Theorem eigorthi 28696

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. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)

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		oveq2 6658	. . . 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 27943	. . . . 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 1424	. . . 4 |- ( A .ih ( D .h B ) ) = ( ( * ` D ) x. ( A .ih B ) )
7	1, 6	syl6eq 2672	. . 3 |- ( ( T ` B ) = ( D .h B ) -> ( A .ih ( T ` B ) ) = ( ( * ` D ) x. ( A .ih B ) ) )
8		oveq1 6657	. . . 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 27941	. . . . 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 1424	. . . 4 |- ( ( C .h A ) .ih B ) = ( C x. ( A .ih B ) )
12	8, 11	syl6eq 2672	. . 3 |- ( ( T ` A ) = ( C .h A ) -> ( ( T ` A ) .ih B ) = ( C x. ( A .ih B ) ) )
13	7, 12	eqeqan12rd 2640	. 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 27938	. . . . . . . 8 |- ( A .ih B ) e. CC
15	2	cjcli 13909	. . . . . . . . 9 |- ( * ` D ) e. CC
16		mulcan2 10665	. . . . . . . . 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 1414	. . . . . . . 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 706	. . . . . . 7 |- ( ( A .ih B ) =/= 0 -> ( ( ( * ` D ) x. ( A .ih B ) ) = ( C x. ( A .ih B ) ) <-> ( * ` D ) = C ) )
19		eqcom 2629	. . . . . . 7 |- ( ( * ` D ) = C <-> C = ( * ` D ) )
20	18, 19	syl6bb 276	. . . . . 6 |- ( ( A .ih B ) =/= 0 -> ( ( ( * ` D ) x. ( A .ih B ) ) = ( C x. ( A .ih B ) ) <-> C = ( * ` D ) ) )
21	20	biimpcd 239	. . . . 5 |- ( ( ( * ` D ) x. ( A .ih B ) ) = ( C x. ( A .ih B ) ) -> ( ( A .ih B ) =/= 0 -> C = ( * ` D ) ) )
22	21	necon1d 2816	. . . 4 |- ( ( ( * ` D ) x. ( A .ih B ) ) = ( C x. ( A .ih B ) ) -> ( C =/= ( * ` D ) -> ( A .ih B ) = 0 ) )
23	22	com12 32	. . 3 |- ( C =/= ( * ` D ) -> ( ( ( * ` D ) x. ( A .ih B ) ) = ( C x. ( A .ih B ) ) -> ( A .ih B ) = 0 ) )
24		oveq2 6658	. . . 4 |- ( ( A .ih B ) = 0 -> ( ( * ` D ) x. ( A .ih B ) ) = ( ( * ` D ) x. 0 ) )
25		oveq2 6658	. . . . 5 |- ( ( A .ih B ) = 0 -> ( C x. ( A .ih B ) ) = ( C x. 0 ) )
26	9	mul01i 10226	. . . . . 6 |- ( C x. 0 ) = 0
27	15	mul01i 10226	. . . . . 6 |- ( ( * ` D ) x. 0 ) = 0
28	26, 27	eqtr4i 2647	. . . . 5 |- ( C x. 0 ) = ( ( * ` D ) x. 0 )
29	25, 28	syl6eq 2672	. . . 4 |- ( ( A .ih B ) = 0 -> ( C x. ( A .ih B ) ) = ( ( * ` D ) x. 0 ) )
30	24, 29	eqtr4d 2659	. . 3 |- ( ( A .ih B ) = 0 -> ( ( * ` D ) x. ( A .ih B ) ) = ( C x. ( A .ih B ) ) )
31	23, 30	impbid1 215	. 2 |- ( C =/= ( * ` D ) -> ( ( ( * ` D ) x. ( A .ih B ) ) = ( C x. ( A .ih B ) ) <-> ( A .ih B ) = 0 ) )
32	13, 31	sylan9bb 736	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 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   *ccj 13836   ~Hchil 27776   .h csm 27778   .ih csp 27779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hfvmul 27862  ax-hfi 27936  ax-his1 27939  ax-his3 27941

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-cj 13839  df-re 13840  df-im 13841

This theorem is referenced by:  eigorth  28697