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
s : Set X
x : X
hx : x ∈ (closure sᶜ)ᶜ
g : C(X, ℝ)
hgs : Set.EqOn (⇑g) 0 (closure sᶜ)
hgx : Set.EqOn (⇑g) 1 {x}
⊢ ∀ ⦃x : X⦄, x ∈ sᶜ → { toFun := fun x ↦ ↑(g x), continuous_toFun := ⋯ } x = 0

Tactic:
by
      simpa only [coe_mk, ofReal_eq_zero] using! fun x hx => hgs (subset_closure hx)

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

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
grind

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

Goal state before:
pp: case insert
A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
⊢ f j * ∑ i ∈ s, f i = 0

Tactic:
simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h0 (by grind)

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

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a b : A
ha : IsStarNormal a
hb : IsStarNormal b
hcomm : Commute a b
hab : a * b = 0
S : NonUnitalStarSubalgebra ℂ A := (adjoin ℂ {a, b}).topologicalClosure
hS : IsClosed ↑S
hcomm₁ : Commute (star a) b
hcomm₂ : Commute a (star b)
⊢ ∀ x ∈ {a, b}, ∀ y ∈ {a, b}, x * y = y * x

Tactic:
grind

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

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : (∀ i ∈ s, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
h : ∀ i ∈ insert j s, IsSelfAdjoint (f i)
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
grind

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

Goal state before:
pp: case insert
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : (∀ i ∈ s, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
h : ∀ i ∈ insert j s, IsSelfAdjoint (f i)
⊢ f j * ∑ i ∈ s, f i = 0

Tactic:
simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h0 (by grind)

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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹ : CommCStarAlgebra A
inst✝ : CommCStarAlgebra B
φ : A →⋆ₐ[ℂ] B
⊢ (↑(gelfandStarTransform B)).comp φ =
    (ContinuousMap.compStarAlgHom' ℂ ℂ (compContinuousMap φ)).comp ↑(gelfandStarTransform A)

Tactic:
rfl

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 : IsSelfAdjoint a
⊢ a⁺ = 0 ↔ a ≤ 0

Tactic:
rw [← negPart_eq_neg, eq_comm (b := -a)]
  nth_rw 2 [← posPart_sub_negPart a]
  simp

Goal state after:
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
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
⊢ σₙ ℝ a ⊆ s ∧ σₙ ℝ b ⊆ s ∧ σₙ ℝ (-c) ⊆ s

Tactic:
grind

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✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
f : γ → α →ᵤ β
K : ℝ≥0
s : Set γ
⊢ (∀ (c : α), LipschitzWith K fun x ↦ toFun (f ↑x) c) ↔ ∀ (c : α), LipschitzWith K (s.domRestrict fun x ↦ toFun (f x) c)

Tactic:
rfl

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

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ β
h✝ : Nonempty α
⊢ (Metric.ediam Set.univ).toReal ∈ upperBounds (Set.range fun x ↦ dist (toFun f x) (toFun g x))

Tactic:
simp +contextual [mem_upperBounds, eq_comm (a := dist _ _), ← edist_dist,
          ← ENNReal.ofReal_le_iff_le_toReal BoundedSpace.bounded_univ.ediam_ne_top,
          Metric.edist_le_ediam_of_mem]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Fintype ↑𝔖
f g : α →ᵤ[𝔖] β
⊢ ⨆ x, ⨆ x_1, edist ((toFun 𝔖) f ↑x_1) ((toFun 𝔖) g ↑x_1) =
    ⨆ b, edist (UniformFun.ofFun ((↑b).domRestrict ((toFun 𝔖) f))) (UniformFun.ofFun ((↑b).domRestrict ((toFun 𝔖) g)))

Tactic:
rfl

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

Goal state before:
pp: α : Type u_1
β : Type u_2
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Finite ↑𝔖
f g : α →ᵤ[𝔖] β
C : ℝ≥0∞
⊢ edist f g ≤ C ↔ ∀ x ∈ ⋃₀ 𝔖, edist ((toFun 𝔖) f x) ((toFun 𝔖) g x) ≤ C

