eigorthi - Hilbert Space Explorer

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Theorem eigorthi 32129

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 7419	. . . 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 31378	. . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
6	2, 3, 4, 5	mp3an 1487	. . . 4 ⊢ (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵))
7	1, 6	eqtrdi 2820	. . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵) → (𝐴 ·_ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
8		oveq1 7418	. . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵))
9		eigorthi.3	. . . . 5 ⊢ 𝐶 ∈ ℂ
10		ax-his3 31376	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
11	9, 3, 4, 10	mp3an 1487	. . . 4 ⊢ ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵))
12	8, 11	eqtrdi 2820	. . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
13	7, 12	eqeqan12rd 2784	. 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 31373	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
15	2	cjcli 15219	. . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ
16		mulcan2 11851	. . . . . . . . 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 1477	. . . . . . . 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 2776	. . . . . . 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 2986	. . . 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 7419	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
25		oveq2 7419	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = (𝐶 · 0))
26	9	mul01i 11399	. . . . . 6 ⊢ (𝐶 · 0) = 0
27	15	mul01i 11399	. . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0
28	26, 27	eqtr4i 2795	. . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0)
29	25, 28	eqtrdi 2820	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
30	24, 29	eqtr4d 2807	. . 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 1567 ∈ wcel 2149 ≠ wne 2964 ‘cfv 6537 (class class class)co 7411 ℂcc 11097 0cc0 11099 · cmul 11104 ∗ccj 15146 ℋchba 31211 ·_ℎ csm 31213 ·_ih csp 31214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-hfvmul 31297 ax-hfi 31371 ax-his1 31374 ax-his3 31376

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-cj 15149 df-re 15150 df-im 15151

This theorem is referenced by: eigorth 32130

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