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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 6557 | . . . 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 27327 | . . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝐷 ·ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))) | |
6 | 2, 3, 4, 5 | mp3an 1416 | . . . 4 ⊢ (𝐴 ·ih (𝐷 ·ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵)) |
7 | 1, 6 | syl6eq 2660 | . . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·ℎ 𝐵) → (𝐴 ·ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))) |
8 | oveq1 6556 | . . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = ((𝐶 ·ℎ 𝐴) ·ih 𝐵)) | |
9 | eigorthi.3 | . . . . 5 ⊢ 𝐶 ∈ ℂ | |
10 | ax-his3 27325 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·ℎ 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))) | |
11 | 9, 3, 4, 10 | mp3an 1416 | . . . 4 ⊢ ((𝐶 ·ℎ 𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵)) |
12 | 8, 11 | syl6eq 2660 | . . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))) |
13 | 7, 12 | eqeqan12rd 2628 | . 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 27322 | . . . . . . . 8 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
15 | 2 | cjcli 13757 | . . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ |
16 | mulcan2 10544 | . . . . . . . . 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 1406 | . . . . . . . 8 ⊢ (((𝐴 ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ≠ 0) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶)) |
18 | 14, 17 | mpan 702 | . . . . . . 7 ⊢ ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (∗‘𝐷) = 𝐶)) |
19 | eqcom 2617 | . . . . . . 7 ⊢ ((∗‘𝐷) = 𝐶 ↔ 𝐶 = (∗‘𝐷)) | |
20 | 18, 19 | syl6bb 275 | . . . . . 6 ⊢ ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ 𝐶 = (∗‘𝐷))) |
21 | 20 | biimpcd 238 | . . . . 5 ⊢ (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) → ((𝐴 ·ih 𝐵) ≠ 0 → 𝐶 = (∗‘𝐷))) |
22 | 21 | necon1d 2804 | . . . 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 6557 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) | |
25 | oveq2 6557 | . . . . 5 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = (𝐶 · 0)) | |
26 | 9 | mul01i 10105 | . . . . . 6 ⊢ (𝐶 · 0) = 0 |
27 | 15 | mul01i 10105 | . . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0 |
28 | 26, 27 | eqtr4i 2635 | . . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0) |
29 | 25, 28 | syl6eq 2660 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) |
30 | 24, 29 | eqtr4d 2647 | . . 3 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵))) |
31 | 23, 30 | impbid1 214 | . 2 ⊢ (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·ih 𝐵)) = (𝐶 · (𝐴 ·ih 𝐵)) ↔ (𝐴 ·ih 𝐵) = 0)) |
32 | 13, 31 | sylan9bb 732 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 · cmul 9820 ∗ccj 13684 ℋchil 27160 ·ℎ csm 27162 ·ih csp 27163 |
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-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hfvmul 27246 ax-hfi 27320 ax-his1 27323 ax-his3 27325 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 |
This theorem is referenced by: eigorth 28081 |
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