theorem extracted_formal_statement_47.{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 → (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
⊢ (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * |c|

Tactic: apply
  mul_le_mul_of_nonneg_left (le_trans' (le_abs_self (max ‖a‖ ‖b‖)) (abs_le_max_abs_abs h.1 h.2))
    (by linarith [abs_nonneg t] : 0 ≤ 2 * |t| + 1)

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|
abs_le_max_abs_abs : ∀ {G : Type u_1} [inst : LinearOrderedAddCommGroup G] {a b c : G}, a ≤ b → b ≤ c → |b| ≤ |a| ⊔ |c|
le_trans' : ∀ {α : Type u_2} [inst : Preorder α] {a b c : α}, b ≤ c → a ≤ b → a ≤ c
mul_le_mul_of_nonneg_left : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α]
  [inst_3 : PosMulMono α], b ≤ c → 0 ≤ a → a * b ≤ a * c
le_abs_self : ∀ {α : Type u_1} [inst : Lattice α] [inst_1 : AddGroup α] (a : α), a ≤ |a|