eigorthi - Hilbert Space Explorer

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Theorem eigorthi 32189

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 7418	. . . 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 31438	. . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
6	2, 3, 4, 5	mp3an 1490	. . . 4 ⊢ (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵))
7	1, 6	eqtrdi 2814	. . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵) → (𝐴 ·_ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
8		oveq1 7417	. . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵))
9		eigorthi.3	. . . . 5 ⊢ 𝐶 ∈ ℂ
10		ax-his3 31436	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
11	9, 3, 4, 10	mp3an 1490	. . . 4 ⊢ ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵))
12	8, 11	eqtrdi 2814	. . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
13	7, 12	eqeqan12rd 2778	. 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 31433	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
15	2	cjcli 15216	. . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ
16		mulcan2 11847	. . . . . . . . 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 1480	. . . . . . . 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 2770	. . . . . . 7 ⊢ ((∗‘𝐷) = 𝐶 ↔ 𝐶 = (∗‘𝐷))
20	18, 19	bitrdi 290	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ 𝐶 = (∗‘𝐷)))
21	20	biimpcd 252	. . . . 5 ⊢ (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) → ((𝐴 ·_ih 𝐵) ≠ 0 → 𝐶 = (∗‘𝐷)))
22	21	necon1d 2980	. . . 4 ⊢ (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) → (𝐶 ≠ (∗‘𝐷) → (𝐴 ·_ih 𝐵) = 0))
23	22	com12 33	. . 3 ⊢ (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) → (𝐴 ·_ih 𝐵) = 0))
24		oveq2 7418	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
25		oveq2 7418	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = (𝐶 · 0))
26	9	mul01i 11395	. . . . . 6 ⊢ (𝐶 · 0) = 0
27	15	mul01i 11395	. . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0
28	26, 27	eqtr4i 2789	. . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0)
29	25, 28	eqtrdi 2814	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
30	24, 29	eqtr4d 2801	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)))
31	23, 30	impbid1 228	. 2 ⊢ (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ (𝐴 ·_ih 𝐵) = 0))
32	13, 31	sylan9bb 518	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 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 0cc0 11095 · cmul 11100 ∗ccj 15143 ℋchba 31271 ·_ℎ csm 31273 ·_ih csp 31274

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-hfvmul 31357 ax-hfi 31431 ax-his1 31434 ax-his3 31436

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-cj 15146 df-re 15147 df-im 15148

This theorem is referenced by: eigorth 32190

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