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Theorem eigvalval 29741
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 29678 . . 3 (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
21fveq1d 6654 . 2 (𝑇: ℋ⟶ ℋ → ((eigval‘𝑇)‘𝐴) = ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴))
3 fveq2 6652 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
4 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
53, 4oveq12d 7158 . . . 4 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑥) = ((𝑇𝐴) ·ih 𝐴))
6 fveq2 6652 . . . . 5 (𝑥 = 𝐴 → (norm𝑥) = (norm𝐴))
76oveq1d 7155 . . . 4 (𝑥 = 𝐴 → ((norm𝑥)↑2) = ((norm𝐴)↑2))
85, 7oveq12d 7158 . . 3 (𝑥 = 𝐴 → (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
9 eqid 2822 . . 3 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
10 ovex 7173 . . 3 (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)) ∈ V
118, 9, 10fvmpt 6750 . 2 (𝐴 ∈ (eigvec‘𝑇) → ((𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
122, 11sylan9eq 2877 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 399   = wceq 1538  wcel 2114  cmpt 5122  wf 6330  cfv 6334  (class class class)co 7140   / cdiv 11286  2c2 11680  cexp 13425  chba 28700   ·ih csp 28703  normcno 28704  eigveccei 28740  eigvalcel 28741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-hilex 28780
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-eigval 29635
This theorem is referenced by:  eigvalcl  29742  eigvec1  29743
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