Theorem: hasDerivAt_integral_of_dominated_loc_of_lip
File: Mathlib/Analysis/Calculus/ParametricIntegral.lean

tactic: set L : E →L[𝕜] 𝕜 →L[𝕜] E := ContinuousLinearMap.smulRightL 𝕜 𝕜 E 1
state_before: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
h_diff : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' a) x₀
⊢ Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
state_after: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
h_diff : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' a) x₀
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
⊢ 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
state_before: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
state_after: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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

tactic: cases'
  hasFDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable
    h_diff with
  hF'_int key
state_before: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
state_after: case intro
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
hF'_int : Integrable (⇑L ∘ F') μ
key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
⊢ Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: refine ⟨hF'_int, ?_⟩
state_before: case intro
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
⊢ Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
state_after: case intro
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
⊢ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

tactic: by_cases hE : CompleteSpace E
state_before: case intro
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
⊢ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
state_after: case pos
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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

case neg
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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: swap
state_before: case pos
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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

case neg
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
state_after: case neg
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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

case pos
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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 ⊢
state_before: case pos
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
state_after: case pos
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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

tactic: rw [← integrable_norm_iff hm] at hF'_int
state_before: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
hF'_int : Integrable (⇑L ∘ F') μ
key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
⊢ Integrable F' μ
state_after: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
hF'_int : Integrable (fun a => ‖(⇑L ∘ F') a‖) μ
key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
⊢ Integrable F' μ

tactic: simp [integral, hE]
state_before: case neg
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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
state_after: case neg
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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 => 0) 0 x₀

tactic: exact hasDerivAt_const x₀ 0
state_before: case neg
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F✝ : H → α → E
x₀✝ : H
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 => 0) 0 x₀
state_after: no goals