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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, quasispectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
hf0 : f 0 = 0
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) Set.univ

Tactic:
]

Goal state after:
X : Type u_1
A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, quasispectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
hf0 : f 0 = 0
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) Set.univ
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✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
a : A
f : X → C(↑(spectrum 𝕜 a), 𝕜)
hf₁ : Integrable f μ
ha : p a
⊢ (cfcL ha) (∫ (x : X), f x ∂μ) = (cfcL ha) (∫ (x : X), f x ∂μ)

Tactic:
]

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

Goal state before:
pp: m : Type u_1
n : Type u_2
R✝ : Type u_3
S : Type u_4
A✝ : Type u_5
B : Type u_6
R : Type ?u.14
A : Type ?u.16
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Module R A
inst✝ : Star A
e : m ≃ n
M : CStarMatrix m m A
i✝ j✝ : n
⊢ star (M (e.symm j✝) (e.symm i✝)) = star (Matrix.submatrix M (⇑e.symm) (⇑e.symm) j✝ i✝)

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
R✝ : Type u_3
S : Type u_4
A✝ : Type u_5
B : Type u_6
R : Type ?u.14
A : Type ?u.16
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Module R A
inst✝ : Star A
e : m ≃ n
M : CStarMatrix m m A
i✝ j✝ : n
⊢ star (M (e.symm j✝) (e.symm i✝)) = star (Matrix.submatrix M (⇑e.symm) (⇑e.symm) j✝ i✝)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), quasispectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
hf0 : f 0 = 0
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) Set.univ

Tactic:
]

Goal state after:
X : Type u_1
A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), quasispectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
hf0 : f 0 = 0
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹³ : NonUnitalNormedRing A
inst✝¹² : StarRing A
inst✝¹¹ : NormedSpace ℝ A
inst✝¹⁰ : IsScalarTower ℝ A A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
hs : s ∈ 𝓝ˢ (⋃ x, quasispectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) Set.univ

Tactic:
]

Goal state after:
X : Type u_1
A : Type u_2
inst✝¹³ : NonUnitalNormedRing A
inst✝¹² : StarRing A
inst✝¹¹ : NormedSpace ℝ A
inst✝¹⁰ : IsScalarTower ℝ A A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
hs : s ∈ 𝓝ˢ (⋃ x, quasispectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
S : Type u_4
A : Type u_5
B : Type u_6
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M₁ M₂ : CStarMatrix m n A
x✝ : C⋆ᵐᵒᵈ(A, m → A)
i✝ : n
⊢ ((WithCStarModule.equiv A (m → A)) x✝ ᵥ* M₁) i✝ + ((WithCStarModule.equiv A (m → A)) x✝ ᵥ* M₂) i✝ =
    ((WithCStarModule.equiv A (m → A)) x✝ ᵥ* M₁) i✝ + ((WithCStarModule.equiv A (m → A)) x✝ ᵥ* M₂) i✝

Tactic:
]

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

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
S : Type u_4
A : Type u_5
B : Type u_6
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
c : ℂ
M : CStarMatrix m n A
x : C⋆ᵐᵒᵈ(A, m → A)
i : n
⊢ c • ((WithCStarModule.equiv A (m → A)) x ᵥ* M) i = c • ((WithCStarModule.equiv A (m → A)) x ᵥ* M) i

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✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(spectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = ∫ (x : X), (mkD ((spectrum 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✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(spectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = ∫ (x : X), (mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) z ∂μ
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
⊢ ∀ (a : CStarMatrix m n A), toCLM a = 0 → a = 0

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
⊢ ∀ (a : CStarMatrix m n A), toCLM a = 0 → a = 0
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✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(spectrum 𝕜 a)
x : X
cont_x : ContinuousOn (f x) (spectrum 𝕜 a)
⊢ f x ↑z = f x ↑z

Tactic:
]

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

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
h : toCLM M = 0
i : m
j : n
⊢ ‖(toCLM M) ((WithCStarModule.equiv A (m → A)).symm (Pi.single i (star (M i j)))) j‖ = 0

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
h : toCLM M = 0
i : m
j : n
⊢ ‖(toCLM M) ((WithCStarModule.equiv A (m → A)).symm (Pi.single i (star (M i j)))) j‖ = 0
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✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = ∫ (x : X), f x ↑z ∂μ

pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ ContinuousOn (fun z ↦ ∫ (x : X), f x z ∂μ) (spectrum 𝕜 a)

Tactic:
]

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ ContinuousOn (fun z ↦ ∫ (x : X), f x z ∂μ) (spectrum 𝕜 a)
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
v : C⋆ᵐᵒᵈ(A, n → A)
w : C⋆ᵐᵒᵈ(A, m → A)
⊢ ∑ y, ∑ x, w x * (M x y * star (v y)) = ∑ x, ∑ i, w i * M i x * star (v x)

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
v : C⋆ᵐᵒᵈ(A, n → A)
w : C⋆ᵐᵒᵈ(A, m → A)
⊢ ∑ y, ∑ x, w x * (M x y * star (v y)) = ∑ x, ∑ i, w i * M i x * star (v x)
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✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ((spectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ)

