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Theorem 0ltat 30090
Description: An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
Hypotheses
Ref Expression
0ltat.z  |-  .0.  =  ( 0. `  K )
0ltat.s  |-  .<  =  ( lt `  K )
0ltat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
0ltat  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  .<  P )

Proof of Theorem 0ltat
StepHypRef Expression
1 simpl 445 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  K  e.  OP )
2 eqid 2437 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 0ltat.z . . . 4  |-  .0.  =  ( 0. `  K )
42, 3op0cl 29983 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
54adantr 453 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  e.  ( Base `  K ) )
6 0ltat.a . . . 4  |-  A  =  ( Atoms `  K )
72, 6atbase 30088 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
87adantl 454 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  P  e.  ( Base `  K ) )
9 eqid 2437 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
103, 9, 6atcvr0 30087 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
11 0ltat.s . . 3  |-  .<  =  ( lt `  K )
122, 11, 9cvrlt 30069 . 2  |-  ( ( ( K  e.  OP  /\  .0.  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K ) P )  ->  .0.  .<  P )
131, 5, 8, 10, 12syl31anc 1188 1  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  .<  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455   Basecbs 13470   ltcplt 14399   0.cp0 14467   OPcops 29971    <o ccvr 30061   Atomscatm 30062
This theorem is referenced by:  2atm2atN  30583  dia2dimlem2  31864  dia2dimlem3  31865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oposet 29975  df-covers 30065  df-ats 30066
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