HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  eigvecval Unicode version

Theorem eigvecval 23360
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 22463 . . . 4  |-  ~H  e.  _V
2 difexg 4319 . . . 4  |-  ( ~H  e.  _V  ->  ( ~H  \  0H )  e. 
_V )
31, 2ax-mp 8 . . 3  |-  ( ~H 
\  0H )  e. 
_V
43rabex 4322 . 2  |-  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) }  e.  _V
5 fveq1 5694 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65eqeq1d 2420 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  =  ( y  .h  x )  <->  ( T `  x )  =  ( y  .h  x ) ) )
76rexbidv 2695 . . 3  |-  ( t  =  T  ->  ( E. y  e.  CC  ( t `  x
)  =  ( y  .h  x )  <->  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) ) )
87rabbidv 2916 . 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 23317 . 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 7017 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 1649    e. wcel 1721   E.wrex 2675   {crab 2678   _Vcvv 2924    \ cdif 3285   -->wf 5417   ` cfv 5421  (class class class)co 6048   CCcc 8952   ~Hchil 22383    .h csm 22385   0Hc0h 22399   eigveccei 22423
This theorem is referenced by:  eleigvec  23421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-hilex 22463
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-eigvec 23317
  Copyright terms: Public domain W3C validator