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Theorem ocel 21860
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 21859 . . 3  |-  ( H 
C_  ~H  ->  ( _|_ `  H )  =  {
y  e.  ~H  |  A. x  e.  H  ( y  .ih  x
)  =  0 } )
21eleq2d 2350 . 2  |-  ( H 
C_  ~H  ->  ( A  e.  ( _|_ `  H
)  <->  A  e.  { y  e.  ~H  |  A. x  e.  H  (
y  .ih  x )  =  0 } ) )
3 oveq1 5865 . . . . 5  |-  ( y  =  A  ->  (
y  .ih  x )  =  ( A  .ih  x ) )
43eqeq1d 2291 . . . 4  |-  ( y  =  A  ->  (
( y  .ih  x
)  =  0  <->  ( A  .ih  x )  =  0 ) )
54ralbidv 2563 . . 3  |-  ( y  =  A  ->  ( A. x  e.  H  ( y  .ih  x
)  =  0  <->  A. x  e.  H  ( A  .ih  x )  =  0 ) )
65elrab 2923 . 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 252 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   ` cfv 5255  (class class class)co 5858   0cc0 8737   ~Hchil 21499    .ih csp 21502   _|_cort 21510
This theorem is referenced by:  shocel  21861  ocsh  21862  ocorth  21870  ococss  21872  occllem  21882  occl  21883  chocnul  21907  h1deoi  22128  h1dei  22129  hmopidmpji  22732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oc 21831
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