Theorem: Real.smul_map_diagonal_volume_pi
File: Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean

Proof steps:
state_before: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
⊢ ofReal |(Matrix.diagonal D).det| • Measure.map (⇑(toLin' (Matrix.diagonal D))) volume = volume
state_after: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ (ofReal |(Matrix.diagonal D).det| • Measure.map (⇑(toLin' (Matrix.diagonal D))) volume) (univ.pi s) =
    ∏ i : ι, volume (s i)
tactic: refine (Measure.pi_eq fun s hs => ?_).symm
state_before: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ (ofReal |(Matrix.diagonal D).det| • Measure.map (⇑(toLin' (Matrix.diagonal D))) volume) (univ.pi s) =
    ∏ i : ι, volume (s i)
state_after: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ofReal |∏ i : ι, D i| * (Measure.map (⇑(toLin' (Matrix.diagonal D))) volume) (univ.pi s) = ∏ x : ι, volume (s x)
tactic: simp only [det_diagonal, Measure.coe_smul, Algebra.id.smul_eq_mul, Pi.smul_apply]
state_before: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ofReal |∏ i : ι, D i| * (Measure.map (⇑(toLin' (Matrix.diagonal D))) volume) (univ.pi s) = ∏ x : ι, volume (s x)
state_after: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ofReal |∏ i : ι, D i| * volume (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi s) = ∏ x : ι, volume (s x)

ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ Measurable ⇑(toLin' (Matrix.diagonal D))
tactic: rw [Measure.map_apply _ (MeasurableSet.univ_pi hs)]
state_before: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ofReal |∏ i : ι, D i| * volume (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi s) = ∏ x : ι, volume (s x)

ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ Measurable ⇑(toLin' (Matrix.diagonal D))
state_after: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ Measurable ⇑(toLin' (Matrix.diagonal D))

ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ofReal |∏ i : ι, D i| * volume (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi s) = ∏ x : ι, volume (s x)
tactic: swap
state_before: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
this : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i => s i) = univ.pi fun i => (fun x => D i * x) ⁻¹' s i
B : ∀ (i : ι), ofReal |D i| * volume ((fun x => D i * x) ⁻¹' s i) = volume (s i)
⊢ ofReal |∏ i : ι, D i| * volume (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi s) = ∏ x : ι, volume (s x)
state_after: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
this : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i => s i) = univ.pi fun i => (fun x => D i * x) ⁻¹' s i
B : ∀ (i : ι), ofReal |D i| * volume ((fun x => D i * x) ⁻¹' s i) = volume (s i)
⊢ ∏ x : ι, ofReal |D x| * volume ((fun x_1 => D x * x_1) ⁻¹' s x) = ∏ x : ι, volume (s x)
tactic: rw [this, volume_pi_pi, Finset.abs_prod,
    ENNReal.ofReal_prod_of_nonneg fun i _ => abs_nonneg (D i), ← Finset.prod_mul_distrib]
state_before: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
this : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i => s i) = univ.pi fun i => (fun x => D i * x) ⁻¹' s i
B : ∀ (i : ι), ofReal |D i| * volume ((fun x => D i * x) ⁻¹' s i) = volume (s i)
⊢ ∏ x : ι, ofReal |D x| * volume ((fun x_1 => D x * x_1) ⁻¹' s x) = ∏ x : ι, volume (s x)
state_after: no goals
tactic: simp only [B]