Theorem latmcl

Description: Closure of meet operation in a lattice. ( incom analog.) (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	latmcl.b	|- B = ( Base ` K )
	latmcl.m	|- ./\ = ( meet ` K )
Assertion	latmcl	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		latmcl.b	|- B = ( Base ` K )
2		latmcl.m	|- ./\ = ( meet ` K )
3		eqid	|- ( join ` K ) = ( join ` K )
4	1 3 2	latlem	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X ( join ` K ) Y ) e. B /\ ( X ./\ Y ) e. B ) )
5	4	simprd	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )