Metamath Proof Explorer


Theorem hlexchb2

Description: A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012)

Ref Expression
Hypotheses hlsuprexch.b
|- B = ( Base ` K )
hlsuprexch.l
|- .<_ = ( le ` K )
hlsuprexch.j
|- .\/ = ( join ` K )
hlsuprexch.a
|- A = ( Atoms ` K )
Assertion hlexchb2
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( Q .\/ X ) <-> ( P .\/ X ) = ( Q .\/ X ) ) )

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 cvlexchb2
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( Q .\/ X ) <-> ( P .\/ X ) = ( Q .\/ X ) ) )
7 5 6 syl3an1
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( Q .\/ X ) <-> ( P .\/ X ) = ( Q .\/ X ) ) )