theorem extracted_formal_statement_39.{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 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (y : Ω) (x : ℝ) (b1 : x ∈ Metric.ball t (|t| + 1)) (p2 : |x| ≤ 2 * |t| + 1) :
  rexp (|x| * |X y|) * |X y| ≤ rexp ((2 * |t| + 1) * |X y|) * |X y| := sorry

State before:
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
bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω|
y : Ω
x : ℝ
b1 : x ∈ Metric.ball t (|t| + 1)
p2 : |x| ≤ 2 * |t| + 1
⊢ rexp (|x| * |X y|) * |X y| ≤ rexp ((2 * |t| + 1) * |X y|) * |X y|

Tactic: exact
  mul_le_mul_of_nonneg_right (exp_le_exp.mpr (mul_le_mul_of_nonneg_right p2 (abs_nonneg (X y)))) (abs_nonneg (X y))

State after:

Relevant lemmas:
abs_nonneg : ∀ {α : Type u_1} [inst : Lattice α] [inst_1 : AddGroup α] [inst_2 : AddLeftMono α]
  [inst_3 : AddRightMono α] (a : α), 0 ≤ |a|
mul_le_mul_of_nonneg_right : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α]
  [inst_3 : MulPosMono α], b ≤ c → 0 ≤ a → b * a ≤ c * a