theorem extracted_formal_statement_97.{u_1} {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))
  (h_original_goal :
    (ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)).toReal • f (X ω) =
      (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω)) :
  0 ≤ rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ ∨ f (X ω) = 0 := sorry

State before:
case 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)
⊢ (ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)).toReal • f (X ω) =
    (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ) * f (X ω)

Tactic: simp only [smul_eq_mul, mul_eq_mul_right_iff, ENNReal.toReal_ofReal_eq_iff]

State after:
[case 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)
 ⊢ 0 ≤ rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ ∨ f (X ω) = 0]

Relevant lemmas:
smul_eq_mul : ∀ {α : Type u_9} [inst : Mul α] (a b : α), a • b = a * b
mul_eq_mul_right_iff : ∀ {M₀ : Type u_1} [inst : CancelMonoidWithZero M₀] {a b c : M₀}, a * c = b * c ↔ a = b ∨ c = 0
ENNReal.toReal_ofReal_eq_iff : ∀ {a : ℝ}, (ENNReal.ofReal a).toReal = a ↔ 0 ≤ a