Theorem latlem12

Description: An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011)

	Ref	Expression
Hypotheses	latmle.b	|- B = ( Base ` K )
	latmle.l	|- .<_ = ( le ` K )
	latmle.m	|- ./\ = ( meet ` K )
Assertion	latlem12	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .<_ Y /\ X .<_ Z ) <-> X .<_ ( Y ./\ Z ) ) )

Proof

Step	Hyp	Ref	Expression
1		latmle.b	|- B = ( Base ` K )
2		latmle.l	|- .<_ = ( le ` K )
3		latmle.m	|- ./\ = ( meet ` K )
4		latpos	|- ( K e. Lat -> K e. Poset )
5	4	adantr	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Poset )
6		simpr2	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
7		simpr3	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
8		simpr1	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
9		eqid	|- ( join ` K ) = ( join ` K )
10		simpl	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
11	1 9 3 10 6 7	latcl2	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( <. Y , Z >. e. dom ( join ` K ) /\ <. Y , Z >. e. dom ./\ ) )
12	11	simprd	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> <. Y , Z >. e. dom ./\ )
13	1 2 3 5 6 7 8 12	meetle	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .<_ Y /\ X .<_ Z ) <-> X .<_ ( Y ./\ Z ) ) )