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Theorem eigvecval 23430
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 22533 . . . 4  |-  ~H  e.  _V
2 difexg 4380 . . . 4  |-  ( ~H  e.  _V  ->  ( ~H  \  0H )  e. 
_V )
31, 2ax-mp 5 . . 3  |-  ( ~H 
\  0H )  e. 
_V
43rabex 4383 . 2  |-  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) }  e.  _V
5 fveq1 5756 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65eqeq1d 2450 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  =  ( y  .h  x )  <->  ( T `  x )  =  ( y  .h  x ) ) )
76rexbidv 2732 . . 3  |-  ( t  =  T  ->  ( E. y  e.  CC  ( t `  x
)  =  ( y  .h  x )  <->  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) ) )
87rabbidv 2954 . 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 23387 . 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 7079 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 1653    e. wcel 1727   E.wrex 2712   {crab 2715   _Vcvv 2962    \ cdif 3303   -->wf 5479   ` cfv 5483  (class class class)co 6110   CCcc 9019   ~Hchil 22453    .h csm 22455   0Hc0h 22469   eigveccei 22493
This theorem is referenced by:  eleigvec  23491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-hilex 22533
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-map 7049  df-eigvec 23387
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