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Theorem opltn0 29673
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 444 . . 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 29667 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
54adantr 452 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
6 simpr 448 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
7 eqid 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
8 opltne0.s . . . 4  |-  .<  =  ( lt `  K )
97, 8pltval 14372 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
101, 5, 6, 9syl3anc 1184 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
11 necom 2648 . . 3  |-  ( X  =/=  .0.  <->  .0.  =/=  X )
122, 7, 3op0le 29669 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
1312biantrurd 495 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  =/=  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
1411, 13syl5rbb 250 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( (  .0.  ( le `  K ) X  /\  .0.  =/=  X
)  <->  X  =/=  .0.  ) )
1510, 14bitrd 245 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   ltcplt 14353   0.cp0 14421   OPcops 29655
This theorem is referenced by:  atle  29918  dalemcea  30142  2atm2atN  30267  dia2dimlem2  31548  dia2dimlem3  31549
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-undef 6502  df-riota 6508  df-plt 14370  df-glb 14387  df-p0 14423  df-oposet 29659
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