theorem extracted_formal_statement_4.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p_1 : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) (ω : Ω)
  (p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) (h_original_goal : ↑‖rexp (t * X ω)‖₊ ≤ ↑‖rexp (t * b)‖₊) :
  ‖rexp (t * X ω)‖₊ ≤ ‖rexp (t * b)‖₊ := sorry

State before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p✝ : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
ω : Ω
p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
⊢ ↑‖rexp (t * X ω)‖₊ ≤ ↑‖rexp (t * b)‖₊

Tactic: simp only [ENNReal.coe_le_coe]

State after:
[case h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t b : ℝ
 ht : t > 0
 hX : AEMeasurable X μ
 hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
 b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
 p✝ : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
 ω : Ω
 p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
 ⊢ ‖rexp (t * X ω)‖₊ ≤ ‖rexp (t * b)‖₊]

Relevant lemmas:
ENNReal.coe_le_coe : ∀ {r q : ℝ≥0}, ↑r ≤ ↑q ↔ r ≤ q