/-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming `F x₀` is
integrable, `‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖` for `x` in a ball around `x₀` for ae `a` with
integrable Lipschitz bound `bound` (with a ball radius independent of `a`), and `F x` is
ae-measurable for `x` in the same ball. See `hasFDerivAt_integral_of_dominated_loc_of_lip` for a
slightly less general but usually more useful version. -/
theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → H →L[𝕜] E} {x₀ : H}
    {bound : α → ℝ} {ε : ℝ} (ε_pos : 0 < ε)
    (hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
    (hF'_meas : AEStronglyMeasurable F' μ)
    (h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
    (bound_integrable : Integrable (bound : α → ℝ) μ)
    (h_diff : ∀ᵐ a ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀) :
    Integrable F' μ ∧ HasFDerivAt (fun x => ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
  have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
  have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x => inv_nonneg.mpr (norm_nonneg _)
  set b : α → ℝ := fun a => |bound a|
  have b_int : Integrable b μ := bound_integrable.norm
  have b_nonneg : ∀ a, 0 ≤ b a := fun a => abs_nonneg _
  replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
  

Proof state:
case h_lipsch
α : Type u_1
inst✝⁷ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁶ : IsROrC 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
F' : α → H →L[𝕜] E
x₀ : H
bound : α → ℝ
ε : ℝ
ε_pos : 0 < ε
hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀)
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖
bound_integrable : Integrable bound
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a => |bound a|
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
⊢ ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
α : Type u_1
inst✝⁷ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁶ : IsROrC 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
F' : α → H →L[𝕜] E
x₀ : H
bound : α → ℝ
ε : ℝ
ε_pos : 0 < ε
hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀)
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a => |bound a|
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Next tactic:
exact h_lipsch.mono fun a ha x hx =>
    (ha x hx).trans <| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _)