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Theorem 0ltat 29928
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 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 0ltat.z . . . 4  |-  .0.  =  ( 0. `  K )
42, 3op0cl 29821 . . 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 29926 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
87adantl 453 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  P  e.  ( Base `  K ) )
9 eqid 2435 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
103, 9, 6atcvr0 29925 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
11 0ltat.s . . 3  |-  .<  =  ( lt `  K )
122, 11, 9cvrlt 29907 . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5445   Basecbs 13457   ltcplt 14386   0.cp0 14454   OPcops 29809    <o ccvr 29899   Atomscatm 29900
This theorem is referenced by:  2atm2atN  30421  dia2dimlem2  31702  dia2dimlem3  31703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-oposet 29813  df-covers 29903  df-ats 29904
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