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Theorem eigvecval 28139
 Description: The set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
eigvecval (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem eigvecval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 27240 . . . 4 ℋ ∈ V
2 difexg 4735 . . . 4 ( ℋ ∈ V → ( ℋ ∖ 0) ∈ V)
31, 2ax-mp 5 . . 3 ( ℋ ∖ 0) ∈ V
43rabex 4740 . 2 {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)} ∈ V
5 fveq1 6102 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65eqeq1d 2612 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) = (𝑦 · 𝑥) ↔ (𝑇𝑥) = (𝑦 · 𝑥)))
76rexbidv 3034 . . 3 (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)))
87rabbidv 3164 . 2 (𝑡 = 𝑇 → {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)} = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
9 df-eigvec 28096 . 2 eigvec = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)})
104, 1, 1, 8, 9fvmptmap 7780 1 (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   ℋchil 27160   ·ℎ csm 27162  0ℋc0h 27176  eigveccei 27200 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-eigvec 28096 This theorem is referenced by:  eleigvec  28200
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