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Theorem eigvalval 31980
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 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))

Proof of Theorem eigvalval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eigvalfval 31917 . . 3 (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
21fveq1d 6907 . 2 (𝑇: ℋ⟶ ℋ → ((eigval‘𝑇)‘𝐴) = ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴))
3 fveq2 6905 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4oveq12d 7450 . . . 4 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑥) = ((𝑇𝐴) ·ih 𝐴))
6 fveq2 6905 . . . . 5 (𝑥 = 𝐴 → (norm𝑥) = (norm𝐴))
76oveq1d 7447 . . . 4 (𝑥 = 𝐴 → ((norm𝑥)↑2) = ((norm𝐴)↑2))
85, 7oveq12d 7450 . . 3 (𝑥 = 𝐴 → (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
9 eqid 2736 . . 3 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
10 ovex 7465 . . 3 (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)) ∈ V
118, 9, 10fvmpt 7015 . 2 (𝐴 ∈ (eigvec‘𝑇) → ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
122, 11sylan9eq 2796 1 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  cmpt 5224  wf 6556  cfv 6560  (class class class)co 7432   / cdiv 11921  2c2 12322  cexp 14103  chba 30939   ·ih csp 30942  normcno 30943  eigveccei 30979  eigvalcel 30980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-hilex 31019
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-eigval 31874
This theorem is referenced by:  eigvalcl  31981  eigvec1  31982
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