-- Mathlib/MeasureTheory/Function/Jacobian.lean
theorem addHaar_image_le_lintegral_abs_det_fderiv (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    μ (f '' s) ≤ ∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
n : ℕ
⊢ μ (f '' (s ∩ u n)) ≤ ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ

Tactic:
apply
  addHaar_image_le_lintegral_abs_det_fderiv_aux2 μ (hs.inter (u_meas n)) _ fun x hx =>
    (hf' x hx.1).mono inter_subset_left

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
n : ℕ
⊢ μ (s ∩ u n) ≠ ⊤

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
⊢ ∀ (n : ℕ), MeasurableSet (u n)

Tactic:
intro n

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
n : ℕ
⊢ MeasurableSet (u n)

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ μ (⋃ i, f '' (s ∩ u i)) ≤ ∑' (n : ℕ), μ (f '' (s ∩ u n))

Tactic:
exact measure_iUnion_le _

Result:
no goals

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
n i : ℕ
⊢ MeasurableSet (spanningSets μ i)

Tactic:
exact measurableSet_spanningSets μ i

Result:
no goals

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
⊢ s = ⋃ n, s ∩ u n

Tactic:
rw [← inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]

Result:
no goals

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
⊢ μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ

Tactic:
have A : s = ⋃ n, s ∩ u n := by
  rw [← inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
n : ℕ
this : μ (u n) < ⊤
⊢ μ (s ∩ u n) ≠ ⊤

Tactic:
exact ne_of_lt (lt_of_le_of_lt (measure_mono inter_subset_right) this)

Result:
no goals

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ =
    ∫⁻ (x : E) in ⋃ n, s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ

Tactic:
rw [lintegral_iUnion]

Result:
case hm
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∀ (i : ℕ), MeasurableSet (s ∩ u i)

case hd
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ Pairwise (Disjoint on fun n => s ∩ u n)

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∑' (n : ℕ), μ (f '' (s ∩ u n)) ≤ ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ

Tactic:
apply ENNReal.tsum_le_tsum fun n => ?_

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
n : ℕ
⊢ μ (f '' (s ∩ u n)) ≤ ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