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Theorem eigvalfval 29311
 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 6446 . . 3 (eigvec‘𝑇) ∈ V
21mptex 6742 . 2 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) ∈ V
3 ax-hilex 28411 . 2 ℋ ∈ V
4 fveq2 6433 . . 3 (𝑡 = 𝑇 → (eigvec‘𝑡) = (eigvec‘𝑇))
5 fveq1 6432 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65oveq1d 6920 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih 𝑥) = ((𝑇𝑥) ·ih 𝑥))
76oveq1d 6920 . . 3 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
84, 7mpteq12dv 4956 . 2 (𝑡 = 𝑇 → (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
9 df-eigval 29268 . 2 eigval = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
102, 3, 3, 8, 9fvmptmap 8159 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 1658   ↦ cmpt 4952  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905   / cdiv 11009  2c2 11406  ↑cexp 13154   ℋchba 28331   ·ih csp 28334  normℎcno 28335  eigveccei 28371  eigvalcel 28372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-hilex 28411 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-eigval 29268 This theorem is referenced by:  eigvalval  29374
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