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Theorem h1deoi 22144
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 3775 . . 3  |-  ( B  e.  ~H  ->  { B }  C_  ~H )
3 ocel 21876 . . 3  |-  ( { B }  C_  ~H  ->  ( A  e.  ( _|_ `  { B } )  <->  ( A  e.  ~H  /\  A. x  e.  { B }  ( A  .ih  x )  =  0 ) ) )
41, 2, 3mp2b 9 . 2  |-  ( A  e.  ( _|_ `  { B } )  <->  ( A  e.  ~H  /\  A. x  e.  { B }  ( A  .ih  x )  =  0 ) )
51elexi 2810 . . . 4  |-  B  e. 
_V
6 oveq2 5882 . . . . 5  |-  ( x  =  B  ->  ( A  .ih  x )  =  ( A  .ih  B
) )
76eqeq1d 2304 . . . 4  |-  ( x  =  B  ->  (
( A  .ih  x
)  =  0  <->  ( A  .ih  B )  =  0 ) )
85, 7ralsn 3687 . . 3  |-  ( A. x  e.  { B }  ( A  .ih  x )  =  0  <-> 
( A  .ih  B
)  =  0 )
98anbi2i 675 . 2  |-  ( ( A  e.  ~H  /\  A. x  e.  { B }  ( A  .ih  x )  =  0 )  <->  ( A  e. 
~H  /\  ( A  .ih  B )  =  0 ) )
104, 9bitri 240 1  |-  ( A  e.  ( _|_ `  { B } )  <->  ( A  e.  ~H  /\  ( A 
.ih  B )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   0cc0 8753   ~Hchil 21515    .ih csp 21518   _|_cort 21526
This theorem is referenced by:  h1dei  22145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oc 21847
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