Description: An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012)
Ref | Expression | ||
---|---|---|---|
Hypotheses | atcvr1.j | |- .\/ = ( join ` K ) |
|
atcvr1.c | |- C = ( |
||
atcvr1.a | |- A = ( Atoms ` K ) |
||
Assertion | atcvr1 | |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atcvr1.j | |- .\/ = ( join ` K ) |
|
2 | atcvr1.c | |- C = ( |
|
3 | atcvr1.a | |- A = ( Atoms ` K ) |
|
4 | hlomcmcv | |- ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. CvLat ) ) |
|
5 | 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 ) ) ) |
6 | 4 5 | syl3an1 | |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) ) |