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Theorem opltn0 33864
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 457 . . 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 33858 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
54adantr 465 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
6 simpr 461 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
7 eqid 2462 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
8 opltne0.s . . . 4  |-  .<  =  ( lt `  K )
97, 8pltval 15438 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
101, 5, 6, 9syl3anc 1223 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
11 necom 2731 . . 3  |-  ( X  =/=  .0.  <->  .0.  =/=  X )
122, 7, 3op0le 33860 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
1312biantrurd 508 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  =/=  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
1411, 13syl5rbb 258 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( (  .0.  ( le `  K ) X  /\  .0.  =/=  X
)  <->  X  =/=  .0.  ) )
1510, 14bitrd 253 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581   Basecbs 14481   lecple 14553   ltcplt 15419   0.cp0 15515   OPcops 33846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-plt 15436  df-glb 15453  df-p0 15517  df-oposet 33850
This theorem is referenced by:  atle  34109  dalemcea  34333  2atm2atN  34458  dia2dimlem2  35739  dia2dimlem3  35740
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