mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.Basic

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
⊢ IsCompact (characterSpace 𝕜 A)

Tactic:
have h : characterSpace 𝕜 A ⊆ toStrongDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ := by
    intro φ hφ
    rw [Set.mem_preimage, mem_closedBall_zero_iff]
    exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ :)

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
h : characterSpace 𝕜 A ⊆ ⇑toStrongDual ⁻¹' Metric.closedBall 0 ‖1‖
⊢ IsCompact (characterSpace 𝕜 A)
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
n : ℕ
x : X
hx : x ∈ c.C
⊢ approx n c x = 0

Tactic:
induction n generalizing c with

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
n : ℕ
x : X
hx : x ∉ c.U
⊢ approx n c x = 1

Tactic:
induction n generalizing c with

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
n : ℕ
x : X
⊢ 0 ≤ approx n c x

Tactic:
induction n generalizing c with

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
n : ℕ
x : X
⊢ approx n c x ≤ 1

Tactic:
induction n generalizing c with

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
n : ℕ
x : X
⊢ approx n c x ∈ Icc (approx n c.right x) (approx n c.left x)

Tactic:
induction n generalizing c with

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
n : ℕ
x : X
⊢ approx n c x ≤ approx (n + 1) c x

Tactic:
induction n generalizing c with

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
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
n : ℕ
x✝ : True
⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n

Tactic:
induction n generalizing c with

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
hI : IsClosed ↑I
⊢ I.IsMaximal ↔ ∃ x, idealOfSet 𝕜 {x}ᶜ = I

Tactic:
by
  refine
    ⟨?_, fun h =>
      let ⟨x, hx⟩ := h
      hx ▸ idealOf_compl_singleton_isMaximal 𝕜 x⟩
  intro hI'
  obtain ⟨x, hx⟩ := setOfIdeal_eq_compl_singleton I
  exact
    ⟨x, by
      simpa only [idealOfSet_ofIdeal_eq_closure, I.closure_eq_of_isClosed hI] using
        congr_arg (idealOfSet 𝕜) hx.symm⟩

Goal state after:
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
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : -x ∈ spectrum ℝ ↑(-a)
⊢ id x = -x⁻

Tactic:
← Unitization.quasispectrum_eq_spectrum_inr ℝ

Goal state after:
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
ha : 0 ≤ -a
x✝ : IsSelfAdjoint a
x : ℝ
hx : -x ∈ quasispectrum ℝ (-a)
⊢ id x = -x⁻
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
⊢ a⁺ = b

Tactic:
let s := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)

Goal state after:
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
⊢ a⁺ = b
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.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x y : ℝ≥0
hx : 0 < x
hy : 0 < y
⊢ a ^ (x + y) = a ^ x * a ^ y

Tactic:
simp only [nnrpow_def]
  rw [← cfcₙ_mul _ _ a]
  congr! 2 with z
  exact mod_cast z.rpow_add' <| ne_of_gt (add_pos hx hy)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ≥0
hx : x ≠ 0
ha : 0 ≤ a
⊢ (a ^ x) ^ x⁻¹ = a

Tactic:
simp [mul_inv_cancel₀ hx, nnrpow_one _ ha]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ (∀ (i : Fin 4), 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i) ∧
    (∀ (i : Fin 4), ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i‖ ≤ ‖a‖) ∧
      a = ∑ i, I ^ ↑i • ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i

Tactic:
rw [← and_assoc, ← forall_and]

Goal state after:
case h
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ (∀ (x : Fin 4),
      0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] x ∧
        ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] x‖ ≤ ‖a‖) ∧
    a = ∑ i, I ^ ↑i • ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ (↑(realPart a))⁺ - (↑(realPart a))⁻ + (I • (↑(imaginaryPart a))⁺ - I • (↑(imaginaryPart a))⁻) =
    ∑ i, I ^ ↑i • ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i

Tactic:
simp only [Fin.sum_univ_four, Fin.coe_ofNat_eq_mod, Matrix.cons_val, Nat.reduceMod, I_sq,
      I_pow_three]

Goal state after:
case h.right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ (↑(realPart a))⁺ - (↑(realPart a))⁻ + (I • (↑(imaginaryPart a))⁺ - I • (↑(imaginaryPart a))⁻) =
    I ^ 0 • (↑(realPart a))⁺ + I ^ 1 • (↑(imaginaryPart a))⁺ + -1 • (↑(realPart a))⁻ + -I • (↑(imaginaryPart 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
a : A
n : ℕ
ha : 0 ≤ a
⊢ a ^ ↑n = cfc (fun x ↦ x ^ n) a

