Theorem eigorth 11401

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.

Assertion

Ref	Expression
eigorth	|- ((((A e. ~H /\ B e. ~H) /\ (C e. CC /\ D e. CC)) /\ (((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		fveq2 4681	. . . . . . 7 |- (A = if(A e. ~H, A, 0h) -> (T` A) = (T` if(A e. ~H, A, 0h)))
2		opreq2 4890	. . . . . . 7 |- (A = if(A e. ~H, A, 0h) -> (C .h A) = (C .h if(A e. ~H, A, 0h)))
3	1, 2	eqeq12d 1899	. . . . . 6 |- (A = if(A e. ~H, A, 0h) -> ((T` A) = (C .h A) <-> (T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h))))
4	3	anbi1d 679	. . . . 5 |- (A = if(A e. ~H, A, 0h) -> (((T` A) = (C .h A) /\ (T` B) = (D .h B)) <-> ((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` B) = (D .h B))))
5	4	anbi1d 679	. . . 4 |- (A = if(A e. ~H, A, 0h) -> ((((T` A) = (C .h A) /\ (T` B) = (D .h B)) /\ C =/= (*` D)) <-> (((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` B) = (D .h B)) /\ C =/= (*` D))))
6		opreq1 4889	. . . . . 6 |- (A = if(A e. ~H, A, 0h) -> (A .ih (T` B)) = (if(A e. ~H, A, 0h) .ih (T` B)))
7	1	opreq1d 4897	. . . . . 6 |- (A = if(A e. ~H, A, 0h) -> ((T` A) .ih B) = ((T` if(A e. ~H, A, 0h)) .ih B))
8	6, 7	eqeq12d 1899	. . . . 5 |- (A = if(A e. ~H, A, 0h) -> ((A .ih (T` B)) = ((T` A) .ih B) <-> (if(A e. ~H, A, 0h) .ih (T` B)) = ((T` if(A e. ~H, A, 0h)) .ih B)))
9		opreq1 4889	. . . . . 6 |- (A = if(A e. ~H, A, 0h) -> (A .ih B) = (if(A e. ~H, A, 0h) .ih B))
10	9	eqeq1d 1892	. . . . 5 |- (A = if(A e. ~H, A, 0h) -> ((A .ih B) = 0 <-> (if(A e. ~H, A, 0h) .ih B) = 0))
11	8, 10	bibi12d 691	. . . 4 |- (A = if(A e. ~H, A, 0h) -> (((A .ih (T` B)) = ((T` A) .ih B) <-> (A .ih B) = 0) <-> ((if(A e. ~H, A, 0h) .ih (T` B)) = ((T` if(A e. ~H, A, 0h)) .ih B) <-> (if(A e. ~H, A, 0h) .ih B) = 0)))
12	5, 11	imbi12d 688	. . 3 |- (A = if(A e. ~H, A, 0h) -> (((((T` A) = (C .h A) /\ (T` B) = (D .h B)) /\ C =/= (*` D)) -> ((A .ih (T` B)) = ((T` A) .ih B) <-> (A .ih B) = 0)) <-> ((((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` B) = (D .h B)) /\ C =/= (*` D)) -> ((if(A e. ~H, A, 0h) .ih (T` B)) = ((T` if(A e. ~H, A, 0h)) .ih B) <-> (if(A e. ~H, A, 0h) .ih B) = 0))))
13		fveq2 4681	. . . . . . 7 |- (B = if(B e. ~H, B, 0h) -> (T` B) = (T` if(B e. ~H, B, 0h)))
14		opreq2 4890	. . . . . . 7 |- (B = if(B e. ~H, B, 0h) -> (D .h B) = (D .h if(B e. ~H, B, 0h)))
15	13, 14	eqeq12d 1899	. . . . . 6 |- (B = if(B e. ~H, B, 0h) -> ((T` B) = (D .h B) <-> (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))))
16	15	anbi2d 678	. . . . 5 |- (B = if(B e. ~H, B, 0h) -> (((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` B) = (D .h B)) <-> ((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h)))))
17	16	anbi1d 679	. . . 4 |- (B = if(B e. ~H, B, 0h) -> ((((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` B) = (D .h B)) /\ C =/= (*` D)) <-> (((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ C =/= (*` D))))
18	13	opreq2d 4898	. . . . . 6 |- (B = if(B e. ~H, B, 0h) -> (if(A e. ~H, A, 0h) .ih (T` B)) = (if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))))
19		opreq2 4890	. . . . . 6 |- (B = if(B e. ~H, B, 0h) -> ((T` if(A e. ~H, A, 0h)) .ih B) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)))
20	18, 19	eqeq12d 1899	. . . . 5 |- (B = if(B e. ~H, B, 0h) -> ((if(A e. ~H, A, 0h) .ih (T` B)) = ((T` if(A e. ~H, A, 0h)) .ih B) <-> (if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h))))
21		opreq2 4890	. . . . . 6 |- (B = if(B e. ~H, B, 0h) -> (if(A e. ~H, A, 0h) .ih B) = (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)))
22	21	eqeq1d 1892	. . . . 5 |- (B = if(B e. ~H, B, 0h) -> ((if(A e. ~H, A, 0h) .ih B) = 0 <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0))
23	20, 22	bibi12d 691	. . . 4 |- (B = if(B e. ~H, B, 0h) -> (((if(A e. ~H, A, 0h) .ih (T` B)) = ((T` if(A e. ~H, A, 0h)) .ih B) <-> (if(A e. ~H, A, 0h) .ih B) = 0) <-> ((if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0)))
24	17, 23	imbi12d 688	. . 3 |- (B = if(B e. ~H, B, 0h) -> (((((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` B) = (D .h B)) /\ C =/= (*` D)) -> ((if(A e. ~H, A, 0h) .ih (T` B)) = ((T` if(A e. ~H, A, 0h)) .ih B) <-> (if(A e. ~H, A, 0h) .ih B) = 0)) <-> ((((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ C =/= (*` D)) -> ((if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0))))
