theorem extracted_formal_statement_102.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (ω : Ω)
  (w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ))
  (q2 : 0 ≤ ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) : 0 ≤ rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ := sorry

State before:
case h.h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t : ℝ
f : ℝ → ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
ω : Ω
w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)
q2 : 0 ≤ ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ
⊢ 0 ≤ rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ

Tactic: exact div_nonneg (exp_nonneg (t * X ω)) q2

State after:

Relevant lemmas:
div_nonneg : ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] [inst_1 : PartialOrder G₀] [inst_2 : ZeroLEOneClass G₀]
  [inst_3 : PosMulReflectLT G₀] {a b : G₀} [inst_4 : PosMulMono G₀], 0 ≤ a → 0 ≤ b → 0 ≤ a / b
Real.exp_nonneg : ∀ (x : ℝ), 0 ≤ rexp x