Metamath Proof Explorer

Theorem cvlatexch2

Description: Atom exchange property. (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlatexch.l 
|- .<_ = ( le ` K )
cvlatexch.j 
|- .\/ = ( join ` K )
cvlatexch.a 
|- A = ( Atoms ` K )
Assertion cvlatexch2 
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) -> Q .<_ ( P .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 cvlatexch.l 
 |-  .<_ = ( le ` K )
2 cvlatexch.j 
 |-  .\/ = ( join ` K )
3 cvlatexch.a 
 |-  A = ( Atoms ` K )
4 1 2 3 cvlatexch1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( R .\/ Q ) -> Q .<_ ( R .\/ P ) ) )
5 cvllat 
 |-  ( K e. CvLat -> K e. Lat )
6 5 3ad2ant1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> K e. Lat )
7 simp22 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> Q e. A )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
10 7 9 syl 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> Q e. ( Base ` K ) )
11 simp23 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> R e. A )
12 8 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
13 11 12 syl 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> R e. ( Base ` K ) )
14 8 2 latjcom 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
15 6 10 13 14 syl3anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( Q .\/ R ) = ( R .\/ Q ) )
16 15 breq2d 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) <-> P .<_ ( R .\/ Q ) ) )
17 simp21 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> P e. A )
18 8 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
19 17 18 syl 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> P e. ( Base ` K ) )
20 8 2 latjcom 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ R ) = ( R .\/ P ) )
21 6 19 13 20 syl3anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .\/ R ) = ( R .\/ P ) )
22 21 breq2d 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( Q .<_ ( P .\/ R ) <-> Q .<_ ( R .\/ P ) ) )
23 4 16 22 3imtr4d 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) -> Q .<_ ( P .\/ R ) ) )