Theorem: MeasureTheory.VectorMeasure.of_diff_of_diff_eq_zero
File: Mathlib/MeasureTheory/Measure/VectorMeasure.lean

tactic: symm
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ ↑v (A \ B) + ↑v B = ↑v A
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ ↑v A = ↑v (A \ B) + ↑v B

tactic: simp only [Set.diff_union_inter]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ ↑v A = ↑v (A \ B ∪ A ∩ B)
state_after: no goals

tactic: rw [of_union]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ ↑v (A \ B ∪ A ∩ B) = ↑v (A \ B) + ↑v (A ∩ B)
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ Disjoint (A \ B) (A ∩ B)

case hA
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (A \ B)

case hB
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

tactic: rw [disjoint_comm]
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ Disjoint (A \ B) (A ∩ B)
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ Disjoint (A ∩ B) (A \ B)

tactic: exact Set.disjoint_of_subset_left A.inter_subset_right disjoint_sdiff_self_right
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ Disjoint (A ∩ B) (A \ B)
state_after: no goals

tactic: exact hA.diff hB
state_before: case hA
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (A \ B)
state_after: no goals

tactic: exact hA.inter hB
state_before: case hB
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)
state_after: no goals

tactic: rw [of_union, h', add_zero]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ ↑v (A \ B) + ↑v (A ∩ B) = ↑v (A \ B) + ↑v (A ∩ B ∪ B \ A)
state_after: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ Disjoint (A ∩ B) (B \ A)

case hA
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

case hB
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (B \ A)

tactic: exact Set.disjoint_of_subset_left A.inter_subset_left disjoint_sdiff_self_right
state_before: case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ Disjoint (A ∩ B) (B \ A)
state_after: no goals

tactic: exact hA.inter hB
state_before: case hA
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)
state_after: no goals

tactic: exact hB.diff hA
state_before: case hB
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ MeasurableSet (B \ A)
state_after: no goals

tactic: rw [Set.union_comm, Set.inter_comm, Set.diff_union_inter]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : ↑v (B \ A) = 0
⊢ ↑v (A \ B) + ↑v (A ∩ B ∪ B \ A) = ↑v (A \ B) + ↑v B
state_after: no goals