Theorem lnopeq0i 25362

Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 25183 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 25361	. . . . . 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 24365	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y +h z ) e. ~H )
5		fveq2 5686	. . . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . 14 |- ( x = ( y +h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) )
8	7	eqeq1d 2446	. . . . . . . . . . . . 13 |- ( x = ( y +h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y +h z ) ) .ih ( y +h z ) ) = 0 ) )
9	8	rspccva 3067	. . . . . . . . . . . 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 24370	. . . . . . . . . . . 12 |- ( ( y e. ~H /\ z e. ~H ) -> ( y -h z ) e. ~H )
12		fveq2 5686	. . . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . 14 |- ( x = ( y -h z ) -> ( ( T ` x ) .ih x ) = ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) )
15	14	eqeq1d 2446	. . . . . . . . . . . . 13 |- ( x = ( y -h z ) -> ( ( ( T ` x ) .ih x ) = 0 <-> ( ( T ` ( y -h z ) ) .ih ( y -h z ) ) = 0 ) )
16	15	rspccva 3067	. . . . . . . . . . . 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 6104	. . . . . . . . . 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 10423	. . . . . . . . . 10 |- ( 0 - 0 ) = 0
20	18, 19	syl6eq 2486	. . . . . . . . 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 9333	. . . . . . . . . . . . . . . 16 |- _i e. CC
22		hvmulcl 24366	. . . . . . . . . . . . . . . 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 24365	. . . . . . . . . . . . . . 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 5686	. . . . . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . . . 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 2446	. . . . . . . . . . . . . . 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 3067	. . . . . . . . . . . . . 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 24370	. . . . . . . . . . . . . . 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 5686	. . . . . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . . . 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 2446	. . . . . . . . . . . . . . 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 3067	. . . . . . . . . . . . . 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 6104	. . . . . . . . . . . 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 2486	. . . . . . . . . . 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 6102	. . . . . . . . . 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 10539	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
44	42, 43	syl6eq 2486	. . . . . . . . 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 6104	. . . . . . . 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 9536	. . . . . . . 8 |- ( 0 + 0 ) = 0
47	45, 46	syl6eq 2486	. . . . . . 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 6101	. . . . . 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 10391	. . . . . . 7 |- 4 e. CC
50		4ne0 10410	. . . . . . 7 |- 4 =/= 0
51	49, 50	div0i 10057	. . . . . 6 |- ( 0 / 4 ) = 0
52	48, 51	syl6eq 2486	. . . . 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 2470	. . . 4 |- ( ( A. x e. ~H ( ( T ` x ) .ih x ) = 0 /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( T ` y ) .ih z ) = 0 )
54	53	ralrimivva 2803	. . 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 25324	. . . 4 |- T : ~H --> ~H
56	55	ho01i 25183	. . 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 5685	. . . . . 6 |- ( T = 0hop -> ( T ` x ) = ( 0hop ` x ) )
59		ho0val 25105	. . . . . 6 |- ( x e. ~H -> ( 0hop ` x ) = 0h )
60	58, 59	sylan9eq 2490	. . . . 5 |- ( ( T = 0hop /\ x e. ~H ) -> ( T ` x ) = 0h )
61	60	oveq1d 6101	. . . 4 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = ( 0h .ih x ) )
62		hi01 24449	. . . . 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 2470	. . 3 |- ( ( T = 0hop /\ x e. ~H ) -> ( ( T ` x ) .ih x ) = 0 )
65	64	ralrimiva 2794	. 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 1369   e. wcel 1756   A.wral 2710   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   _ici 9276   + caddc 9277   x. cmul 9279   - cmin 9587   / cdiv 9985   4c4 10365   ~Hchil 24272   +h cva 24273   .h csm 24274   .ih csp 24275   0hc0v 24277   -h cmv 24278   0hopch0o 24296   LinOpclo 24300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437  ax-his4 24438  ax-hcompl 24555

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-cn 18806  df-cnp 18807  df-lm 18808  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cfil 20741  df-cau 20742  df-cmet 20743  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-subgo 23740  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-dip 24047  df-ssp 24071  df-ph 24164  df-cbn 24215  df-hnorm 24321  df-hba 24322  df-hvsub 24324  df-hlim 24325  df-hcau 24326  df-sh 24560  df-ch 24575  df-oc 24606  df-ch0 24607  df-shs 24662  df-pjh 24749  df-h0op 25103  df-lnop 25196

This theorem is referenced by:  lnopeqi  25363