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

Theorem eleigvec 27608
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 27547 . . 3  |-  ( T : ~H --> ~H  ->  (
eigvec `  T )  =  { y  e.  ( ~H  \  0H )  |  E. x  e.  CC  ( T `  y )  =  ( x  .h  y ) } )
21eleq2d 2492 . 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 3446 . . . . 5  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  -.  A  e.  0H ) )
4 elch0 26905 . . . . . . 7  |-  ( A  e.  0H  <->  A  =  0h )
54necon3bbii 2681 . . . . . 6  |-  ( -.  A  e.  0H  <->  A  =/=  0h )
65anbi2i 698 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  e.  0H ) 
<->  ( A  e.  ~H  /\  A  =/=  0h )
)
73, 6bitri 252 . . . 4  |-  ( A  e.  ( ~H  \  0H )  <->  ( A  e. 
~H  /\  A  =/=  0h ) )
87anbi1i 699 . . 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 5881 . . . . . 6  |-  ( y  =  A  ->  ( T `  y )  =  ( T `  A ) )
10 oveq2 6313 . . . . . 6  |-  ( y  =  A  ->  (
x  .h  y )  =  ( x  .h  A ) )
119, 10eqeq12d 2444 . . . . 5  |-  ( y  =  A  ->  (
( T `  y
)  =  ( x  .h  y )  <->  ( T `  A )  =  ( x  .h  A ) ) )
1211rexbidv 2936 . . . 4  |-  ( y  =  A  ->  ( E. x  e.  CC  ( T `  y )  =  ( x  .h  y )  <->  E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) )
1312elrab 3228 . . 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 984 . . 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 280 . 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 264 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   {crab 2775    \ cdif 3433   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9544   ~Hchil 26570    .h csm 26572   0hc0v 26575   0Hc0h 26586   eigveccei 26610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-hilex 26650  ax-hv0cl 26654
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7485  df-ch0 26904  df-eigvec 27504
This theorem is referenced by:  eleigvec2  27609  eigvalcl  27612
  Copyright terms: Public domain W3C validator