-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
theorem integral_comp_smul_deriv''' {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))) (hg1 : IntegrableOn g (f '' [[a, b]]))
    (hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]) :
    (∫ x in a..b, f' x • (g ∘ f) x) = ∫ u in f a..f b, g u

Proof state:
case intro
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (a ⊓ b) (a ⊔ b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Tactic:
obtain ⟨d, hd⟩ := nonempty_Ioo.mpr hx.2

Result:
case intro.intro
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (a ⊓ b) (a ⊔ b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
d : ℝ
hd : d ∈ Ioo x (a ⊔ b)
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Proof state:
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
d : ℝ
hd : d ∈ Ioo x (a ⊔ b)
cdsub : [[c, d]] ⊆ Ioo (a ⊓ b) (a ⊔ b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
⊢ f x ∈ J

Tactic:
rw [← hJ]

Result:
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
d : ℝ
hd : d ∈ Ioo x (a ⊔ b)
cdsub : [[c, d]] ⊆ Ioo (a ⊓ b) (a ⊔ b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
⊢ f x ∈ f '' [[c, d]]

Proof state:
case intro.intro
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
d : ℝ
hd : d ∈ Ioo x (a ⊔ b)
cdsub : [[c, d]] ⊆ Ioo (a ⊓ b) (a ⊔ b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
h2x : f x ∈ J
h2g : IntervalIntegrable g volume (f a) (f x)
h3g : StronglyMeasurableAtFilter g (𝓝[J] f x) volume
this✝ : Fact (f x ∈ J)
this : HasDerivWithinAt (fun u => ∫ (x : ℝ) in f a..u, g x) (g (f x)) J (f x)
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Tactic:
refine (this.scomp x ((hff' x hx).Ioo_of_Ioi hd.1) ?_).Ioi_of_Ioo hd.1

Result:
case intro.intro
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
d : ℝ
hd : d ∈ Ioo x (a ⊔ b)
cdsub : [[c, d]] ⊆ Ioo (a ⊓ b) (a ⊔ b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
h2x : f x ∈ J
h2g : IntervalIntegrable g volume (f a) (f x)
h3g : StronglyMeasurableAtFilter g (𝓝[J] f x) volume
this✝ : Fact (f x ∈ J)
this : HasDerivWithinAt (fun u => ∫ (x : ℝ) in f a..u, g x) (g (f x)) J (f x)
⊢ MapsTo f (Ioo x d) J

Proof state:
case pos
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (a ⊓ b) (a ⊔ b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Tactic:
have h_der :
  ∀ x ∈ Ioo (min a b) (max a b),
    HasDerivWithinAt (fun u => ∫ t in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x := by
  intro x hx
  obtain ⟨c, hc⟩ := nonempty_Ioo.mpr hx.1
  obtain ⟨d, hd⟩ := nonempty_Ioo.mpr hx.2
  have cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b) := by
    rw [uIcc_of_le (hc.2.trans hd.1).le]
    exact Icc_subset_Ioo hc.1 hd.2
  replace hg_cont := hg_cont.mono (image_subset f cdsub)
  let J := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
  have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans Ioo_subset_Icc_self)).image_uIcc
  rw [hJ] at hg_cont
  have h2x : f x ∈ J := by rw [← hJ]; exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.le)
  have h2g : IntervalIntegrable g volume (f a) (f x) := by
    refine hg1.mono_set ?_
    rw [← hf.image_uIcc]
    exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx)
  have h3g : StronglyMeasurableAtFilter g (𝓝[J] f x) :=
    hg_cont.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc (f x)
  haveI : Fact (f x ∈ J) := ⟨h2x⟩
  have : HasDerivWithinAt (fun u => ∫ x in f a..u, g x) (g (f x)) J (f x) :=
    intervalIntegral.integral_hasDerivWithinAt_right h2g h3g (hg_cont (f x) h2x)
  refine (this.scomp x ((hff' x hx).Ioo_of_Ioi hd.1) ?_).Ioi_of_Ioo hd.1
  rw [← hJ]
  refine (mapsTo_image _ _).mono ?_ Subset.rfl
  exact Ioo_subset_Icc_self.trans ((Icc_subset_Icc_left hc.2.le).trans Icc_subset_uIcc)

Result:
case pos
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (a ⊓ b) (a ⊔ b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
h_der : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u