Step |
Hyp |
Ref |
Expression |
1 |
|
atnem0.m |
|- ./\ = ( meet ` K ) |
2 |
|
atnem0.z |
|- .0. = ( 0. ` K ) |
3 |
|
atnem0.a |
|- A = ( Atoms ` K ) |
4 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
5 |
4 3
|
atncmp |
|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( -. P ( le ` K ) Q <-> P =/= Q ) ) |
6 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
7 |
6 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
8 |
6 4 1 2 3
|
atnle |
|- ( ( K e. AtLat /\ P e. A /\ Q e. ( Base ` K ) ) -> ( -. P ( le ` K ) Q <-> ( P ./\ Q ) = .0. ) ) |
9 |
7 8
|
syl3an3 |
|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( -. P ( le ` K ) Q <-> ( P ./\ Q ) = .0. ) ) |
10 |
5 9
|
bitr3d |
|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ./\ Q ) = .0. ) ) |