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

tactic: rw [IntegrableOn, ← restrict_map_withDensity_abs_det_fderiv_eq_addHaar μ hs hf' hf,
  (measurableEmbedding_of_fderivWithin hs hf' hf).integrable_map_iff]
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ IntegrableOn g (f '' s) ↔ IntegrableOn (fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)) s

tactic: change Integrable ((g ∘ f) ∘ ((↑) : s → E)) _ ↔ _
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ Integrable (g ∘ Set.restrict s f) ↔ IntegrableOn (fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)) s

tactic: rw [← (MeasurableEmbedding.subtype_coe hs).integrable_map_iff, map_comap_subtype_coe hs]
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ Integrable ((g ∘ f) ∘ Subtype.val) ↔ IntegrableOn (fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)) s

tactic: simp only [ENNReal.ofReal]
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ Integrable (g ∘ f) ↔ IntegrableOn (fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)) s

tactic: rw [restrict_withDensity hs, integrable_withDensity_iff_integrable_coe_smul₀, IntegrableOn]
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ Integrable (g ∘ f) ↔ IntegrableOn (fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)) s

tactic: rw [iff_iff_eq]
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ (Integrable fun x => ↑(Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))) • (g ∘ f) x) ↔
    Integrable fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)

tactic: congr 2 with x
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ (Integrable fun x => ↑(Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))) • (g ∘ f) x) =
    Integrable fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)

tactic: rw [Real.coe_toNNReal _ (abs_nonneg _)]
state_before: case e_f.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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
x : E
⊢ ↑(Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))) • (g ∘ f) x = abs (ContinuousLinearMap.det (f' x)) • g (f x)

tactic: rfl
state_before: case e_f.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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
x : E
⊢ abs (ContinuousLinearMap.det (f' x)) • (g ∘ f) x = abs (ContinuousLinearMap.det (f' x)) • g (f x)

tactic: exact aemeasurable_toNNReal_abs_det_fderivWithin μ hs hf'
state_before: case hf
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✝ : IsAddHaarMeasure μ
hs : MeasurableSet s
hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
g : E → F
⊢ AEMeasurable fun x => Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))