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Theorem eleigvec 22537
Description: Membership in 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
eleigvec  |-  ( T : ~H --> ~H  ->  ( A  e.  ( eigvec `  T )  <->  ( A  e.  ~H  /\  A  =/= 
0h  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) ) )
Distinct variable groups:    x, A    x, T

Proof of Theorem eleigvec
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eigvecval 22476 . . 3  |-  ( T : ~H --> ~H  ->  (
eigvec `  T )  =  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) } )
21eleq2d 2350 . 2  |-  ( T : ~H --> ~H  ->  ( A  e.  ( eigvec `  T )  <->  A  e.  { y  e.  ( ~H 
\  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) } ) )
3 eldif 3162 . . . . 5  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  -.  A  e.  0H ) )
4 elch0 21833 . . . . . . 7  |-  ( A  e.  0H  <->  A  =  0h )
54necon3bbii 2477 . . . . . 6  |-  ( -.  A  e.  0H  <->  A  =/=  0h )
65anbi2i 675 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  e.  0H ) 
<->  ( A  e.  ~H  /\  A  =/=  0h )
)
73, 6bitri 240 . . . 4  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  A  =/=  0h ) )
87anbi1i 676 . . 3  |-  ( ( A  e.  ( ~H 
\  0H )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) )  <->  ( ( A  e.  ~H  /\  A  =/=  0h )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
9 fveq2 5525 . . . . . 6  |-  ( y  =  A  ->  ( T `  y )  =  ( T `  A ) )
10 oveq2 5866 . . . . . 6  |-  ( y  =  A  ->  (
x  .h  y )  =  ( x  .h  A ) )
119, 10eqeq12d 2297 . . . . 5  |-  ( y  =  A  ->  (
( T `  y
)  =  ( x  .h  y )  <->  ( T `  A )  =  ( x  .h  A ) ) )
1211rexbidv 2564 . . . 4  |-  ( y  =  A  ->  ( E. x  e.  CC  ( T `  y )  =  ( x  .h  y )  <->  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
1312elrab 2923 . . 3  |-  ( A  e.  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) }  <->  ( A  e.  ( ~H  \  0H )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
14 df-3an 936 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) )  <->  ( ( A  e.  ~H  /\  A  =/=  0h )  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
158, 13, 143bitr4i 268 . 2  |-  ( A  e.  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) }  <->  ( A  e. 
~H  /\  A  =/=  0h 
/\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
162, 15syl6bb 252 1  |-  ( T : ~H --> ~H  ->  ( A  e.  ( eigvec `  T )  <->  ( A  e.  ~H  /\  A  =/= 
0h  /\  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    \ cdif 3149   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    .h csm 21501   0hc0v 21504   0Hc0h 21515   eigveccei 21539
This theorem is referenced by:  eleigvec2  22538  eigvalcl  22541
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  ax-hv0cl 21583
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-ch0 21832  df-eigvec 22433
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