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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 7420 | . . . 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 30771 | . . . . 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 2787 | . . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·ℎ 𝐵) → (𝐴 ·ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·ih 𝐵))) |
8 | oveq1 7419 | . . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = ((𝐶 ·ℎ 𝐴) ·ih 𝐵)) | |
9 | eigorthi.3 | . . . . 5 ⊢ 𝐶 ∈ ℂ | |
10 | ax-his3 30769 | . . . . 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 2787 | . . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐶 · (𝐴 ·ih 𝐵))) |
13 | 7, 12 | eqeqan12rd 2746 | . 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 30766 | . . . . . . . 8 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
15 | 2 | cjcli 15123 | . . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ |
16 | mulcan2 11859 | . . . . . . . . 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 2738 | . . . . . . 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 2961 | . . . 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 7420 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) | |
25 | oveq2 7420 | . . . . 5 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = (𝐶 · 0)) | |
26 | 9 | mul01i 11411 | . . . . . 6 ⊢ (𝐶 · 0) = 0 |
27 | 15 | mul01i 11411 | . . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0 |
28 | 26, 27 | eqtr4i 2762 | . . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0) |
29 | 25, 28 | eqtrdi 2787 | . . . 4 ⊢ ((𝐴 ·ih 𝐵) = 0 → (𝐶 · (𝐴 ·ih 𝐵)) = ((∗‘𝐷) · 0)) |
30 | 24, 29 | eqtr4d 2774 | . . 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 509 | 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 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 0cc0 11116 · cmul 11121 ∗ccj 15050 ℋchba 30604 ·ℎ csm 30606 ·ih csp 30607 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-hfvmul 30690 ax-hfi 30764 ax-his1 30767 ax-his3 30769 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-2 12282 df-cj 15053 df-re 15054 df-im 15055 |
This theorem is referenced by: eigorth 31523 |
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