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Theorem riotaocN 29692
Description: The orthocomplement of the unique poset element such that 
ps. (riotaneg 9939 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 2540 . . 3  |-  F/_ y  ._|_
2 nfriota1 6516 . . 3  |-  F/_ y
( iota_ y  e.  B ps )
31, 2nffv 5694 . 2  |-  F/_ y
(  ._|_  `  ( iota_ y  e.  B ps ) )
4 riotaoc.b . . 3  |-  B  =  ( Base `  K
)
5 riotaoc.o . . 3  |-  ._|_  =  ( oc `  K )
64, 5opoccl 29677 . 2  |-  ( ( K  e.  OP  /\  y  e.  B )  ->  (  ._|_  `  y )  e.  B )
74, 5opoccl 29677 . 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 5687 . 2  |-  ( y  =  ( iota_ y  e.  B ps )  -> 
(  ._|_  `  y )  =  (  ._|_  `  ( iota_ y  e.  B ps ) ) )
104, 5opoccl 29677 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  (  ._|_  `  x )  e.  B )
114, 5opcon2b 29680 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B  /\  y  e.  B )  ->  ( x  =  ( 
._|_  `  y )  <->  y  =  (  ._|_  `  x )
) )
1210, 11reuhypd 4709 . 2  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  E! y  e.  B  x  =  (  ._|_  `  y ) )
133, 6, 7, 8, 9, 12riotaxfrd 6540 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   E!wreu 2668   ` cfv 5413   iota_crio 6501   Basecbs 13424   occoc 13492   OPcops 29655
This theorem is referenced by:  glbconN  29859
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-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  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-riota 6508  df-oposet 29659
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