mathlib-initiative/mathlib-tactics huggingface dataset:
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)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
g : C(X, ℝ≥0)
hgI : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I
hg : ∀ (x : X), g x ≤ 1
hgt : Set.EqOn (⇑g) 1 t
x : X
hx : x ∉ t
⊢ ‖1 - (algebraMap ℝ≥0 𝕜) (g x)‖₊ = ‖(algebraMap ℝ≥0 𝕜) (1 - g x)‖₊

Tactic:
simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, NNReal.coe_sub (hg x),
              NNReal.coe_one, sub_smul, one_smul]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
⊢ Bornology.IsBounded Set.univ

Tactic:
convert! lipschitzWith_one_ofFun_toFun (𝔖 := 𝔖) (β := β) |>.isBounded_image (.all Set.univ)
    ext f
    simp only [Set.mem_univ, Function.comp_apply, Set.image_univ, Set.mem_range, true_iff]
    exact ⟨UniformFun.ofFun (toFun 𝔖 f), by simp⟩

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

Goal state before:
pp: case mp
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := ⋯
x y : A
⊢ x ≤ y → ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), y = x + p

Tactic:
intro h

Goal state after:
case mp
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x y : A
h : x ≤ y
⊢ ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), y = x + p
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]

Goal state after:
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
⊢ cfcₙ (fun x_1 ↦ x_1.nnrpow (x + y)) a = cfcₙ (fun x_1 ↦ x_1.nnrpow x) a * cfcₙ (fun x ↦ x.nnrpow y) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case neg.inl
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ¬(ContinuousOn f (σₙ R a) ∧ p a)
hf : ¬ContinuousOn f (σₙ R a)
⊢ 0 = cfc f ↑a

Tactic:
cfc_apply_of_not_continuousOn _ (quasispectrum_eq_spectrum_inr' R ℂ a ▸ hf)

Goal state after:
case neg.inl
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ¬(ContinuousOn f (σₙ R a) ∧ p a)
hf : ¬ContinuousOn f (σₙ R a)
⊢ 0 = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: case inr.inr
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 y : ℝ≥0
ha : 0 ≤ a
hx : 0 < x
hy : 0 < y
⊢ (a ^ x) ^ y = a ^ (x * y)

Tactic:
simp only [nnrpow_def]

Goal state after:
case inr.inr
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 y : ℝ≥0
ha : 0 ≤ a
hx : 0 < x
hy : 0 < y
⊢ cfcₙ (fun x ↦ x.nnrpow y) (cfcₙ (fun x_1 ↦ x_1.nnrpow x) a) = cfcₙ (fun x_1 ↦ x_1.nnrpow (x * y)) a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝²⁵ : PartialOrder A
inst✝²⁴ : NonUnitalRing A
inst✝²³ : TopologicalSpace A
inst✝²² : StarRing A
inst✝²¹ : Module ℝ A
inst✝²⁰ : SMulCommClass ℝ A A
inst✝¹⁹ : IsScalarTower ℝ A A
inst✝¹⁸ : StarOrderedRing A
inst✝¹⁷ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹⁶ : NonnegSpectrumClass ℝ A
inst✝¹⁵ : IsSemitopologicalRing A
inst✝¹⁴ : T2Space A
B : Type u_2
inst✝¹³ : PartialOrder B
inst✝¹² : NonUnitalRing B
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : StarRing B
inst✝⁹ : Module ℝ B
inst✝⁸ : SMulCommClass ℝ B B
inst✝⁷ : IsScalarTower ℝ B B
inst✝⁶ : NonUnitalContinuousFunctionalCalculus ℝ B IsSelfAdjoint
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint
inst✝⁴ : IsSemitopologicalRing B
inst✝³ : T2Space B
inst✝² : NonnegSpectrumClass ℝ B
inst✝¹ : NonnegSpectrumClass ℝ (A × B)
inst✝ : StarOrderedRing B
a : A
b : B
x : ℝ≥0
ha : 0 ≤ a
hb : 0 ≤ b
⊢ nnrpow (a, b) x = (a ^ x, b ^ x)

Tactic:
simp only [nnrpow_def]