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
a : A
n : ℕ
ha : 0 ≤ a
⊢ cfc (fun x ↦ x ^ ↑n) a = cfc (fun x ↦ x ^ n) 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
a : Aˣ
n : ℤ
ha : 0 ≤ ↑a
⊢ ↑a ^ ↑n = cfc (fun x ↦ x ^ n) ↑a

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
a : Aˣ
n : ℤ
ha : 0 ≤ ↑a
⊢ cfc (fun x ↦ x ^ ↑n) ↑a = cfc (fun x ↦ x ^ n) ↑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
a : A
y : ℝ
hy : y ≠ 0
ha : 0 ≤ a
a✝ : Nontrivial A
h : IsUnit (a ^ y)
⊢ IsUnit a

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
a : A
y : ℝ
hy : y ≠ 0
ha : 0 ≤ a
a✝ : Nontrivial A
h : IsUnit (cfc (fun x ↦ x ^ y) a)
⊢ IsUnit 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
y : ℝ≥0
hy : y ≠ 0
ha : 0 ≤ a
⊢ IsUnit (a ^ y) ↔ IsUnit a

Tactic:
nnrpow_eq_rpow (pos_of_ne_zero hy)

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
y : ℝ≥0
hy : y ≠ 0
ha : 0 ≤ a
⊢ IsUnit (a ^ ↑y) ↔ IsUnit a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum 𝕜 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), p (a x)
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ cfc f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
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 𝕜 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), p (a x)
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), p (a x)
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ cfc f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), p (a x)
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

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 : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
hs : s ∈ 𝓝ˢ (⋃ x, spectrum 𝕜 (a x))
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha' : ∀ (x : X), p (a x)
hf : ContinuousOn f s
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ cfc f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
hs : s ∈ 𝓝ˢ (⋃ x, spectrum 𝕜 (a x))
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), p (a x)
hf : ContinuousOn f s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ
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 : 1 ≤ ↑a
⊢ ↑a ^ (-1) ≤ 1

Tactic:
rw [rpow_neg_one_eq_inv a (zero_le_one.trans ha)]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : Aˣ
ha : 1 ≤ ↑a
⊢ ↑a⁻¹ ≤ 1
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:
simp only [norm_eq_sqrt_norm_inner_self (A := A), norm_nonneg, Real.sq_sqrt]

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 simpa [ha.star_eq, he.isSelfAdjoint.star_eq] using congr(star $this)

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:
have ha' : 0 ≤ a := zero_le_one.trans ha

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
⊢ Monotone fun x ↦ a ^ x
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 y z : C⋆ᵐᵒᵈ(A, E × F)
⊢ ⟪x.1, (y + z).1⟫_A + ⟪x.2, (y + z).2⟫_A = ⟪x.1, y.1⟫_A + ⟪x.2, y.2⟫_A + (⟪x.1, z.1⟫_A + ⟪x.2, z.2⟫_A)

Tactic:
simpa using add_add_add_comm ..

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
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
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
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ cfc f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

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.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 : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), 0 ≤ a x
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ cfc f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

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.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 : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ cfc f (a x)

Tactic:
rw [← continuousOn_univ] at ha_cont ⊢

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.Matrix.Normed

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike α
A : Matrix l m α
B : Matrix m n α
⊢ ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊

Tactic:
by
  simp_rw [frobenius_nnnorm_def, Matrix.mul_apply]
  rw [← NNReal.mul_rpow, @Finset.sum_comm _ _ m, Finset.sum_mul_sum]
  gcongr with i _ j
  rw [← NNReal.rpow_le_rpow_iff one_half_pos, ← NNReal.rpow_mul,
    mul_div_cancel₀ (1 : ℝ) two_ne_zero, NNReal.rpow_one, NNReal.mul_rpow]
  simpa only [PiLp.toLp_apply, PiLp.inner_apply, RCLike.inner_apply', starRingEnd_apply,
    Pi.nnnorm_def, PiLp.nnnorm_eq_of_L2, star_star, nnnorm_star, NNReal.sqrt_eq_rpow,
    NNReal.rpow_two] using nnnorm_inner_le_nnnorm (𝕜 := α) (toLp 2 (star <| A i ·)) (toLp 2 (B · j))

Goal state after: