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Theorem eigvalfval 29666
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 6676 . . 3 (eigvec‘𝑇) ∈ V
21mptex 6978 . 2 (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) ∈ V
3 ax-hilex 28768 . 2 ℋ ∈ V
4 fveq2 6663 . . 3 (𝑡 = 𝑇 → (eigvec‘𝑡) = (eigvec‘𝑇))
5 fveq1 6662 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65oveq1d 7163 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) ·ih 𝑥) = ((𝑇𝑥) ·ih 𝑥))
76oveq1d 7163 . . 3 (𝑡 = 𝑇 → (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2)) = (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2)))
84, 7mpteq12dv 5142 . 2 (𝑡 = 𝑇 → (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
9 df-eigval 29623 . 2 eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
102, 3, 3, 8, 9fvmptmap 8437 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 1530  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7148   / cdiv 11289  2c2 11684  cexp 13421  chba 28688   ·ih csp 28691  normcno 28692  eigveccei 28728  eigvalcel 28729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-hilex 28768
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-eigval 29623
This theorem is referenced by:  eigvalval  29729
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