Theorem lnopeq0i 26602

Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 26423 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form ( T ` x ) .ih x ). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
lnopeq0.1	|- T e. LinOp

Assertion

Ref	Expression
lnopeq0i	|- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 <-> T = 0hop )

Distinct variable group:   x, T

Proof of Theorem lnopeq0i

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnopeq0.1	. . . . . . 7 |- T e. LinOp
2	1	lnopeq0lem2 26601	. . . . . 6 |- ( ( y e. ~H /\ z e. ~H ) -> ( ( T ` y ) .ih z ) = ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) )
3	2	adantl 466	. . . . 5 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` y ) .ih z ) = ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) )
4		hvaddcl 25605	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y +h z ) e. ~H )
5		fveq2 5864	. . . . . . . . . . . . . . 15 |- ( x = ( y +h z ) -> ( T ` x ) = ( T ` ( y +h z ) ) )
6		id 22	. . . . . . . . . . . . . . 15 |- ( x = ( y +h z ) -> x = ( y +h z ) )
7	5, 6	oveq12d 6300	. . . . . . . . . . . . . 14 |- ( x = ( y +h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) )
8	7	eqeq1d 2469	. . . . . . . . . . . . 13 |- ( x = ( y +h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 ) )
9	8	rspccva 3213	. . . . . . . . . . . 12 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y +h z ) e. ~H ) -> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 )
10	4, 9	sylan2 474	. . . . . . . . . . 11 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 )
11		hvsubcl 25610	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y -h z ) e. ~H )
12		fveq2 5864	. . . . . . . . . . . . . . 15 |- ( x = ( y -h z ) -> ( T ` x ) = ( T ` ( y -h z ) ) )
13		id 22	. . . . . . . . . . . . . . 15 |- ( x = ( y -h z ) -> x = ( y -h z ) )
14	12, 13	oveq12d 6300	. . . . . . . . . . . . . 14 |- ( x = ( y -h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) )
15	14	eqeq1d 2469	. . . . . . . . . . . . 13 |- ( x = ( y -h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 ) )
16	15	rspccva 3213	. . . . . . . . . . . 12 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y -h z ) e. ~H ) -> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 )
17	11, 16	sylan2 474	. . . . . . . . . . 11 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 )
18	10, 17	oveq12d 6300	. . . . . . . . . 10 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) = ( 0 - 0 ) )
19		0m0e0 10641	. . . . . . . . . 10 |- ( 0 - 0 ) = 0
20	18, 19	syl6eq 2524	. . . . . . . . 9 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) = 0 )
21		ax-icn 9547	. . . . . . . . . . . . . . . 16 |- _i e. CC
22		hvmulcl 25606	. . . . . . . . . . . . . . . 16 |- ( ( _i e. CC /\ z e. ~H ) -> ( _i .h z ) e. ~H )
23	21, 22	mpan 670	. . . . . . . . . . . . . . 15 |- ( z e. ~H -> ( _i .h z ) e. ~H )
24		hvaddcl 25605	. . . . . . . . . . . . . . 15 |- ( ( y e. ~H /\ ( _i .h z ) e. ~H ) -> ( y +h ( _i .h z ) ) e. ~H )
25	23, 24	sylan2 474	. . . . . . . . . . . . . 14 |- ( ( y e. ~H /\ z e. ~H ) -> ( y +h ( _i .h z ) ) e. ~H )
26		fveq2 5864	. . . . . . . . . . . . . . . . 17 |- ( x = ( y +h ( _i .h z ) ) -> ( T ` x ) = ( T ` ( y +h ( _i .h z ) ) ) )
27		id 22	. . . . . . . . . . . . . . . . 17 |- ( x = ( y +h ( _i .h z ) ) -> x = ( y +h ( _i .h z ) ) )
28	26, 27	oveq12d 6300	. . . . . . . . . . . . . . . 16 |- ( x = ( y +h ( _i .h z ) ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) )
29	28	eqeq1d 2469	. . . . . . . . . . . . . . 15 |- ( x = ( y +h ( _i .h z ) ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) = 0 ) )
30	29	rspccva 3213	. . . . . . . . . . . . . 14 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y +h ( _i .h z ) ) e. ~H ) -> ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) = 0 )
31	25, 30	sylan2 474	. . . . . . . . . . . . 13 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) = 0 )
32		hvsubcl 25610	. . . . . . . . . . . . . . 15 |- ( ( y e. ~H /\ ( _i .h z ) e. ~H ) -> ( y -h ( _i .h z ) ) e. ~H )
33	23, 32	sylan2 474	. . . . . . . . . . . . . 14 |- ( ( y e. ~H /\ z e. ~H ) -> ( y -h ( _i .h z ) ) e. ~H )
34		fveq2 5864	. . . . . . . . . . . . . . . . 17 |- ( x = ( y -h ( _i .h z ) ) -> ( T ` x ) = ( T ` ( y -h ( _i .h z ) ) ) )
