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Theorem eigvecval 22584
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 21687 . . . 4  |-  ~H  e.  _V
2 difexg 4241 . . . 4  |-  ( ~H  e.  _V  ->  ( ~H  \  0H )  e. 
_V )
31, 2ax-mp 8 . . 3  |-  ( ~H 
\  0H )  e. 
_V
43rabex 4244 . 2  |-  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) }  e.  _V
5 fveq1 5604 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65eqeq1d 2366 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  =  ( y  .h  x )  <->  ( T `  x )  =  ( y  .h  x ) ) )
76rexbidv 2640 . . 3  |-  ( t  =  T  ->  ( E. y  e.  CC  ( t `  x
)  =  ( y  .h  x )  <->  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) ) )
87rabbidv 2856 . 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 22541 . 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 6889 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 1642    e. wcel 1710   E.wrex 2620   {crab 2623   _Vcvv 2864    \ cdif 3225   -->wf 5330   ` cfv 5334  (class class class)co 5942   CCcc 8822   ~Hchil 21607    .h csm 21609   0Hc0h 21623   eigveccei 21647
This theorem is referenced by:  eleigvec  22645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-hilex 21687
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-map 6859  df-eigvec 22541
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