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Theorem riotaocN 29399
Description: The orthocomplement of the unique poset element such that 
ps. (riotaneg 9729 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
riotaoc.b  |-  B  =  ( Base `  K
)
riotaoc.o  |-  ._|_  =  ( oc `  K )
riotaoc.a  |-  ( x  =  (  ._|_  `  y
)  ->  ( ph  <->  ps ) )
Assertion
Ref Expression
riotaocN  |-  ( ( K  e.  OP  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  B ph )  =  (  ._|_  `  ( iota_ y  e.  B ps ) ) )
Distinct variable groups:    x, y, B    x, K, y    x,  ._|_ ,
y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem riotaocN
StepHypRef Expression
1 nfcv 2419 . . 3  |-  F/_ y  ._|_
2 nfriota1 6312 . . 3  |-  F/_ y
( iota_ y  e.  B ps )
31, 2nffv 5532 . 2  |-  F/_ y
(  ._|_  `  ( iota_ y  e.  B ps ) )
4 riotaoc.b . . 3  |-  B  =  ( Base `  K
)
5 riotaoc.o . . 3  |-  ._|_  =  ( oc `  K )
64, 5opoccl 29384 . 2  |-  ( ( K  e.  OP  /\  y  e.  B )  ->  (  ._|_  `  y )  e.  B )
74, 5opoccl 29384 . 2  |-  ( ( K  e.  OP  /\  ( iota_ y  e.  B ps )  e.  B
)  ->  (  ._|_  `  ( iota_ y  e.  B ps ) )  e.  B
)
8 riotaoc.a . 2  |-  ( x  =  (  ._|_  `  y
)  ->  ( ph  <->  ps ) )
9 fveq2 5525 . 2  |-  ( y  =  ( iota_ y  e.  B ps )  -> 
(  ._|_  `  y )  =  (  ._|_  `  ( iota_ y  e.  B ps ) ) )
104, 5opoccl 29384 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  (  ._|_  `  x )  e.  B )
114, 5opcon2b 29387 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B  /\  y  e.  B )  ->  ( x  =  ( 
._|_  `  y )  <->  y  =  (  ._|_  `  x )
) )
1210, 11reuhypd 4561 . 2  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  E! y  e.  B  x  =  (  ._|_  `  y ) )
133, 6, 7, 8, 9, 12riotaxfrd 6336 1  |-  ( ( K  e.  OP  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  B ph )  =  (  ._|_  `  ( iota_ y  e.  B ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E!wreu 2545   ` cfv 5255   iota_crio 6297   Basecbs 13148   occoc 13216   OPcops 29362
This theorem is referenced by:  glbconN  29566
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-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-riota 6304  df-oposet 29366
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