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Theorem shocel 22789
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 22717 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
2 ocel 22788 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ` cfv 5457  (class class class)co 6084   0cc0 8995   ~Hchil 22427    .ih csp 22430   SHcsh 22436   _|_cort 22438
This theorem is referenced by:  ocin  22803  choc0  22833  choc1  22834  pjhthlem2  22899  pjclem4  23707  pj3si  23715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-hilex 22507
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-sh 22714  df-oc 22759
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