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Theorem op0le 29984
Description: Orthoposet zero is less than or equal to any element. (ch0le 22943 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 29982 . . 3  |-  ( K  e.  OP  ->  .0.  e.  B )
43adantr 452 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
5 opposet 29980 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
6 op0le.l . . . . 5  |-  .<_  =  ( le `  K )
71, 6, 2p0le 14472 . . . 4  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
85, 7syl3an1 1217 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
983com23 1159 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  .<_  X )
104, 9mpd3an3 1280 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397   0.cp0 14466   OPcops 29970
This theorem is referenced by:  ople0  29985  opnlen0  29986  lub0N  29987  opltn0  29988  olj01  30023  olm01  30034  leatb  30090  1cvratex  30270  llnn0  30313  lplnn0N  30344  lvoln0N  30388  dalemcea  30457  ltrnatb  30934  ltrnmw  30948  tendo0tp  31586  cdlemk39s-id  31737  dia0eldmN  31838  dib0  31962  dih0  32078  dihmeetlem18N  32122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-undef 6543  df-riota 6549  df-glb 14432  df-p0 14468  df-oposet 29974
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