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Theorem eigvalfval 22591
Description: The eigenvalues of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
eigvalfval  |-  ( T : ~H --> ~H  ->  (
eigval `  T )  =  ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) )
Distinct variable group:    x, T

Proof of Theorem eigvalfval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fvex 5622 . . 3  |-  ( eigvec `  T )  e.  _V
21mptex 5832 . 2  |-  ( x  e.  ( eigvec `  T
)  |->  ( ( ( T `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) )  e.  _V
3 ax-hilex 21693 . 2  |-  ~H  e.  _V
4 fveq2 5608 . . 3  |-  ( t  =  T  ->  ( eigvec `
 t )  =  ( eigvec `  T )
)
5 fveq1 5607 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65oveq1d 5960 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  .ih  x )  =  ( ( T `
 x )  .ih  x ) )
76oveq1d 5960 . . 3  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) )  =  ( ( ( T `
 x )  .ih  x )  /  (
( normh `  x ) ^ 2 ) ) )
84, 7mpteq12dv 4179 . 2  |-  ( t  =  T  ->  (
x  e.  ( eigvec `  t )  |->  ( ( ( t `  x
)  .ih  x )  /  ( ( normh `  x ) ^ 2 ) ) )  =  ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) )
9 df-eigval 22548 . 2  |-  eigval  =  ( t  e.  ( ~H 
^m  ~H )  |->  ( x  e.  ( eigvec `  t
)  |->  ( ( ( t `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) ) )
102, 3, 3, 8, 9fvmptmap 6892 1  |-  ( T : ~H --> ~H  ->  (
eigval `  T )  =  ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. cmpt 4158   -->wf 5333   ` cfv 5337  (class class class)co 5945    / cdiv 9513   2c2 9885   ^cexp 11197   ~Hchil 21613    .ih csp 21616   normhcno 21617   eigveccei 21653   eigvalcel 21654
This theorem is referenced by:  eigvalval  22654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-hilex 21693
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-map 6862  df-eigval 22548
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