theorem extracted_formal_statement_8 (x : ℝ) (hx : |x| ≤ 1) (hx_le_1 : x ≤ 1) (u : ℝ) (hu : u ∈ interior (Ici 0))
  (h_original_goal : rexp (u * x) ≤ rexp u) : u * x ≤ u := sorry

State before:
case a.a
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
hf'0 : f' 0 = 0
t : ℝ
ht : t ∈ interior (Ici 0)
u : ℝ
hu : u ∈ interior (Ici 0)
⊢ rexp (u * x) ≤ rexp u

Tactic: apply exp_monotone

State after:
[case a.a.a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
 f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
 f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
 hf0 : f 0 = 0
 hf'0 : f' 0 = 0
 t : ℝ
 ht : t ∈ interior (Ici 0)
 u : ℝ
 hu : u ∈ interior (Ici 0)
 ⊢ u * x ≤ u]

Relevant lemmas:
Real.exp_monotone : Monotone rexp