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Theorem eleigvec 31976
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 31915 . . 3 (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)})
21eleq2d 2827 . 2 (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)}))
3 eldif 3961 . . . . 5 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0))
4 elch0 31273 . . . . . . 7 (𝐴 ∈ 0𝐴 = 0)
54necon3bbii 2988 . . . . . 6 𝐴 ∈ 0𝐴 ≠ 0)
65anbi2i 623 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
73, 6bitri 275 . . . 4 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
87anbi1i 624 . . 3 ((𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
9 fveq2 6906 . . . . . 6 (𝑦 = 𝐴 → (𝑇𝑦) = (𝑇𝐴))
10 oveq2 7439 . . . . . 6 (𝑦 = 𝐴 → (𝑥 · 𝑦) = (𝑥 · 𝐴))
119, 10eqeq12d 2753 . . . . 5 (𝑦 = 𝐴 → ((𝑇𝑦) = (𝑥 · 𝑦) ↔ (𝑇𝐴) = (𝑥 · 𝐴)))
1211rexbidv 3179 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
1312elrab 3692 . . 3 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
14 df-3an 1089 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
158, 13, 143bitr4i 303 . 2 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
162, 15bitrdi 287 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  cdif 3948  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  chba 30938   · csm 30940  0c0v 30943  0c0h 30954  eigveccei 30978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-hilex 31018  ax-hv0cl 31022
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-ch0 31272  df-eigvec 31872
This theorem is referenced by:  eleigvec2  31977  eigvalcl  31980
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