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Theorem eigvalval 22540
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 22477 . . 3  |-  ( T : ~H --> ~H  ->  (
eigval `  T )  =  ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) )
21fveq1d 5527 . 2  |-  ( T : ~H --> ~H  ->  ( ( eigval `  T ) `  A )  =  ( ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) `
 A ) )
3 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
4 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
53, 4oveq12d 5876 . . . 4  |-  ( x  =  A  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 A )  .ih  A ) )
6 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  ( normh `  x )  =  ( normh `  A )
)
76oveq1d 5873 . . . 4  |-  ( x  =  A  ->  (
( normh `  x ) ^ 2 )  =  ( ( normh `  A
) ^ 2 ) )
85, 7oveq12d 5876 . . 3  |-  ( x  =  A  ->  (
( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) )  =  ( ( ( T `
 A )  .ih  A )  /  ( (
normh `  A ) ^
2 ) ) )
9 eqid 2283 . . 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 5883 . . 3  |-  ( ( ( T `  A
)  .ih  A )  /  ( ( normh `  A ) ^ 2 ) )  e.  _V
118, 9, 10fvmpt 5602 . 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 2335 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 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    / cdiv 9423   2c2 9795   ^cexp 11104   ~Hchil 21499    .ih csp 21502   normhcno 21503   eigveccei 21539   eigvalcel 21540
This theorem is referenced by:  eigvalcl  22541  eigvec1  22542
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-eigval 22434
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