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Theorem op0cl 29667
Description: An orthoposet has a zero element. (h0elch 22710 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 2404 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2404 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
4 eqid 2404 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2404 . . 3  |-  ( meet `  K )  =  (
meet `  K )
6 op0cl.z . . 3  |-  .0.  =  ( 0. `  K )
7 eqid 2404 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7isopos 29663 . 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 961 . 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 188 1  |-  ( K  e.  OP  ->  .0.  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   occoc 13492   Posetcpo 14352   joincjn 14356   meetcmee 14357   0.cp0 14421   1.cp1 14422   OPcops 29655
This theorem is referenced by:  op0le  29669  ople0  29670  lub0N  29672  opltn0  29673  opoc1  29685  opoc0  29686  olj01  29708  olj02  29709  olm01  29719  olm02  29720  0ltat  29774  leatb  29775  hlhgt2  29871  hl0lt1N  29872  hl2at  29887  atcvr0eq  29908  lnnat  29909  atle  29918  athgt  29938  1cvratex  29955  ps-2  29960  dalemcea  30142  pmapeq0  30248  2atm2atN  30267  lhp0lt  30485  lhpn0  30486  ltrnatb  30619  ltrnmw  30633  cdleme3c  30712  cdleme7e  30729  dia0eldmN  31523  dia2dimlem2  31548  dia2dimlem3  31549  dib0  31647  dih0  31763  dih0bN  31764  dih0rn  31767  dihlspsnssN  31815  dihlspsnat  31816  dihatexv  31821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-oposet 29659
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