mathlib-initiative/mathlib-tactics huggingface dataset:
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 : ∀ x ∈ spectrum ℝ a, 1 ≤ x
ha' : 0 ≤ a
n m : ℕ
hnm : n ≤ m
⊢ ∀ x ∈ spectrum ℝ a, x ^ n ≤ x ^ m

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : ∀ x ∈ spectrum ℝ a, 1 ≤ x
ha' : 0 ≤ a
n m : ℕ
hnm : n ≤ m
⊢ ∀ x ∈ spectrum ℝ a, x ^ n ≤ x ^ m
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
⊢ ∀ x ∈ spectrum ℝ a, x ^ m ≤ x ^ n

Tactic:
]

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
⊢ ∀ x ∈ spectrum ℝ a, x ^ m ≤ x ^ n
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 : ∀ x ∈ spectrum ℝ a, x ≤ 1
n m : ℕ
hnm : n ≤ m
⊢ ∀ x ∈ spectrum ℝ a, x ^ m ≤ x ^ n

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha₀ : 0 ≤ a
ha₁ : ∀ x ∈ spectrum ℝ a, x ≤ 1
n m : ℕ
hnm : n ≤ m
⊢ ∀ x ∈ spectrum ℝ a, x ^ m ≤ x ^ n
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
⊢ ‖x‖ ^ 2 = √‖x * star x‖ ^ 2

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
⊢ ‖x‖ ^ 2 = √‖x * star x‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalCStarAlgebra A
inst✝⁸ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : Module ℂ E
inst✝⁵ : SMul A E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : Module ℂ F
inst✝² : SMul A F
inst✝¹ : CStarModule A E
inst✝ : CStarModule A F
x : C⋆ᵐᵒᵈ(A, E × F)
⊢ ‖x.1‖ ^ 2 + 2 * ‖x.1‖ * ‖x.2‖ + ‖x.2‖ ^ 2 = (‖x.1‖ + ‖x.2‖) ^ 2

Tactic:
by ring

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

Goal state before:
pp: A : Type u_1
inst✝¹⁰ : NonUnitalCStarAlgebra A
inst✝⁹ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : Module ℂ E
inst✝⁶ : SMul A E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : Module ℂ F
inst✝³ : SMul A F
inst✝² : CStarModule A E
inst✝¹ : CStarModule A F
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, E × F)
⊢ max ‖x.1‖ ‖x.2‖ ≤ √‖⟪x.1, x.1⟫_A + ⟪x.2, x.2⟫_A‖

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹⁰ : NonUnitalCStarAlgebra A
inst✝⁹ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : Module ℂ E
inst✝⁶ : SMul A E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : Module ℂ F
inst✝³ : SMul A F
inst✝² : CStarModule A E
inst✝¹ : CStarModule A F
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, E × F)
⊢ max ‖x.1‖ ‖x.2‖ ≤ √‖⟪x.1, x.1⟫_A + ⟪x.2, x.2⟫_A‖
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ a
hb : 0 ≤ b
⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ a
hb : 0 ≤ b
⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b
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
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ

Tactic:
]

Goal state after:
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
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → Module ℂ (E i)
inst✝¹ : (i : ι) → SMul A (E i)
inst✝ : (i : ι) → CStarModule A (E i)
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
⊢ ‖x‖ ≤ ∑ i, ‖x i‖

Tactic:
refine abs_le_of_sq_le_sq' ?_ (by positivity) |>.2
  calc ‖x‖ ^ 2 ≤ ∑ i, ‖⟪x i, x i⟫_A‖ := pi_norm_sq x ▸ norm_sum_le _ _
    _ = ∑ i, ‖x i‖ ^ 2 := by simp only [norm_sq_eq A]
    _ ≤ (∑ i, ‖x i‖) ^ 2 := sum_sq_le_sq_sum_of_nonneg (fun _ _ ↦ norm_nonneg _)

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
⊢ 1 - cfc (fun x ↦ x⁻¹) (1 + c) = 1 - cfc (fun x ↦ x⁻¹) (1 + c)

pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
⊢ ContinuousOn (fun x ↦ x⁻¹) ((fun x ↦ 1 + x) '' σ ℝ≥0 c)

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
⊢ ContinuousOn (fun x ↦ x⁻¹) ((fun x ↦ 1 + x) '' σ ℝ≥0 c)
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
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ

Tactic:
]

Goal state after:
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
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
h_cfc_one_sub :
  ∀ (c : Unitization ℂ A),
    autoParam (0 ≤ c) monotoneOn_one_sub_one_add_inv._auto_1_6 →
      cfc (fun x ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (fun x ↦ x⁻¹) (1 + c)
⊢ 1 - cfc (fun x ↦ x⁻¹) (1 + ↑a) ≤ 1 - cfc (fun x ↦ x⁻¹) (1 + ↑b)

