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Theorem shocel 26401
Description: Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shocel  |-  ( H  e.  SH  ->  ( A  e.  ( _|_ `  H )  <->  ( A  e.  ~H  /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
Distinct variable groups:    x, H    x, A

Proof of Theorem shocel
StepHypRef Expression
1 shss 26328 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
2 ocel 26400 . 2  |-  ( H 
C_  ~H  ->  ( A  e.  ( _|_ `  H
)  <->  ( A  e. 
~H  /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
31, 2syl 16 1  |-  ( H  e.  SH  ->  ( 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 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   ` cfv 5570  (class class class)co 6270   0cc0 9481   ~Hchil 26037    .ih csp 26040   SHcsh 26046   _|_cort 26048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-hilex 26117
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-sh 26325  df-oc 26371
This theorem is referenced by:  ocin  26415  choc0  26445  choc1  26446  pjhthlem2  26511  pjclem4  27319  pj3si  27327
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