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Theorem eleigvec 23460
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 23399 . . 3  |-  ( T : ~H --> ~H  ->  (
eigvec `  T )  =  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) } )
21eleq2d 2503 . 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 3330 . . . . 5  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  -.  A  e.  0H ) )
4 elch0 22756 . . . . . . 7  |-  ( A  e.  0H  <->  A  =  0h )
54necon3bbii 2632 . . . . . 6  |-  ( -.  A  e.  0H  <->  A  =/=  0h )
65anbi2i 676 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  e.  0H ) 
<->  ( A  e.  ~H  /\  A  =/=  0h )
)
73, 6bitri 241 . . . 4  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  A  =/=  0h ) )
87anbi1i 677 . . 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 5728 . . . . . 6  |-  ( y  =  A  ->  ( T `  y )  =  ( T `  A ) )
10 oveq2 6089 . . . . . 6  |-  ( y  =  A  ->  (
x  .h  y )  =  ( x  .h  A ) )
119, 10eqeq12d 2450 . . . . 5  |-  ( y  =  A  ->  (
( T `  y
)  =  ( x  .h  y )  <->  ( T `  A )  =  ( x  .h  A ) ) )
1211rexbidv 2726 . . . 4  |-  ( y  =  A  ->  ( E. x  e.  CC  ( T `  y )  =  ( x  .h  y )  <->  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
1312elrab 3092 . . 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 938 . . 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 269 . 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 253 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709    \ cdif 3317   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   ~Hchil 22422    .h csm 22424   0hc0v 22427   0Hc0h 22438   eigveccei 22462
This theorem is referenced by:  eleigvec2  23461  eigvalcl  23464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-hilex 22502  ax-hv0cl 22506
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-ch0 22755  df-eigvec 23356
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