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Theorem eigvecval 27525
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 26628 . . . 4  |-  ~H  e.  _V
2 difexg 4565 . . . 4  |-  ( ~H  e.  _V  ->  ( ~H  \  0H )  e. 
_V )
31, 2ax-mp 5 . . 3  |-  ( ~H 
\  0H )  e. 
_V
43rabex 4568 . 2  |-  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) }  e.  _V
5 fveq1 5872 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65eqeq1d 2422 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  =  ( y  .h  x )  <->  ( T `  x )  =  ( y  .h  x ) ) )
76rexbidv 2937 . . 3  |-  ( t  =  T  ->  ( E. y  e.  CC  ( t `  x
)  =  ( y  .h  x )  <->  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) ) )
87rabbidv 3070 . 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 27482 . 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 7508 1  |-  ( T : ~H --> ~H  ->  (
eigvec `  T )  =  { x  e.  ( ~H  \  0H )  |  E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   E.wrex 2774   {crab 2777   _Vcvv 3078    \ cdif 3430   -->wf 5589   ` cfv 5593  (class class class)co 6297   CCcc 9533   ~Hchil 26548    .h csm 26550   0Hc0h 26564   eigveccei 26588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-hilex 26628
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-map 7474  df-eigvec 27482
This theorem is referenced by:  eleigvec  27586
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