ι : Type ?u.2199955
inst✝ : Fintype ι
a : ℝ
h : a ≠ 0
p q : ℚ
⊢ ↑↑volume (Ioo ↑p ↑q) = ↑↑(ofReal (abs a) • Measure.map (fun x => a * x) volume) (Ioo ↑p ↑q)
ι : Type ?u.2199955
inst✝ : Fintype ι
a : ℝ
h : a ≠ 0
⊢ ofReal (abs a) • Measure.map (fun x => a * x) volume = volume
refine' (Real.measure_ext_Ioo_rat fun p q => _).symm
case inl
ι : Type ?u.2199955
inst✝ : Fintype ι
a : ℝ
h✝ : a ≠ 0
p q : ℚ
h : a < 0
⊢ ↑↑volume (Ioo ↑p ↑q) = ↑↑(ofReal (abs a) • Measure.map (fun x => a * x) volume) (Ioo ↑p ↑q)

case inr
ι : Type ?u.2199955
inst✝ : Fintype ι
a : ℝ
h✝ : a ≠ 0
p q : ℚ
h : a > 0
⊢ ↑↑volume (Ioo ↑p ↑q) = ↑↑(ofReal (abs a) • Measure.map (fun x => a * x) volume) (Ioo ↑p ↑q)
ι : Type ?u.2199955
inst✝ : Fintype ι
a : ℝ
h : a ≠ 0
p q : ℚ
⊢ ↑↑volume (Ioo ↑p ↑q) = ↑↑(ofReal (abs a) • Measure.map (fun x => a * x) volume) (Ioo ↑p ↑q)
cases' lt_or_gt_of_ne h with h h
no goals
case inl
ι : Type ?u.2199955
inst✝ : Fintype ι
a : ℝ
h✝ : a ≠ 0
p q : ℚ
h : a < 0
⊢ ↑↑volume (Ioo ↑p ↑q) = ↑↑(ofReal (abs a) • Measure.map (fun x => a * x) volume) (Ioo ↑p ↑q)
simp only [Real.volume_Ioo, Measure.smul_apply, ← ENNReal.ofReal_mul (le_of_lt <| neg_pos.2 h),
  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, neg_sub_neg, neg_mul,
  preimage_const_mul_Ioo_of_neg _ _ h, abs_of_neg h, mul_sub, smul_eq_mul,
  mul_div_cancel' _ (ne_of_lt h)]
no goals
case inr
ι : Type ?u.2199955
inst✝ : Fintype ι
a : ℝ
h✝ : a ≠ 0
p q : ℚ
h : a > 0
⊢ ↑↑volume (Ioo ↑p ↑q) = ↑↑(ofReal (abs a) • Measure.map (fun x => a * x) volume) (Ioo ↑p ↑q)
simp only [Real.volume_Ioo, Measure.smul_apply, ← ENNReal.ofReal_mul (le_of_lt h),
  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, preimage_const_mul_Ioo _ _ h,
  abs_of_pos h, mul_sub, mul_div_cancel' _ (ne_of_gt h), smul_eq_mul]