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Theorem riotaocN 32484
Description: The orthocomplement of the unique poset element such that 
ps. (riotaneg 10586 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 2591 . . 3  |-  F/_ y  ._|_
2 nfriota1 6274 . . 3  |-  F/_ y
( iota_ y  e.  B  ps )
31, 2nffv 5888 . 2  |-  F/_ y
(  ._|_  `  ( iota_ y  e.  B  ps ) )
4 riotaoc.b . . 3  |-  B  =  ( Base `  K
)
5 riotaoc.o . . 3  |-  ._|_  =  ( oc `  K )
64, 5opoccl 32469 . 2  |-  ( ( K  e.  OP  /\  y  e.  B )  ->  (  ._|_  `  y )  e.  B )
74, 5opoccl 32469 . 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 5881 . 2  |-  ( y  =  ( iota_ y  e.  B  ps )  -> 
(  ._|_  `  y )  =  (  ._|_  `  ( iota_ y  e.  B  ps ) ) )
104, 5opoccl 32469 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  (  ._|_  `  x )  e.  B )
114, 5opcon2b 32472 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B  /\  y  e.  B )  ->  ( x  =  ( 
._|_  `  y )  <->  y  =  (  ._|_  `  x )
) )
1210, 11reuhypd 4649 . 2  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  E! y  e.  B  x  =  (  ._|_  `  y ) )
133, 6, 7, 8, 9, 12riotaxfrd 6297 1  |-  ( ( K  e.  OP  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  B  ph )  =  (  ._|_  `  ( iota_ y  e.  B  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   E!wreu 2784   ` cfv 5601   iota_crio 6266   Basecbs 15084   occoc 15160   OPcops 32447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-dm 4864  df-iota 5565  df-fv 5609  df-riota 6267  df-ov 6308  df-oposet 32451
This theorem is referenced by:  glbconN  32651
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