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Theorem eleigvec 28662
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 28601 . . 3 (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)})
21eleq2d 2684 . 2 (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)}))
3 eldif 3565 . . . . 5 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0))
4 elch0 27957 . . . . . . 7 (𝐴 ∈ 0𝐴 = 0)
54necon3bbii 2837 . . . . . 6 𝐴 ∈ 0𝐴 ≠ 0)
65anbi2i 729 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
73, 6bitri 264 . . . 4 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
87anbi1i 730 . . 3 ((𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
9 fveq2 6148 . . . . . 6 (𝑦 = 𝐴 → (𝑇𝑦) = (𝑇𝐴))
10 oveq2 6612 . . . . . 6 (𝑦 = 𝐴 → (𝑥 · 𝑦) = (𝑥 · 𝐴))
119, 10eqeq12d 2636 . . . . 5 (𝑦 = 𝐴 → ((𝑇𝑦) = (𝑥 · 𝑦) ↔ (𝑇𝐴) = (𝑥 · 𝐴)))
1211rexbidv 3045 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
1312elrab 3346 . . 3 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
14 df-3an 1038 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
158, 13, 143bitr4i 292 . 2 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
162, 15syl6bb 276 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  cdif 3552  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  chil 27622   · csm 27624  0c0v 27627  0c0h 27638  eigveccei 27662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-hilex 27702  ax-hv0cl 27706
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-ch0 27956  df-eigvec 28558
This theorem is referenced by:  eleigvec2  28663  eigvalcl  28666
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