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Theorem riotaocN 29468
Description: The orthocomplement of the unique poset element such that 
ps. (riotaneg 9819 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 2494 . . 3  |-  F/_ y  ._|_
2 nfriota1 6399 . . 3  |-  F/_ y
( iota_ y  e.  B ps )
31, 2nffv 5615 . 2  |-  F/_ y
(  ._|_  `  ( iota_ y  e.  B ps ) )
4 riotaoc.b . . 3  |-  B  =  ( Base `  K
)
5 riotaoc.o . . 3  |-  ._|_  =  ( oc `  K )
64, 5opoccl 29453 . 2  |-  ( ( K  e.  OP  /\  y  e.  B )  ->  (  ._|_  `  y )  e.  B )
74, 5opoccl 29453 . 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 5608 . 2  |-  ( y  =  ( iota_ y  e.  B ps )  -> 
(  ._|_  `  y )  =  (  ._|_  `  ( iota_ y  e.  B ps ) ) )
104, 5opoccl 29453 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  (  ._|_  `  x )  e.  B )
114, 5opcon2b 29456 . . 3  |-  ( ( K  e.  OP  /\  x  e.  B  /\  y  e.  B )  ->  ( x  =  ( 
._|_  `  y )  <->  y  =  (  ._|_  `  x )
) )
1210, 11reuhypd 4643 . 2  |-  ( ( K  e.  OP  /\  x  e.  B )  ->  E! y  e.  B  x  =  (  ._|_  `  y ) )
133, 6, 7, 8, 9, 12riotaxfrd 6423 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 1642    e. wcel 1710   E!wreu 2621   ` cfv 5337   iota_crio 6384   Basecbs 13245   occoc 13313   OPcops 29431
This theorem is referenced by:  glbconN  29635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948  df-riota 6391  df-oposet 29435
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