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Theorem eigvalfval 31917
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 6918 . . 3 (eigvec‘𝑇) ∈ V
21mptex 7244 . 2 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) ∈ V
3 ax-hilex 31019 . 2 ℋ ∈ V
4 fveq2 6905 . . 3 (𝑡 = 𝑇 → (eigvec‘𝑡) = (eigvec‘𝑇))
5 fveq1 6904 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65oveq1d 7447 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih 𝑥) = ((𝑇𝑥) ·ih 𝑥))
76oveq1d 7447 . . 3 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
84, 7mpteq12dv 5232 . 2 (𝑡 = 𝑇 → (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
9 df-eigval 31874 . 2 eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
102, 3, 3, 8, 9fvmptmap 8922 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 1539  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:  eigvalval  31980
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