-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
theorem hasFDerivAt_integral_of_dominated_of_fderiv_le
    {F : H → ℝ → E} {F' : H → ℝ → H →L[𝕜] E} {x₀ : H} (ε_pos : 0 < ε)
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b)))
    (hF_int : IntervalIntegrable (F x₀) μ a b)
    (hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b)))
    (h_bound : ∀ᵐ t ∂μ, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, ‖F' x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_diff : ∀ᵐ t ∂μ, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x t) (F' x t) x) :
    HasFDerivAt (fun x => ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' x₀ t ∂μ) x₀

Proof state:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, ‖F' x_1 x‖ ≤ bound x
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
⊢ HasFDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)
    ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' x₀ t ∂μ) x₀

Tactic:
exact (hasFDerivAt_integral_of_dominated_of_fderiv_le ε_pos hF_meas hF_int hF'_meas h_bound
  bound_integrable h_diff).const_smul _

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ∀ 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 ∈ ball x₀ ε, HasFDerivAt (fun x => F x t) (F' x t) x
⊢ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Tactic:
rw [← ae_restrict_iff' measurableSet_uIoc] at h_bound h_diff

Result:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, ‖F' x_1 x‖ ≤ bound x
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
⊢ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Proof state:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, ‖F' x_1 x‖ ≤ bound x
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
⊢ HasFDerivAt (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]

Result:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, ‖F' x_1 x‖ ≤ bound x
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
⊢ HasFDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)
    ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' x₀ t ∂μ) x₀

Proof state:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, ‖F' x_1 x‖ ≤ bound x
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
⊢ HasFDerivAt (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

Result:
𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
a b ε : ℝ
bound : ℝ → ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[𝕜] E
x₀ : H
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, ‖F' x_1 x‖ ≤ bound x
h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1
hF_int : IntegrableOn (F x₀) (Ι a b) μ
bound_integrable : IntegrableOn bound (Ι a b) μ
⊢ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