Theorem: MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive
File: Mathlib/MeasureTheory/Decomposition/Jordan.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0
state_after: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u ∆ v) = 0
⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet v

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet u
tactic: rw [restrict_le_restrict_iff] at hsu hsv
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u ∆ v) = 0
⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet v

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet u
state_after: case left
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u \ v) + ↑s (v \ u) = 0
a : 0 ≤ ↑s (u \ v)
b : 0 ≤ ↑s (v \ u)
⊢ ↑s (u \ v) = 0

case right
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u \ v) + ↑s (v \ u) = 0
a : 0 ≤ ↑s (u \ v)
b : 0 ≤ ↑s (v \ u)
⊢ ↑s (v \ u) = 0

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet v

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet u
tactic: on_goal 1 =>
    have a := hsu (hu.diff hv) diff_subset
    have b := hsv (hv.diff hu) diff_subset
    erw [of_union (Set.disjoint_of_subset_left diff_subset disjoint_sdiff_self_right)
        (hu.diff hv) (hv.diff hu)] at hs
    rw [zero_apply] at a b
    constructor
state_before: case left
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u \ v) + ↑s (v \ u) = 0
a : 0 ≤ ↑s (u \ v)
b : 0 ≤ ↑s (v \ u)
⊢ ↑s (u \ v) = 0

case right
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u \ v) + ↑s (v \ u) = 0
a : 0 ≤ ↑s (u \ v)
b : 0 ≤ ↑s (v \ u)
⊢ ↑s (v \ u) = 0

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet v

case hi
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ MeasurableSet u
state_after: no goals
tactic: all_goals first | linarith | assumption