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Theorem h1deoi 23052
Description: Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)
Hypothesis
Ref Expression
h1deot.1  |-  B  e. 
~H
Assertion
Ref Expression
h1deoi  |-  ( A  e.  ( _|_ `  { B } )  <->  ( A  e.  ~H  /\  ( A 
.ih  B )  =  0 ) )

Proof of Theorem h1deoi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 h1deot.1 . . 3  |-  B  e. 
~H
2 snssi 3943 . . 3  |-  ( B  e.  ~H  ->  { B }  C_  ~H )
3 ocel 22784 . . 3  |-  ( { B }  C_  ~H  ->  ( A  e.  ( _|_ `  { B } )  <->  ( A  e.  ~H  /\  A. x  e.  { B }  ( A  .ih  x )  =  0 ) ) )
41, 2, 3mp2b 10 . 2  |-  ( A  e.  ( _|_ `  { B } )  <->  ( A  e.  ~H  /\  A. x  e.  { B }  ( A  .ih  x )  =  0 ) )
51elexi 2966 . . . 4  |-  B  e. 
_V
6 oveq2 6090 . . . . 5  |-  ( x  =  B  ->  ( A  .ih  x )  =  ( A  .ih  B
) )
76eqeq1d 2445 . . . 4  |-  ( x  =  B  ->  (
( A  .ih  x
)  =  0  <->  ( A  .ih  B )  =  0 ) )
85, 7ralsn 3850 . . 3  |-  ( A. x  e.  { B }  ( A  .ih  x )  =  0  <-> 
( A  .ih  B
)  =  0 )
98anbi2i 677 . 2  |-  ( ( A  e.  ~H  /\  A. x  e.  { B }  ( A  .ih  x )  =  0 )  <->  ( A  e. 
~H  /\  ( A  .ih  B )  =  0 ) )
104, 9bitri 242 1  |-  ( A  e.  ( _|_ `  { B } )  <->  ( A  e.  ~H  /\  ( A 
.ih  B )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706    C_ wss 3321   {csn 3815   ` cfv 5455  (class class class)co 6082   0cc0 8991   ~Hchil 22423    .ih csp 22426   _|_cort 22434
This theorem is referenced by:  h1dei  23053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-hilex 22503
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oc 22755
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