-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le
    {F : 𝕜 → ℝ → E} {F' : 𝕜 → ℝ → E} {x₀ : 𝕜}
    (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, HasDerivAt (fun x => F x t) (F' x t) x) :
    IntervalIntegrable (F' x₀) μ a b ∧
      HasDerivAt (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✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
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) (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, HasDerivAt (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
⊢ IntegrableOn (F' x₀) (Ι a b) μ ∧ 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]

Result:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
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) (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, HasDerivAt (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
⊢ IntegrableOn (F' x₀) (Ι a b) μ ∧
    HasDerivAt (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✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
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) (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, 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:
rw [← ae_restrict_iff' measurableSet_uIoc] at h_bound h_diff

Result:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
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) (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, 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

Proof state:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
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) (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, HasDerivAt (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
this :
  Integrable (F' x₀) (μ.restrict (Ι a b)) ∧
    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) μ ∧
    HasDerivAt (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 ⟨this.1, this.2.const_smul _⟩

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
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) (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, HasDerivAt (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
⊢ IntegrableOn (F' x₀) (Ι a b) μ ∧
    HasDerivAt (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:
have := hasDerivAt_integral_of_dominated_loc_of_deriv_le ε_pos hF_meas hF_int hF'_meas h_bound
  bound_integrable h_diff

Result:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
μ : Measure ℝ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
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) (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, HasDerivAt (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
this :
  Integrable (F' x₀) (μ.restrict (Ι a b)) ∧
    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) μ ∧
    HasDerivAt (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