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Theorem eigvecval 29341
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 28442 . . . 4 ℋ ∈ V
2 difexg 5045 . . . 4 ( ℋ ∈ V → ( ℋ ∖ 0) ∈ V)
31, 2ax-mp 5 . . 3 ( ℋ ∖ 0) ∈ V
43rabex 5049 . 2 {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)} ∈ V
5 fveq1 6445 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65eqeq1d 2779 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) = (𝑦 · 𝑥) ↔ (𝑇𝑥) = (𝑦 · 𝑥)))
76rexbidv 3236 . . 3 (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)))
87rabbidv 3385 . 2 (𝑡 = 𝑇 → {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)} = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
9 df-eigvec 29298 . 2 eigvec = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)})
104, 1, 1, 8, 9fvmptmap 8178 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 1601  wcel 2106  wrex 3090  {crab 3093  Vcvv 3397  cdif 3788  wf 6131  cfv 6135  (class class class)co 6922  cc 10270  chba 28362   · csm 28364  0c0h 28378  eigveccei 28402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-hilex 28442
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-eigvec 29298
This theorem is referenced by:  eleigvec  29402
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