Theorem latmle1

Description: A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011)

	Ref	Expression
Hypotheses	latmle.b	|- B = ( Base ` K )
	latmle.l	|- .<_ = ( le ` K )
	latmle.m	|- ./\ = ( meet ` K )
Assertion	latmle1	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) .<_ X )

Proof

Step	Hyp	Ref	Expression
1		latmle.b	|- B = ( Base ` K )
2		latmle.l	|- .<_ = ( le ` K )
3		latmle.m	|- ./\ = ( meet ` 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	|- ( join ` K ) = ( join ` K )
8	1 7 3 4 5 6	latcl2	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( <. X , Y >. e. dom ( join ` K ) /\ <. X , Y >. e. dom ./\ ) )
9	8	simprd	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. dom ./\ )
10	1 2 3 4 5 6 9	lemeet1	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) .<_ X )