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Theorem eigvecval 31705
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 30808 . . . 4 ℋ ∈ V
2 difexg 5329 . . . 4 ( ℋ ∈ V → ( ℋ ∖ 0) ∈ V)
31, 2ax-mp 5 . . 3 ( ℋ ∖ 0) ∈ V
43rabex 5334 . 2 {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)} ∈ V
5 fveq1 6896 . . . . 5 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
65eqeq1d 2730 . . . 4 (𝑡 = 𝑇 → ((𝑡𝑥) = (𝑦 · 𝑥) ↔ (𝑇𝑥) = (𝑦 · 𝑥)))
76rexbidv 3175 . . 3 (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)))
87rabbidv 3437 . 2 (𝑡 = 𝑇 → {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)} = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
9 df-eigvec 31662 . 2 eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑡𝑥) = (𝑦 · 𝑥)})
104, 1, 1, 8, 9fvmptmap 8899 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 1534  wcel 2099  wrex 3067  {crab 3429  Vcvv 3471  cdif 3944  wf 6544  cfv 6548  (class class class)co 7420  cc 11136  chba 30728   · csm 30730  0c0h 30744  eigveccei 30768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-hilex 30808
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-eigvec 31662
This theorem is referenced by:  eleigvec  31766
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