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Metamath Proof Explorer

Theorem cvlatcvr1

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 cvlatcvr1 
|- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Proof

Step Hyp Ref Expression
1 cvlatcvr1.j 
 |-  .\/ = ( join ` K )
2 cvlatcvr1.c 
 |-  C = (
3 cvlatcvr1.a 
 |-  A = ( Atoms ` K )
4 simp13 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. CvLat )
5 cvlatl 
 |-  ( K e. CvLat -> K e. AtLat )
6 4 5 syl 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. AtLat )
7 eqid 
 |-  ( meet ` K ) = ( meet ` K )
8 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
9 7 8 3 atnem0 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
10 6 9 syld3an1 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
13 11 1 7 8 2 3 cvlcvrp 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. ( Base ` K ) /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
14 12 13 syl3an2 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
15 10 14 bitrd 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )