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Theorem opltn0 32221
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 32215 . . . 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 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
8 opltne0.s . . . 4  |-  .<  =  ( lt `  K )
97, 8pltval 15916 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
101, 5, 6, 9syl3anc 1232 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
11 necom 2674 . . 3  |-  ( X  =/=  .0.  <->  .0.  =/=  X )
122, 7, 3op0le 32217 . . . 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 260 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( (  .0.  ( le `  K ) X  /\  .0.  =/=  X
)  <->  X  =/=  .0.  ) )
1510, 14bitrd 255 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   class class class wbr 4397   ` cfv 5571   Basecbs 14843   lecple 14918   ltcplt 15896   0.cp0 15993   OPcops 32203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-plt 15914  df-glb 15931  df-p0 15995  df-oposet 32207
This theorem is referenced by:  atle  32466  dalemcea  32690  2atm2atN  32815  dia2dimlem2  34098  dia2dimlem3  34099
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