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Theorem ocel 26341
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 26340 . . 3  |-  ( H 
C_  ~H  ->  ( _|_ `  H )  =  {
y  e.  ~H  |  A. x  e.  H  ( y  .ih  x
)  =  0 } )
21eleq2d 2466 . 2  |-  ( H 
C_  ~H  ->  ( A  e.  ( _|_ `  H
)  <->  A  e.  { y  e.  ~H  |  A. x  e.  H  (
y  .ih  x )  =  0 } ) )
3 oveq1 6225 . . . . 5  |-  ( y  =  A  ->  (
y  .ih  x )  =  ( A  .ih  x ) )
43eqeq1d 2398 . . . 4  |-  ( y  =  A  ->  (
( y  .ih  x
)  =  0  <->  ( A  .ih  x )  =  0 ) )
54ralbidv 2835 . . 3  |-  ( y  =  A  ->  ( A. x  e.  H  ( y  .ih  x
)  =  0  <->  A. x  e.  H  ( A  .ih  x )  =  0 ) )
65elrab 3199 . 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 261 1  |-  ( H 
C_  ~H  ->  ( A  e.  ( _|_ `  H
)  <->  ( A  e. 
~H  /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746   {crab 2750    C_ wss 3406   ` cfv 5513  (class class class)co 6218   0cc0 9425   ~Hchil 25978    .ih csp 25981   _|_cort 25989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618  ax-hilex 26058
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-iota 5477  df-fun 5515  df-fv 5521  df-ov 6221  df-oc 26312
This theorem is referenced by:  shocel  26342  ocsh  26343  ocorth  26351  ococss  26353  occllem  26363  occl  26364  chocnul  26388  h1deoi  26609  h1dei  26610  hmopidmpji  27212
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