mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.AbsConvexOpen

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : ContinuousSMul ℝ E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
⊢ 0 ∈ (t.sup (gaugeSeminormFamily 𝕜 E)).ball 0 r ∧
    IsOpen[inst✝⁷] ((t.sup (gaugeSeminormFamily 𝕜 E)).ball 0 r) ∧
      AbsConvex 𝕜 ((t.sup (gaugeSeminormFamily 𝕜 E)).ball 0 r)

Tactic:
Seminorm.ball_finset_sup_eq_iInter _ _ _ hr

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : ContinuousSMul ℝ E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
⊢ 0 ∈ ⋂ i ∈ t, (gaugeSeminormFamily 𝕜 E i).ball 0 r ∧
    IsOpen[inst✝⁷] (⋂ i ∈ t, (gaugeSeminormFamily 𝕜 E i).ball 0 r) ∧
      AbsConvex 𝕜 (⋂ i ∈ t, (gaugeSeminormFamily 𝕜 E i).ball 0 r)
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂

Tactic:
by
  rw [dist_comm, dist_midpoint_right]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
⊢ 0 < 2⁻¹ ∧ 2⁻¹ < 3 / 4 ∧ 3 / 4 < 1

Tactic:
norm_num

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

Goal state before:
pp: A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
⊢ spectrum ℂ ((gelfandTransform ℂ A) a) = spectrum ℂ a

Tactic:
ext z

Goal state after:
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
z : ℂ
⊢ z ∈ spectrum ℂ ((gelfandTransform ℂ A) a) ↔ z ∈ spectrum ℂ a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ a⁺ = a ↔ 0 ≤ a

Tactic:
by
  refine ⟨fun ha ↦ ha ▸ posPart_nonneg a, fun ha ↦ ?_⟩
  conv_rhs => rw [← cfcₙ_id ℝ a]
  rw [posPart_def]
  refine cfcₙ_congr (fun x hx ↦ ?_)
  simpa [_root_.posPart_def] using quasispectrum_nonneg_of_nonneg a ha x hx

Goal state after:
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : PseudoEMetricSpace β
s : Set α
⊢ Isometry (⇑UniformFun.ofFun ∘ s.domRestrict ∘ ⇑(toFun {s}))

Tactic:
by
  simp [Isometry, edist_def, UniformFun.edist_def, iSup_subtype]

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

Goal state before:
pp: case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
z : ℂ
hz : z ∈ spectrum ℂ a
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩

Tactic:
have hz' := (StarSubalgebra.spectrum_eq (hS := isClosed ℂ a)
      (a := ⟨a, self_mem ℂ a⟩) ▸ hz)

Goal state after:
case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
z : ℂ
hz : z ∈ spectrum ℂ a
hz' : z ∈ spectrum ℂ ⟨a, ⋯⟩
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
⊢ ((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
      (ContinuousMap.restrict (spectrum ℂ a) (ContinuousMap.id ℂ)) =
    a

Tactic:
simp [continuousFunctionalCalculus_map_id a]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℂ ((algebraMap ℝ A) ↑t - a) ≤ ↑t

pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)

pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : ↑‖a‖₊ ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint a

Tactic:
← SpectrumRestricts.spectralRadius_eq (f := Complex.reCLM)

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℝ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t

A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℝ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ SpectrumRestricts ((algebraMap ℝ A) ↑t - a) ⇑Complex.reCLM

A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ SpectrumRestricts a ⇑Complex.reCLM

A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : spectralRadius ℂ a ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)

A : Type u_1
inst✝ : CStarAlgebra A
a : A
t : ℝ≥0
ha : IsSelfAdjoint a
ht : ↑‖a‖₊ ≤ ↑t
this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)
⊢ IsSelfAdjoint a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
x : ℝ≥0
y : ℝ
⊢ (algebraMap ℝ≥0 A) x ^ y = (algebraMap ℝ≥0 A) (x ^ y)

Tactic:
rpow_def,

Goal state after:
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
x : ℝ≥0
y : ℝ
⊢ cfc (fun x ↦ x ^ y) ((algebraMap ℝ≥0 A) x) = (algebraMap ℝ≥0 A) (x ^ y)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : Aˣ
x : ℝ
ha' : 0 ≤ ↑a
h₁ : ContinuousOn (fun z ↦ z ^ x) (Inv.inv '' spectrum ℝ≥0 ↑a)
⊢ ↑a ^ (-x) = cfc (fun x ↦ x⁻¹) ↑a ^ x

Tactic:
rpow_def,

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : Aˣ
x : ℝ
ha' : 0 ≤ ↑a
h₁ : ContinuousOn (fun z ↦ z ^ x) (Inv.inv '' spectrum ℝ≥0 ↑a)
⊢ cfc (fun x_1 ↦ x_1 ^ (-x)) ↑a = cfc (fun x ↦ x⁻¹) ↑a ^ x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : Aˣ
x : ℝ
ha' : 0 ≤ ↑a
h₁ : ContinuousOn (fun z ↦ z ^ x) (Inv.inv '' spectrum ℝ≥0 ↑a)
⊢ cfc (fun x_1 ↦ x_1 ^ (-x)) ↑a = cfc (fun x ↦ x⁻¹) ↑a ^ x

