-- Mathlib/Analysis/Calculus/ParametricIntegral.lean
theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0 < ε)
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
    (hF'_meas : AEStronglyMeasurable F' μ)
    (h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
    (bound_integrable : Integrable (bound : α → ℝ) μ)
    (h_diff : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' a) x₀) :
    Integrable F' μ ∧ HasDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀

Proof state:
case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
ε : ℝ
F : 𝕜 → α → E
x₀ : 𝕜
F' : α → E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
hF'_int : Integrable F' μ
hE : CompleteSpace E
⊢ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Tactic:
simp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff ⊢

Result:
case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
ε : ℝ
F : 𝕜 → α → E
x₀ : 𝕜
F' : α → E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
hF'_int : Integrable F' μ
hE : CompleteSpace E
⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (ContinuousLinearMap.smulRight 1 (∫ (a : α), F' a ∂μ)) x₀

Proof state:
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
ε : ℝ
F : 𝕜 → α → E
x₀ : 𝕜
F' : α → E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀
⊢ Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Tactic:
have hm : AEStronglyMeasurable (L ∘ F') μ := L.continuous.comp_aestronglyMeasurable hF'_meas

Result:
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
ε : ℝ
F : 𝕜 → α → E
x₀ : 𝕜
F' : α → E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
⊢ Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