Step |
Hyp |
Ref |
Expression |
1 |
|
pmap0.z |
|- .0. = ( 0. ` K ) |
2 |
|
pmap0.m |
|- M = ( pmap ` K ) |
3 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
4 |
3 1
|
atl0cl |
|- ( K e. AtLat -> .0. e. ( Base ` K ) ) |
5 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
6 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
7 |
3 5 6 2
|
pmapval |
|- ( ( K e. AtLat /\ .0. e. ( Base ` K ) ) -> ( M ` .0. ) = { a e. ( Atoms ` K ) | a ( le ` K ) .0. } ) |
8 |
4 7
|
mpdan |
|- ( K e. AtLat -> ( M ` .0. ) = { a e. ( Atoms ` K ) | a ( le ` K ) .0. } ) |
9 |
5 1 6
|
atnle0 |
|- ( ( K e. AtLat /\ a e. ( Atoms ` K ) ) -> -. a ( le ` K ) .0. ) |
10 |
9
|
nrexdv |
|- ( K e. AtLat -> -. E. a e. ( Atoms ` K ) a ( le ` K ) .0. ) |
11 |
|
rabn0 |
|- ( { a e. ( Atoms ` K ) | a ( le ` K ) .0. } =/= (/) <-> E. a e. ( Atoms ` K ) a ( le ` K ) .0. ) |
12 |
10 11
|
sylnibr |
|- ( K e. AtLat -> -. { a e. ( Atoms ` K ) | a ( le ` K ) .0. } =/= (/) ) |
13 |
|
nne |
|- ( -. { a e. ( Atoms ` K ) | a ( le ` K ) .0. } =/= (/) <-> { a e. ( Atoms ` K ) | a ( le ` K ) .0. } = (/) ) |
14 |
12 13
|
sylib |
|- ( K e. AtLat -> { a e. ( Atoms ` K ) | a ( le ` K ) .0. } = (/) ) |
15 |
8 14
|
eqtrd |
|- ( K e. AtLat -> ( M ` .0. ) = (/) ) |