Metamath Proof Explorer


Theorem hlatexch2

Description: Atom exchange property. (Contributed by NM, 8-Jan-2012)

Ref Expression
Hypotheses hlatexchb.l
|- .<_ = ( le ` K )
hlatexchb.j
|- .\/ = ( join ` K )
hlatexchb.a
|- A = ( Atoms ` K )
Assertion hlatexch2
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) -> Q .<_ ( P .\/ R ) ) )

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 cvlatexch2
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) -> Q .<_ ( P .\/ R ) ) )
6 4 5 syl3an1
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) -> Q .<_ ( P .\/ R ) ) )