eigorthi - Hilbert Space Explorer

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Theorem eigorthi 32040

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 7404	. . . 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 31289	. . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
6	2, 3, 4, 5	mp3an 1482	. . . 4 ⊢ (𝐴 ·_ih (𝐷 ·_ℎ 𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵))
7	1, 6	eqtrdi 2813	. . 3 ⊢ ((𝑇‘𝐵) = (𝐷 ·_ℎ 𝐵) → (𝐴 ·_ih (𝑇‘𝐵)) = ((∗‘𝐷) · (𝐴 ·_ih 𝐵)))
8		oveq1 7403	. . . 4 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵))
9		eigorthi.3	. . . . 5 ⊢ 𝐶 ∈ ℂ
10		ax-his3 31287	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
11	9, 3, 4, 10	mp3an 1482	. . . 4 ⊢ ((𝐶 ·_ℎ 𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵))
12	8, 11	eqtrdi 2813	. . 3 ⊢ ((𝑇‘𝐴) = (𝐶 ·_ℎ 𝐴) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐶 · (𝐴 ·_ih 𝐵)))
13	7, 12	eqeqan12rd 2777	. 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 31284	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
15	2	cjcli 15196	. . . . . . . . 9 ⊢ (∗‘𝐷) ∈ ℂ
16		mulcan2 11825	. . . . . . . . 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 1472	. . . . . . . 8 ⊢ (((𝐴 ·_ih 𝐵) ∈ ℂ ∧ (𝐴 ·_ih 𝐵) ≠ 0) → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ (∗‘𝐷) = 𝐶))
18	14, 17	mpan 700	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ (∗‘𝐷) = 𝐶))
19		eqcom 2769	. . . . . . 7 ⊢ ((∗‘𝐷) = 𝐶 ↔ 𝐶 = (∗‘𝐷))
20	18, 19	bitrdi 289	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) ≠ 0 → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ 𝐶 = (∗‘𝐷)))
21	20	biimpcd 251	. . . . 5 ⊢ (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) → ((𝐴 ·_ih 𝐵) ≠ 0 → 𝐶 = (∗‘𝐷)))
22	21	necon1d 2979	. . . 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 7404	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
25		oveq2 7404	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = (𝐶 · 0))
26	9	mul01i 11373	. . . . . 6 ⊢ (𝐶 · 0) = 0
27	15	mul01i 11373	. . . . . 6 ⊢ ((∗‘𝐷) · 0) = 0
28	26, 27	eqtr4i 2788	. . . . 5 ⊢ (𝐶 · 0) = ((∗‘𝐷) · 0)
29	25, 28	eqtrdi 2813	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐶 · (𝐴 ·_ih 𝐵)) = ((∗‘𝐷) · 0))
30	24, 29	eqtr4d 2800	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)))
31	23, 30	impbid1 227	. 2 ⊢ (𝐶 ≠ (∗‘𝐷) → (((∗‘𝐷) · (𝐴 ·_ih 𝐵)) = (𝐶 · (𝐴 ·_ih 𝐵)) ↔ (𝐴 ·_ih 𝐵) = 0))
32	13, 31	sylan9bb 517	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 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 0cc0 11073 · cmul 11078 ∗ccj 15123 ℋchba 31122 ·_ℎ csm 31124 ·_ih csp 31125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-hfvmul 31208 ax-hfi 31282 ax-his1 31285 ax-his3 31287

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-cj 15126 df-re 15127 df-im 15128

This theorem is referenced by: eigorth 32041

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