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 )