Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p).rpowIntegrand₀₁ x 0 = 0
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ

Tactic:
refine ⟨?integrable, ?integral⟩

Goal state after:
case integrable
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p).rpowIntegrand₀₁ x 0 = 0
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ

case integral
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p).rpowIntegrand₀₁ x 0 = 0
⊢ a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