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
h_cfc_one_sub :
  ∀ (c : Unitization ℂ A),
    autoParam (0 ≤ c) monotoneOn_one_sub_one_add_inv._auto_1_6 →
      cfc (fun x ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (fun x ↦ x⁻¹) (1 + c)
⊢ 1 - cfc (fun x ↦ x⁻¹) (1 + ↑a) ≤ 1 - cfc (fun x ↦ x⁻¹) (1 + ↑b)
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ⟪Pi.single i y i, x i⟫_A = ⟪y, x i⟫_A

pp: case h₀
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ∀ b ∈ Finset.univ, b ≠ i → ⟪Pi.single i y b, x b⟫_A = 0

pp: case h₁
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ i ∉ Finset.univ → ⟪Pi.single i y i, x i⟫_A = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ⟪Pi.single i y i, x i⟫_A = ⟪y, x i⟫_A

case h₀
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ∀ b ∈ Finset.univ, b ≠ i → ⟪Pi.single i y b, x b⟫_A = 0

case h₁
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ i ∉ Finset.univ → ⟪Pi.single i y i, x i⟫_A = 0
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case h
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
h_cfc_one_sub :
  ∀ (c : Unitization ℂ A),
    autoParam (0 ≤ c) monotoneOn_one_sub_one_add_inv._auto_1_6 →
      cfc (fun x ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (fun x ↦ x⁻¹) (1 + c)
⊢ (1 + ↑b) ^ (-1) ≤ (1 + ↑a) ^ (-1)

Tactic:
]

Goal state after:
case h
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
h_cfc_one_sub :
  ∀ (c : Unitization ℂ A),
    autoParam (0 ≤ c) monotoneOn_one_sub_one_add_inv._auto_1_6 →
      cfc (fun x ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (fun x ↦ x⁻¹) (1 + c)
⊢ (1 + ↑b) ^ (-1) ≤ (1 + ↑a) ^ (-1)
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ⟪x i, Pi.single i y i⟫_A = ⟪x i, y⟫_A

pp: case h₀
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ∀ b ∈ Finset.univ, b ≠ i → ⟪x b, Pi.single i y b⟫_A = 0

pp: case h₁
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ i ∉ Finset.univ → ⟪x i, Pi.single i y i⟫_A = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ⟪x i, Pi.single i y i⟫_A = ⟪x i, y⟫_A

case h₀
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ ∀ b ∈ Finset.univ, b ≠ i → ⟪x b, Pi.single i y b⟫_A = 0

case h₁
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
y : E i
⊢ i ∉ Finset.univ → ⟪x i, Pi.single i y i⟫_A = 0
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
hs : s ∈ 𝓝ˢ (⋃ x, spectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ

Tactic:
]

Goal state after:
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
hs : s ∈ 𝓝ˢ (⋃ x, spectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) a = cfcₙ (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) a

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) a = cfcₙ (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) a
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ cfcₙ (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) b = cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ cfcₙ (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) b = cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
i : ι
y : E i
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ‖(equiv A ((j : ι) → E j)).symm (Pi.single i y)‖ ^ 2 = ‖y‖ ^ 2

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
i : ι
y : E i
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ‖(equiv A ((j : ι) → E j)).symm (Pi.single i y)‖ ^ 2 = ‖y‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: this : (fun x ↦ x * (1 + x)⁻¹) = fun x ↦ 1 - (1 + x)⁻¹
⊢ InvOn (fun x ↦ x * (1 + x)⁻¹) (fun x ↦ x * (1 - x)⁻¹) {x | x < 1} {x | x < 1}

Tactic:
]

Goal state after:
this : (fun x ↦ x * (1 + x)⁻¹) = fun x ↦ 1 - (1 + x)⁻¹
⊢ InvOn (fun x ↦ x * (1 + x)⁻¹) (fun x ↦ x * (1 - x)⁻¹) {x | x < 1} {x | x < 1}
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ‖⟪x i, x i⟫_A‖ ≤ ‖∑ i, ⟪x i, x i⟫_A‖

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ‖⟪x i, x i⟫_A‖ ≤ ‖∑ i, ⟪x i, x i⟫_A‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))

Tactic:
have hf := h_tendsto.continuousOn hF.frequently

Goal state after:
case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ∀ (i : ι), ‖(equiv A ((i : ι) → E i)) x i‖ ≤ ‖x‖

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ∀ (i : ι), ‖(equiv A ((i : ι) → E i)) x i‖ ≤ ‖x‖