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Theorem eigvalfval 27526
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 5883 . . 3  |-  ( eigvec `  T )  e.  _V
21mptex 6143 . 2  |-  ( x  e.  ( eigvec `  T
)  |->  ( ( ( T `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) )  e.  _V
3 ax-hilex 26628 . 2  |-  ~H  e.  _V
4 fveq2 5873 . . 3  |-  ( t  =  T  ->  ( eigvec `
 t )  =  ( eigvec `  T )
)
5 fveq1 5872 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65oveq1d 6312 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  .ih  x )  =  ( ( T `
 x )  .ih  x ) )
76oveq1d 6312 . . 3  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) )  =  ( ( ( T `
 x )  .ih  x )  /  (
( normh `  x ) ^ 2 ) ) )
84, 7mpteq12dv 4496 . 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 27483 . 2  |-  eigval  =  ( t  e.  ( ~H 
^m  ~H )  |->  ( x  e.  ( eigvec `  t
)  |->  ( ( ( t `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) ) )
102, 3, 3, 8, 9fvmptmap 7508 1  |-  ( T : ~H --> ~H  ->  (
eigval `  T )  =  ( x  e.  (
eigvec `  T )  |->  ( ( ( T `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    |-> cmpt 4476   -->wf 5589   ` cfv 5593  (class class class)co 6297    / cdiv 10265   2c2 10655   ^cexp 12265   ~Hchil 26548    .ih csp 26551   normhcno 26552   eigveccei 26588   eigvalcel 26589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-hilex 26628
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-map 7474  df-eigval 27483
This theorem is referenced by:  eigvalval  27589
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