Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 29673 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)}) | |
2 | 1 | eleq2d 2898 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ 𝐴 ∈ {𝑦 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦)})) |
3 | eldif 3946 | . . . . 5 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ)) | |
4 | elch0 29031 | . . . . . . 7 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) | |
5 | 4 | necon3bbii 3063 | . . . . . 6 ⊢ (¬ 𝐴 ∈ 0ℋ ↔ 𝐴 ≠ 0ℎ) |
6 | 5 | anbi2i 624 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 ∈ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)) |
7 | 3, 6 | bitri 277 | . . . 4 ⊢ (𝐴 ∈ ( ℋ ∖ 0ℋ) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)) |
8 | 7 | anbi1i 625 | . . 3 ⊢ ((𝐴 ∈ ( ℋ ∖ 0ℋ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)) ↔ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
9 | fveq2 6670 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑇‘𝑦) = (𝑇‘𝐴)) | |
10 | oveq2 7164 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝑥 ·ℎ 𝐴)) | |
11 | 9, 10 | eqeq12d 2837 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
12 | 11 | rexbidv 3297 | . . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℂ (𝑇‘𝑦) = (𝑥 ·ℎ 𝑦) ↔ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))) |
13 | 12 | elrab 3680 | . . 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 1085 | . . 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 305 | . 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 289 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 ∖ cdif 3933 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℋchba 28696 ·ℎ csm 28698 0ℎc0v 28701 0ℋc0h 28712 eigveccei 28736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-hilex 28776 ax-hv0cl 28780 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-ch0 29030 df-eigvec 29630 |
This theorem is referenced by: eleigvec2 29735 eigvalcl 29738 |
Copyright terms: Public domain | W3C validator |