Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0ltat Unicode version

Theorem 0ltat 30103
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 443 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  K  e.  OP )
2 eqid 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 0ltat.z . . . 4  |-  .0.  =  ( 0. `  K )
42, 3op0cl 29996 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
54adantr 451 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  e.  ( Base `  K ) )
6 0ltat.a . . . 4  |-  A  =  ( Atoms `  K )
72, 6atbase 30101 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
87adantl 452 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  P  e.  ( Base `  K ) )
9 eqid 2296 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
103, 9, 6atcvr0 30100 . 2  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
11 0ltat.s . . 3  |-  .<  =  ( lt `  K )
122, 11, 9cvrlt 30082 . 2  |-  ( ( ( K  e.  OP  /\  .0.  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K ) P )  ->  .0.  .<  P )
131, 5, 8, 10, 12syl31anc 1185 1  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  .<  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   ltcplt 14091   0.cp0 14159   OPcops 29984    <o ccvr 30074   Atomscatm 30075
This theorem is referenced by:  2atm2atN  30596  dia2dimlem2  31877  dia2dimlem3  31878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oposet 29988  df-covers 30078  df-ats 30079
  Copyright terms: Public domain W3C validator