Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opltn0 Unicode version

Theorem opltn0 30002
Description: A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
opltne0.b  |-  B  =  ( Base `  K
)
opltne0.s  |-  .<  =  ( lt `  K )
opltne0.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
opltn0  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )

Proof of Theorem opltn0
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  OP )
2 opltne0.b . . . . 5  |-  B  =  ( Base `  K
)
3 opltne0.z . . . . 5  |-  .0.  =  ( 0. `  K )
42, 3op0cl 29996 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
54adantr 451 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
6 simpr 447 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
7 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
8 opltne0.s . . . 4  |-  .<  =  ( lt `  K )
97, 8pltval 14110 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
101, 5, 6, 9syl3anc 1182 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
11 necom 2540 . . 3  |-  ( X  =/=  .0.  <->  .0.  =/=  X )
122, 7, 3op0le 29998 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
1312biantrurd 494 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  =/=  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
1411, 13syl5rbb 249 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( (  .0.  ( le `  K ) X  /\  .0.  =/=  X
)  <->  X  =/=  .0.  ) )
1510, 14bitrd 244 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   ltcplt 14091   0.cp0 14159   OPcops 29984
This theorem is referenced by:  atle  30247  dalemcea  30471  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-rep 4147  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-plt 14108  df-glb 14125  df-p0 14161  df-oposet 29988
  Copyright terms: Public domain W3C validator