l3lab/ntp-mathlib huggingface dataset:
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  

Proof state:
case pos
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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]]
hG : CompleteSpace G
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Next tactic:
rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  

Proof state:
case pos
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Next tactic:
have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; exact mapsTo_image _ _
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    

Proof state:
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
⊢ ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]

Next tactic:
refine' (continuousOn_primitive_interval' hg1 _).comp hf _
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · 

Proof state:
case refine'_1
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
⊢ f a ∈ [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]

Next tactic:
rw [← hf.image_uIcc]
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; 

Proof state:
case refine'_1
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
⊢ f a ∈ f '' [[a, b]]

Next tactic:
exact mem_image_of_mem f left_mem_uIcc
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · 

Proof state:
case refine'_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
⊢ MapsTo (fun u => f u) [[a, b]] [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]

Next tactic:
rw [← hf.image_uIcc]
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; 

Proof state:
case refine'_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
⊢ MapsTo (fun u => f u) [[a, b]] (f '' [[a, b]])

Next tactic:
exact mapsTo_image _ _
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; exact mapsTo_image _ _
  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
    

Proof state:
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
⊢ ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Next tactic:
intro x hx
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; exact mapsTo_image _ _
  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
    

Proof state:
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Next tactic:
obtain ⟨c, hc⟩ := nonempty_Ioo.mpr hx.1
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; exact mapsTo_image _ _
  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
    

Proof state:
case intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Next tactic:
obtain ⟨d, hd⟩ := nonempty_Ioo.mpr hx.2
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; exact mapsTo_image _ _
  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
    

Proof state:
case intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Next tactic:
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
/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
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 := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; exact mapsTo_image _ _
  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
      

Proof state:
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
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 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
⊢ [[c, d]] ⊆ Ioo (min a b) (max a b)

Next tactic:
rw [uIcc_of_le (hc.2.trans hd.1).le]