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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 7283 | . . . 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 29448 | . . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝐷 ·ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))) | |
6 | 2, 3, 4, 5 | mp3an 1460 | . . . 4 ⊢ (𝐴 ·ih (𝐷 ·ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵)) |
7 | 1, 6 | eqtrdi 2794 | . . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·ℎ 𝐵) → (𝐴 ·ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))) |
8 | oveq1 7282 | . . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = ((𝐶 ·ℎ 𝐴) ·ih 𝐵)) | |
9 | eigorthi.3 | . . . . 5 ⊢ 𝐶 ∈ ℂ | |
10 | ax-his3 29446 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·ℎ 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))) | |
11 | 9, 3, 4, 10 | mp3an 1460 | . . . 4 ⊢ ((𝐶 ·ℎ 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵)) |
12 | 8, 11 | eqtrdi 2794 | . . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))) |
13 | 7, 12 | eqeqan12rd 2753 | . 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 29443 | . . . . . . . 8 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
15 | 2 | cjcli 14880 | . . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ |
16 | mulcan2 11613 | . . . . . . . . 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 1450 | . . . . . . . 8 ⊢ (((𝐴 ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ≠ 0) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶)) |
18 | 14, 17 | mpan 687 | . . . . . . 7 ⊢ ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶)) |
19 | eqcom 2745 | . . . . . . 7 ⊢ ((∗‘𝐷) = 𝐶 ↔ 𝐶 = (∗‘𝐷)) | |
20 | 18, 19 | bitrdi 287 | . . . . . 6 ⊢ ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ 𝐶 = (∗‘𝐷))) |
21 | 20 | biimpcd 248 | . . . . 5 ⊢ (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → ((𝐴 ·ih 𝐵) ≠ 0 → 𝐶 = (∗‘𝐷))) |
22 | 21 | necon1d 2965 | . . . 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 7283 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) | |
25 | oveq2 7283 | . . . . 5 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = (𝐶 · 0)) | |
26 | 9 | mul01i 11165 | . . . . . 6 ⊢ (𝐶 · 0) = 0 |
27 | 15 | mul01i 11165 | . . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0 |
28 | 26, 27 | eqtr4i 2769 | . . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0) |
29 | 25, 28 | eqtrdi 2794 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) |
30 | 24, 29 | eqtr4d 2781 | . . 3 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵))) |
31 | 23, 30 | impbid1 224 | . 2 ⊢ (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (𝐴 ·ih 𝐵) = 0)) |
32 | 13, 31 | sylan9bb 510 | 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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 · cmul 10876 ∗ccj 14807 ℋchba 29281 ·ℎ csm 29283 ·ih csp 29284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-hfvmul 29367 ax-hfi 29441 ax-his1 29444 ax-his3 29446 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-2 12036 df-cj 14810 df-re 14811 df-im 14812 |
This theorem is referenced by: eigorth 30200 |
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