Theorem: MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image
File: Mathlib/MeasureTheory/Function/Jacobian.lean

tactic: let u n := disjointed (spanningSets μ) n
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)

tactic: have u_meas : ∀ n, MeasurableSet (u n) := by
  intro n
  apply MeasurableSet.disjointed fun i => ?_
  exact measurable_spanningSets μ i
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)

tactic: have A : s = ⋃ n, s ∩ u n := by
  rw [← inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)

tactic: intro n
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
⊢ ∀ (n : ℕ), MeasurableSet (u n)
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
n : ℕ
⊢ MeasurableSet (u n)

tactic: apply MeasurableSet.disjointed fun i => ?_
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
n : ℕ
⊢ MeasurableSet (u n)
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
n i : ℕ
⊢ MeasurableSet (spanningSets μ i)

tactic: exact measurable_spanningSets μ i
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
n i : ℕ
⊢ MeasurableSet (spanningSets μ i)
state_after: no goals

tactic: rw [← inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
⊢ s = ⋃ n, s ∩ u n
state_after: no goals

tactic: conv_lhs => rw [A]
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∫⁻ (x : E) in ⋃ n, s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ =
    ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ

tactic: rw [lintegral_iUnion]
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
⊢ ∫⁻ (x : E) in ⋃ n, s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ =
    ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ
state_after: case hm
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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)

tactic: intro n
state_before: case hm
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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)
state_after: case hm
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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
⊢ MeasurableSet (s ∩ u n)

tactic: exact hs.inter (u_meas n)
state_before: case hm
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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
⊢ MeasurableSet (s ∩ u n)
state_after: no goals

tactic: exact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right
state_before: case hd
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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)
state_after: no goals

tactic: apply ENNReal.tsum_le_tsum fun n => ?_
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 ≤ ∑' (n : ℕ), μ (f '' (s ∩ u n))
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 ≤ μ (f '' (s ∩ u n))

tactic: apply
  lintegral_abs_det_fderiv_le_addHaar_image_aux2 μ (hs.inter (u_meas n)) _
    (fun x hx => (hf' x hx.1).mono inter_subset_left) (hf.mono inter_subset_left)
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 ≤ μ (f '' (s ∩ u n))
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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) ≠ ⊤

tactic: have : μ (u n) < ∞ :=
  lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top μ n)
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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) ≠ ⊤
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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)
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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) ≠ ⊤
state_after: no goals

tactic: conv_rhs => rw [A, image_iUnion]
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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)) = μ (f '' s)
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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)) = μ (⋃ i, f '' (s ∩ u i))

tactic: rw [measure_iUnion]
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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)) = μ (⋃ i, f '' (s ∩ u i))
state_after: case hn
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 i => f '' (s ∩ u i))

case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 (f '' (s ∩ u i))

tactic: intro i j hij
state_before: case hn
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 i => f '' (s ∩ u i))
state_after: case hn
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
i j : ℕ
hij : i ≠ j
⊢ (Disjoint on fun i => f '' (s ∩ u i)) i j

tactic: apply Disjoint.image _ hf inter_subset_left inter_subset_left
state_before: case hn
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
i j : ℕ
hij : i ≠ j
⊢ (Disjoint on fun i => f '' (s ∩ u i)) i j
state_after: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
i j : ℕ
hij : i ≠ j
⊢ Disjoint (s ∩ u i) (s ∩ u j)

tactic: exact
  Disjoint.mono inter_subset_right inter_subset_right
    (disjoint_disjointed _ hij)
state_before: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
u : ℕ → Set E := fun n => disjointed (spanningSets μ) n
u_meas : ∀ (n : ℕ), MeasurableSet (u n)
A : s = ⋃ n, s ∩ u n
i j : ℕ
hij : i ≠ j
⊢ Disjoint (s ∩ u i) (s ∩ u j)
state_after: no goals

tactic: intro i
state_before: case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 (f '' (s ∩ u i))
state_after: case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 (f '' (s ∩ u i))

tactic: exact
  measurable_image_of_fderivWithin (hs.inter (u_meas i))
    (fun x hx => (hf' x hx.1).mono inter_subset_left)
    (hf.mono inter_subset_left)
state_before: case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
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
hf : InjOn f s
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 (f '' (s ∩ u i))
state_after: no goals