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Theorem eigvalfval 29065
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 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
Distinct variable group:   𝑥,𝑇

Proof of Theorem eigvalfval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fvex 6362 . . 3 (eigvec‘𝑇) ∈ V
21mptex 6650 . 2 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) ∈ V
3 ax-hilex 28165 . 2 ℋ ∈ V
4 fveq2 6352 . . 3 (𝑡 = 𝑇 → (eigvec‘𝑡) = (eigvec‘𝑇))
5 fveq1 6351 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65oveq1d 6828 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih 𝑥) = ((𝑇𝑥) ·ih 𝑥))
76oveq1d 6828 . . 3 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
84, 7mpteq12dv 4885 . 2 (𝑡 = 𝑇 → (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
9 df-eigval 29022 . 2 eigval = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
102, 3, 3, 8, 9fvmptmap 8060 1 (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813   / cdiv 10876  2c2 11262  cexp 13054  chil 28085   ·ih csp 28088  normcno 28089  eigveccei 28125  eigvalcel 28126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-hilex 28165
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-eigval 29022
This theorem is referenced by:  eigvalval  29128
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