Goal state after:
A : Type u_1
inst✝²⁵ : PartialOrder A
inst✝²⁴ : NonUnitalRing A
inst✝²³ : TopologicalSpace A
inst✝²² : StarRing A
inst✝²¹ : Module ℝ A
inst✝²⁰ : SMulCommClass ℝ A A
inst✝¹⁹ : IsScalarTower ℝ A A
inst✝¹⁸ : StarOrderedRing A
inst✝¹⁷ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹⁶ : NonnegSpectrumClass ℝ A
inst✝¹⁵ : IsSemitopologicalRing A
inst✝¹⁴ : T2Space A
B : Type u_2
inst✝¹³ : PartialOrder B
inst✝¹² : NonUnitalRing B
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : StarRing B
inst✝⁹ : Module ℝ B
inst✝⁸ : SMulCommClass ℝ B B
inst✝⁷ : IsScalarTower ℝ B B
inst✝⁶ : NonUnitalContinuousFunctionalCalculus ℝ B IsSelfAdjoint
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint
inst✝⁴ : IsSemitopologicalRing B
inst✝³ : T2Space B
inst✝² : NonnegSpectrumClass ℝ B
inst✝¹ : NonnegSpectrumClass ℝ (A × B)
inst✝ : StarOrderedRing B
a : A
b : B
x : ℝ≥0
ha : 0 ≤ a
hb : 0 ≤ b
⊢ nnrpow (a, b) x = (cfcₙ (fun x_1 ↦ x_1.nnrpow x) a, cfcₙ (fun x_1 ↦ x_1.nnrpow x) b)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: ι : Type u_2
C : ι → Type u_3
inst✝¹⁴ : (i : ι) → PartialOrder (C i)
inst✝¹³ : (i : ι) → NonUnitalRing (C i)
inst✝¹² : (i : ι) → TopologicalSpace (C i)
inst✝¹¹ : (i : ι) → StarRing (C i)
inst✝¹⁰ : ∀ (i : ι), StarOrderedRing (C i)
inst✝⁹ : StarOrderedRing ((i : ι) → C i)
inst✝⁸ : (i : ι) → Module ℝ (C i)
inst✝⁷ : ∀ (i : ι), SMulCommClass ℝ (C i) (C i)
inst✝⁶ : ∀ (i : ι), IsScalarTower ℝ (C i) (C i)
inst✝⁵ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus ℝ (C i) IsSelfAdjoint
inst✝⁴ : NonUnitalContinuousFunctionalCalculus ℝ ((i : ι) → C i) IsSelfAdjoint
inst✝³ : ∀ (i : ι), IsSemitopologicalRing (C i)
inst✝² : ∀ (i : ι), T2Space (C i)
inst✝¹ : NonnegSpectrumClass ℝ ((i : ι) → C i)
inst✝ : ∀ (i : ι), NonnegSpectrumClass ℝ (C i)
c : (i : ι) → C i
x : ℝ≥0
hc : ∀ (i : ι), 0 ≤ c i
⊢ nnrpow c x = fun i ↦ c i ^ x

Tactic:
simp only [nnrpow_def]

Goal state after:
ι : Type u_2
C : ι → Type u_3
inst✝¹⁴ : (i : ι) → PartialOrder (C i)
inst✝¹³ : (i : ι) → NonUnitalRing (C i)
inst✝¹² : (i : ι) → TopologicalSpace (C i)
inst✝¹¹ : (i : ι) → StarRing (C i)
inst✝¹⁰ : ∀ (i : ι), StarOrderedRing (C i)
inst✝⁹ : StarOrderedRing ((i : ι) → C i)
inst✝⁸ : (i : ι) → Module ℝ (C i)
inst✝⁷ : ∀ (i : ι), SMulCommClass ℝ (C i) (C i)
inst✝⁶ : ∀ (i : ι), IsScalarTower ℝ (C i) (C i)
inst✝⁵ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus ℝ (C i) IsSelfAdjoint
inst✝⁴ : NonUnitalContinuousFunctionalCalculus ℝ ((i : ι) → C i) IsSelfAdjoint
inst✝³ : ∀ (i : ι), IsSemitopologicalRing (C i)
inst✝² : ∀ (i : ι), T2Space (C i)
inst✝¹ : NonnegSpectrumClass ℝ ((i : ι) → C i)
inst✝ : ∀ (i : ι), NonnegSpectrumClass ℝ (C i)
c : (i : ι) → C i
x : ℝ≥0
hc : ∀ (i : ι), 0 ≤ c i
⊢ nnrpow c x = fun i ↦ cfcₙ (fun x_1 ↦ x_1.nnrpow x) (c i)
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
ha : 0 ≤ a
⊢ sqrt a = 0 ↔ a = 0

