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Theorem 0ltat 29406
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 444 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  K  e.  OP )
2 eqid 2387 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 0ltat.z . . . 4  |-  .0.  =  ( 0. `  K )
42, 3op0cl 29299 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
54adantr 452 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  e.  ( Base `  K ) )
6 0ltat.a . . . 4  |-  A  =  ( Atoms `  K )
72, 6atbase 29404 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
87adantl 453 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  P  e.  ( Base `  K ) )
9 eqid 2387 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
103, 9, 6atcvr0 29403 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
11 0ltat.s . . 3  |-  .<  =  ( lt `  K )
122, 11, 9cvrlt 29385 . 2  |-  ( ( ( K  e.  OP  /\  .0.  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K ) P )  ->  .0.  .<  P )
131, 5, 8, 10, 12syl31anc 1187 1  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  .<  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394   Basecbs 13396   ltcplt 14325   0.cp0 14393   OPcops 29287    <o ccvr 29377   Atomscatm 29378
This theorem is referenced by:  2atm2atN  29899  dia2dimlem2  31180  dia2dimlem3  31181
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oposet 29291  df-covers 29381  df-ats 29382
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