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

Theorem eigvalval 23463
Description: The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
eigvalval  |-  ( ( T : ~H --> ~H  /\  A  e.  ( eigvec `  T ) )  -> 
( ( eigval `  T
) `  A )  =  ( ( ( T `  A ) 
.ih  A )  / 
( ( normh `  A
) ^ 2 ) ) )

Proof of Theorem eigvalval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eigvalfval 23400 . . 3  |-  ( T : ~H --> ~H  ->  (
eigval `  T )  =  ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) )
21fveq1d 5730 . 2  |-  ( T : ~H --> ~H  ->  ( ( eigval `  T ) `  A )  =  ( ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) `
 A ) )
3 fveq2 5728 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
4 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4oveq12d 6099 . . . 4  |-  ( x  =  A  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 A )  .ih  A ) )
6 fveq2 5728 . . . . 5  |-  ( x  =  A  ->  ( normh `  x )  =  ( normh `  A )
)
76oveq1d 6096 . . . 4  |-  ( x  =  A  ->  (
( normh `  x ) ^ 2 )  =  ( ( normh `  A
) ^ 2 ) )
85, 7oveq12d 6099 . . 3  |-  ( x  =  A  ->  (
( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) )  =  ( ( ( T `
 A )  .ih  A )  /  ( (
normh `  A ) ^
2 ) ) )
9 eqid 2436 . . 3  |-  ( x  e.  ( eigvec `  T
)  |->  ( ( ( T `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) )  =  ( x  e.  ( eigvec `  T )  |->  ( ( ( T `  x
)  .ih  x )  /  ( ( normh `  x ) ^ 2 ) ) )
10 ovex 6106 . . 3  |-  ( ( ( T `  A
)  .ih  A )  /  ( ( normh `  A ) ^ 2 ) )  e.  _V
118, 9, 10fvmpt 5806 . 2  |-  ( A  e.  ( eigvec `  T
)  ->  ( (
x  e.  ( eigvec `  T )  |->  ( ( ( T `  x
)  .ih  x )  /  ( ( normh `  x ) ^ 2 ) ) ) `  A )  =  ( ( ( T `  A )  .ih  A
)  /  ( (
normh `  A ) ^
2 ) ) )
122, 11sylan9eq 2488 1  |-  ( ( T : ~H --> ~H  /\  A  e.  ( eigvec `  T ) )  -> 
( ( eigval `  T
) `  A )  =  ( ( ( T `  A ) 
.ih  A )  / 
( ( normh `  A
) ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081    / cdiv 9677   2c2 10049   ^cexp 11382   ~Hchil 22422    .ih csp 22425   normhcno 22426   eigveccei 22462   eigvalcel 22463
This theorem is referenced by:  eigvalcl  23464  eigvec1  23465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-eigval 23357
  Copyright terms: Public domain W3C validator