theorem extracted_formal_statement_46.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (ω : Ω),
    X ω ∈ Set.Icc a b →
      ‖rexp ((2 * |t| + 1) * |X ω|)‖ ≤ rexp ((2 * |t| + 1) * |c|) → (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * |c| := sorry

State before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h✝ : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
ω : Ω
h : X ω ∈ Set.Icc a b
⊢ ‖rexp ((2 * |t| + 1) * |X ω|)‖ ≤ rexp ((2 * |t| + 1) * |c|)

Tactic: simp only [norm_eq_abs, abs_exp, exp_le_exp]

State after:
[case h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 t a b : ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 h✝ : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 c : ℝ := ‖a‖ ⊔ ‖b‖
 e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
 ω : Ω
 h : X ω ∈ Set.Icc a b
 ⊢ (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * |c|]

Relevant lemmas:
Real.abs_exp : ∀ (x : ℝ), |rexp x| = rexp x
Real.norm_eq_abs : ∀ (r : ℝ), ‖r‖ = |r|
Real.exp_le_exp : ∀ {x y : ℝ}, rexp x ≤ rexp y ↔ x ≤ y