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Theorem op0le 29376
Description: Orthoposet zero is less than or equal to any element. (ch0le 22020 analog.) (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
op0le.b  |-  B  =  ( Base `  K
)
op0le.l  |-  .<_  =  ( le `  K )
op0le.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
op0le  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )

Proof of Theorem op0le
StepHypRef Expression
1 op0le.b . . . 4  |-  B  =  ( Base `  K
)
2 op0le.z . . . 4  |-  .0.  =  ( 0. `  K )
31, 2op0cl 29374 . . 3  |-  ( K  e.  OP  ->  .0.  e.  B )
43adantr 451 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
5 opposet 29372 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
6 op0le.l . . . . 5  |-  .<_  =  ( le `  K )
71, 6, 2p0le 14149 . . . 4  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
85, 7syl3an1 1215 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
983com23 1157 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  .<_  X )
104, 9mpd3an3 1278 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   0.cp0 14143   OPcops 29362
This theorem is referenced by:  ople0  29377  opnlen0  29378  lub0N  29379  opltn0  29380  olj01  29415  olm01  29426  leatb  29482  1cvratex  29662  llnn0  29705  lplnn0N  29736  lvoln0N  29780  dalemcea  29849  ltrnatb  30326  ltrnmw  30340  tendo0tp  30978  cdlemk39s-id  31129  dia0eldmN  31230  dib0  31354  dih0  31470  dihmeetlem18N  31514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-undef 6298  df-riota 6304  df-glb 14109  df-p0 14145  df-oposet 29366
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