35		id 22	. . . . . . . . . . . . . . . . 17 |- ( x = ( y -h ( _i .h z ) ) -> x = ( y -h ( _i .h z ) ) )
36	34, 35	oveq12d 6300	. . . . . . . . . . . . . . . 16 |- ( x = ( y -h ( _i .h z ) ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) )
37	36	eqeq1d 2469	. . . . . . . . . . . . . . 15 |- ( x = ( y -h ( _i .h z ) ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) = 0 ) )
38	37	rspccva 3213	. . . . . . . . . . . . . 14 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y -h ( _i .h z ) ) e. ~H ) -> ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) = 0 )
39	33, 38	sylan2 474	. . . . . . . . . . . . 13 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) = 0 )
40	31, 39	oveq12d 6300	. . . . . . . . . . . 12 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) = ( 0 - 0 ) )
41	40, 19	syl6eq 2524	. . . . . . . . . . 11 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) = 0 )
42	41	oveq2d 6298	. . . . . . . . . 10 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) = ( _i x. 0 ) )
43		it0e0 10757	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
44	42, 43	syl6eq 2524	. . . . . . . . 9 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) = 0 )
45	20, 44	oveq12d 6300	. . . . . . . 8 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) = ( 0 + 0 ) )
46		00id 9750	. . . . . . . 8 |- ( 0 + 0 ) = 0
47	45, 46	syl6eq 2524	. . . . . . 7 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) = 0 )
48	47	oveq1d 6297	. . . . . 6 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) = ( 0 / 4 ) )
49		4cn 10609	. . . . . . 7 |- 4 e. CC
50		4ne0 10628	. . . . . . 7 |- 4 =/= 0
51	49, 50	div0i 10274	. . . . . 6 |- ( 0 / 4 ) = 0
52	48, 51	syl6eq 2524	. . . . 5 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) - ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) ) + ( _i x. ( ( ( T ` ( y +h ( _i .h z ) ) ) .ih ( y +h ( _i .h z ) ) ) - ( ( T ` ( y -h ( _i .h z ) ) ) .ih ( y -h ( _i .h z ) ) ) ) ) ) / 4 ) = 0 )
53	3, 52	eqtrd 2508	. . . 4 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` y ) .ih z ) = 0 )
54	53	ralrimivva 2885	. . 3 |- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 -> A. y e. ~H A. z e. ~H ( ( T ` y ) .ih z ) = 0 )
55	1	lnopfi 26564	. . . 4 |- T : ~H --> ~H
56	55	ho01i 26423	. . 3 |- ( A. y e. ~H A. z e. ~H ( ( T ` y ) .ih z ) = 0 <-> T = 0hop )
57	54, 56	sylib 196	. 2 |- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 -> T = 0hop )
58		fveq1 5863	. . . . . 6 |- ( T = 0hop -> ( T ` x ) = ( 0hop ` x ) )
59		ho0val 26345	. . . . . 6 |- ( x e. ~H -> ( 0hop ` x ) = 0h )
60	58, 59	sylan9eq 2528	. . . . 5 |- ( ( T = 0hop /\ x e. ~H ) -> ( T ` x ) = 0h )
61	60	oveq1d 6297	. . . 4 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = ( 0h .ih x ) )
62		hi01 25689	. . . . 5 |- ( x e. ~H -> ( 0h .ih x ) = 0 )
63	62	adantl 466	. . . 4 |- ( ( T = 0hop /\ x e. ~H ) -> ( 0h .ih x ) = 0 )
64	61, 63	eqtrd 2508	. . 3 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = 0 )
65	64	ralrimiva 2878	. 2 |- ( T = 0hop -> A. x e. ~H ( ( T ` x ) .ih x ) = 0 )
66	57, 65	impbii 188	1 |- ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 <-> T = 0hop )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   _ici 9490   + caddc 9491   x. cmul 9493   - cmin 9801   / cdiv 10202   4c4 10583   ~Hchil 25512   +h cva 25513   .h csm 25514   .ih csp 25515   0hc0v 25517   -h cmv 25518   0hopch0o 25536   LinOpclo 25540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568  ax-hilex 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603  ax-hfi 25672  ax-his1 25675  ax-his2 25676  ax-his3 25677  ax-his4 25678  ax-hcompl 25795

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-cnp 19495  df-lm 19496  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cfil 21429  df-cau 21430  df-cmet 21431  df-grpo 24869  df-gid 24870  df-ginv 24871  df-gdiv 24872  df-ablo 24960  df-subgo 24980  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-vs 25168  df-nmcv 25169  df-ims 25170  df-dip 25287  df-ssp 25311  df-ph 25404  df-cbn 25455  df-hnorm 25561  df-hba 25562  df-hvsub 25564  df-hlim 25565  df-hcau 25566  df-sh 25800  df-ch 25815  df-oc 25846  df-ch0 25847  df-shs 25902  df-pjh 25989  df-h0op 26343  df-lnop 26436

This theorem is referenced by:  lnopeqi  26603