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Theorem atnle0 29792
Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
atnle0.l  |-  .<_  =  ( le `  K )
atnle0.z  |-  .0.  =  ( 0. `  K )
atnle0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atnle0  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  .0.  )

Proof of Theorem atnle0
StepHypRef Expression
1 atlpos 29784 . . 3  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
21adantr 452 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  K  e.  Poset )
3 eqid 2404 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 atnle0.z . . . 4  |-  .0.  =  ( 0. `  K )
53, 4atl0cl 29786 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  ( Base `  K )
)
65adantr 452 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  e.  ( Base `  K
) )
7 atnle0.a . . . 4  |-  A  =  ( Atoms `  K )
83, 7atbase 29772 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
98adantl 453 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  e.  ( Base `  K
) )
10 eqid 2404 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
114, 10, 7atcvr0 29771 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  (  <o  `  K ) P )
12 atnle0.l . . 3  |-  .<_  =  ( le `  K )
133, 12, 10cvrnle 29763 . 2  |-  ( ( ( K  e.  Poset  /\  .0.  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K ) P )  ->  -.  P  .<_  .0.  )
142, 6, 9, 11, 13syl31anc 1187 1  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   0.cp0 14421    <o ccvr 29745   Atomscatm 29746   AtLatcal 29747
This theorem is referenced by:  pmap0  30247  trlnle  30668  cdlemg27b  31178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-poset 14358  df-plt 14370  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781
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