Theorem eigorth 9680

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 e. H~
eigorth.2	|- B e. H~
eigorth.3	|- C e. CC
eigorth.4	|- D e. CC

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 3954	. . . 4 |- ((T` B) = (D .h B) -> (A .ih (T` B)) = (A .ih (D .h B)))
2		eigorth.4	. . . . 5 |- D e. CC
3		eigorth.1	. . . . 5 |- A e. H~
4		eigorth.2	. . . . 5 |- B e. H~
5		his5t 8874	. . . . 5 |- ((D e. CC /\ A e. H~ /\ B e. H~) -> (A .ih (D .h B)) = ((*` D) x. (A .ih B)))
6	2, 3, 4, 5	mp3an 913	. . . 4 |- (A .ih (D .h B)) = ((*` D) x. (A .ih B))
7	1, 6	syl6eq 1515	. . 3 |- ((T` B) = (D .h B) -> (A .ih (T` B)) = ((*` D) x. (A .ih B)))
8		opreq1 3953	. . . 4 |- ((T` A) = (C .h A) -> ((T` A) .ih B) = ((C .h A) .ih B))
9		eigorth.3	. . . . 5 |- C e. CC
10		ax-his3 8872	. . . . 5 |- ((C e. CC /\ A e. H~ /\ B e. H~) -> ((C .h A) .ih B) = (C x. (A .ih B)))
11	9, 3, 4, 10	mp3an 913	. . . 4 |- ((C .h A) .ih B) = (C x. (A .ih B))
12	8, 11	syl6eq 1515	. . 3 |- ((T` A) = (C .h A) -> ((T` A) .ih B) = (C x. (A .ih B)))
13	7, 12	eqeqan12rd 1483	. 2 |- (((T` A) = (C .h A) /\ (T` B) = (D .h B)) -> ((A .ih (T` B)) = ((T` A) .ih B) <-> ((*` D) x. (A .ih B)) = (C x. (A .ih B))))
14	2	cjcl 6699	. . . . . . . 8 |- (*` D) e. CC
15	3, 4	hicl 8869	. . . . . . . 8 |- (A .ih B) e. CC
16		mulcan2t 5662	. . . . . . . . . 10 |- ((((*` D) e. CC /\ C e. CC /\ (A .ih B) e. CC) /\ (A .ih B) =/= 0) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (*` D) = C))
17		df-ne 1579	. . . . . . . . . 10 |- ((A .ih B) =/= 0 <-> -. (A .ih B) = 0)
18	16, 17	sylan2br 453	. . . . . . . . 9 |- ((((*` D) e. CC /\ C e. CC /\ (A .ih B) e. CC) /\ -. (A .ih B) = 0) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (*` D) = C))
19	18	ex 373	. . . . . . . 8 |- (((*` D) e. CC /\ C e. CC /\ (A .ih B) e. CC) -> (-. (A .ih B) = 0 -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (*` D) = C)))
20	14, 9, 15, 19	mp3an 913	. . . . . . 7 |- (-. (A .ih B) = 0 -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (*` D) = C))
21		eqcom 1469	. . . . . . 7 |- ((*` D) = C <-> C = (*` D))
22	20, 21	syl6bb 534	. . . . . 6 |- (-. (A .ih B) = 0 -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> C = (*` D)))
23	22	biimpcd 155	. . . . 5 |- (((*` D) x. (A .ih B)) = (C x. (A .ih B)) -> (-. (A .ih B) = 0 -> C = (*` D)))
24	23	con1d 93	. . . 4 |- (((*` D) x. (A .ih B)) = (C x. (A .ih B)) -> (-. C = (*` D) -> (A .ih B) = 0))
25	24	com12 11	. . 3 |- (-. C = (*` D) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) -> (A .ih B) = 0))
26		opreq2 3954	. . . 4 |- ((A .ih B) = 0 -> ((*` D) x. (A .ih B)) = ((*` D) x. 0))
27		opreq2 3954	. . . . 5 |- ((A .ih B) = 0 -> (C x. (A .ih B)) = (C x. 0))
28	9	mul01 5403	. . . . . 6 |- (C x. 0) = 0
29	14	mul01 5403	. . . . . 6 |- ((*` D) x. 0) = 0
30	28, 29	eqtr4 1490	. . . . 5 |- (C x. 0) = ((*` D) x. 0)
31	27, 30	syl6eq 1515	. . . 4 |- ((A .ih B) = 0 -> (C x. (A .ih B)) = ((*` D) x. 0))
32	26, 31	eqtr4d 1502	. . 3 |- ((A .ih B) = 0 -> ((*` D) x. (A .ih B)) = (C x. (A .ih B)))
33	25, 32	impbid1 515	. 2 |- (-. C = (*` D) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (A .ih B) = 0))
34	13, 33	sylan9bb 538	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 773   = wceq 953   e. wcel 955   =/= wne 1577  ` cfv 3172  (class class class)co 3948  CCcc 5204  0cc0 5206   x. cmul 5211  *ccj 6680  H~chil 8727   .h csm 8729   .ih csp 8732

This theorem is referenced by:  eigortht 9681

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hfvmul 8796  ax-hfi 8867  ax-his1 8870  ax-his3 8872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-re 6682  df-im 6683  df-cj 6684

Copyright terms: Public domain