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Theorem eigvalfval 23405
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 5745 . . 3  |-  ( eigvec `  T )  e.  _V
21mptex 5969 . 2  |-  ( x  e.  ( eigvec `  T
)  |->  ( ( ( T `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) )  e.  _V
3 ax-hilex 22507 . 2  |-  ~H  e.  _V
4 fveq2 5731 . . 3  |-  ( t  =  T  ->  ( eigvec `
 t )  =  ( eigvec `  T )
)
5 fveq1 5730 . . . . 5  |-  ( t  =  T  ->  (
t `  x )  =  ( T `  x ) )
65oveq1d 6099 . . . 4  |-  ( t  =  T  ->  (
( t `  x
)  .ih  x )  =  ( ( T `
 x )  .ih  x ) )
76oveq1d 6099 . . 3  |-  ( t  =  T  ->  (
( ( t `  x )  .ih  x
)  /  ( (
normh `  x ) ^
2 ) )  =  ( ( ( T `
 x )  .ih  x )  /  (
( normh `  x ) ^ 2 ) ) )
84, 7mpteq12dv 4290 . 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 23362 . 2  |-  eigval  =  ( t  e.  ( ~H 
^m  ~H )  |->  ( x  e.  ( eigvec `  t
)  |->  ( ( ( t `  x ) 
.ih  x )  / 
( ( normh `  x
) ^ 2 ) ) ) )
102, 3, 3, 8, 9fvmptmap 7053 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 1653    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084    / cdiv 9682   2c2 10054   ^cexp 11387   ~Hchil 22427    .ih csp 22430   normhcno 22431   eigveccei 22467   eigvalcel 22468
This theorem is referenced by:  eigvalval  23468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-hilex 22507
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-eigval 23362
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