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Theorem eigvecval 22476
Description: 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
eigvecval  |-  ( T : ~H --> ~H  ->  (
eigvec `  T )  =  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) } )
Distinct variable group:    x, y, T

Proof of Theorem eigvecval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 21579 . . . 4  |-  ~H  e.  _V
2 difexg 4162 . . . 4  |-  ( ~H  e.  _V  ->  ( ~H  \  0H )  e. 
_V )
31, 2ax-mp 8 . . 3  |-  ( ~H 
\  0H )  e. 
_V
43rabex 4165 . 2  |-  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) }  e.  _V
5 fveq1 5524 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65eqeq1d 2291 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  =  ( y  .h  x )  <->  ( T `  x )  =  ( y  .h  x ) ) )
76rexbidv 2564 . . 3  |-  ( t  =  T  ->  ( E. y  e.  CC  ( t `  x
)  =  ( y  .h  x )  <->  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) ) )
87rabbidv 2780 . 2  |-  ( t  =  T  ->  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( t `  x )  =  ( y  .h  x ) }  =  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) } )
9 df-eigvec 22433 . 2  |-  eigvec  =  ( t  e.  ( ~H 
^m  ~H )  |->  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( t `  x )  =  ( y  .h  x ) } )
104, 1, 1, 8, 9fvmptmap 6804 1  |-  ( T : ~H --> ~H  ->  (
eigvec `  T )  =  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    .h csm 21501   0Hc0h 21515   eigveccei 21539
This theorem is referenced by:  eleigvec  22537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-eigvec 22433
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