Theorem: MeasureTheory.withDensityᵥ_add
File: Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
⊢ μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ ↑(μ.withDensityᵥ (f + g)) i = ↑(μ.withDensityᵥ f + μ.withDensityᵥ g) i
tactic: ext1 i hi
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ ↑(μ.withDensityᵥ (f + g)) i = ↑(μ.withDensityᵥ f + μ.withDensityᵥ g) i
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, (f + g) x ∂μ = ∫ (x : α) in i, f x ∂μ + ∫ (x : α) in i, g x ∂μ
tactic: rw [withDensityᵥ_apply (hf.add hg) hi, VectorMeasure.add_apply, withDensityᵥ_apply hf hi,
    withDensityᵥ_apply hg hi]
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, (f + g) x ∂μ = ∫ (x : α) in i, f x ∂μ + ∫ (x : α) in i, g x ∂μ
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, f x + g x ∂μ = ∫ (x : α) in i, f x ∂μ + ∫ (x : α) in i, g x ∂μ
tactic: simp_rw [Pi.add_apply]
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, f x + g x ∂μ = ∫ (x : α) in i, f x ∂μ + ∫ (x : α) in i, g x ∂μ
state_after: case h.hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ IntegrableOn f Set.univ (μ.restrict i)

case h.hg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
i : Set α
hi : MeasurableSet i
⊢ IntegrableOn g Set.univ (μ.restrict i)
tactic: rw [integral_add] <;> rw [← integrableOn_univ]