Theorem latlej2

Description: A join's second argument is less than or equal to the join. ( chub2 analog.) (Contributed by NM, 17-Sep-2011)

	Ref	Expression
Hypotheses	latlej.b	|- B = ( Base ` K )
	latlej.l	|- .<_ = ( le ` K )
	latlej.j	|- .\/ = ( join ` K )
Assertion	latlej2	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> Y .<_ ( X .\/ Y ) )

Proof

Step	Hyp	Ref	Expression
1		latlej.b	|- B = ( Base ` K )
2		latlej.l	|- .<_ = ( le ` K )
3		latlej.j	|- .\/ = ( join ` K )
4		simp1	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> K e. Lat )
5		simp2	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X e. B )
6		simp3	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> Y e. B )
7		eqid	|- ( meet ` K ) = ( meet ` K )
8	1 3 7 4 5 6	latcl2	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( <. X , Y >. e. dom .\/ /\ <. X , Y >. e. dom ( meet ` K ) ) )
9	8	simpld	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. dom .\/ )
10	1 2 3 4 5 6 9	lejoin2	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> Y .<_ ( X .\/ Y ) )