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Mirrors > Home > HSE Home > Th. List > eigorthi | Structured version Visualization version GIF version |
Description: A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eigorthi.1 | ⊢ 𝐴 ∈ ℋ |
eigorthi.2 | ⊢ 𝐵 ∈ ℋ |
eigorthi.3 | ⊢ 𝐶 ∈ ℂ |
eigorthi.4 | ⊢ 𝐷 ∈ ℂ |
Ref | Expression |
---|---|
eigorthi | ⊢ ((((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7143 | . . . 4 ⊢ ((𝑇‘𝐵) = (𝐷 ·ℎ 𝐵) → (𝐴 ·ih (𝑇‘𝐵)) = (𝐴 ·ih (𝐷 ·ℎ 𝐵))) | |
2 | eigorthi.4 | . . . . 5 ⊢ 𝐷 ∈ ℂ | |
3 | eigorthi.1 | . . . . 5 ⊢ 𝐴 ∈ ℋ | |
4 | eigorthi.2 | . . . . 5 ⊢ 𝐵 ∈ ℋ | |
5 | his5 28869 | . . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝐷 ·ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))) | |
6 | 2, 3, 4, 5 | mp3an 1458 | . . . 4 ⊢ (𝐴 ·ih (𝐷 ·ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵)) |
7 | 1, 6 | eqtrdi 2849 | . . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·ℎ 𝐵) → (𝐴 ·ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))) |
8 | oveq1 7142 | . . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = ((𝐶 ·ℎ 𝐴) ·ih 𝐵)) | |
9 | eigorthi.3 | . . . . 5 ⊢ 𝐶 ∈ ℂ | |
10 | ax-his3 28867 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·ℎ 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))) | |
11 | 9, 3, 4, 10 | mp3an 1458 | . . . 4 ⊢ ((𝐶 ·ℎ 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵)) |
12 | 8, 11 | eqtrdi 2849 | . . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))) |
13 | 7, 12 | eqeqan12rd 2817 | . 2 ⊢ (((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)))) |
14 | 3, 4 | hicli 28864 | . . . . . . . 8 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
15 | 2 | cjcli 14520 | . . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ |
16 | mulcan2 11267 | . . . . . . . . 9 ⊢ (((∗‘𝐷) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ((𝐴 ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ≠ 0)) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶)) | |
17 | 15, 9, 16 | mp3an12 1448 | . . . . . . . 8 ⊢ (((𝐴 ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ≠ 0) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶)) |
18 | 14, 17 | mpan 689 | . . . . . . 7 ⊢ ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶)) |
19 | eqcom 2805 | . . . . . . 7 ⊢ ((∗‘𝐷) = 𝐶 ↔ 𝐶 = (∗‘𝐷)) | |
20 | 18, 19 | syl6bb 290 | . . . . . 6 ⊢ ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ 𝐶 = (∗‘𝐷))) |
21 | 20 | biimpcd 252 | . . . . 5 ⊢ (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → ((𝐴 ·ih 𝐵) ≠ 0 → 𝐶 = (∗‘𝐷))) |
22 | 21 | necon1d 3009 | . . . 4 ⊢ (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → (𝐶 ≠ (∗‘𝐷) → (𝐴 ·ih 𝐵) = 0)) |
23 | 22 | com12 32 | . . 3 ⊢ (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → (𝐴 ·ih 𝐵) = 0)) |
24 | oveq2 7143 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) | |
25 | oveq2 7143 | . . . . 5 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = (𝐶 · 0)) | |
26 | 9 | mul01i 10819 | . . . . . 6 ⊢ (𝐶 · 0) = 0 |
27 | 15 | mul01i 10819 | . . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0 |
28 | 26, 27 | eqtr4i 2824 | . . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0) |
29 | 25, 28 | eqtrdi 2849 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) |
30 | 24, 29 | eqtr4d 2836 | . . 3 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵))) |
31 | 23, 30 | impbid1 228 | . 2 ⊢ (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (𝐴 ·ih 𝐵) = 0)) |
32 | 13, 31 | sylan9bb 513 | 1 ⊢ ((((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 · cmul 10531 ∗ccj 14447 ℋchba 28702 ·ℎ csm 28704 ·ih csp 28705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-hfvmul 28788 ax-hfi 28862 ax-his1 28865 ax-his3 28867 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-cj 14450 df-re 14451 df-im 14452 |
This theorem is referenced by: eigorth 29621 |
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