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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra 𝕜 A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarModule 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : ContinuousStar A
a : A
ha : p a
⊢ ↑(elemental 𝕜 a) ⊆ ↑(elemental 𝕜 a)

Tactic:
]

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

Goal state before:
pp: case h
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
M : CStarMatrix m n A
v : C⋆ᵐᵒᵈ(A, m → A)
i : n
a✝¹ : i ∈ Finset.univ
j : m
a✝ : j ∈ Finset.univ
⊢ ‖M j i‖ * ‖v j‖ ≤ ‖M j i‖ * ‖v‖

Tactic:
]

Goal state after:
case h
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
M : CStarMatrix m n A
v : C⋆ᵐᵒᵈ(A, m → A)
i : n
a✝¹ : i ∈ Finset.univ
j : m
a✝ : j ∈ Finset.univ
⊢ ‖M j i‖ * ‖v j‖ ≤ ‖M j i‖ * ‖v‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℂ A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : SMulCommClass ℂ A A
inst✝² : StarModule ℂ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMapZero.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.re) '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ (fun x ↦ f ↑x.re) a = cfcₙ (fun x ↦ f ↑x.re) a

Tactic:
]

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

Goal state before:
pp: A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℂ A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : SMulCommClass ℂ A A
inst✝² : StarModule ℂ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMapZero.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ f ↑(ℑ a) = cfcₙ (fun x ↦ f ↑x.im) a

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℂ A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : SMulCommClass ℂ A A
inst✝² : StarModule ℂ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMapZero.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ f ↑(ℑ a) = cfcₙ (fun x ↦ f ↑x.im) a
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : (fun x ↦ (ofMatrix.symm x.1, ofMatrix.symm x.2)) = id
⊢ 𝓤 (Matrix m n A) = 𝓤 (CStarMatrix m n A)

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : (fun x ↦ (ofMatrix.symm x.1, ofMatrix.symm x.2)) = id
⊢ 𝓤 (Matrix m n A) = 𝓤 (CStarMatrix m n A)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℂ A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : SMulCommClass ℂ A A
inst✝² : StarModule ℂ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMapZero.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ (fun x ↦ f ↑x.im) a = cfcₙ (fun x ↦ f ↑x.im) a

Tactic:
]

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
a : A
f : X → C(↑(quasispectrum 𝕜 a), 𝕜)₀
hf₁ : Integrable f μ
ha : p a
⊢ (cfcₙL ha) (∫ (x : X), f x ∂μ) = (cfcₙL ha) (∫ (x : X), f x ∂μ)

Tactic:
]

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (re '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.re) '' quasispectrum ℂ a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (re '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.re) '' quasispectrum ℂ a)
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
⊢ ∀ (x : C⋆ᵐᵒᵈ(A, n → A)), ‖(toCLM M) x‖ ≤ √‖M * star M‖ * ‖x‖

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
⊢ ∀ (x : C⋆ᵐᵒᵈ(A, n → A)), ‖(toCLM M) x‖ ≤ √‖M * star M‖ * ‖x‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: A : Type u_1
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra ℝ A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : IsTopologicalRing A
inst✝⁶ : T2Space A
inst✝⁵ : PartialOrder A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : StarOrderedRing A
inst✝² : ContinuousStar A
inst✝¹ : StarModule ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x} ⊆ (fun x ↦ cfc x a) '' (fun x ↦ cfc x a) ⁻¹' {a | 0 ≤ a}

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra ℝ A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : IsTopologicalRing A
inst✝⁶ : T2Space A
inst✝⁵ : PartialOrder A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : StarOrderedRing A
inst✝² : ContinuousStar A
inst✝¹ : StarModule ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x} ⊆ (fun x ↦ cfc x a) '' (fun x ↦ cfc x a) ⁻¹' {a | 0 ≤ a}
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ √‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ √‖M * star M‖ * ‖v‖

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ √‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ √‖M * star M‖ * ‖v‖
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ ‖v‖ * ‖(toCLM Mᴴ) ((toCLM M) v)‖ ≤ ‖v‖ * (‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ ‖v‖ * ‖(toCLM Mᴴ) ((toCLM M) v)‖ ≤ ‖v‖ * (‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖)
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case hbc
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ ‖(toCLM Mᴴ ∘SL toCLM M) v‖ ≤ ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖

Tactic:
]

Goal state after:
case hbc
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ ‖(toCLM Mᴴ ∘SL toCLM M) v‖ ≤ ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: A : Type u_1
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra ℝ A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : IsTopologicalRing A
inst✝⁶ : T2Space A
inst✝⁵ : PartialOrder A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : StarOrderedRing A
inst✝² : ContinuousStar A
inst✝¹ : StarModule ℝ A
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfc x a) '' (fun x ↦ cfc x a) ⁻¹' {a | 0 ≤ a} ⊆ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x}

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra ℝ A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : IsTopologicalRing A
inst✝⁶ : T2Space A
inst✝⁵ : PartialOrder A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : StarOrderedRing A
inst✝² : ContinuousStar A
inst✝¹ : StarModule ℝ A
inst✝ : ClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfc x a) '' (fun x ↦ cfc x a) ⁻¹' {a | 0 ≤ a} ⊆ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x}
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ ‖v‖ * ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖ = ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖ ^ 2

