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Theorem ocel 22784
Description: Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
ocel  |-  ( H 
C_  ~H  ->  ( A  e.  ( _|_ `  H
)  <->  ( A  e. 
~H  /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
Distinct variable groups:    x, H    x, A

Proof of Theorem ocel
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ocval 22783 . . 3  |-  ( H 
C_  ~H  ->  ( _|_ `  H )  =  {
y  e.  ~H  |  A. x  e.  H  ( y  .ih  x
)  =  0 } )
21eleq2d 2504 . 2  |-  ( H 
C_  ~H  ->  ( A  e.  ( _|_ `  H
)  <->  A  e.  { y  e.  ~H  |  A. x  e.  H  (
y  .ih  x )  =  0 } ) )
3 oveq1 6089 . . . . 5  |-  ( y  =  A  ->  (
y  .ih  x )  =  ( A  .ih  x ) )
43eqeq1d 2445 . . . 4  |-  ( y  =  A  ->  (
( y  .ih  x
)  =  0  <->  ( A  .ih  x )  =  0 ) )
54ralbidv 2726 . . 3  |-  ( y  =  A  ->  ( A. x  e.  H  ( y  .ih  x
)  =  0  <->  A. x  e.  H  ( A  .ih  x )  =  0 ) )
65elrab 3093 . 2  |-  ( A  e.  { y  e. 
~H  |  A. x  e.  H  ( y  .ih  x )  =  0 }  <->  ( A  e. 
~H  /\  A. x  e.  H  ( A  .ih  x )  =  0 ) )
72, 6syl6bb 254 1  |-  ( H 
C_  ~H  ->  ( A  e.  ( _|_ `  H
)  <->  ( A  e. 
~H  /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   {crab 2710    C_ wss 3321   ` cfv 5455  (class class class)co 6082   0cc0 8991   ~Hchil 22423    .ih csp 22426   _|_cort 22434
This theorem is referenced by:  shocel  22785  ocsh  22786  ocorth  22794  ococss  22796  occllem  22806  occl  22807  chocnul  22831  h1deoi  23052  h1dei  23053  hmopidmpji  23656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-hilex 22503
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oc 22755
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