theorem extracted_formal_statement_2.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      ∀ (i : Fin n) (ω : Ω), |X i ω| ≤ 1 → ‖rexp (τ * X i ω)‖ ≤ rexp τ → τ * X i ω ≤ τ := sorry

State before:
case h
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
⊢ ‖rexp (τ * X i ω)‖ ≤ rexp τ

Tactic: rw [norm_eq_abs, abs_of_nonneg (exp_pos _).le]
    rw [exp_le_exp]
    have h := mul_le_mul_of_nonneg_left (abs_le.mp hω).2 (hlam)

State after:
[case h
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 i : Fin n
 ω : Ω
 hω : |X i ω| ≤ 1
 h : τ * X i ω ≤ τ * 1
 ⊢ τ * X i ω ≤ τ]

Relevant lemmas:
Real.exp_pos : ∀ (x : ℝ), 0 < rexp x
Real.norm_eq_abs : ∀ (r : ℝ), ‖r‖ = |r|
mul_le_mul_of_nonneg_left : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_2 : Zero α] [inst_3 : Preorder α]
  [inst_4 : PosMulMono α], b ≤ c → 0 ≤ a → a * b ≤ a * c
Real.exp_le_exp : ∀ {x y : ℝ}, rexp x ≤ rexp y ↔ x ≤ y
abs_of_nonneg : ∀ {α : Type u_1} [inst : Lattice α] [inst_2 : AddGroup α] {a : α} [inst_3 : AddLeftMono α],
  0 ≤ a → |a| = a