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Theorem eigvalval 29391
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 29328 . . 3 (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
21fveq1d 6448 . 2 (𝑇: ℋ⟶ ℋ → ((eigval‘𝑇)‘𝐴) = ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴))
3 fveq2 6446 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4oveq12d 6940 . . . 4 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑥) = ((𝑇𝐴) ·ih 𝐴))
6 fveq2 6446 . . . . 5 (𝑥 = 𝐴 → (norm𝑥) = (norm𝐴))
76oveq1d 6937 . . . 4 (𝑥 = 𝐴 → ((norm𝑥)↑2) = ((norm𝐴)↑2))
85, 7oveq12d 6940 . . 3 (𝑥 = 𝐴 → (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
9 eqid 2778 . . 3 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
10 ovex 6954 . . 3 (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)) ∈ V
118, 9, 10fvmpt 6542 . 2 (𝐴 ∈ (eigvec‘𝑇) → ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
122, 11sylan9eq 2834 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 386   = wceq 1601  wcel 2107  cmpt 4965  wf 6131  cfv 6135  (class class class)co 6922   / cdiv 11032  2c2 11430  cexp 13178  chba 28348   ·ih csp 28351  normcno 28352  eigveccei 28388  eigvalcel 28389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-hilex 28428
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-eigval 29285
This theorem is referenced by:  eigvalcl  29392  eigvec1  29393
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