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Theorem op0le 29194
Description: Orthoposet zero is less than or equal to any element. (ch0le 22075 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 29192 . . 3  |-  ( K  e.  OP  ->  .0.  e.  B )
43adantr 451 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
5 opposet 29190 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
6 op0le.l . . . . 5  |-  .<_  =  ( le `  K )
71, 6, 2p0le 14198 . . . 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 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292   Basecbs 13195   lecple 13262   Posetcpo 14123   0.cp0 14192   OPcops 29180
This theorem is referenced by:  ople0  29195  opnlen0  29196  lub0N  29197  opltn0  29198  olj01  29233  olm01  29244  leatb  29300  1cvratex  29480  llnn0  29523  lplnn0N  29554  lvoln0N  29598  dalemcea  29667  ltrnatb  30144  ltrnmw  30158  tendo0tp  30796  cdlemk39s-id  30947  dia0eldmN  31048  dib0  31172  dih0  31288  dihmeetlem18N  31332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-undef 6340  df-riota 6346  df-glb 14158  df-p0 14194  df-oposet 29184
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