set.inclusion h_spec, continuous_toFun := ⋯, map_zero' := ⋯ }
this : ψ.comp (cfcₙHom ha) = (cfcₙHom hψa).comp (ContinuousMapZero.nonUnitalStarAlgHom_precomp R ι)
⊢ φ (cfcₙ f a) = cfcₙ f (φ a)

Tactic:
by

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

Goal state before:
pp: 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
map_one' : ∀ (φ : ↑(characterSpace 𝕜 A)), φ 1 = 1
φ : ↑(characterSpace 𝕜 A)
r : 𝕜
⊢ φ ((algebraMap 𝕜 A) r) = (algebraMap 𝕜 𝕜) r

Tactic:
by

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

Goal state before:
pp: 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:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : NoZeroDivisors 𝕜
inst✝⁸ : TopologicalSpace 𝕜
inst✝⁷ : ContinuousAdd 𝕜
inst✝⁶ : ContinuousConstSMul 𝕜 𝕜
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Semiring A
inst✝³ : Algebra 𝕜 A
inst✝² : Nontrivial 𝕜
inst✝¹ : T2Space 𝕜
inst✝ : ContinuousMul 𝕜
⊢ IsClosed (characterSpace 𝕜 A)

Tactic:
by

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

Goal state before:
pp: 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
⊢ ‖x‖ * ‖x‖ ≤ ‖x⋆ * x‖

Tactic:
by

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

Goal state before:
pp: 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 : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
x : A
⊢ x - (algebraMap 𝕜 A) (ψ x) ∈ RingHom.ker φ

Tactic:
by

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

Goal state before:
pp: 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 : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
⊢ φ = ψ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : Ring A
inst✝¹ : TopologicalSpace A
inst✝ : Algebra 𝕜 A
φ : ↑(characterSpace 𝕜 A)
z : 𝕜
⊢ φ ((algebraMap 𝕜 A) z) = z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
⊢ { toFun := fun φ ↦ φ 1, continuous_toFun := ⋯ } = 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
a b : A
⊢ { toFun := fun φ ↦ φ (a * b), continuous_toFun := ⋯ } =
    { toFun := fun φ ↦ φ a, continuous_toFun := ⋯ } * { toFun := fun φ ↦ φ b, continuous_toFun := ⋯ }

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
⊢ { toFun := fun φ ↦ φ 0, continuous_toFun := ⋯ } = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
a b : A
⊢ { toFun := fun φ ↦ φ (a + b), continuous_toFun := ⋯ } =
    { toFun := fun φ ↦ φ a, continuous_toFun := ⋯ } + { toFun := fun φ ↦ φ b, continuous_toFun := ⋯ }

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
k : 𝕜
⊢ { toFun := fun φ ↦ φ ((algebraMap 𝕜 A) k), continuous_toFun := ⋯ } = (algebraMap 𝕜 C(↑(characterSpace 𝕜 A), 𝕜)) k

Tactic:
by

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

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

Tactic:
by

Goal state after:
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)
⊢ φ (cfcₙ f a) = cfcₙ f (φ a)

Tactic:
by

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

Goal state before:
pp: 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
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
⊢ Continuous[inst✝⁸, inst✝⁸] fun y ↦ y - φ y • 1

Tactic:
by

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

Goal state before:
pp: 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
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
⊢ Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 ↑s) (𝓝 x)

Tactic:
by

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

Goal state before:
pp: 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
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
this : Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 ↑s) (𝓝 x)
g : A
hg : g ∈ ↑s
⊢ g - φ g • 1 ∈ ↑(RingHom.ker φ)

Tactic:
by

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

Goal state before:
pp: 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:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
inst✝ : CompactSpace ↑s
⊢ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)) = closure ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)})

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
⊢ Dense ↑(adjoin 𝕜 {ContinuousMapZero.id s})

Tactic:
by

Goal state after:
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 := StarAlgHom.restrictScalars R ↑φ
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)) := { toFun := Set.inclusion h_spec, continuous_toFun := ⋯ }
this : ψ.comp (cfcHom ha) = (cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι)
⊢ φ (cfc f a) = cfc f (φ a)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Fact (0 ∈ s)
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id s)
star_id : p (star (ContinuousMapZero.id s))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id s}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f : C(↑s, 𝕜)₀
⊢ p f

Tactic:
by

Goal state after: