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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
ha : 0 ≤ a
hab : a ≤ b
this : ∀ (a b : Unitization ℂ A), 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖
⊢ ‖a‖ ≤ ‖b‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
ha : 0 ≤ a
hab : a ≤ b
⊢ ‖a‖ ≤ ‖b‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hb : IsSelfAdjoint b
this : ∀ (a b : Unitization ℂ A), IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a)
⊢ IsSelfAdjoint (star a * b * a)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
a : A
⊢ 0 ≤ ⟪a •> x - ‖x‖ ^ 2 •> y, a •> x - ‖x‖ ^ 2 •> y⟫_A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hb : IsSelfAdjoint b
this : ∀ (a b : Unitization ℂ A), IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a)
⊢ star a * b * a ≤ ‖b‖ • (star a * a)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ b✝ : A
hb✝ : IsSelfAdjoint b✝
a b : Unitization ℂ A
hb : IsSelfAdjoint b
⊢ star a * (algebraMap ℝ (Unitization ℂ A)) ‖b‖ * a = ‖b‖ • (star a * a)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hb : IsSelfAdjoint b
⊢ star a * b * a ≤ ‖b‖ • (star a * a)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hb : IsSelfAdjoint b
⊢ a * b * star a ≤ ‖b‖ • (a * star a)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
a : A
⊢ ⟪a •> x - ‖x‖ ^ 2 •> y, a •> x - ‖x‖ ^ 2 •> y⟫_A =
    a * ⟪x, x⟫_A * star a - ‖x‖ ^ 2 •> (a * ⟪y, x⟫_A) - ‖x‖ ^ 2 •> (⟪x, y⟫_A * star a) + ‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : IsClosed {a | 0 ≤ a}
a : Unitization ℂ A
x : A
h : ↑x = a
⊢ 0 ≤ x ↔ 0 ≤ a

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : IsClosed {a | 0 ≤ a}
⊢ IsClosed {a | 0 ≤ a}

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ IsClosed {a | 0 ≤ a}

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConvexOn ℝ (inr '' s) (cfc f)
inrl : A →ₗ[ℝ] Unitization ℂ A := inrHom ℝ ℂ A
hf₀ : ¬f 0 = 0
⊢ cfcₙ f = fun x ↦ 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
a : A
⊢ ‖⟪x, x⟫_A‖ •> (a * star a) = √‖⟪x, x⟫_A‖ ^ 2 •> (a * star a)

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
⊢ ContinuousOn (fun fa ↦ cfc ((toFun {s}) fa.1) fa.2)
    ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)})

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
a : A
⊢ √‖⟪x, x⟫_A‖ ^ 2 •> (a * star a) = ‖x‖ ^ 2 •> (a * star a)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
a : A
⊢ a * ⟪x, x⟫_A * star a - ‖x‖ ^ 2 •> (a * ⟪y, x⟫_A) - ‖x‖ ^ 2 •> (⟪x, y⟫_A * star a) + ‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A ≤
    ‖x‖ ^ 2 •> (a * star a) - ‖x‖ ^ 2 •> (a * ⟪y, x⟫_A) - ‖x‖ ^ 2 •> (⟪x, y⟫_A * star a) +
      ‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
a : A
⊢ 0 ≤
    ‖x‖ ^ 2 •> (a * star a) - ‖x‖ ^ 2 •> (a * ⟪y, x⟫_A) - ‖x‖ ^ 2 •> (⟪x, y⟫_A * star a) +
      ‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConvexOn ℝ (inr '' s) (cfc f)
inrl : A →ₗ[ℝ] Unitization ℂ A := inrHom ℝ ℂ A
hf₀ : f 0 = 0
h₁ : inr ∘ cfcₙ f = fun x ↦ ↑(cfcₙ f x)
⊢ (fun x ↦ ↑(cfcₙ f x)) = fun x ↦ cfc f ↑x

Tactic:
by

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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝⁹ : NormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedAlgebra ℝ A
inst✝⁶ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : ContinuousStar A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, spectrum ℝ≥0 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, 0 ≤ a x
ha₀ : spectrum ℝ≥0 a₀ ⊆ s
ha₀' : 0 ≤ a₀
hf : ContinuousOn f s
⊢ Tendsto (fun x ↦ cfc f (a x)) l (𝓝 (cfc f a₀))

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ⟪x, y⟫_A * ⟪y, x⟫_A ≤ ‖x‖ ^ 2 •> ⟪y, y⟫_A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, y⟫_A‖ ^ 2 = ‖⟪x, y⟫_A * ⟪y, x⟫_A‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, y⟫_A * ⟪y, x⟫_A‖ ≤ ‖‖x‖ ^ 2 •> ⟪y, y⟫_A‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConvexOn ℝ (inr '' s) (cfc f)
⊢ ConvexOn ℝ s (cfcₙ f)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖‖x‖ ^ 2 •> ⟪y, y⟫_A‖ = ‖x‖ ^ 2 * ‖⟪y, y⟫_A‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖x‖ ^ 2 * ‖⟪y, y⟫_A‖ = ‖x‖ ^ 2 * ‖y‖ ^ 2

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖x‖ ^ 2 * ‖y‖ ^ 2 = (‖x‖ * ‖y‖) ^ 2

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
this : ‖⟪x, y⟫_A‖ ^ 2 ≤ (‖x‖ * ‖y‖) ^ 2
⊢ 2 ≠ 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConcaveOn ℝ (inr '' s) (cfc f)
⊢ ConcaveOn ℝ s (- -cfcₙ f)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, y⟫_A‖ ≤ ‖x‖ * ‖y‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConcaveOn ℝ (inr '' s) (cfc f)
⊢ ConcaveOn ℝ s (cfcₙ f)

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖x + y‖ ^ 2 ≤ ‖⟪x, x⟫ + ⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
⊢ ↑x ∈ Icc 0 1 ↔ 0 ≤ x ∧ ‖x‖ ≤ 1

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, x⟫ + ⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖ ≤ ‖⟪x, x⟫‖ + ‖⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ inr ⁻¹' Icc 0 1 = {x | 0 ≤ x} ∩ closedBall 0 1

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ inr '' Ici 0 ⊆ Ici 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, x⟫‖ + ‖⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖ ≤ ‖⟪x, x⟫‖ + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖⟪y, y⟫‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a e : A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
this : a * e = a
⊢ e * a = a

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, x⟫‖ + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖⟪y, y⟫‖ = ‖x‖ ^ 2 + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖y‖ ^ 2

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖x‖ ^ 2 + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖x + y‖ ^ 2 ≤ (‖x‖ + ‖y‖) ^ 2

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ e✝ : A
he✝ : IsStarProjection e✝
ha✝ : 0 ≤ a✝
hae✝ : a✝ ≤ e✝
a e : Unitization ℂ A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
this : sqrt a * (1 - e) = 0
⊢ a * e = a

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : ‖x + y‖ ^ 2 ≤ (‖x‖ + ‖y‖) ^ 2
⊢ 2 ≠ 0

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖x + y‖ ≤ ‖x‖ + ‖y‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
c : ℂ
x : E
⊢ ‖c •> x‖ = ‖c‖ * ‖x‖

Tactic:
by

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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
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
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a e : A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
⊢ a * e = a ∧ e * a = a

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a e : A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
⊢ e * a * e = a

Tactic:
by

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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
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
t : Set X
ha_cont : ContinuousOn a t
ha : ∀ x ∈ t, spectrum ℝ≥0 (a x) ⊆ s
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
n : ℕ
⊢ 0 ≤ a ^ n

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 1 ≤ a
ha' : 0 ≤ a
n m : ℕ
hnm : n ≤ m
⊢ (fun x ↦ a ^ x) n ≤ (fun x ↦ a ^ x) m

Tactic:
simp only

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 1 ≤ a
ha' : 0 ≤ a
n m : ℕ
hnm : n ≤ m
⊢ a ^ n ≤ a ^ m
Module: Mathlib.Analysis.CStarAlgebra.Module.Defs

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
v : E
instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯
instNS : NormedSpace ℂ E := NormedSpace.ofCore ⋯
⊢ 0 ≤ ‖v‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 1 ≤ a
⊢ Monotone fun x ↦ a ^ x

Tactic:
by

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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹¹ : NormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedAlgebra ℝ A
inst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁷ : ContinuousStar A
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
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
x : ↑t
⊢ ∃ S, IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum ℝ≥0 x' ⊆ S) ∧ S ⊆ s

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
v : E
instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯
instNS : NormedSpace ℂ E := NormedSpace.ofCore ⋯
w : E
hw : ‖w‖ ≤ 1
⊢ ‖w‖ * ‖v‖ ≤ 1 * ‖v‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
v : E
instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯
instNS : NormedSpace ℂ E := NormedSpace.ofCore ⋯
w : E
hw : ‖w‖ ≤ 1
⊢ 1 * ‖v‖ = ‖v‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
n m : ℕ
hnm : n ≤ m
⊢ (fun x ↦ a ^ x) m ≤ (fun x ↦ a ^ x) n

Tactic:
simp only

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
n m : ℕ
hnm : n ≤ m
⊢ a ^ m ≤ a ^ n
Module: Mathlib.Analysis.CStarAlgebra.Module.Defs

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
v : E
⊢ ‖v‖ = sSup {x | ∃ w, ∃ (_ : ‖w‖ ≤ 1), ‖⟪w, v⟫_A‖ = x}

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : SMul A E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CStarModule A E
x y : E
⊢ ‖(innerₛₗ x) y‖ ≤ 1 * ‖x‖ * ‖y‖

Tactic:
by

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : SMul A E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CStarModule A E
⊢ Continuous fun x ↦ ⟪x.1, x.2⟫_A

Tactic:
by

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
⊢ Antitone fun x ↦ a ^ x

Tactic:
by

Goal state after: