Theorem: exists_ratio_hasDerivAt_eq_ratio_slope'
File: Mathlib/Analysis/Calculus/MeanValue.lean

tactic: let h x := (lgb - lga) * f x - (lfb - lfa) * g x
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

tactic: have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by
  have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) :=
    (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)
  convert this using 2
  ring
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

tactic: have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by
  have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) :=
    (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)
  convert this using 2
  ring
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

tactic: let h' x := (lgb - lga) * f' x - (lfb - lfa) * g' x
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

tactic: have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := by
  intro x hx
  exact ((hff' x hx).const_mul _).sub ((hgg' x hx).const_mul _)
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

tactic: rcases exists_hasDerivAt_eq_zero' hab hha hhb hhh' with ⟨c, cmem, hc⟩
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
state_after: case intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
c : ℝ
cmem : c ∈ Ioo a b
hc : h' c = 0
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

tactic: exact ⟨c, cmem, sub_eq_zero.1 hc⟩
state_before: case intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
c : ℝ
cmem : c ∈ Ioo a b
hc : h' c = 0
⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
state_after: no goals

tactic: have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) :=
  (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
⊢ Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
this : Tendsto h (𝓝[>] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))
⊢ Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))

tactic: convert this using 2
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
this : Tendsto h (𝓝[>] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))
⊢ Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
state_after: case h.e'_5.h.e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
this : Tendsto h (𝓝[>] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))
⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfa - (lfb - lfa) * lga

tactic: ring
state_before: case h.e'_5.h.e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
this : Tendsto h (𝓝[>] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))
⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfa - (lfb - lfa) * lga
state_after: no goals

tactic: have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) :=
  (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
⊢ Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
this : Tendsto h (𝓝[<] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))
⊢ Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))

tactic: convert this using 2
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
this : Tendsto h (𝓝[<] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))
⊢ Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
state_after: case h.e'_5.h.e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
this : Tendsto h (𝓝[<] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))
⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfb - (lfb - lfa) * lgb

tactic: ring
state_before: case h.e'_5.h.e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
this : Tendsto h (𝓝[<] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))
⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfb - (lfb - lfa) * lgb
state_after: no goals

tactic: intro x hx
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
⊢ ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
x : ℝ
hx : x ∈ Ioo a b
⊢ HasDerivAt h (h' x) x

tactic: exact ((hff' x hx).const_mul _).sub ((hgg' x hx).const_mul _)
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hfd : DifferentiableOn ℝ f (Ioo a b)
g g' : ℝ → ℝ
hgc : ContinuousOn g (Icc a b)
hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hgd : DifferentiableOn ℝ g (Ioo a b)
lfa lga lfb lgb : ℝ
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)
hga : Tendsto g (𝓝[>] a) (𝓝 lga)
hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)
hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)
h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x
hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))
hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))
h' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x
x : ℝ
hx : x ∈ Ioo a b
⊢ HasDerivAt h (h' x) x
state_after: no goals