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Theorem op0cl 29374
Description: An orthoposet has a zero element. (h0elch 21834 analog.) (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
op0cl.b  |-  B  =  ( Base `  K
)
op0cl.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
op0cl  |-  ( K  e.  OP  ->  .0.  e.  B )

Proof of Theorem op0cl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 op0cl.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2283 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
4 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2283 . . 3  |-  ( meet `  K )  =  (
meet `  K )
6 op0cl.z . . 3  |-  .0.  =  ( 0. `  K )
7 eqid 2283 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7isopos 29370 . 2  |-  ( K  e.  OP  <->  ( ( K  e.  Poset  /\  .0.  e.  B  /\  ( 1. `  K )  e.  B )  /\  A. x  e.  B  A. y  e.  B  (
( ( ( oc
`  K ) `  x )  e.  B  /\  ( ( oc `  K ) `  (
( oc `  K
) `  x )
)  =  x  /\  ( x ( le
`  K ) y  ->  ( ( oc
`  K ) `  y ) ( le
`  K ) ( ( oc `  K
) `  x )
) )  /\  (
x ( join `  K
) ( ( oc
`  K ) `  x ) )  =  ( 1. `  K
)  /\  ( x
( meet `  K )
( ( oc `  K ) `  x
) )  =  .0.  ) ) )
9 simpl2 959 . 2  |-  ( ( ( K  e.  Poset  /\  .0.  e.  B  /\  ( 1. `  K )  e.  B )  /\  A. x  e.  B  A. y  e.  B  (
( ( ( oc
`  K ) `  x )  e.  B  /\  ( ( oc `  K ) `  (
( oc `  K
) `  x )
)  =  x  /\  ( x ( le
`  K ) y  ->  ( ( oc
`  K ) `  y ) ( le
`  K ) ( ( oc `  K
) `  x )
) )  /\  (
x ( join `  K
) ( ( oc
`  K ) `  x ) )  =  ( 1. `  K
)  /\  ( x
( meet `  K )
( ( oc `  K ) `  x
) )  =  .0.  ) )  ->  .0.  e.  B )
108, 9sylbi 187 1  |-  ( K  e.  OP  ->  .0.  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   Posetcpo 14074   joincjn 14078   meetcmee 14079   0.cp0 14143   1.cp1 14144   OPcops 29362
This theorem is referenced by:  op0le  29376  ople0  29377  lub0N  29379  opltn0  29380  opoc1  29392  opoc0  29393  olj01  29415  olj02  29416  olm01  29426  olm02  29427  0ltat  29481  leatb  29482  hlhgt2  29578  hl0lt1N  29579  hl2at  29594  atcvr0eq  29615  lnnat  29616  atle  29625  athgt  29645  1cvratex  29662  ps-2  29667  dalemcea  29849  pmapeq0  29955  2atm2atN  29974  lhp0lt  30192  lhpn0  30193  ltrnatb  30326  ltrnmw  30340  cdleme3c  30419  cdleme7e  30436  dia0eldmN  31230  dia2dimlem2  31255  dia2dimlem3  31256  dib0  31354  dih0  31470  dih0bN  31471  dih0rn  31474  dihlspsnssN  31522  dihlspsnat  31523  dihatexv  31528
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-oposet 29366
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