Theorem eigorth 9720

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 3964	. . . 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 8908	. . . . 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 915	. . . 4 |- (A .ih (D .h B)) = ((*` D) x. (A .ih B))
7	1, 6	syl6eq 1521	. . 3 |- ((T` B) = (D .h B) -> (A .ih (T` B)) = ((*` D) x. (A .ih B)))
8		opreq1 3963	. . . 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 8906	. . . . 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 915	. . . 4 |- ((C .h A) .ih B) = (C x. (A .ih B))
12	8, 11	syl6eq 1521	. . 3 |- ((T` A) = (C .h A) -> ((T` A) .ih B) = (C x. (A .ih B)))
13	7, 12	eqeqan12rd 1489	. 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 6714	. . . . . . . 8 |- (*` D) e. CC
15	3, 4	hicl 8903	. . . . . . . 8 |- (A .ih B) e. CC
16		mulcan2t 5672	. . . . . . . . . 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 1585	. . . . . . . . . 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 915	. . . . . . 7 |- (-. (A .ih B) = 0 -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (*` D) = C))
21		eqcom 1475	. . . . . . 7 |- ((*` D) = C <-> C = (*` D))
22	20, 21	syl6bb 535	. . . . . 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 3964	. . . 4 |- ((A .ih B) = 0 -> ((*` D) x. (A .ih B)) = ((*` D) x. 0))
27		opreq2 3964	. . . . 5 |- ((A .ih B) = 0 -> (C x. (A .ih B)) = (C x. 0))
28	9	mul01 5414	. . . . . 6 |- (C x. 0) = 0
29	14	mul01 5414	. . . . . 6 |- ((*` D) x. 0) = 0
30	28, 29	eqtr4 1496	. . . . 5 |- (C x. 0) = ((*` D) x. 0)
31	27, 30	syl6eq 1521	. . . 4 |- ((A .ih B) = 0 -> (C x. (A .ih B)) = ((*` D) x. 0))
32	26, 31	eqtr4d 1508	. . 3 |- ((A .ih B) = 0 -> ((*` D) x. (A .ih B)) = (C x. (A .ih B)))
33	25, 32	impbid1 516	. 2 |- (-. C = (*` D) -> (((*` D) x. (A .ih B)) = (C x. (A .ih B)) <-> (A .ih B) = 0))
34	13, 33	sylan9bb 539	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 774   = wceq 955   e. wcel 957   =/= wne 1583  ` cfv 3178  (class class class)co 3958  CCcc 5215  0cc0 5217   x. cmul 5222  *ccj 6695  H~chil 8743   .h csm 8745   .ih csp 8748

This theorem is referenced by:  eigortht 9721

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608  ax-hfvmul 8830  ax-hfi 8901  ax-his1 8904  ax-his3 8906

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682  df-re 6697  df-im 6698  df-cj 6699

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