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Theorem op1cl 29375
Description: An orthoposet has a unit element. (helch 21823 analog.) (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
op1cl.b  |-  B  =  ( Base `  K
)
op1cl.u  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
op1cl  |-  ( K  e.  OP  ->  .1.  e.  B )

Proof of Theorem op1cl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 op1cl.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 eqid 2283 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 op1cl.u . . 3  |-  .1.  =  ( 1. `  K )
81, 2, 3, 4, 5, 6, 7isopos 29370 . 2  |-  ( K  e.  OP  <->  ( ( K  e.  Poset  /\  ( 0. `  K )  e.  B  /\  .1.  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.  /\  ( x ( meet `  K
) ( ( oc
`  K ) `  x ) )  =  ( 0. `  K
) ) ) )
9 simpl3 960 . 2  |-  ( ( ( K  e.  Poset  /\  ( 0. `  K
)  e.  B  /\  .1.  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.  /\  ( x ( meet `  K
) ( ( oc
`  K ) `  x ) )  =  ( 0. `  K
) ) )  ->  .1.  e.  B )
108, 9sylbi 187 1  |-  ( K  e.  OP  ->  .1.  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:  ople1  29381  op1le  29382  glb0N  29383  opoc1  29392  opoc0  29393  olm11  29417  olm12  29418  ncvr1  29462  hlhgt2  29578  hl0lt1N  29579  hl2at  29594  athgt  29645  1cvrco  29661  1cvrjat  29664  pmap1N  29956  pol1N  30099  lhp2lt  30190  lhpexnle  30195  dih1  31476  dih1rn  31477  dih1cnv  31478  dihglb2  31532  dochocss  31556  dihjatc  31607
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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