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Theorem op0cl 29443
Description: An orthoposet has a zero element. (h0elch 21948 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 2358 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2358 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
4 eqid 2358 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2358 . . 3  |-  ( meet `  K )  =  (
meet `  K )
6 op0cl.z . . 3  |-  .0.  =  ( 0. `  K )
7 eqid 2358 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7isopos 29439 . 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 1642    e. wcel 1710   A.wral 2619   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   lecple 13312   occoc 13313   Posetcpo 14173   joincjn 14177   meetcmee 14178   0.cp0 14242   1.cp1 14243   OPcops 29431
This theorem is referenced by:  op0le  29445  ople0  29446  lub0N  29448  opltn0  29449  opoc1  29461  opoc0  29462  olj01  29484  olj02  29485  olm01  29495  olm02  29496  0ltat  29550  leatb  29551  hlhgt2  29647  hl0lt1N  29648  hl2at  29663  atcvr0eq  29684  lnnat  29685  atle  29694  athgt  29714  1cvratex  29731  ps-2  29736  dalemcea  29918  pmapeq0  30024  2atm2atN  30043  lhp0lt  30261  lhpn0  30262  ltrnatb  30395  ltrnmw  30409  cdleme3c  30488  cdleme7e  30505  dia0eldmN  31299  dia2dimlem2  31324  dia2dimlem3  31325  dib0  31423  dih0  31539  dih0bN  31540  dih0rn  31543  dihlspsnssN  31591  dihlspsnat  31592  dihatexv  31597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948  df-oposet 29435
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