25		opreq1 4889	. . . . . . 7 |- (C = if(C e. CC, C, 0) -> (C .h if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)))
26	25	eqeq2d 1895	. . . . . 6 |- (C = if(C e. CC, C, 0) -> ((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) <-> (T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h))))
27	26	anbi1d 679	. . . . 5 |- (C = if(C e. CC, C, 0) -> (((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) <-> ((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h)))))
28		neeq1 2024	. . . . 5 |- (C = if(C e. CC, C, 0) -> (C =/= (*` D) <-> if(C e. CC, C, 0) =/= (*` D)))
29	27, 28	anbi12d 690	. . . 4 |- (C = if(C e. CC, C, 0) -> ((((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ C =/= (*` D)) <-> (((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ if(C e. CC, C, 0) =/= (*` D))))
30	29	imbi1d 675	. . 3 |- (C = if(C e. CC, C, 0) -> (((((T` if(A e. ~H, A, 0h)) = (C .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ C =/= (*` D)) -> ((if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0)) <-> ((((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ if(C e. CC, C, 0) =/= (*` D)) -> ((if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0))))
31		opreq1 4889	. . . . . . 7 |- (D = if(D e. CC, D, 0) -> (D .h if(B e. ~H, B, 0h)) = (if(D e. CC, D, 0) .h if(B e. ~H, B, 0h)))
32	31	eqeq2d 1895	. . . . . 6 |- (D = if(D e. CC, D, 0) -> ((T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h)) <-> (T` if(B e. ~H, B, 0h)) = (if(D e. CC, D, 0) .h if(B e. ~H, B, 0h))))
33	32	anbi2d 678	. . . . 5 |- (D = if(D e. CC, D, 0) -> (((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) <-> ((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (if(D e. CC, D, 0) .h if(B e. ~H, B, 0h)))))
34		fveq2 4681	. . . . . 6 |- (D = if(D e. CC, D, 0) -> (*` D) = (*` if(D e. CC, D, 0)))
35	34	neeq2d 2029	. . . . 5 |- (D = if(D e. CC, D, 0) -> (if(C e. CC, C, 0) =/= (*` D) <-> if(C e. CC, C, 0) =/= (*` if(D e. CC, D, 0))))
36	33, 35	anbi12d 690	. . . 4 |- (D = if(D e. CC, D, 0) -> ((((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ if(C e. CC, C, 0) =/= (*` D)) <-> (((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (if(D e. CC, D, 0) .h if(B e. ~H, B, 0h))) /\ if(C e. CC, C, 0) =/= (*` if(D e. CC, D, 0)))))
37	36	imbi1d 675	. . 3 |- (D = if(D e. CC, D, 0) -> (((((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (D .h if(B e. ~H, B, 0h))) /\ if(C e. CC, C, 0) =/= (*` D)) -> ((if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0)) <-> ((((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (if(D e. CC, D, 0) .h if(B e. ~H, B, 0h))) /\ if(C e. CC, C, 0) =/= (*` if(D e. CC, D, 0))) -> ((if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0))))
38		ax-hv0cl 10505	. . . . 5 |- 0h e. ~H
39	38	elimel 3025	. . . 4 |- if(A e. ~H, A, 0h) e. ~H
40	38	elimel 3025	. . . 4 |- if(B e. ~H, B, 0h) e. ~H
41		0cn 6481	. . . . 5 |- 0 e. CC
42	41	elimel 3025	. . . 4 |- if(C e. CC, C, 0) e. CC
43	41	elimel 3025	. . . 4 |- if(D e. CC, D, 0) e. CC
44	39, 40, 42, 43	eigorthi 11400	. . 3 |- ((((T` if(A e. ~H, A, 0h)) = (if(C e. CC, C, 0) .h if(A e. ~H, A, 0h)) /\ (T` if(B e. ~H, B, 0h)) = (if(D e. CC, D, 0) .h if(B e. ~H, B, 0h))) /\ if(C e. CC, C, 0) =/= (*` if(D e. CC, D, 0))) -> ((if(A e. ~H, A, 0h) .ih (T` if(B e. ~H, B, 0h))) = ((T` if(A e. ~H, A, 0h)) .ih if(B e. ~H, B, 0h)) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = 0))
45	12, 24, 30, 37, 44	dedth4h 3017	. 2 |- (((A e. ~H /\ B e. ~H) /\ (C e. CC /\ D e. CC)) -> ((((T` A) = (C .h A) /\ (T` B) = (D .h B)) /\ C =/= (*` D)) -> ((A .ih (T` B)) = ((T` A) .ih B) <-> (A .ih B) = 0)))
46	45	imp 377	1 |- ((((A e. ~H /\ B e. ~H) /\ (C e. CC /\ D e. CC)) /\ (((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  ifcif 2982  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  *ccj 7999  ~Hchil 10420   .h csm 10422  0hc0v 10423   .ih csp 10425

This theorem is referenced by:  eighmorth 11525

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-hv0cl 10505  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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