eigorthi - Hilbert Space Explorer

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Theorem eigorthi 31816

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.)

**Hypotheses**
Ref	Expression
eigorthi.1	⊢ 𝐴 ∈ ℋ
eigorthi.2	⊢ 𝐵 ∈ ℋ
eigorthi.3	⊢ 𝐶 ∈ ℂ
eigorthi.4	⊢ 𝐷 ∈ ℂ

**Assertion**
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))

**Proof of Theorem eigorthi**
Step	Hyp	Ref	Expression
1		oveq2 7377	. . . 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 31065	. . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
6	2, 3, 4, 5	mp3an 1463	. . . 4 ⊢ (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵))
7	1, 6	eqtrdi 2780	. . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵) → (𝐴 ·_ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
8		oveq1 7376	. . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵))
9		eigorthi.3	. . . . 5 ⊢ 𝐶 ∈ ℂ
10		ax-his3 31063	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
11	9, 3, 4, 10	mp3an 1463	. . . 4 ⊢ ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵))
12	8, 11	eqtrdi 2780	. . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
13	7, 12	eqeqan12rd 2744	. 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 31060	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
15	2	cjcli 15111	. . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ
16		mulcan2 11792	. . . . . . . . 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 1453	. . . . . . . 8 ⊢ (((𝐴 ·_ih 𝐵) ∈ ℂ ∧ (𝐴 ·_ih 𝐵) ≠ 0) → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ (∗‘𝐷) = 𝐶))
18	14, 17	mpan 690	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ (∗‘𝐷) = 𝐶))
19		eqcom 2736	. . . . . . 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 249	. . . . 5 ⊢ (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) → ((𝐴 ·_ih 𝐵) ≠ 0 → 𝐶 = (∗‘𝐷)))
22	21	necon1d 2947	. . . 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 7377	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
25		oveq2 7377	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = (𝐶 · 0))
26	9	mul01i 11340	. . . . . 6 ⊢ (𝐶 · 0) = 0
27	15	mul01i 11340	. . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0
28	26, 27	eqtr4i 2755	. . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0)
29	25, 28	eqtrdi 2780	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
30	24, 29	eqtr4d 2767	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)))
31	23, 30	impbid1 225	. 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 0cc0 11044 · cmul 11049 ∗ccj 15038 ℋchba 30898 ·_ℎ csm 30900 ·_ih csp 30901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-hfvmul 30984 ax-hfi 31058 ax-his1 31061 ax-his3 31063

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-cj 15041 df-re 15042 df-im 15043

This theorem is referenced by: eigorth 31817

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