Theorem: MeasureTheory.VectorMeasure.MutuallySingular.neg_left
File: Mathlib/MeasureTheory/Measure/VectorMeasure.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M✝ : Type u_4
N : Type u_5
inst✝⁸ : AddCommMonoid L
inst✝⁷ : TopologicalSpace L
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : TopologicalSpace M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M✝
w✝ w₁ w₂ : VectorMeasure α N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
h : v ⟂ᵥ w
⊢ -v ⟂ᵥ w
state_after: case intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M✝ : Type u_4
N : Type u_5
inst✝⁸ : AddCommMonoid L
inst✝⁷ : TopologicalSpace L
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : TopologicalSpace M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M✝
w✝ w₁ w₂ : VectorMeasure α N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, ↑v t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
⊢ -v ⟂ᵥ w
tactic: obtain ⟨u, hmu, hu₁, hu₂⟩ := h
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M✝ : Type u_4
N : Type u_5
inst✝⁸ : AddCommMonoid L
inst✝⁷ : TopologicalSpace L
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : TopologicalSpace M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M✝
w✝ w₁ w₂ : VectorMeasure α N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, ↑v t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
⊢ -v ⟂ᵥ w
state_after: case intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M✝ : Type u_4
N : Type u_5
inst✝⁸ : AddCommMonoid L
inst✝⁷ : TopologicalSpace L
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : TopologicalSpace M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M✝
w✝ w₁ w₂ : VectorMeasure α N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, ↑v t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
s : Set α
hs : s ⊆ u
⊢ ↑(-v) s = 0
tactic: refine ⟨u, hmu, fun s hs => ?_, hu₂⟩
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M✝ : Type u_4
N : Type u_5
inst✝⁸ : AddCommMonoid L
inst✝⁷ : TopologicalSpace L
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : TopologicalSpace M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M✝
w✝ w₁ w₂ : VectorMeasure α N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, ↑v t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
s : Set α
hs : s ⊆ u
⊢ ↑(-v) s = 0
state_after: case intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M✝ : Type u_4
N : Type u_5
inst✝⁸ : AddCommMonoid L
inst✝⁷ : TopologicalSpace L
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : TopologicalSpace M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M✝
w✝ w₁ w₂ : VectorMeasure α N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, ↑v t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
s : Set α
hs : s ⊆ u
⊢ ↑v s = 0
tactic: rw [neg_apply v s, neg_eq_zero]
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M✝ : Type u_4
N : Type u_5
inst✝⁸ : AddCommMonoid L
inst✝⁷ : TopologicalSpace L
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : TopologicalSpace M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M✝
w✝ w₁ w₂ : VectorMeasure α N
M : Type u_6
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalAddGroup M
v : VectorMeasure α M
w : VectorMeasure α N
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, ↑v t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
s : Set α
hs : s ⊆ u
⊢ ↑v s = 0
state_after: no goals
tactic: exact hu₁ s hs