Theorem eigorth 9758

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.

Hypotheses

Ref	Expression
eigorth.1	⊢ A ∈ ℋ
eigorth.2	⊢ B ∈ ℋ
eigorth.3	⊢ C ∈ ℂ
eigorth.4	⊢ D ∈ ℂ

Assertion

Ref	Expression
eigorth	⊢ ((((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 eigorth

Step	Hyp	Ref	Expression
1		opreq2 3975	. . . 4 ⊢ ((T ‘B) = (D ·h B) → (A ·ih (T ‘B)) = (A ·ih (D ·h B)))
2		eigorth.4	. . . . 5 ⊢ D ∈ ℂ
3		eigorth.1	. . . . 5 ⊢ A ∈ ℋ
4		eigorth.2	. . . . 5 ⊢ B ∈ ℋ
5		his5t 8948	. . . . 5 ⊢ ((D ∈ ℂ ⋀ A ∈ ℋ ⋀ B ∈ ℋ ) → (A ·ih (D ·h B)) = ((∗ ‘D) · (A ·ih B)))
6	2, 3, 4, 5	mp3an 918	. . . 4 ⊢ (A ·ih (D ·h B)) = ((∗ ‘D) · (A ·ih B))
7	1, 6	syl6eq 1526	. . 3 ⊢ ((T ‘B) = (D ·h B) → (A ·ih (T ‘B)) = ((∗ ‘D) · (A ·ih B)))
8		opreq1 3974	. . . 4 ⊢ ((T ‘A) = (C ·h A) → ((T ‘A) ·ih B) = ((C ·h A) ·ih B))
9		eigorth.3	. . . . 5 ⊢ C ∈ ℂ
10		ax-his3 8946	. . . . 5 ⊢ ((C ∈ ℂ ⋀ A ∈ ℋ ⋀ B ∈ ℋ ) → ((C ·h A) ·ih B) = (C · (A ·ih B)))
11	9, 3, 4, 10	mp3an 918	. . . 4 ⊢ ((C ·h A) ·ih B) = (C · (A ·ih B))
12	8, 11	syl6eq 1526	. . 3 ⊢ ((T ‘A) = (C ·h A) → ((T ‘A) ·ih B) = (C · (A ·ih B)))
13	7, 12	eqeqan12rd 1494	. 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	2	cjcl 6768	. . . . . . . 8 ⊢ (∗ ‘D) ∈ ℂ
15	3, 4	hicl 8943	. . . . . . . 8 ⊢ (A ·ih B) ∈ ℂ
16		mulcan2tOLD 5705	. . . . . . . . . 10 ⊢ ((((∗ ‘D) ∈ ℂ ⋀ C ∈ ℂ ⋀ (A ·ih B) ∈ ℂ) ⋀ (A ·ih B) ≠ 0) → (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) ↔ (∗ ‘D) = C))
17		df-ne 1590	. . . . . . . . . 10 ⊢ ((A ·ih B) ≠ 0 ↔ ¬ (A ·ih B) = 0)
18	16, 17	sylan2br 455	. . . . . . . . 9 ⊢ ((((∗ ‘D) ∈ ℂ ⋀ C ∈ ℂ ⋀ (A ·ih B) ∈ ℂ) ⋀ ¬ (A ·ih B) = 0) → (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) ↔ (∗ ‘D) = C))
19	18	ex 373	. . . . . . . 8 ⊢ (((∗ ‘D) ∈ ℂ ⋀ C ∈ ℂ ⋀ (A ·ih B) ∈ ℂ) → (¬ (A ·ih B) = 0 → (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) ↔ (∗ ‘D) = C)))
20	14, 9, 15, 19	mp3an 918	. . . . . . 7 ⊢ (¬ (A ·ih B) = 0 → (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) ↔ (∗ ‘D) = C))
21		eqcom 1480	. . . . . . 7 ⊢ ((∗ ‘D) = C ↔ C = (∗ ‘D))
22	20, 21	syl6bb 538	. . . . . 6 ⊢ (¬ (A ·ih B) = 0 → (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) ↔ C = (∗ ‘D)))
23	22	biimpcd 155	. . . . 5 ⊢ (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) → (¬ (A ·ih B) = 0 → C = (∗ ‘D)))
24	23	con1d 93	. . . 4 ⊢ (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) → (¬ C = (∗ ‘D) → (A ·ih B) = 0))
25	24	com12 11	. . 3 ⊢ (¬ C = (∗ ‘D) → (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) → (A ·ih B) = 0))
26		opreq2 3975	. . . 4 ⊢ ((A ·ih B) = 0 → ((∗ ‘D) · (A ·ih B)) = ((∗ ‘D) · 0))
27		opreq2 3975	. . . . 5 ⊢ ((A ·ih B) = 0 → (C · (A ·ih B)) = (C · 0))
28	9	mul01 5443	. . . . . 6 ⊢ (C · 0) = 0
29	14	mul01 5443	. . . . . 6 ⊢ ((∗ ‘D) · 0) = 0
30	28, 29	eqtr4 1501	. . . . 5 ⊢ (C · 0) = ((∗ ‘D) · 0)
31	27, 30	syl6eq 1526	. . . 4 ⊢ ((A ·ih B) = 0 → (C · (A ·ih B)) = ((∗ ‘D) · 0))
32	26, 31	eqtr4d 1513	. . 3 ⊢ ((A ·ih B) = 0 → ((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)))
33	25, 32	impbid1 519	. 2 ⊢ (¬ C = (∗ ‘D) → (((∗ ‘D) · (A ·ih B)) = (C · (A ·ih B)) ↔ (A ·ih B) = 0))
34	13, 33	sylan9bb 542	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))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960   ≠ wne 1588   ‘cfv 3188  (class class class)co 3969  ℂcc 5244  0cc0 5246   · cmul 5251  ∗ccj 6750   ℋ chil 8783   ·_h csm 8785   ·_ih csp 8788

This theorem is referenced by:  eigortht 9759

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hfvmul 8870  ax-hfi 8941  ax-his1 8944  ax-his3 8946

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-re 6752  df-im 6753  df-cj 6754

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