Metamath Proof Explorer


Theorem latnlej2r

Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012)

Ref Expression
Hypotheses latlej.b
|- B = ( Base ` K )
latlej.l
|- .<_ = ( le ` K )
latlej.j
|- .\/ = ( join ` K )
Assertion latnlej2r
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ -. X .<_ ( Y .\/ Z ) ) -> -. X .<_ Z )

Proof

Step Hyp Ref Expression
1 latlej.b
 |-  B = ( Base ` K )
2 latlej.l
 |-  .<_ = ( le ` K )
3 latlej.j
 |-  .\/ = ( join ` K )
4 1 2 3 latnlej2
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ -. X .<_ ( Y .\/ Z ) ) -> ( -. X .<_ Y /\ -. X .<_ Z ) )
5 4 simprd
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ -. X .<_ ( Y .\/ Z ) ) -> -. X .<_ Z )