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Theorem eigvalfval 23361
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 5709 . . 3  |-  ( eigvec `  T )  e.  _V
21mptex 5933 . 2  |-  ( x  e.  ( eigvec `  T
)  |->  ( ( ( T `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) )  e.  _V
3 ax-hilex 22463 . 2  |-  ~H  e.  _V
4 fveq2 5695 . . 3  |-  ( t  =  T  ->  ( eigvec `
 t )  =  ( eigvec `  T )
)
5 fveq1 5694 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65oveq1d 6063 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  .ih  x )  =  ( ( T `
 x )  .ih  x ) )
76oveq1d 6063 . . 3  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) )  =  ( ( ( T `
 x )  .ih  x )  /  (
( normh `  x ) ^ 2 ) ) )
84, 7mpteq12dv 4255 . 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 23318 . 2  |-  eigval  =  ( t  e.  ( ~H 
^m  ~H )  |->  ( x  e.  ( eigvec `  t
)  |->  ( ( ( t `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) ) )
102, 3, 3, 8, 9fvmptmap 7017 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 1649    e. cmpt 4234   -->wf 5417   ` cfv 5421  (class class class)co 6048    / cdiv 9641   2c2 10013   ^cexp 11345   ~Hchil 22383    .ih csp 22386   normhcno 22387   eigveccei 22423   eigvalcel 22424
This theorem is referenced by:  eigvalval  23424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-hilex 22463
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-eigval 23318
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