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

Theorem opltn0 30050
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 445 . . 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 30044 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
54adantr 453 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
6 simpr 449 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
7 eqid 2438 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
8 opltne0.s . . . 4  |-  .<  =  ( lt `  K )
97, 8pltval 14419 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
101, 5, 6, 9syl3anc 1185 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
11 necom 2687 . . 3  |-  ( X  =/=  .0.  <->  .0.  =/=  X )
122, 7, 3op0le 30046 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
1312biantrurd 496 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  =/=  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
1411, 13syl5rbb 251 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( (  .0.  ( le `  K ) X  /\  .0.  =/=  X
)  <->  X  =/=  .0.  ) )
1510, 14bitrd 246 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456   Basecbs 13471   lecple 13538   ltcplt 14400   0.cp0 14468   OPcops 30032
This theorem is referenced by:  atle  30295  dalemcea  30519  2atm2atN  30644  dia2dimlem2  31925  dia2dimlem3  31926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-undef 6545  df-riota 6551  df-plt 14417  df-glb 14434  df-p0 14470  df-oposet 30036
  Copyright terms: Public domain W3C validator