Metamath Proof Explorer

Theorem cvlatcvr2

Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlatcvr1.j 
|- .\/ = ( join ` K )
cvlatcvr1.c 
|- C = (
cvlatcvr1.a 
|- A = ( Atoms ` K )
Assertion cvlatcvr2 
|- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( Q .\/ P ) ) )

Proof

Step Hyp Ref Expression
1 cvlatcvr1.j 
 |-  .\/ = ( join ` K )
2 cvlatcvr1.c 
 |-  C = (
3 cvlatcvr1.a 
 |-  A = ( Atoms ` K )
4 1 2 3 cvlatcvr1 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )
5 simp13 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. CvLat )
6 cvllat 
 |-  ( K e. CvLat -> K e. Lat )
7 5 6 syl 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. Lat )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
10 9 3ad2ant2 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> P e. ( Base ` K ) )
11 8 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
12 11 3ad2ant3 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> Q e. ( Base ` K ) )
13 8 1 latjcom 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
14 7 10 12 13 syl3anc 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
15 14 breq2d 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P C ( P .\/ Q ) <-> P C ( Q .\/ P ) ) )
16 4 15 bitrd 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( Q .\/ P ) ) )