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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 28139 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)}) | |
2 | 1 | eleq2d 2673 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)})) |
3 | eldif 3550 | . . . . 5 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ)) | |
4 | elch0 27495 | . . . . . . 7 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) | |
5 | 4 | necon3bbii 2829 | . . . . . 6 ⊢ (¬ 𝐴 ∈ 0ℋ ↔ 𝐴 ≠ 0ℎ) |
6 | 5 | anbi2i 726 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)) |
7 | 3, 6 | bitri 263 | . . . 4 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)) |
8 | 7 | anbi1i 727 | . . 3 ⊢ ((𝐴 ∈ ( ℋ ∖ 0ℋ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
9 | fveq2 6103 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑇‘𝑦) = (𝑇‘𝐴)) | |
10 | oveq2 6557 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 𝐴)) | |
11 | 9, 10 | eqeq12d 2625 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
12 | 11 | rexbidv 3034 | . . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
13 | 12 | elrab 3331 | . . 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 1033 | . . 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 291 | . 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 275 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 ∖ cdif 3537 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℋchil 27160 ·ℎ csm 27162 0ℎc0v 27165 0ℋc0h 27176 eigveccei 27200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-hilex 27240 ax-hv0cl 27244 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-ch0 27494 df-eigvec 28096 |
This theorem is referenced by: eleigvec2 28201 eigvalcl 28204 |
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