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

tactic: by_cases hf : Integrable f μ
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
⊢ μ.withDensityᵥ (-f) = -μ.withDensityᵥ f
state_after: case pos
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
⊢ μ.withDensityᵥ (-f) = -μ.withDensityᵥ f

case neg
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
⊢ μ.withDensityᵥ (-f) = -μ.withDensityᵥ f

tactic: ext1 i hi
state_before: case pos
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
⊢ μ.withDensityᵥ (-f) = -μ.withDensityᵥ f
state_after: case pos.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
i : Set α
hi : MeasurableSet i
⊢ ↑(μ.withDensityᵥ (-f)) i = ↑(-μ.withDensityᵥ f) i

tactic: rw [VectorMeasure.neg_apply, withDensityᵥ_apply hf hi, ← integral_neg,
  withDensityᵥ_apply hf.neg hi]
state_before: case pos.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
i : Set α
hi : MeasurableSet i
⊢ ↑(μ.withDensityᵥ (-f)) i = ↑(-μ.withDensityᵥ f) i
state_after: case pos.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
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, (-f) x ∂μ = ∫ (a : α) in i, -f a ∂μ

tactic: rfl
state_before: case pos.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
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, (-f) x ∂μ = ∫ (a : α) in i, -f a ∂μ
state_after: no goals

tactic: rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg, neg_zero]
state_before: case neg
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
⊢ μ.withDensityᵥ (-f) = -μ.withDensityᵥ f
state_after: case neg.hnc
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
⊢ ¬Integrable (-f) μ

tactic: rwa [integrable_neg_iff]
state_before: case neg.hnc
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
⊢ ¬Integrable (-f) μ
state_after: no goals