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

tactic: obtain ⟨u, hmu, hu₁, hu₂⟩ := h₁
state_before: α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
h₁ : v₁ ⟂ᵥ w
h₂ : v₂ ⟂ᵥ w
⊢ v₁ + v₂ ⟂ᵥ w

tactic: obtain ⟨v, hmv, hv₁, hv₂⟩ := h₂
state_before: case intro.intro.intro
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
h₂ : v₂ ⟂ᵥ w
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
⊢ v₁ + v₂ ⟂ᵥ w

tactic: refine' mk (u ∩ v) (hmu.inter hmv) (fun t ht _ => _) fun t ht hmt => _
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
⊢ v₁ + v₂ ⟂ᵥ w

tactic: rw [add_apply, hu₁ _ (Set.subset_inter_iff.1 ht).1, hv₁ _ (Set.subset_inter_iff.1 ht).2,
  zero_add]
state_before: case intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ u ∩ v
x✝ : MeasurableSet t
⊢ ↑(v₁ + v₂) t = 0

tactic: rw [Set.compl_inter] at ht
state_before: case intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ (u ∩ v)ᶜ
hmt : MeasurableSet t
⊢ ↑w t = 0

tactic: rw [(_ : t = uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t),
  of_union _ (hmu.compl.inter hmt) ((hmv.compl.diff hmu.compl).inter hmt), hu₂, hv₂, add_zero]
state_before: case intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ ↑w t = 0

tactic: exact Set.Subset.trans (Set.inter_subset_left _ _) (Set.diff_subset _ _)
state_before: case intro.intro.intro.intro.intro.intro.refine'_2.x
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ vᶜ \ uᶜ ∩ t ⊆ vᶜ

tactic: exact Set.inter_subset_left _ _
state_before: case intro.intro.intro.intro.intro.intro.refine'_2.x
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ uᶜ ∩ t ⊆ uᶜ

tactic: exact disjoint_sdiff_self_right.mono (Set.inter_subset_left _ _) (Set.inter_subset_left _ _)
state_before: α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ Disjoint (uᶜ ∩ t) (vᶜ \ uᶜ ∩ t)

tactic: apply Set.Subset.antisymm <;> intro x hx
state_before: α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ t = uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t

tactic: by_cases hxu' : x ∈ uᶜ
state_before: case h₁
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
x : α
hx : x ∈ t
⊢ x ∈ uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t

tactic: rcases ht hx with (hxu | hxv)
state_before: case neg
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
x : α
hx : x ∈ t
hxu' : ¬x ∈ uᶜ
⊢ x ∈ uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t

tactic: exacts [False.elim (hxu' hxu), Or.inr ⟨⟨hxv, hxu'⟩, hx⟩]
state_before: case neg.inl
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
x : α
hx : x ∈ t
hxu' : ¬x ∈ uᶜ
hxu : x ∈ uᶜ
⊢ x ∈ uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t

case neg.inr
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
x : α
hx : x ∈ t
hxu' : ¬x ∈ uᶜ
hxv : x ∈ vᶜ
⊢ x ∈ uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t

tactic: exact Or.inl ⟨hxu', hx⟩
state_before: case pos
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
x : α
hx : x ∈ t
hxu' : x ∈ uᶜ
⊢ x ∈ uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t

tactic: cases' hx with hx hx <;> exact hx.2
state_before: case h₂
α : Type u_3
β : Type ?u.661519
m : MeasurableSpace α
L : Type ?u.661525
M : Type u_2
N : Type u_1
inst✝⁷ : AddCommMonoid L
inst✝⁶ : TopologicalSpace L
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v✝ v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ (t : Set α), t ⊆ u → ↑v₁ t = 0
hu₂ : ∀ (t : Set α), t ⊆ uᶜ → ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ (t : Set α), t ⊆ v → ↑v₂ t = 0
hv₂ : ∀ (t : Set α), t ⊆ vᶜ → ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
x : α
hx : x ∈ uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t
⊢ x ∈ t