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Theorem riotaocN 32851
Description: The orthocomplement of the unique poset element such that 
ps. (riotaneg 10303 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 2577 . . 3  |-  F/_ y  ._|_
2 nfriota1 6057 . . 3  |-  F/_ y
( iota_ y  e.  B  ps )
31, 2nffv 5696 . 2  |-  F/_ y
(  ._|_  `  ( iota_ y  e.  B  ps ) )
4 riotaoc.b . . 3  |-  B  =  ( Base `  K
)
5 riotaoc.o . . 3  |-  ._|_  =  ( oc `  K )
64, 5opoccl 32836 . 2  |-  ( ( K  e.  OP  /\  y  e.  B )  ->  (  ._|_  `  y )  e.  B )
74, 5opoccl 32836 . 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 5689 . 2  |-  ( y  =  ( iota_ y  e.  B  ps )  -> 
(  ._|_  `  y )  =  (  ._|_  `  ( iota_ y  e.  B  ps ) ) )
104, 5opoccl 32836 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  (  ._|_  `  x )  e.  B )
114, 5opcon2b 32839 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B  /\  y  e.  B )  ->  ( x  =  ( 
._|_  `  y )  <->  y  =  (  ._|_  `  x )
) )
1210, 11reuhypd 4517 . 2  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  E! y  e.  B  x  =  (  ._|_  `  y ) )
133, 6, 7, 8, 9, 12riotaxfrd 6081 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   E!wreu 2715   ` cfv 5416   iota_crio 6049   Basecbs 14172   occoc 14244   OPcops 32814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-nul 4419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-dm 4848  df-iota 5379  df-fv 5424  df-riota 6050  df-ov 6092  df-oposet 32818
This theorem is referenced by:  glbconN  33018
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