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

tactic: constructor
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
⊢ v.restrict univ ≤ w.restrict univ ↔ v ≤ w
state_after: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
⊢ v.restrict univ ≤ w.restrict univ → v ≤ w

case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
⊢ v ≤ w → v.restrict univ ≤ w.restrict univ

tactic: intro h s hs
state_before: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
⊢ v.restrict univ ≤ w.restrict univ → v ≤ w
state_after: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v.restrict univ ≤ w.restrict univ
s : Set α
hs : MeasurableSet s
⊢ ↑v s ≤ ↑w s

tactic: have := h s hs
state_before: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v.restrict univ ≤ w.restrict univ
s : Set α
hs : MeasurableSet s
⊢ ↑v s ≤ ↑w s
state_after: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v.restrict univ ≤ w.restrict univ
s : Set α
hs : MeasurableSet s
this : ↑(v.restrict univ) s ≤ ↑(w.restrict univ) s
⊢ ↑v s ≤ ↑w s

tactic: rwa [restrict_apply _ MeasurableSet.univ hs, Set.inter_univ,
  restrict_apply _ MeasurableSet.univ hs, Set.inter_univ] at this
state_before: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v.restrict univ ≤ w.restrict univ
s : Set α
hs : MeasurableSet s
this : ↑(v.restrict univ) s ≤ ↑(w.restrict univ) s
⊢ ↑v s ≤ ↑w s
state_after: no goals

tactic: intro h s hs
state_before: case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
⊢ v ≤ w → v.restrict univ ≤ w.restrict univ
state_after: case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v ≤ w
s : Set α
hs : MeasurableSet s
⊢ ↑(v.restrict univ) s ≤ ↑(w.restrict univ) s

tactic: rw [restrict_apply _ MeasurableSet.univ hs, Set.inter_univ,
  restrict_apply _ MeasurableSet.univ hs, Set.inter_univ]
state_before: case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v ≤ w
s : Set α
hs : MeasurableSet s
⊢ ↑(v.restrict univ) s ≤ ↑(w.restrict univ) s
state_after: case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v ≤ w
s : Set α
hs : MeasurableSet s
⊢ ↑v s ≤ ↑w s

tactic: exact h s hs
state_before: case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v w : VectorMeasure α M
h : v ≤ w
s : Set α
hs : MeasurableSet s
⊢ ↑v s ≤ ↑w s
state_after: no goals