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Theorem eleigvec 32275
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 32214 . . 3 (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)})
21eleq2d 2847 . 2 (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)}))
3 eldif 3914 . . . . 5 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0))
4 elch0 31572 . . . . . . 7 (𝐴 ∈ 0𝐴 = 0)
54necon3bbii 3003 . . . . . 6 𝐴 ∈ 0𝐴 ≠ 0)
65anbi2i 634 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
73, 6bitri 278 . . . 4 (𝐴 ∈ ( ℋ ∖ 0) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0))
87anbi1i 635 . . 3 ((𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
9 fveq2 6881 . . . . . 6 (𝑦 = 𝐴 → (𝑇𝑦) = (𝑇𝐴))
10 oveq2 7418 . . . . . 6 (𝑦 = 𝐴 → (𝑥 · 𝑦) = (𝑥 · 𝐴))
119, 10eqeq12d 2777 . . . . 5 (𝑦 = 𝐴 → ((𝑇𝑦) = (𝑥 · 𝑦) ↔ (𝑇𝐴) = (𝑥 · 𝐴)))
1211rexbidv 3187 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
1312elrab 3649 . . 3 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ( ℋ ∖ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
14 df-3an 1103 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
158, 13, 143bitr4i 306 . 2 (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0) ∣ ∃𝑥 ∈ ℂ (𝑇𝑦) = (𝑥 · 𝑦)} ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴)))
162, 15bitrdi 290 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 209  wa 400  w3a 1101   = wceq 1568  wcel 2141  wne 2956  wrex 3087  {crab 3414  cdif 3901  wf 6532  cfv 6536  (class class class)co 7410  cc 11097  chba 31237   · csm 31239  0c0v 31242  0c0h 31253  eigveccei 31277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-hilex 31317  ax-hv0cl 31321
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8825  df-ch0 31571  df-eigvec 32171
This theorem is referenced by:  eleigvec2  32276  eigvalcl  32279
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