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

Proof steps:
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
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
state_after: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
tactic: obtain ⟨u, u_meas, uf⟩ : ∃ u, Measurable u ∧ EqOn u f s := by
    classical
    refine ⟨piecewise s f 0, ?_, piecewise_eqOn _ _ _⟩
    refine ContinuousOn.measurable_piecewise ?_ continuous_zero.continuousOn hs
    have : DifferentiableOn ℝ f s := fun x hx => (hf' x hx).differentiableWithinAt
    exact this.continuousOn
state_before: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
state_after: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
tactic: have u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x := fun x hx =>
    (hf' x hx).congr (fun y hy => uf hy) (uf hx)
state_before: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
state_after: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
tactic: set F : s → E := u ∘ (↑) with hF
state_before: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
state_after: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
A : Measure.map F (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) = μ.restrict (u '' s)
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
tactic: have A :
    Measure.map F (comap (↑) (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
      μ.restrict (u '' s) := by
    rw [hF, ← Measure.map_map u_meas measurable_subtype_coe, map_comap_subtype_coe hs,
      restrict_withDensity hs]
    exact map_withDensity_abs_det_fderiv_eq_addHaar μ hs u' (hf.congr uf.symm) u_meas
state_before: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
A : Measure.map F (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) = μ.restrict (u '' s)
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
state_after: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
A : Measure.map F (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) = μ.restrict (f '' s)
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
tactic: rw [uf.image_eq] at A
state_before: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
A : Measure.map F (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) = μ.restrict (f '' s)
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
state_after: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
A : Measure.map F (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) = μ.restrict (f '' s)
this : F = s.restrict f
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
tactic: have : F = s.restrict f := by
    ext x
    exact uf x.2
state_before: case intro.intro
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 : E → E
u_meas : Measurable u
uf : EqOn u f s
u' : ∀ x ∈ s, HasFDerivWithinAt u (f' x) s x
F : ↑s → E := u ∘ Subtype.val
hF : F = u ∘ Subtype.val
A : Measure.map F (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) = μ.restrict (f '' s)
this : F = s.restrict f
⊢ Measure.map (s.restrict f) (Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
    μ.restrict (f '' s)
state_after: no goals
tactic: rwa [this] at A