eigorthi - Hilbert Space Explorer

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Theorem eigorthi 31869

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 7456	. . . 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 31118	. . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
6	2, 3, 4, 5	mp3an 1461	. . . 4 ⊢ (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵))
7	1, 6	eqtrdi 2796	. . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵) → (𝐴 ·_ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
8		oveq1 7455	. . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵))
9		eigorthi.3	. . . . 5 ⊢ 𝐶 ∈ ℂ
10		ax-his3 31116	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
11	9, 3, 4, 10	mp3an 1461	. . . 4 ⊢ ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵))
12	8, 11	eqtrdi 2796	. . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
13	7, 12	eqeqan12rd 2755	. 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 31113	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
15	2	cjcli 15218	. . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ
16		mulcan2 11928	. . . . . . . . 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 1451	. . . . . . . 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 2747	. . . . . . 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 2968	. . . 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 7456	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
25		oveq2 7456	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = (𝐶 · 0))
26	9	mul01i 11480	. . . . . 6 ⊢ (𝐶 · 0) = 0
27	15	mul01i 11480	. . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0
28	26, 27	eqtr4i 2771	. . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0)
29	25, 28	eqtrdi 2796	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
30	24, 29	eqtr4d 2783	. . 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 1537 ∈ wcel 2108 ≠ wne 2946 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 · cmul 11189 ∗ccj 15145 ℋchba 30951 ·_ℎ csm 30953 ·_ih csp 30954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-hfvmul 31037 ax-hfi 31111 ax-his1 31114 ax-his3 31116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-2 12356 df-cj 15148 df-re 15149 df-im 15150

This theorem is referenced by: eigorth 31870

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