Theorem latjcl

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

	Ref	Expression
Hypotheses	latjcl.b	|- B = ( Base ` K )
	latjcl.j	|- .\/ = ( join ` K )
Assertion	latjcl	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )

Proof

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