Tactic:
edist_def,

Goal state after:
α : Type u_1
β : Type u_2
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Finite ↑𝔖
f g : α →ᵤ[𝔖] β
C : ℝ≥0∞
⊢ ⨆ x ∈ ⋃₀ 𝔖, edist ((toFun 𝔖) f x) ((toFun 𝔖) g x) ≤ C ↔ ∀ x ∈ ⋃₀ 𝔖, edist ((toFun 𝔖) f x) ((toFun 𝔖) g x) ≤ C
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Finite ↑𝔖
f : γ → α →ᵤ[𝔖] β
K : ℝ≥0
s : Set γ
⊢ (∀ (c : α), ∀ x ∈ 𝔖, c ∈ x → LipschitzWith K fun x ↦ (toFun 𝔖) (f ↑x) c) ↔
    ∀ (c : α), ∀ x ∈ 𝔖, c ∈ x → LipschitzWith K (s.domRestrict fun x ↦ (toFun 𝔖) (f x) c)

Tactic:
rfl

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

Goal state before:
pp: case h
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
f g : α →ᵤ[𝔖] β
h✝ : Nonempty ↑(⋃₀ 𝔖)
⊢ (Metric.ediam Set.univ).toReal ∈ upperBounds (Set.range fun x ↦ dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))

Tactic:
simp +contextual [mem_upperBounds, eq_comm (a := dist _ _), ← edist_dist,
          ← ENNReal.ofReal_le_iff_le_toReal BoundedSpace.bounded_univ.ediam_ne_top,
          Metric.edist_le_ediam_of_mem]

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

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
f : C(↑(σₙ ℂ a), ℂ)₀
⊢ ‖toContinuousMapHom f‖ = ‖f‖

Tactic:
rfl

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

Goal state before:
pp: case h₁
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal

Tactic:
rfl

Goal state after:
case h₁
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case h₁.ha
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint a

Tactic:
rfl

Goal state after:
case h₁.ha
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case h₁.ht
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ ‖a‖₊ ≤ ‖a‖₊

Tactic:
rfl

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

Goal state before:
pp: case h₂
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts b ⇑ContinuousMap.realToNNReal

Tactic:
rfl

Goal state after:
case h₂
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts b ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case h₂.ha
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint b

Tactic:
rfl

Goal state after:
case h₂.ha
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint b
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case h₂.ht
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ ‖b‖₊ ≤ ‖b‖₊

Tactic:
rfl

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

Goal state before:
pp: A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
r : ℝ
hr : 0 ≤ r
⊢ SpectrumRestricts (r • a) ⇑ContinuousMap.realToNNReal

Tactic:
by
  rw [SpectrumRestricts.nnreal_iff] at ha ⊢
  nontriviality A
  intro x hx
  by_cases hr' : r = 0
  · simp only [hr', zero_smul, spectrum.zero_eq, Set.mem_singleton_iff] at hx ⊢
    exact hx.symm.le
  · lift r to ℝˣ using IsUnit.mk0 r hr'
    rw [← Units.smul_def, spectrum.unit_smul_eq_smul, Set.mem_smul_set_iff_inv_smul_mem] at hx
    refine le_of_smul_le_smul_left ?_ (inv_pos.mpr <| lt_of_le_of_ne hr <| ne_comm.mpr hr')
    simpa [Units.smul_def] using ha _ hx

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 : T2Space A
inst✝ : IsSemitopologicalRing A
a : A
y : ℝ≥0
ha : 0 ≤ a
x : ℝ
hx : x ∈ quasispectrum ℝ a
⊢ 0 ≤ x

Tactic:
grind

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
ha : 0 ≤ a
⊢ a ^ 2 = a * a

Tactic:
by
  simp only [nnrpow_def, NNReal.nnrpow_def, NNReal.coe_ofNat, NNReal.rpow_ofNat, pow_two]
  change cfcₙ (fun z : ℝ≥0 => id z * id z) a = a * a
  rw [cfcₙ_mul id id a, cfcₙ_id ℝ≥0 a]

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
x : A
hx : x ∈ {a | 0 ≤ a}
hx_pos : 0 < ‖x‖
this : (↑r * ↑‖x‖⁻¹)⁻¹ • (↑r * ↑‖x‖⁻¹) • x = x
⊢ x ∈ ↑(span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 r))

Tactic:
rw [← this]
      refine smul_mem _ _ (subset_span <| Set.mem_inter ?_ ?_)
      · norm_cast
        exact smul_nonneg (by positivity) hx
      · simp [mul_smul, norm_smul, abs_of_pos hr, inv_mul_cancel₀ hx_pos.ne']

