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Theorem eleigvec 22553
Description: Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
eleigvec  |-  ( T : ~H --> ~H  ->  ( A  e.  ( eigvec `  T )  <->  ( A  e.  ~H  /\  A  =/= 
0h  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) ) )
Distinct variable groups:    x, A    x, T

Proof of Theorem eleigvec
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eigvecval 22492 . . 3  |-  ( T : ~H --> ~H  ->  (
eigvec `  T )  =  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) } )
21eleq2d 2363 . 2  |-  ( T : ~H --> ~H  ->  ( A  e.  ( eigvec `  T )  <->  A  e.  { y  e.  ( ~H 
\  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) } ) )
3 eldif 3175 . . . . 5  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  -.  A  e.  0H ) )
4 elch0 21849 . . . . . . 7  |-  ( A  e.  0H  <->  A  =  0h )
54necon3bbii 2490 . . . . . 6  |-  ( -.  A  e.  0H  <->  A  =/=  0h )
65anbi2i 675 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  e.  0H ) 
<->  ( A  e.  ~H  /\  A  =/=  0h )
)
73, 6bitri 240 . . . 4  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  A  =/=  0h ) )
87anbi1i 676 . . 3  |-  ( ( A  e.  ( ~H 
\  0H )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) )  <->  ( ( A  e.  ~H  /\  A  =/=  0h )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
9 fveq2 5541 . . . . . 6  |-  ( y  =  A  ->  ( T `  y )  =  ( T `  A ) )
10 oveq2 5882 . . . . . 6  |-  ( y  =  A  ->  (
x  .h  y )  =  ( x  .h  A ) )
119, 10eqeq12d 2310 . . . . 5  |-  ( y  =  A  ->  (
( T `  y
)  =  ( x  .h  y )  <->  ( T `  A )  =  ( x  .h  A ) ) )
1211rexbidv 2577 . . . 4  |-  ( y  =  A  ->  ( E. x  e.  CC  ( T `  y )  =  ( x  .h  y )  <->  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
1312elrab 2936 . . 3  |-  ( A  e.  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) }  <->  ( A  e.  ( ~H  \  0H )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
14 df-3an 936 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) )  <->  ( ( A  e.  ~H  /\  A  =/=  0h )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
158, 13, 143bitr4i 268 . 2  |-  ( A  e.  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) }  <->  ( A  e. 
~H  /\  A  =/=  0h 
/\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
162, 15syl6bb 252 1  |-  ( T : ~H --> ~H  ->  ( A  e.  ( eigvec `  T )  <->  ( A  e.  ~H  /\  A  =/= 
0h  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    \ cdif 3162   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   ~Hchil 21515    .h csm 21517   0hc0v 21520   0Hc0h 21531   eigveccei 21555
This theorem is referenced by:  eleigvec2  22554  eigvalcl  22557
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595  ax-hv0cl 21599
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-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ch0 21848  df-eigvec 22449
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