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Theorem op1cl 29997
Description: An orthoposet has a unit element. (helch 21839 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 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 eqid 2296 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 op1cl.u . . 3  |-  .1.  =  ( 1. `  K )
81, 2, 3, 4, 5, 6, 7isopos 29992 . 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 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:  ople1  30003  op1le  30004  glb0N  30005  opoc1  30014  opoc0  30015  olm11  30039  olm12  30040  ncvr1  30084  hlhgt2  30200  hl0lt1N  30201  hl2at  30216  athgt  30267  1cvrco  30283  1cvrjat  30286  pmap1N  30578  pol1N  30721  lhp2lt  30812  lhpexnle  30817  dih1  32098  dih1rn  32099  dih1cnv  32100  dihglb2  32154  dochocss  32178  dihjatc  32229
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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