Goal state after:
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
ha : 0 ≤ a
x : ℝ
hx : x ∈ spectrum ℝ a
⊢ max x 0 ^ y = x ^ y

Tactic:
grind

Goal state after:
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
f : ℝ → ℝ
hf₁ : ∀ x ∈ spectrum ℝ a, 0 < f x
hf₂ : ContinuousOn f (spectrum ℝ a)
ha : IsSelfAdjoint a
⊢ ∀ x ∈ f '' spectrum ℝ a, id x ≠ 0 ∨ 0 ≤ y

Tactic:
grind

Goal state after:
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
f : ℝ → ℝ
hf₁ : ∀ x ∈ spectrum ℝ a, 0 < f x
hf₂ : ContinuousOn f (spectrum ℝ a)
ha : IsSelfAdjoint a
hg : ContinuousOn (fun r ↦ r ^ y) (f '' spectrum ℝ a)
⊢ cfc ((fun x ↦ x ^ y) ∘ f) a = cfc (fun r ↦ f r ^ y) a

Tactic:
rfl

Goal state after:
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
⊢ sqrt 1 = 1

Tactic:
simp [sqrt_eq_cfc]

Goal state after:
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
h : IsUnit a
hx : x ≠ 0
hnonneg : 0 ≤ a
⊢ IsStrictlyPositive a

Tactic:
grind

Goal state after:
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
h : IsUnit a
ha : 0 ≤ a
⊢ IsStrictlyPositive a

Tactic:
grind

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

Goal state before:
pp: 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✝ : CompleteSpace A
s : Set 𝕜
x✝¹ : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
f : 𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (spectrum 𝕜 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : p (f, a).2
has : s ∈ 𝓝ˢ (spectrum 𝕜 (f, a).2)
hs : IsCompact (spectrum 𝕜 a)
k : Set 𝕜
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum 𝕜 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum 𝕜 x)}
fa : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
⊢ cfc ((toFun {s}) fa.1) = cfc ((toFun {k}) ((ofFun {k}) ((toFun {t | IsCompact t ∧ t ⊆ s}) fa.1)))

Tactic:
rfl

Goal state after:
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
ha : IsStrictlyPositive a
⊢ IsUnit (sqrt a)

Tactic:
grind

Goal state after:
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 : IsStrictlyPositive a
⊢ IsStrictlyPositive (a ^ y)

Tactic:
grind

Goal state after:
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
ha : IsStrictlyPositive a
⊢ IsStrictlyPositive (CFC.sqrt a)

Tactic:
grind

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
a : A
y : ℝ
ha : IsStrictlyPositive a
⊢ IsStrictlyPositive (a ^ y)

Tactic:
grind

Goal state after:
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
hx : x ≠ 0
ha : IsStrictlyPositive a
y✝ : ℝ
⊢ IsStrictlyPositive a

Tactic:
grind

Goal state after:
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
hx : x ≠ 0
ha : IsStrictlyPositive a
this : a ^ (-x) = (a ^ x) ^ (-1)
⊢ IsStrictlyPositive (a ^ x)

Tactic:
grind

Goal state after:
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
ha : IsStrictlyPositive a
⊢ 1 / 2 ≠ 0

Tactic:
grind

Goal state after:
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
ha : IsStrictlyPositive a
⊢ -1 ≠ 0

Tactic:
grind

Goal state after:
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
ha : ¬IsStrictlyPositive a
ha' : ¬IsUnit (sqrt a)
⊢ ¬0 ≤ a ∨ ¬IsUnit a

Tactic:
grind

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

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (spectrum ℝ≥0 (f, a).2)
hs : IsCompact (spectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum ℝ≥0 a)
hk : IsCompact k
⊢ ∃ t,
    IsOpen[instTopologicalSpaceProd] t ∧
      (f, a) ∈ t ∧
        ContinuousOn (fun fa ↦ cfc ((toFun {s}) fa.1) fa.2)
          ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)} ∩ t)

Tactic:
have := (upperHemicontinuous_spectrum_nnreal A).isOpen k

Goal state after:
A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (spectrum ℝ≥0 (f, a).2)
hs : IsCompact (spectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)}
⊢ ∃ t,
    IsOpen[instTopologicalSpaceProd] t ∧
      (f, a) ∈ t ∧
        ContinuousOn (fun fa ↦ cfc ((toFun {s}) fa.1) fa.2)
          ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)} ∩ t)