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Theorem op0cl 29996
Description: An orthoposet has a zero element. (h0elch 21850 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 2296 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2296 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
4 eqid 2296 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2296 . . 3  |-  ( meet `  K )  =  (
meet `  K )
6 op0cl.z . . 3  |-  .0.  =  ( 0. `  K )
7 eqid 2296 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7isopos 29992 . 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 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   Posetcpo 14090   joincjn 14094   meetcmee 14095   0.cp0 14159   1.cp1 14160   OPcops 29984
This theorem is referenced by:  op0le  29998  ople0  29999  lub0N  30001  opltn0  30002  opoc1  30014  opoc0  30015  olj01  30037  olj02  30038  olm01  30048  olm02  30049  0ltat  30103  leatb  30104  hlhgt2  30200  hl0lt1N  30201  hl2at  30216  atcvr0eq  30237  lnnat  30238  atle  30247  athgt  30267  1cvratex  30284  ps-2  30289  dalemcea  30471  pmapeq0  30577  2atm2atN  30596  lhp0lt  30814  lhpn0  30815  ltrnatb  30948  ltrnmw  30962  cdleme3c  31041  cdleme7e  31058  dia0eldmN  31852  dia2dimlem2  31877  dia2dimlem3  31878  dib0  31976  dih0  32092  dih0bN  32093  dih0rn  32096  dihlspsnssN  32144  dihlspsnat  32145  dihatexv  32150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-oposet 29988
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