Theorem hlexchb1

Description: A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011)

	Ref	Expression
Hypotheses	hlsuprexch.b	|- B = ( Base ` K )
	hlsuprexch.l	|- .<_ = ( le ` K )
	hlsuprexch.j	|- .\/ = ( join ` K )
	hlsuprexch.a	|- A = ( Atoms ` K )
Assertion	hlexchb1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlsuprexch.b	|- B = ( Base ` K )
2		hlsuprexch.l	|- .<_ = ( le ` K )
3		hlsuprexch.j	|- .\/ = ( join ` K )
4		hlsuprexch.a	|- A = ( Atoms ` K )
5		hlcvl	|- ( K e. HL -> K e. CvLat )
6	1 2 3 4	cvlexchb1	|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) )
7	5 6	syl3an1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) )