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Theorem shocel 24806
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 24733 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
2 ocel 24805 . 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 369    = wceq 1370    e. wcel 1757   A.wral 2792    C_ wss 3412   ` cfv 5502  (class class class)co 6176   0cc0 9369   ~Hchil 24442    .ih csp 24445   SHcsh 24451   _|_cort 24453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615  ax-hilex 24522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fv 5510  df-ov 6179  df-sh 24730  df-oc 24776
This theorem is referenced by:  ocin  24820  choc0  24850  choc1  24851  pjhthlem2  24916  pjclem4  25724  pj3si  25732
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