Step |
Hyp |
Ref |
Expression |
1 |
|
atnle0.l |
|- .<_ = ( le ` K ) |
2 |
|
atnle0.z |
|- .0. = ( 0. ` K ) |
3 |
|
atnle0.a |
|- A = ( Atoms ` K ) |
4 |
|
atlpos |
|- ( K e. AtLat -> K e. Poset ) |
5 |
4
|
adantr |
|- ( ( K e. AtLat /\ P e. A ) -> K e. Poset ) |
6 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
7 |
6 2
|
atl0cl |
|- ( K e. AtLat -> .0. e. ( Base ` K ) ) |
8 |
7
|
adantr |
|- ( ( K e. AtLat /\ P e. A ) -> .0. e. ( Base ` K ) ) |
9 |
6 3
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
10 |
9
|
adantl |
|- ( ( K e. AtLat /\ P e. A ) -> P e. ( Base ` K ) ) |
11 |
|
eqid |
|- ( |
12 |
2 11 3
|
atcvr0 |
|- ( ( K e. AtLat /\ P e. A ) -> .0. ( |
13 |
6 1 11
|
cvrnle |
|- ( ( ( K e. Poset /\ .0. e. ( Base ` K ) /\ P e. ( Base ` K ) ) /\ .0. ( -. P .<_ .0. ) |
14 |
5 8 10 12 13
|
syl31anc |
|- ( ( K e. AtLat /\ P e. A ) -> -. P .<_ .0. ) |