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Theorem eleigvec 29661
Description: Membership in 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
eleigvec (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem eleigvec
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eigvecval 29600 . . 3 (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)})
21eleq2d 2895 . 2 (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)}))
3 eldif 3943 . . . . 5 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0))
4 elch0 28958 . . . . . . 7 (𝐴 ∈ 0𝐴 = 0)
54necon3bbii 3060 . . . . . 6 𝐴 ∈ 0𝐴 ≠ 0)
65anbi2i 622 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
73, 6bitri 276 . . . 4 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
87anbi1i 623 . . 3 ((𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
9 fveq2 6663 . . . . . 6 (𝑦 = 𝐴 → (𝑇𝑦) = (𝑇𝐴))
10 oveq2 7153 . . . . . 6 (𝑦 = 𝐴 → (𝑥 · 𝑦) = (𝑥 · 𝐴))
119, 10eqeq12d 2834 . . . . 5 (𝑦 = 𝐴 → ((𝑇𝑦) = (𝑥 · 𝑦) ↔ (𝑇𝐴) = (𝑥 · 𝐴)))
1211rexbidv 3294 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
1312elrab 3677 . . 3 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
14 df-3an 1081 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
158, 13, 143bitr4i 304 . 2 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
162, 15syl6bb 288 1 (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  {crab 3139  cdif 3930  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  chba 28623   · csm 28625  0c0v 28628  0c0h 28639  eigveccei 28663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-hilex 28703  ax-hv0cl 28707
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  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-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-ch0 28957  df-eigvec 29557
This theorem is referenced by:  eleigvec2  29662  eigvalcl  29665
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