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

tactic: obtain ⟨u, hmu, hu₁, hu₂⟩ := h₁
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✝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
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✝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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
⊢ v₁ + v₂ ⟂ᵥ w

tactic: obtain ⟨v, hmv, hv₁, hv₂⟩ := h₂
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✝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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
⊢ v₁ + v₂ ⟂ᵥ w
state_after: case intro.intro.intro.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✝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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑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_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
⊢ v₁ + v₂ ⟂ᵥ w
state_after: case intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ u ∩ v
x✝ : MeasurableSet t
⊢ ↑(v₁ + v₂) t = 0

case intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ (u ∩ v)ᶜ
hmt : MeasurableSet t
⊢ ↑w t = 0

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_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ u ∩ v
x✝ : MeasurableSet t
⊢ ↑(v₁ + v₂) t = 0
state_after: no goals

tactic: rw [Set.compl_inter] at ht
state_before: case intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ (u ∩ v)ᶜ
hmt : MeasurableSet t
⊢ ↑w t = 0
state_after: case intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
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_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ ↑w t = 0
state_after: case intro.intro.intro.intro.intro.intro.refine_2.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ vᶜ \ uᶜ ∩ t ⊆ vᶜ

case intro.intro.intro.intro.intro.intro.refine_2.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ uᶜ ∩ t ⊆ uᶜ

α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ Disjoint (uᶜ ∩ t) (vᶜ \ uᶜ ∩ t)

α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ t = uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t

tactic: exact Set.Subset.trans Set.inter_subset_left diff_subset
state_before: case intro.intro.intro.intro.intro.intro.refine_2.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ vᶜ \ uᶜ ∩ t ⊆ vᶜ
state_after: no goals

tactic: exact Set.inter_subset_left
state_before: case intro.intro.intro.intro.intro.intro.refine_2.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ uᶜ ∩ t ⊆ uᶜ
state_after: no goals

tactic: exact disjoint_sdiff_self_right.mono Set.inter_subset_left Set.inter_subset_left
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✝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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ Disjoint (uᶜ ∩ t) (vᶜ \ uᶜ ∩ t)
state_after: no goals

tactic: apply Set.Subset.antisymm <;> intro x hx
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✝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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, ↑w t = 0
t : Set α
ht : t ⊆ uᶜ ∪ vᶜ
hmt : MeasurableSet t
⊢ t = uᶜ ∩ t ∪ vᶜ \ uᶜ ∩ t
state_after: case h₁
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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

case h₂
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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

tactic: by_cases hxu' : x ∈ uᶜ
state_before: case h₁
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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
state_after: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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: rcases ht hx with (hxu | hxv)
state_before: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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
state_after: case neg.inl
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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: exacts [False.elim (hxu' hxu), Or.inr ⟨⟨hxv, hxu'⟩, hx⟩]
state_before: case neg.inl
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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
state_after: no goals

tactic: exact Or.inl ⟨hxu', hx⟩
state_before: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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
state_after: no goals

tactic: cases' hx with hx hx <;> exact hx.2
state_before: case h₂
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
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 ⊆ u, ↑v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, ↑w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, ↑v₂ t = 0
hv₂ : ∀ 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
state_after: no goals