Theorem hlatjcom

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

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

Proof

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