-- Mathlib/MeasureTheory/Function/Jacobian.lean
theorem lintegral_image_eq_lintegral_abs_det_fderiv_mul (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → ℝ≥0∞) :
    ∫⁻ x in f '' s, g x ∂μ = ∫⁻ x in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ AEMeasurable (fun x => ENNReal.ofReal |(f' x).det|) (μ.restrict s)

Tactic:
exact aemeasurable_ofReal_abs_det_fderivWithin μ hs hf'

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (a : ↑s), g (s.restrict f a) ∂Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|) =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂μ

Tactic:
simp only [Set.restrict_apply, ← Function.comp_apply (f := g)]

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (a : ↑s), (g ∘ f) ↑a ∂Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|) =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * (g ∘ f) x ∂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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (a : E) in s, ((fun x => ENNReal.ofReal |(f' x).det|) * g ∘ f) a ∂μ =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * (g ∘ f) x ∂μ

Tactic:
simp only [Pi.mul_apply]

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (x : E) in f '' s, g x ∂μ = ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂μ

Tactic:
rw [← restrict_map_withDensity_abs_det_fderiv_eq_addHaar μ hs hf' hf,
  (measurableEmbedding_of_fderivWithin hs hf' hf).lintegral_map]

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (a : ↑s), g (s.restrict f a) ∂Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|) =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (a : ↑s), (g ∘ f) ↑a ∂Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|) =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * (g ∘ f) x ∂μ

Tactic:
rw [← (MeasurableEmbedding.subtype_coe hs).lintegral_map, map_comap_subtype_coe hs,
  setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀ _ _ _ hs]

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∫⁻ (a : E) in s, ((fun x => ENNReal.ofReal |(f' x).det|) * g ∘ f) a ∂μ =
    ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * (g ∘ f) x ∂μ

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∀ᵐ (x : E) ∂μ.restrict s, ENNReal.ofReal |(f' x).det| < ⊤

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ AEMeasurable (fun x => ENNReal.ofReal |(f' x).det|) (μ.restrict s)

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
hf : InjOn f s
g : E → ℝ≥0∞
⊢ ∀ᵐ (x : E) ∂μ.restrict s, ENNReal.ofReal |(f' x).det| < ⊤

Tactic:
simp only [eventually_true, ENNReal.ofReal_lt_top]

Result:
no goals