Theorem eigorthi 11400

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
eigorthi.1	|- A e. ~H
eigorthi.2	|- B e. ~H
eigorthi.3	|- C e. CC
eigorthi.4	|- D e. CC

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		opreq2 4890	. . . 4 |- ((T` B) = (D .h B) -> (A .ih (T` B)) = (A .ih (D .h B)))
2		eigorthi.4	. . . . 5 |- D e. CC
3		eigorthi.1	. . . . 5 |- A e. ~H
4		eigorthi.2	. . . . 5 |- B e. ~H
5		his5 10586	. . . . 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 1191	. . . 4 |- (A .ih (D .h B)) = ((*` D) x. (A .ih B))
7	1, 6	syl6eq 1944	. . 3 |- ((T` B) = (D .h B) -> (A .ih (T` B)) = ((*` D) x. (A .ih B)))
8		opreq1 4889	. . . 4 |- ((T` A) = (C .h A) -> ((T` A) .ih B) = ((C .h A) .ih B))
9		eigorthi.3	. . . . 5 |- C e. CC
10		ax-his3 10584	. . . . 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 1191	. . . 4 |- ((C .h A) .ih B) = (C x. (A .ih B))
12	8, 11	syl6eq 1944	. . 3 |- ((T` A) = (C .h A) -> ((T` A) .ih B) = (C x. (A .ih B)))
13	7, 12	eqeqan12rd 1903	. 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	3, 4	hicli 10581	. . . . . . . 8 |- (A .ih B) e. CC
15	2	cjcli 8017	. . . . . . . . 9 |- (*` D) e. CC
16		mulcan2 6881	. . . . . . . . 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))
17	15, 9, 16	mp3an12 1181	. . . . . . . 8 |- (((A .ih B) e. CC /\ (A .ih B) =/= 0) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (*` D) = C))
18	14, 17	mpan 759	. . . . . . 7 |- ((A .ih B) =/= 0 -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (*` D) = C))
19		eqcom 1886	. . . . . . 7 |- ((*` D) = C <-> C = (*` D))
20	18, 19	syl6bb 595	. . . . . 6 |- ((A .ih B) =/= 0 -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> C = (*` D)))
21	20	biimpcd 172	. . . . 5 |- (((*` D) x. (A .ih B)) = (C x. (A .ih B)) -> ((A .ih B) =/= 0 -> C = (*` D)))
22	21	necon1d 2082	. . . 4 |- (((*` D) x. (A .ih B)) = (C x. (A .ih B)) -> (C =/= (*` D) -> (A .ih B) = 0))
23	22	com12 14	. . 3 |- (C =/= (*` D) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) -> (A .ih B) = 0))
24		opreq2 4890	. . . 4 |- ((A .ih B) = 0 -> ((*` D) x. (A .ih B)) = ((*` D) x. 0))
25		opreq2 4890	. . . . 5 |- ((A .ih B) = 0 -> (C x. (A .ih B)) = (C x. 0))
26	9	mul01i 6594	. . . . . 6 |- (C x. 0) = 0
27	15	mul01i 6594	. . . . . 6 |- ((*` D) x. 0) = 0
28	26, 27	eqtr4i 1911	. . . . 5 |- (C x. 0) = ((*` D) x. 0)
29	25, 28	syl6eq 1944	. . . 4 |- ((A .ih B) = 0 -> (C x. (A .ih B)) = ((*` D) x. 0))
30	24, 29	eqtr4d 1928	. . 3 |- ((A .ih B) = 0 -> ((*` D) x. (A .ih B)) = (C x. (A .ih B)))
31	23, 30	impbid1 575	. 2 |- (C =/= (*` D) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (A .ih B) = 0))
32	13, 31	sylan9bb 599	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:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386   x. cmul 6391  *ccj 7999  ~Hchil 10420   .h csm 10422   .ih csp 10425

This theorem is referenced by:  eigorth 11401

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-hfvmul 10507  ax-hfi 10579  ax-his1 10582  ax-his3 10584

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-re 8001  df-im 8002  df-cj 8003

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