Theorem: hasFDerivAt_integral_of_dominated_loc_of_lip'
File: Mathlib/Analysis/Calculus/ParametricIntegral.lean

tactic: have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 : H), 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
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x => inv_nonneg.mpr (norm_nonneg _)
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 : H), 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
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: set b : α → ℝ := fun a => |bound a|
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 : H), 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
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: have b_int : Integrable b μ := bound_integrable.norm
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 : H), 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 => abs (bound a)
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: have b_nonneg : ∀ a, 0 ≤ b a := fun a => abs_nonneg _
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 : H), 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 => abs (bound a)
b_int : Integrable b
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

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 _)
state_before: case h_lipsch
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
⊢ ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖

α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), 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

tactic: have hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ := fun x x_in ↦ by
  have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by
    simp only [norm_sub_rev (F x₀ _)]
    refine' h_lipsch.mono fun a ha => (ha x x_in).trans _
    rw [mul_comm ε]
    rw [mem_ball, dist_eq_norm] at x_in 
    exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)
  exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int
    (bound_integrable.norm.const_mul ε) this
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), 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

tactic: have hF'_int : Integrable F' μ :=
  have : ∀ᵐ a ∂μ, ‖F' a‖ ≤ b a := by
    apply (h_diff.and h_lipsch).mono
    rintro a ⟨ha_diff, ha_lip⟩
    refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)
  b_int.mono' hF'_meas this
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: refine' ⟨hF'_int, _⟩
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: have h_ball : ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: have : ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖((∫ a, F x a ∂μ) - ∫ a, F x₀ a ∂μ) - (∫ a, F' a ∂μ) (x - x₀)‖ =
    ‖∫ a, ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ‖ := by
  apply mem_of_superset (ball_mem_nhds _ ε_pos)
  intro x x_in; simp only
  rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub,
    ← ContinuousLinearMap.integral_apply hF'_int]
  exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
    hF'_int.apply_continuousLinearMap _]
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←
  show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp]
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: apply tendsto_integral_filter_of_dominated_convergence
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ) (𝓝 x₀)
    (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)) ∂μ))

tactic: have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by
  simp only [norm_sub_rev (F x₀ _)]
  refine' h_lipsch.mono fun a ha => (ha x x_in).trans _
  rw [mul_comm ε]
  rw [mem_ball, dist_eq_norm] at x_in 
  exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
x : H
x_in : x ∈ ball x₀ ε
⊢ Integrable (F x)

tactic: exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int
  (bound_integrable.norm.const_mul ε) this
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
x : H
x_in : x ∈ ball x₀ ε
this : ∀ᵐ (a : α) ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a
⊢ Integrable (F x)

tactic: simp only [norm_sub_rev (F x₀ _)]
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
x : H
x_in : x ∈ ball x₀ ε
⊢ ∀ᵐ (a : α) ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a

tactic: refine' h_lipsch.mono fun a ha => (ha x x_in).trans _
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
x : H
x_in : x ∈ ball x₀ ε
⊢ ∀ᵐ (a : α) ∂μ, ‖F x a - F x₀ a‖ ≤ ε * abs (bound a)

tactic: rw [mul_comm ε]
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
x : H
x_in : x ∈ ball x₀ ε
a : α
ha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ b a * ‖x - x₀‖ ≤ ε * abs (bound a)

tactic: rw [mem_ball, dist_eq_norm] at x_in
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
x : H
x_in : x ∈ ball x₀ ε
a : α
ha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ b a * ‖x - x₀‖ ≤ abs (bound a) * ε

tactic: exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
x : H
x_in : ‖x - x₀‖ < ε
a : α
ha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ b a * ‖x - x₀‖ ≤ abs (bound a) * ε

tactic: apply (h_diff.and h_lipsch).mono
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
⊢ ∀ᵐ (a : α) ∂μ, ‖F' a‖ ≤ b a

tactic: rintro a ⟨ha_diff, ha_lip⟩
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
⊢ ∀ (x : α),
    (HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ ∧
        ∀ (x_1 : H), x_1 ∈ ball x₀ ε → ‖F x_1 x - F x₀ x‖ ≤ b x * ‖x_1 - x₀‖) →
      ‖F' x‖ ≤ b x

tactic: refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)
state_before: case intro
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
a : α
ha_diff : HasFDerivAt (fun x => F x a) (F' a) x₀
ha_lip : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖F' a‖ ≤ b a

tactic: apply mem_of_superset (ball_mem_nhds _ ε_pos)
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
⊢ ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖

tactic: intro x x_in
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
⊢ ball x₀ ε ⊆
    {x |
      (fun x =>
          ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
            ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖)
        x}

tactic: simp only
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
x : H
x_in : x ∈ ball x₀ ε
⊢ x ∈
    {x |
      (fun x =>
          ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
            ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖)
        x}

tactic: rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub,
  ← ContinuousLinearMap.integral_apply hF'_int]
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
x : H
x_in : x ∈ ball x₀ ε
⊢ x ∈
    {x |
      ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
        ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖}

tactic: exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
  hF'_int.apply_continuousLinearMap _]
state_before: case hf
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
x : H
x_in : x ∈ ball x₀ ε
⊢ Integrable fun a => F x a

case hg
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
x : H
x_in : x ∈ ball x₀ ε
⊢ Integrable fun a => F x₀ a

case hf
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
x : H
x_in : x ∈ ball x₀ ε
⊢ Integrable fun a => F x a - F x₀ a

