Theorem lnopeq0i 25556

Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 25377 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 25555	. . . . . 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 24559	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y +h z ) e. ~H )
5		fveq2 5792	. . . . . . . . . . . . . . 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 6211	. . . . . . . . . . . . . 14 |- ( x = ( y +h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) )
8	7	eqeq1d 2453	. . . . . . . . . . . . 13 |- ( x = ( y +h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 ) )
9	8	rspccva 3171	. . . . . . . . . . . 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 24564	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y -h z ) e. ~H )
12		fveq2 5792	. . . . . . . . . . . . . . 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 6211	. . . . . . . . . . . . . 14 |- ( x = ( y -h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) )
15	14	eqeq1d 2453	. . . . . . . . . . . . 13 |- ( x = ( y -h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 ) )
16	15	rspccva 3171	. . . . . . . . . . . 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 6211	. . . . . . . . . 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 10535	. . . . . . . . . 10 |- ( 0 - 0 ) = 0
20	18, 19	syl6eq 2508	. . . . . . . . 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 9445	. . . . . . . . . . . . . . . 16 |- _i e. CC
22		hvmulcl 24560	. . . . . . . . . . . . . . . 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 24559	. . . . . . . . . . . . . . 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 5792	. . . . . . . . . . . . . . . . 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 6211	. . . . . . . . . . . . . . . 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 2453	. . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . 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 24564	. . . . . . . . . . . . . . 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 5792	. . . . . . . . . . . . . . . . 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 6211	. . . . . . . . . . . . . . . 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 2453	. . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . 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 6211	. . . . . . . . . . . 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 2508	. . . . . . . . . . 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 6209	. . . . . . . . . 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 10651	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
44	42, 43	syl6eq 2508	. . . . . . . . 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 6211	. . . . . . . 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 9648	. . . . . . . 8 |- ( 0 + 0 ) = 0
47	45, 46	syl6eq 2508	. . . . . . 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 6208	. . . . . 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 10503	. . . . . . 7 |- 4 e. CC
50		4ne0 10522	. . . . . . 7 |- 4 =/= 0
51	49, 50	div0i 10169	. . . . . 6 |- ( 0 / 4 ) = 0
52	48, 51	syl6eq 2508	. . . . 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 2492	. . . 4 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` y ) .ih z ) = 0 )
54	53	ralrimivva 2907	. . 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 25518	. . . 4 |- T : ~H --> ~H
56	55	ho01i 25377	. . 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 5791	. . . . . 6 |- ( T = 0hop -> ( T ` x ) = ( 0hop ` x ) )
59		ho0val 25299	. . . . . 6 |- ( x e. ~H -> ( 0hop ` x ) = 0h )
60	58, 59	sylan9eq 2512	. . . . 5 |- ( ( T = 0hop /\ x e. ~H ) -> ( T ` x ) = 0h )
61	60	oveq1d 6208	. . . 4 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = ( 0h .ih x ) )
62		hi01 24643	. . . . 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 2492	. . 3 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = 0 )
65	64	ralrimiva 2825	. 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 1370   e. wcel 1758   A.wral 2795   ` cfv 5519  (class class class)co 6193   CCcc 9384   0cc0 9386   _ici 9388   + caddc 9389   x. cmul 9391   - cmin 9699   / cdiv 10097   4c4 10477   ~Hchil 24466   +h cva 24467   .h csm 24468   .ih csp 24469   0hc0v 24471   -h cmv 24472   0hopch0o 24490   LinOpclo 24494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cc 8708  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466  ax-hilex 24546  ax-hfvadd 24547  ax-hvcom 24548  ax-hvass 24549  ax-hv0cl 24550  ax-hvaddid 24551  ax-hfvmul 24552  ax-hvmulid 24553  ax-hvmulass 24554  ax-hvdistr1 24555  ax-hvdistr2 24556  ax-hvmul0 24557  ax-hfi 24626  ax-his1 24629  ax-his2 24630  ax-his3 24631  ax-his4 24632  ax-hcompl 24749

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-rlim 13078  df-sum 13275  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-cn 18956  df-cnp 18957  df-lm 18958  df-haus 19044  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cfil 20891  df-cau 20892  df-cmet 20893  df-grpo 23823  df-gid 23824  df-ginv 23825  df-gdiv 23826  df-ablo 23914  df-subgo 23934  df-vc 24069  df-nv 24115  df-va 24118  df-ba 24119  df-sm 24120  df-0v 24121  df-vs 24122  df-nmcv 24123  df-ims 24124  df-dip 24241  df-ssp 24265  df-ph 24358  df-cbn 24409  df-hnorm 24515  df-hba 24516  df-hvsub 24518  df-hlim 24519  df-hcau 24520  df-sh 24754  df-ch 24769  df-oc 24800  df-ch0 24801  df-shs 24856  df-pjh 24943  df-h0op 25297  df-lnop 25390

This theorem is referenced by:  lnopeqi  25557