Tactic:
by
  rw [sqrt_eq_iff a _, mul_zero, eq_comm]

Goal state after:
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
⊢ Set.EqOn (fun a ↦ a ^ 0) (fun x ↦ 1) (Set.Ici 0)

Tactic:
by
  intro a ha
  simp [rpow_zero a ha]

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

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : CommSemiring R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
h_cont : Continuous[inst✝, topologicalSpace R R {spectrum R a}] fun x ↦ (ofFun {spectrum R a}) (f x)
hf : ∀ (x : X), ContinuousOn (f x) (spectrum R a)
⊢ Continuous[inst✝, inst✝³] fun x ↦ cfc (f x) a

Tactic:
← continuousOn_univ

Goal state after:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : CommSemiring R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
h_cont : ContinuousOn (fun x ↦ (ofFun {spectrum R a}) (f x)) Set.univ
hf : ∀ (x : X), ContinuousOn (f x) (spectrum R a)
⊢ ContinuousOn (fun x ↦ cfc (f x) a) 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 : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x✝ : X
hx✝ : x✝ ∈ t
x : X
hx : spectrum 𝕜 (a x) ⊆ s x✝
hxt : x ∈ t
⊢ spectrum 𝕜 (a x) ⊆ s x✝

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
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x✝ : X
hx✝ : x✝ ∈ t
x : X
hx : spectrum 𝕜 (a x) ⊆ s x✝
hxt : x ∈ t
⊢ p (a x)

Tactic:
exacts [hx, ha' x hxt]

Goal state after:
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:
← continuousOn_univ

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:
← continuousOn_univ

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:
← continuousOn_univ

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.Module.Defs

Goal state before:
pp: case inr
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
h₁ : ‖x‖ ^ 2 •> (⟪x, y⟫_A * ⟪y, x⟫_A) ≤ ‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A
⊢ ⟪x, y⟫_A * ⟪y, x⟫_A ≤ ‖x‖ ^ 2 •> ⟪y, y⟫_A

Tactic:
smul_le_smul_iff_of_pos_left (pow_pos (CStarModule.norm_pos A h) _)

Goal state after:
case inr
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
h₁ : ⟪x, y⟫_A * ⟪y, x⟫_A ≤ ‖x‖ ^ 2 •> ⟪y, y⟫_A
⊢ ⟪x, y⟫_A * ⟪y, x⟫_A ≤ ‖x‖ ^ 2 •> ⟪y, y⟫_A
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 : ∀ x ∈ t, ContinuousWithinAt a t x
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)
x✝ : X
hx✝ : x✝ ∈ t
x : X
hx : spectrum ℝ≥0 (a x) ⊆ s x✝
hxt : x ∈ t
⊢ spectrum ℝ≥0 (a x) ⊆ s x✝

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 : ∀ x ∈ t, ContinuousWithinAt a t x
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)
x✝ : X
hx✝ : x✝ ∈ t
x : X
hx : spectrum ℝ≥0 (a x) ⊆ s x✝
hxt : x ∈ t
⊢ 0 ≤ a x

Tactic:
exacts [hx, ha' x hxt]

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:
← continuousOn_univ

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.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 ↦ (1 + x)⁻¹) c = 1 - cfc (fun x ↦ x⁻¹) (1 + c)

Tactic:
cfc_comp' (·⁻¹) (1 + ·) c ?_,

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
⊢ 1 - cfc (fun x ↦ x⁻¹) (cfc (fun x ↦ 1 + x) c) = 1 - cfc (fun x ↦ x⁻¹) (1 + c)

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 : 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:
← continuousOn_univ

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:
← continuousOn_univ

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