case hg
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
x : H
x_in : x ∈ ball x₀ ε
⊢ Integrable fun a => ↑(F' a) (x - x₀)

tactic: simp
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)) ∂μ) = 0

tactic: filter_upwards [h_ball] with _ x_in
state_before: case hF_meas
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ ∀ᶠ (n : H) in 𝓝 x₀, AEStronglyMeasurable (fun a => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) μ

tactic: apply AEStronglyMeasurable.const_smul
state_before: case h
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a✝ : H
x_in : a✝ ∈ ball x₀ ε
⊢ AEStronglyMeasurable (fun a => ‖a✝ - x₀‖⁻¹ • (F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀))) μ

tactic: exact ((hF_meas _ x_in).sub (hF_meas _ x₀_in)).sub (hF'_meas.apply_continuousLinearMap _)
state_before: case h.hf
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a✝ : H
x_in : a✝ ∈ ball x₀ ε
⊢ AEStronglyMeasurable (fun a => F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀)) μ

tactic: refine mem_of_superset h_ball fun x hx ↦ ?_
state_before: case h_bound
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ ∀ᶠ (n : H) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a

tactic: apply (h_diff.and h_lipsch).mono
state_before: case h_bound
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
⊢ x ∈ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a) x}

tactic: rintro a ⟨-, ha_bound⟩
state_before: case h_bound
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
⊢ ∀ (x_1 : α),
    (HasFDerivAt (fun x => F x x_1) (F' x_1) x₀ ∧ ∀ (x : H), x ∈ ball x₀ ε → ‖F x x_1 - F x₀ x_1‖ ≤ b x_1 * ‖x - x₀‖) →
      ‖‖x - x₀‖⁻¹ • (F x x_1 - F x₀ x_1 - ↑(F' x_1) (x - x₀))‖ ≤ ?bound x_1

case bound
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ α → ℝ

tactic: show ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ ≤ b a + ‖F' a‖
state_before: case h_bound.intro
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ ?bound a

case bound
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ α → ℝ

tactic: replace ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ := ha_bound x hx
state_before: case h_bound.intro
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖

tactic: calc
  ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ =
      ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • F' a (x - x₀)‖ :=
    by rw [smul_sub]
  _ ≤ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := (norm_sub_le _ _)
  _ = ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖F' a (x - x₀)‖ := by
    rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _
  _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := by
    gcongr; exact (F' a).le_op_norm _
  _ ≤ b a + ‖F' a‖ := ?_
state_before: case h_bound.intro
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖

tactic: simp only [← div_eq_inv_mul]
state_before: case h_bound.intro
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) ≤ b a + ‖F' a‖

tactic: apply_rules [add_le_add, div_le_of_nonneg_of_le_mul] <;> first | rfl | positivity
state_before: case h_bound.intro
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ abs (bound a) * ‖x - x₀‖ / ‖x - x₀‖ + ‖F' a‖ * ‖x - x₀‖ / ‖x - x₀‖ ≤ abs (bound a) + ‖F' a‖

tactic: rw [smul_sub]
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • ↑(F' a) (x - x₀)‖

tactic: rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • ↑(F' a) (x - x₀)‖ =
    ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖↑(F' a) (x - x₀)‖

tactic: gcongr
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖↑(F' a) (x - x₀)‖ ≤
    ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖)

tactic: exact (F' a).le_op_norm _
state_before: case h₂.h
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖↑(F' a) (x - x₀)‖ ≤ ‖F' a‖ * ‖x - x₀‖

tactic: rfl
state_before: case h_bound.intro.h₂.h
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖F' a‖ * ‖x - x₀‖ ≤ ‖F' a‖ * ‖x - x₀‖

tactic: positivity
state_before: case h_bound.intro.h₂.hc
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖F' a‖

tactic: exact b_int.add hF'_int.norm
state_before: case bound_integrable
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ Integrable fun a => b a + ‖F' a‖

tactic: apply h_diff.mono
state_before: case h_lim
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ ∀ᵐ (a : α) ∂μ,
    Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)
      (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))

tactic: intro a ha
state_before: case h_lim
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
⊢ ∀ (x : α),
    HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ →
      Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n x - F x₀ x - ↑(F' x) (n - x₀))) (𝓝 x₀)
        (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ x - F x₀ x - ↑(F' x) (x₀ - x₀))))

tactic: suffices Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0) by simpa
state_before: case h_lim
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
⊢ Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)
    (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))

tactic: rw [tendsto_zero_iff_norm_tendsto_zero]
state_before: case h_lim
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
⊢ Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))) (𝓝 x₀) (𝓝 0)

tactic: have : (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - F' a (x - x₀)‖) = fun x =>
    ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ := by
  ext x
  rw [norm_smul_of_nonneg (nneg _)]
state_before: case h_lim
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0)

tactic: rwa [hasFDerivAt_iff_tendsto, this] at ha
state_before: case h_lim
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this✝ :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
this :
  (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖) = fun x =>
    ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖
⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0)

tactic: simpa
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this✝ :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
this : Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))) (𝓝 x₀) (𝓝 0)
⊢ Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)
    (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))

tactic: ext x
state_before: α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
⊢ (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖) = fun x =>
    ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖

tactic: rw [norm_smul_of_nonneg (nneg _)]
state_before: case h
α : Type u_4
inst✝⁷ : MeasurableSpace α
μ : MeasureTheory.Measure α
𝕜 : Type u_1
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_2
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 : H), 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 => abs (bound a)
b_int : Integrable b
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)
hF'_int : Integrable F'
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖
a : α
ha : HasFDerivAt (fun x => F x a) (F' a) x₀
x : H
⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