Tactic:
rpow_def,

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : Aˣ
x : ℝ
ha' : 0 ≤ ↑a
h₁ : ContinuousOn (fun z ↦ z ^ x) (Inv.inv '' spectrum ℝ≥0 ↑a)
⊢ cfc (fun x_1 ↦ x_1 ^ (-x)) ↑a = cfc (fun x_1 ↦ x_1 ^ x) (cfc (fun x ↦ x⁻¹) ↑a)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ≥0
hx : 0 < x
⊢ cfcₙ (fun x_1 ↦ x_1.nnrpow x) a = a ^ ↑x

Tactic:
rpow_def,

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ≥0
hx : 0 < x
⊢ cfcₙ (fun x_1 ↦ x_1.nnrpow x) a = cfc (fun x_1 ↦ x_1 ^ ↑x) a
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
n : ℕ
ha : 0 ≤ a
⊢ ‖a‖₊ ≤ ↑n ↔ a ≤ ↑n

Tactic:
by
  simpa using nnnorm_le_iff_of_nonneg a n

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 : Aˣ
ha : IsStrictlyPositive ↑a
⊢ ↑a ^ (-(1 / 2) + 1 + -(1 / 2)) = 1

Tactic:
norm_num

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : Aˣ
ha : IsStrictlyPositive ↑a
⊢ ↑a ^ 0 = 1
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
b : Aˣ
hb : IsStrictlyPositive ↑b
hbab : 0 ≤ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2))
h : ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2)) ≤ 1
⊢ a = ↑b ^ (1 / 2 + -(1 / 2)) * a * ↑b ^ (-(1 / 2) + 1 / 2)

Tactic:
norm_num

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : Aˣ
hb : IsStrictlyPositive ↑b
hbab : 0 ≤ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2))
h : ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2)) ≤ 1
⊢ a = ↑b ^ 0 * a * ↑b ^ 0
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 simp only [mul_pow]

Goal state after:
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
⊢ ‖(equiv A ((j : ι) → E j)).symm (Pi.single i y)‖ = ‖y‖

Tactic:
by
  let _ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, Π i, E i) := normedAddCommGroup A
  rw [← sq_eq_sq₀ (by positivity) (by positivity)]
  simp [norm_sq_eq A]

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
m✝ m : A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
ε : ℝ≥0
hε : 0 < ↑ε
⊢ ∃ ia, (0 ≤ ia ∧ ‖ia‖ < 1) ∧ ∀ x ∈ {x | ia ≤ x} ∩ closedBall 0 1, x * m ∈ closedBall m ↑ε

Tactic:
rw [coe_pos] at hε

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝ m : A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
ε : ℝ≥0
hε : 0 < ε
⊢ ∃ ia, (0 ≤ ia ∧ ‖ia‖ < 1) ∧ ∀ x ∈ {x | ia ≤ x} ∩ closedBall 0 1, x * m ∈ closedBall m ↑ε
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : SeminormedAddCommGroup α
A : Matrix m n α
⊢ ‖Aᵀ‖₊ = ‖A‖₊

Tactic:
rw [frobenius_nnnorm_def, frobenius_nnnorm_def, Finset.sum_comm]
  simp_rw [Matrix.transpose_apply]

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

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
⊢ 2⁻¹ < 1

Tactic:
norm_num

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
c : ℂ
M : CStarMatrix m n A
⊢ ‖c • toCLM M‖ = ‖c‖ * ‖toCLM M‖

Tactic:
norm_smul _ (toCLM M)

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
c : ℂ
M : CStarMatrix m n A
⊢ ‖c‖ * ‖toCLM M‖ = ‖c‖ * ‖toCLM M‖
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
⊢ mkD ((spectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0 =
    ∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ

Tactic:
ext z

Goal state after:
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 =
    (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
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✝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
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
key₂ :
  ∀ (z : ↑(quasispectrum 𝕜 a)),
    ∫ (x : X), f x ↑z ∂μ = (mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z
⊢ mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0 =
    ∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ

Tactic:
ext z

Goal state after:
case e_6
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
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
key₂ :
  ∀ (z : ↑(quasispectrum 𝕜 a)),
    ∫ (x : X), f x ↑z ∂μ = (mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z
z : ↑(quasispectrum 𝕜 a)
⊢ (mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z =
    (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
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✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedSpace 𝕜 A
inst✝⁷ : IsScalarTower 𝕜 A A
inst✝⁶ : SMulCommClass 𝕜 A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(quasispectrum 𝕜 a), 𝕜)
hf : ContinuousOn (uncurry f) (s ×ˢ quasispectrum 𝕜 a)
f_zero : ∀ᵐ (x : X) ∂μ.restrict s, f x 0 = 0
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ quasispectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
⊢ cfcₙ (fun r ↦ ∫ (x : X) in s, f x r ∂μ) a = ∫ (x : X) in s, cfcₙ (f x) a ∂μ

Tactic:
by
  have : ∀ᵐ (x : X) ∂(μ.restrict s), ContinuousOn (f x) (quasispectrum 𝕜 a) :=
    ae_restrict_of_forall_mem hs fun x hx ↦
      hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨hx, hz⟩
  refine cfcₙ_setIntegral' _ _ this f_zero ⟨?_, ?_⟩ ha
  · exact aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry hs _ _ hf f_zero
  · exact hasFiniteIntegral_mkD_restrict_of_bound f _ this f_zero bound bound_int bound_ge

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ((innerₛₗ 𝕜) (x + y)).toAddMonoidHom = ((innerₛₗ 𝕜) x).toAddMonoidHom + ((innerₛₗ 𝕜) y).toAddMonoidHom

Tactic:
ext z

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y z : E
⊢ ((innerₛₗ 𝕜) (x + y)).toAddMonoidHom z = (((innerₛₗ 𝕜) x).toAddMonoidHom + ((innerₛₗ 𝕜) y).toAddMonoidHom) z
Module: Mathlib.Analysis.CStarAlgebra.Hom

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CStarAlgebra A
inst✝³ : CStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
a : A
ha : IsSelfAdjoint a
φ : F
hφ : Function.Injective ⇑φ
h_spec : spectrum ℝ ((StarAlgHom.restrictScalars ℝ ↑φ) a) ⊆ spectrum ℝ a
x : ℝ
hx : x ∈ spectrum ℝ a
hx' : x ∉ spectrum ℝ (φ a)
f : C(ℝ, ℝ)
h_eqOn : Set.EqOn (⇑f) 0 (spectrum ℝ (φ a))
h_eqOn_x : Set.EqOn (⇑f) 1 {x}
this : φ (cfc (⇑f) a) = 0
⊢ False

Tactic:
map_eq_zero_iff φ hφ,

Goal state after:
F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CStarAlgebra A
inst✝³ : CStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
a : A
ha : IsSelfAdjoint a
φ : F
hφ : Function.Injective ⇑φ
h_spec : spectrum ℝ ((StarAlgHom.restrictScalars ℝ ↑φ) a) ⊆ spectrum ℝ a
x : ℝ
hx : x ∈ spectrum ℝ a
hx' : x ∉ spectrum ℝ (φ a)
f : C(ℝ, ℝ)
h_eqOn : Set.EqOn (⇑f) 0 (spectrum ℝ (φ a))
h_eqOn_x : Set.EqOn (⇑f) 1 {x}
this : cfc (⇑f) a = 0
⊢ False
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
⊢ (e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) = e.symm.trans e'

Tactic:
ext z

Goal state after:
case h
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ ↑((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)) z = ↑(e.symm.trans e') z

case hinv
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ ↑((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).symm z = ↑(e.symm.trans e').symm z

case hs
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ z ∈ ((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).source ↔ z ∈ (e.symm.trans e').source
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: B : Type u_2
F : Type u_3
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
E : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F E)
inst✝¹ : (b : B) → TopologicalSpace (E b)
inst✝ : FiberBundle F E
s : (x : B) → E x
x₀ : B
⊢ ContinuousAt (fun x ↦ TotalSpace.mk' F x (s x)) x₀ ↔
    ContinuousAt (fun x ↦ (↑(trivializationAt F E x₀) { proj := x, snd := s x }).2) x₀

Tactic:
simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_section F

Goal state after:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case refine_1
ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
v : F
h : ∀ (i j : ι), ∀ x ∈ Z.baseSet i ∩ Z.baseSet j, Z.coordChange i j x v = v
x : B
A : Z.baseSet (Z.indexAt x) ∈ 𝓝 x
⊢ (have this := fun x ↦ { proj := x, snd := v };
      this) ⁻¹'
      (Z.localTrivAt x).source ∈
    𝓝 x

pp: case refine_2
ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
v : F
h : ∀ (i j : ι), ∀ x ∈ Z.baseSet i ∩ Z.baseSet j, Z.coordChange i j x v = v
x : B
A : Z.baseSet (Z.indexAt x) ∈ 𝓝 x
⊢ ContinuousAt
    (↑(Z.localTrivAt x).toOpenPartialHomeomorph ∘
      have this := fun x ↦ { proj := x, snd := v };
      this)
    x

Tactic:
· exact A

Goal state after:
case refine_2
ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
v : F
h : ∀ (i j : ι), ∀ x ∈ Z.baseSet i ∩ Z.baseSet j, Z.coordChange i j x v = v
x : B
A : Z.baseSet (Z.indexAt x) ∈ 𝓝 x
⊢ ContinuousAt
    (↑(Z.localTrivAt x).toOpenPartialHomeomorph ∘
      have this := fun x ↦ { proj := x, snd := v };
      this)
    x