Tactic:
]

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ((spectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ)
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
c : ℂ
M : CStarMatrix m n A
⊢ ‖c‖ * ‖toCLM M‖ = ‖c‖ * ‖toCLM M‖

Tactic:
]

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

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
i : m
j : n
⊢ ‖(toCLM M) ((equiv A (m → A)).symm (Pi.single i (star (M i j)))) j‖ ≤ ‖M‖ * ‖M i j‖

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
i : m
j : n
⊢ ‖(toCLM M) ((equiv A (m → A)).symm (Pi.single i (star (M i j)))) j‖ ≤ ‖M‖ * ‖M i j‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
x : ℂ
hx : x ∈ quasispectrum ℂ a
⊢ (x + (starRingEnd ℂ) x) / 2 = (2⁻¹ • 1) • (x + star x)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
x : ℂ
hx : x ∈ quasispectrum ℂ a
⊢ (x + (starRingEnd ℂ) x) / 2 = (2⁻¹ • 1) • (x + star x)
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case inl
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
C : ℝ≥0
h : ∀ (v : C⋆ᵐᵒᵈ(A, m → A)) (w : C⋆ᵐᵒᵈ(A, n → A)), ‖⟪w, (toCLM M) v⟫_A‖ ≤ ↑C * ‖v‖ * ‖w‖
v : C⋆ᵐᵒᵈ(A, m → A)
h₀ : 0 = ‖(toCLM M) v‖
⊢ 0 ≤ ↑C * ‖v‖

Tactic:
]

Goal state after:
case inl
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
C : ℝ≥0
h : ∀ (v : C⋆ᵐᵒᵈ(A, m → A)) (w : C⋆ᵐᵒᵈ(A, n → A)), ‖⟪w, (toCLM M) v⟫_A‖ ≤ ↑C * ‖v‖ * ‖w‖
v : C⋆ᵐᵒᵈ(A, m → A)
h₀ : 0 = ‖(toCLM M) v‖
⊢ 0 ≤ ↑C * ‖v‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
this : ↑(ℜ a) + I • cfcₙ (fun z ↦ ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)
⊢ cfcₙ (fun x ↦ ↑x.im) a = ↑(ℑ a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
this : ↑(ℜ a) + I • cfcₙ (fun z ↦ ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)
⊢ cfcₙ (fun x ↦ ↑x.im) a = ↑(ℑ a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

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

Tactic:
]

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

Goal state before:
pp: case e_6
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 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) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
key₂ :
  ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (mkD ((spectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z
z : ↑(spectrum 𝕜 a)
⊢ (mkD ((spectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z =
    (mkD ((spectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z

Tactic:
]

Goal state after:
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 : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : ContinuousStar A
a : A
ha : p a
⊢ (adjoin 𝕜 (⇑(cfcHom ha) '' {ContinuousMap.restrict (spectrum 𝕜 a) (ContinuousMap.id 𝕜)})).topologicalClosure =
    elemental 𝕜 a

Tactic:
]

Goal state after:
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 : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : ContinuousStar A
a : A
ha : p a
⊢ (adjoin 𝕜 (⇑(cfcHom ha) '' {ContinuousMap.restrict (spectrum 𝕜 a) (ContinuousMap.id 𝕜)})).topologicalClosure =
    elemental 𝕜 a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

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

Tactic:
]

Goal state after:
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
M : CStarMatrix m n A
⊢ ∀ (i : m) (j : n), ‖ofMatrixₗ.symm M i j‖ ≤ ‖M‖

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
M : CStarMatrix m n A
⊢ ∀ (i : m) (j : n), ‖ofMatrixₗ.symm M i j‖ ≤ ‖M‖
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 : ClosedEmbeddingContinuousFunctionalCalculus 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.ContinuousFunctionalCalculus.RealImaginaryPart

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

Tactic:
by
  rw [← cfcₙ_im_id a, cfcₙ_map_quasispectrum ..]

Goal state after:
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
M : CStarMatrix m n A
⊢ ‖toCLM M‖ ≤ ∑ j, ∑ i, ‖M i j‖

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
M : CStarMatrix m n A
⊢ ‖toCLM M‖ ≤ ∑ j, ∑ i, ‖M i j‖
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
⊢ (adjoin 𝕜 (⇑(cfcHom ha) '' {ContinuousMap.restrict (spectrum 𝕜 a) (ContinuousMap.id 𝕜)})).topologicalClosure ≤
    elemental 𝕜 a

Tactic:
]

Goal state after:
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
⊢ (adjoin 𝕜 (⇑(cfcHom ha) '' {ContinuousMap.restrict (spectrum 𝕜 a) (ContinuousMap.id 𝕜)})).topologicalClosure ≤
    elemental 𝕜 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.re) '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ f ↑(ℜ a) = cfcₙ (fun x ↦ f ↑x.re) 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.re) '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ f ↑(ℜ a) = cfcₙ (fun x ↦ f ↑x.re) a