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

Goal state before:
pp: 𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₁
hx : x ∈ spectrum 𝕜₁ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₂ a, ↑‖k‖₊

pp: 𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₂
hx : x ∈ spectrum 𝕜₂ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊

Tactic:
·

Goal state after:
𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₁
hx : x ∈ spectrum 𝕜₁ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₂ a, ↑‖k‖₊
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₂
hx : x ∈ spectrum 𝕜₂ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊

Tactic:
·

Goal state after:
𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₂
hx : x ∈ spectrum 𝕜₂ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case refine_1
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsSelfAdjoint a
x : ℂ
hx : x ∈ σₙ ℂ a
⊢ (algebraMap ℝ ℂ) (Complex.reCLM x) = x

pp: case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
⊢ IsStarNormal a ∧ QuasispectrumRestricts a ⇑Complex.reCLM → IsSelfAdjoint a

Tactic:
·

Goal state after:
case refine_1
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsSelfAdjoint a
x : ℂ
hx : x ∈ σₙ ℂ a
⊢ (algebraMap ℝ ℂ) (Complex.reCLM x) = x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
⊢ IsStarNormal a ∧ QuasispectrumRestricts a ⇑Complex.reCLM → IsSelfAdjoint a

Tactic:
·

Goal state after:
case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
⊢ IsStarNormal a ∧ QuasispectrumRestricts a ⇑Complex.reCLM → IsSelfAdjoint a
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case refine_1
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, 0 ≤ x
x : ℝ
hx : x ∈ spectrum ℝ ((algebraMap ℝ A) ↑t - a)
⊢ ↑‖x‖₊ ≤ ↑t

pp: case refine_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
·

Goal state after:
case refine_1
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, 0 ≤ x
x : ℝ
hx : x ∈ spectrum ℝ ((algebraMap ℝ A) ↑t - a)
⊢ ↑‖x‖₊ ≤ ↑t
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_4.a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖ ≤ (fun x ↦ ‖f x‖) x

pp: case a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ (fun x ↦ ‖f x‖) x ≤ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖

Tactic:
·

Goal state after:
case e'_4.a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖ ≤ (fun x ↦ ‖f x‖) x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ (fun x ↦ ‖f x‖) x ≤ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖

Tactic:
·

Goal state after:
case a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ (fun x ↦ ‖f x‖) x ≤ ‖{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case right.left
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ QuasispectrumRestricts (cfcₙ (fun x ↦ √x) a) ⇑ContinuousMap.realToNNReal

pp: case right.right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ cfcₙ (fun x ↦ √x) a * cfcₙ (fun x ↦ √x) a = a

Tactic:
·

Goal state after:
case right.left
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ QuasispectrumRestricts (cfcₙ (fun x ↦ √x) a) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case right.right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ cfcₙ (fun x ↦ √x) a * cfcₙ (fun x ↦ √x) a = a

Tactic:
·

Goal state after:
case right.right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ cfcₙ (fun x ↦ √x) a * cfcₙ (fun x ↦ √x) a = a
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case refine_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
·

Goal state after:
case refine_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hw : ¬w.Finite
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
hw : w.Finite
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hw : ¬w.Finite
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
⊢ ‖0‖ ≤ c

pp: case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ ≤ c

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
⊢ ‖0‖ ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ ≤ c

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
⊢ ‖0‖ < c

pp: case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ < c

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
⊢ ‖0‖ < c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ < c

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ < c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : T2Space A
a : A
f : ℝ → ℝ
ha : IsSelfAdjoint a
⊢ cfc f a = cfc (fun x ↦ ↑(f x.re)) a

Tactic:
exact ha.spectrumRestricts.cfc_eq_restrict (f := Complex.reCLM)
    Complex.isometry_ofReal.isClosedEmbedding ha ha.isStarNormal f

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ : A
⊢ ∀ᶠ (x : A) in 𝓝 a₀, spectrum 𝕜 x ⊆ Metric.closedBall 0 ((‖a₀‖ + 1) * ‖1‖)

pp: case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ : A
a : ℕ → A
ha : Tendsto a atTop (𝓝 a₀)
x : ℕ → 𝕜
hx_mem : ∀ (n : ℕ), x n ∈ spectrum 𝕜 (a n)
x₀ : 𝕜
hx : Tendsto x atTop (𝓝 x₀)
⊢ x₀ ∈ spectrum 𝕜 a₀

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ : A
⊢ ∀ᶠ (x : A) in 𝓝 a₀, spectrum 𝕜 x ⊆ Metric.closedBall 0 ((‖a₀‖ + 1) * ‖1‖)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ : A
a : ℕ → A
ha : Tendsto a atTop (𝓝 a₀)
x : ℕ → 𝕜
hx_mem : ∀ (n : ℕ), x n ∈ spectrum 𝕜 (a n)
x₀ : 𝕜
hx : Tendsto x atTop (𝓝 x₀)
⊢ x₀ ∈ spectrum 𝕜 a₀

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ : A
a : ℕ → A
ha : Tendsto a atTop (𝓝 a₀)
x : ℕ → 𝕜
hx_mem : ∀ (n : ℕ), x n ∈ spectrum 𝕜 (a n)
x₀ : 𝕜
hx : Tendsto x atTop (𝓝 x₀)
⊢ x₀ ∈ spectrum 𝕜 a₀
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
hw : w.Finite
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              u ⊆ w →
                ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                  K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
huw : ¬u ⊆ w
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
hw : w.Finite
huw : u ⊆ w
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
hw : w.Finite
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            w.Finite →
              u ⊆ w →
                ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
                  K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
huw : ¬u ⊆ w
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}