Theorem hlatjcl

Description: Closure of join operation. Frequently-used special case of latjcl for atoms. (Contributed by NM, 15-Jun-2012)

	Ref	Expression
Hypotheses	hlatjcl.b	|- B = ( Base ` K )
	hlatjcl.j	|- .\/ = ( join ` K )
	hlatjcl.a	|- A = ( Atoms ` K )
Assertion	hlatjcl	|- ( ( K e. HL /\ X e. A /\ Y e. A ) -> ( X .\/ Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		hlatjcl.b	|- B = ( Base ` K )
2		hlatjcl.j	|- .\/ = ( join ` K )
3		hlatjcl.a	|- A = ( Atoms ` K )
4		hllat	|- ( K e. HL -> K e. Lat )
5	1 3	atbase	|- ( X e. A -> X e. B )
6	1 3	atbase	|- ( Y e. A -> Y e. B )
7	1 2	latjcl	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
8	4 5 6 7	syl3an	|- ( ( K e. HL /\ X e. A /\ Y e. A ) -> ( X .\/ Y ) e. B )