Metamath Proof Explorer

Theorem atnle0

Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011)

Ref Expression
Hypotheses atnle0.l 
|- .<_ = ( le ` K )
atnle0.z 
|- .0. = ( 0. ` K )
atnle0.a 
|- A = ( Atoms ` K )
Assertion atnle0 
|- ( ( K e. AtLat /\ P e. A ) -> -. P .<_ .0. )

Proof

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. )