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Theorem eleigvec 31985
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 31924 . . 3 (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)})
21eleq2d 2824 . 2 (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)}))
3 eldif 3972 . . . . 5 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0))
4 elch0 31282 . . . . . . 7 (𝐴 ∈ 0𝐴 = 0)
54necon3bbii 2985 . . . . . 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 7438 . . . . . 6 (𝑦 = 𝐴 → (𝑥 · 𝑦) = (𝑥 · 𝐴))
119, 10eqeq12d 2750 . . . . 5 (𝑦 = 𝐴 → ((𝑇𝑦) = (𝑥 · 𝑦) ↔ (𝑇𝐴) = (𝑥 · 𝐴)))
1211rexbidv 3176 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
1312elrab 3694 . . 3 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
14 df-3an 1088 . . 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 1086   = wceq 1536  wcel 2105  wne 2937  wrex 3067  {crab 3432  cdif 3959  wf 6558  cfv 6562  (class class class)co 7430  cc 11150  chba 30947   · csm 30949  0c0v 30952  0c0h 30963  eigveccei 30987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-hilex 31027  ax-hv0cl 31031
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-ch0 31281  df-eigvec 31881
This theorem is referenced by:  eleigvec2  31986  eigvalcl  31989
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