Theorem: intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
File: Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean

tactic: rw [← ae_restrict_iff' measurableSet_uIoc] at h_bound h_diff
state_before: 𝕜 : Type u_1
inst✝⁶ : IsROrC 𝕜
μ : MeasureTheory.Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type ?u.684895
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))
h_bound : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x t) (F' x t) x
⊢ IntervalIntegrable (F' x₀) μ a b ∧
    HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

tactic: simp only [intervalIntegrable_iff] at hF_int bound_integrable ⊢
state_before: 𝕜 : Type u_1
inst✝⁶ : IsROrC 𝕜
μ : MeasureTheory.Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type ?u.684895
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x
bound_integrable : IntervalIntegrable bound μ a b
h_diff :
  ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
⊢ IntervalIntegrable (F' x₀) μ a b ∧
    HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

tactic: simp only [intervalIntegral_eq_integral_uIoc]
state_before: 𝕜 : Type u_1
inst✝⁶ : IsROrC 𝕜
μ : MeasureTheory.Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type ?u.684895
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x
h_diff :
  ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b)
bound_integrable : IntegrableOn bound (Ι a b)
⊢ IntegrableOn (F' x₀) (Ι a b) ∧ HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

tactic: have := hasDerivAt_integral_of_dominated_loc_of_deriv_le ε_pos hF_meas hF_int hF'_meas h_bound
  bound_integrable h_diff
state_before: 𝕜 : Type u_1
inst✝⁶ : IsROrC 𝕜
μ : MeasureTheory.Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type ?u.684895
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x
h_diff :
  ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b)
bound_integrable : IntegrableOn bound (Ι a b)
⊢ IntegrableOn (F' x₀) (Ι a b) ∧
    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (x_1 : ℝ) in Ι a b, F x x_1 ∂μ)
      ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F' x₀ x ∂μ) x₀

tactic: exact ⟨this.1, this.2.const_smul _⟩
state_before: 𝕜 : Type u_1
inst✝⁶ : IsROrC 𝕜
μ : MeasureTheory.Measure ℝ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type ?u.684895
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F F' : 𝕜 → ℝ → E
x₀ : 𝕜
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (Measure.restrict μ (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (Measure.restrict μ (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x
h_diff :
  ∀ᵐ (x : ℝ) ∂Measure.restrict μ (Ι a b), ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b)
bound_integrable : IntegrableOn bound (Ι a b)
this : Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : ℝ) in Ι a b, F n a ∂μ) (∫ (a : ℝ) in Ι a b, F' x₀ a ∂μ) x₀
⊢ IntegrableOn (F' x₀) (Ι a b) ∧
    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (x_1 : ℝ) in Ι a b, F x x_1 ∂μ)
      ((if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, F' x₀ x ∂μ) x₀