Theorem hlatexch1

Description: Atom exchange property. (Contributed by NM, 7-Jan-2012)

	Ref	Expression
Hypotheses	hlatexchb.l	|- .<_ = ( le ` K )
	hlatexchb.j	|- .\/ = ( join ` K )
	hlatexchb.a	|- A = ( Atoms ` K )
Assertion	hlatexch1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( R .\/ Q ) -> Q .<_ ( R .\/ P ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlatexchb.l	|- .<_ = ( le ` K )
2		hlatexchb.j	|- .\/ = ( join ` K )
3		hlatexchb.a	|- A = ( Atoms ` K )
4		hlcvl	|- ( K e. HL -> K e. CvLat )
5	1 2 3	cvlatexch1	|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( R .\/ Q ) -> Q .<_ ( R .\/ P ) ) )
6	4 5	syl3an1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( R .\/ Q ) -> Q .<_ ( R .\/ P ) ) )