Tactic:
by ring

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : Module 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousConstSMul 𝕜 A
inst✝³ : StarModule 𝕜 A
inst✝² : IsTopologicalRing A
inst✝¹ : ContinuousStar A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
⊢ (adjoin 𝕜 (⇑(cfcₙHom ha) '' {ContinuousMapZero.id (quasispectrum 𝕜 a)})).topologicalClosure = elemental 𝕜 a

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : Module 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousConstSMul 𝕜 A
inst✝³ : StarModule 𝕜 A
inst✝² : IsTopologicalRing A
inst✝¹ : ContinuousStar A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
⊢ (adjoin 𝕜 (⇑(cfcₙHom ha) '' {ContinuousMapZero.id (quasispectrum 𝕜 a)})).topologicalClosure = elemental 𝕜 a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : Module 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousConstSMul 𝕜 A
inst✝³ : StarModule 𝕜 A
inst✝² : IsTopologicalRing A
inst✝¹ : ContinuousStar A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
⊢ ↑(elemental 𝕜 a) = ↑(elemental 𝕜 a)

Tactic:
]

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(quasispectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = ∫ (x : X), (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) z ∂μ

Tactic:
]

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(quasispectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = ∫ (x : X), (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) z ∂μ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : Module 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousConstSMul 𝕜 A
inst✝³ : StarModule 𝕜 A
inst✝² : IsTopologicalRing A
inst✝¹ : ContinuousStar A
inst✝ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
⊢ (adjoin 𝕜 (⇑(cfcₙHom ha) '' {ContinuousMapZero.id (quasispectrum 𝕜 a)})).topologicalClosure ≤ elemental 𝕜 a

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : Module 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousConstSMul 𝕜 A
inst✝³ : StarModule 𝕜 A
inst✝² : IsTopologicalRing A
inst✝¹ : ContinuousStar A
inst✝ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
⊢ (adjoin 𝕜 (⇑(cfcₙHom ha) '' {ContinuousMapZero.id (quasispectrum 𝕜 a)})).topologicalClosure ≤ elemental 𝕜 a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : Module 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : ContinuousConstSMul 𝕜 A
inst✝³ : StarModule 𝕜 A
inst✝² : IsTopologicalRing A
inst✝¹ : ContinuousStar A
inst✝ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
⊢ ↑(elemental 𝕜 a) ⊆ ↑(elemental 𝕜 a)

Tactic:
]

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(quasispectrum 𝕜 a)
x : X
cont_x : ContinuousOn (f x) (quasispectrum 𝕜 a)
zero_x : f x 0 = 0
⊢ f x ↑z = f x ↑z

Tactic:
]

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

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ √‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ √(‖M * star M‖ * ‖v‖ ^ 2)

pp: case hx
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ 0 ≤ ‖M * star M‖

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ √‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ √(‖M * star M‖ * ‖v‖ ^ 2)

case hx
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ 0 ≤ ‖M * star M‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

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 : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
x : ℂ
hx : x ∈ spectrum ℂ a
⊢ (x + (starRingEnd ℂ) x) / 2 = (2⁻¹ • 1) • (x + star x)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
x : ℂ
hx : x ∈ spectrum ℂ a
⊢ (x + (starRingEnd ℂ) x) / 2 = (2⁻¹ • 1) • (x + star x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

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 : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
this : ↑(ℜ a) + I • cfc (fun z ↦ ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)
⊢ cfc (fun x ↦ ↑x.im) a = ↑(ℑ a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
this : ↑(ℜ a) + I • cfc (fun z ↦ ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)
⊢ cfc (fun x ↦ ↑x.im) a = ↑(ℑ a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

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 : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
⊢ (fun x ↦ ↑x.re) '' spectrum ℂ a = (fun x ↦ ↑x.re) '' spectrum ℂ a

Tactic:
]

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

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 : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
⊢ (fun x ↦ ↑x.im) '' spectrum ℂ a = (fun x ↦ ↑x.im) '' spectrum ℂ a

Tactic:
]

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

Goal state before:
pp: A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.re) '' spectrum ℂ a)
ha : IsStarNormal a
⊢ cfc f ↑(ℜ a) = cfc (fun x ↦ f ↑x.re) a

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.re) '' spectrum ℂ a)
ha : IsStarNormal a
⊢ cfc f ↑(ℜ a) = cfc (fun x ↦ f ↑x.re) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.re) '' spectrum ℂ a)
ha : IsStarNormal a
⊢ cfc (fun x ↦ f ↑x.re) a = cfc (fun x ↦ f ↑x.re) a

Tactic:
]

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

Goal state before:
pp: A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' spectrum ℂ a)
ha : IsStarNormal a
⊢ cfc f ↑(ℑ a) = cfc (fun x ↦ f ↑x.im) a

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' spectrum ℂ a)
ha : IsStarNormal a
⊢ cfc f ↑(ℑ a) = cfc (fun x ↦ f ↑x.im) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' spectrum ℂ a)
ha : IsStarNormal a
⊢ cfc (fun x ↦ f ↑x.im) a = cfc (fun x ↦ f ↑x.im) a

Tactic:
]

Goal state after: