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Mirrors > Home > HSE Home > Th. List > eleigvec | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
eleigvec | ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eigvecval 29600 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)}) | |
2 | 1 | eleq2d 2895 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)})) |
3 | eldif 3943 | . . . . 5 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ)) | |
4 | elch0 28958 | . . . . . . 7 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) | |
5 | 4 | necon3bbii 3060 | . . . . . 6 ⊢ (¬ 𝐴 ∈ 0ℋ ↔ 𝐴 ≠ 0ℎ) |
6 | 5 | anbi2i 622 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)) |
7 | 3, 6 | bitri 276 | . . . 4 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)) |
8 | 7 | anbi1i 623 | . . 3 ⊢ ((𝐴 ∈ ( ℋ ∖ 0ℋ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
9 | fveq2 6663 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑇‘𝑦) = (𝑇‘𝐴)) | |
10 | oveq2 7153 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 𝐴)) | |
11 | 9, 10 | eqeq12d 2834 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
12 | 11 | rexbidv 3294 | . . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
13 | 12 | elrab 3677 | . . 3 ⊢ (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)} ↔ (𝐴 ∈ ( ℋ ∖ 0ℋ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
14 | df-3an 1081 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) | |
15 | 8, 13, 14 | 3bitr4i 304 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)} ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
16 | 2, 15 | syl6bb 288 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 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 ℂcc 10523 ℋchba 28623 ·ℎ csm 28625 0ℎc0v 28628 0ℋc0h 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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