Theorem leopnmid 28997

Description: A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
leopnmid	|- ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) -> T <_op ( ( normop ` T ) .op Iop ) )

Proof of Theorem leopnmid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmopre 28782	. . . . 5 |- ( ( T e. HrmOp /\ x e. ~H ) -> ( ( T ` x ) .ih x ) e. RR )
2	1	adantlr 751	. . . 4 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( T ` x ) .ih x ) e. RR )
3	1	recnd 10068	. . . . . 6 |- ( ( T e. HrmOp /\ x e. ~H ) -> ( ( T ` x ) .ih x ) e. CC )
4	3	abscld 14175	. . . . 5 |- ( ( T e. HrmOp /\ x e. ~H ) -> ( abs ` ( ( T ` x ) .ih x ) ) e. RR )
5	4	adantlr 751	. . . 4 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( abs ` ( ( T ` x ) .ih x ) ) e. RR )
6		idhmop 28841	. . . . . . 7 |- Iop e. HrmOp
7		hmopm 28880	. . . . . . 7 |- ( ( ( normop ` T ) e. RR /\ Iop e. HrmOp ) -> ( ( normop ` T ) .op Iop ) e. HrmOp )
8	6, 7	mpan2 707	. . . . . 6 |- ( ( normop ` T ) e. RR -> ( ( normop ` T ) .op Iop ) e. HrmOp )
9		hmopre 28782	. . . . . 6 |- ( ( ( ( normop ` T ) .op Iop ) e. HrmOp /\ x e. ~H ) -> ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) e. RR )
10	8, 9	sylan 488	. . . . 5 |- ( ( ( normop ` T ) e. RR /\ x e. ~H ) -> ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) e. RR )
11	10	adantll 750	. . . 4 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) e. RR )
12	1	leabsd 14153	. . . . 5 |- ( ( T e. HrmOp /\ x e. ~H ) -> ( ( T ` x ) .ih x ) <_ ( abs ` ( ( T ` x ) .ih x ) ) )
13	12	adantlr 751	. . . 4 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( T ` x ) .ih x ) <_ ( abs ` ( ( T ` x ) .ih x ) ) )
14		hmopf 28733	. . . . . . . 8 |- ( T e. HrmOp -> T : ~H --> ~H )
15		ffvelrn 6357	. . . . . . . . 9 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( T ` x ) e. ~H )
16		normcl 27982	. . . . . . . . 9 |- ( ( T ` x ) e. ~H -> ( normh ` ( T ` x ) ) e. RR )
17	15, 16	syl 17	. . . . . . . 8 |- ( ( T : ~H --> ~H /\ x e. ~H ) -> ( normh ` ( T ` x ) ) e. RR )
18	14, 17	sylan 488	. . . . . . 7 |- ( ( T e. HrmOp /\ x e. ~H ) -> ( normh ` ( T ` x ) ) e. RR )
19	18	adantlr 751	. . . . . 6 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( normh ` ( T ` x ) ) e. RR )
20		normcl 27982	. . . . . . 7 |- ( x e. ~H -> ( normh ` x ) e. RR )
21	20	adantl 482	. . . . . 6 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( normh ` x ) e. RR )
22	19, 21	remulcld 10070	. . . . 5 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( normh ` ( T ` x ) ) x. ( normh ` x ) ) e. RR )
23	14, 15	sylan 488	. . . . . . 7 |- ( ( T e. HrmOp /\ x e. ~H ) -> ( T ` x ) e. ~H )
24		bcs 28038	. . . . . . 7 |- ( ( ( T ` x ) e. ~H /\ x e. ~H ) -> ( abs ` ( ( T ` x ) .ih x ) ) <_ ( ( normh ` ( T ` x ) ) x. ( normh ` x ) ) )
25	23, 24	sylancom 701	. . . . . 6 |- ( ( T e. HrmOp /\ x e. ~H ) -> ( abs ` ( ( T ` x ) .ih x ) ) <_ ( ( normh ` ( T ` x ) ) x. ( normh ` x ) ) )
26	25	adantlr 751	. . . . 5 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( abs ` ( ( T ` x ) .ih x ) ) <_ ( ( normh ` ( T ` x ) ) x. ( normh ` x ) ) )
27		remulcl 10021	. . . . . . . . 9 |- ( ( ( normop ` T ) e. RR /\ ( normh ` x ) e. RR ) -> ( ( normop ` T ) x. ( normh ` x ) ) e. RR )
28	20, 27	sylan2 491	. . . . . . . 8 |- ( ( ( normop ` T ) e. RR /\ x e. ~H ) -> ( ( normop ` T ) x. ( normh ` x ) ) e. RR )
29	28	adantll 750	. . . . . . 7 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( normop ` T ) x. ( normh ` x ) ) e. RR )
30		normge0 27983	. . . . . . . . 9 |- ( x e. ~H -> 0 <_ ( normh ` x ) )
31	20, 30	jca 554	. . . . . . . 8 |- ( x e. ~H -> ( ( normh ` x ) e. RR /\ 0 <_ ( normh ` x ) ) )
32	31	adantl 482	. . . . . . 7 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( normh ` x ) e. RR /\ 0 <_ ( normh ` x ) ) )
33		hmoplin 28801	. . . . . . . . 9 |- ( T e. HrmOp -> T e. LinOp )
34		elbdop2 28730	. . . . . . . . . 10 |- ( T e. BndLinOp <-> ( T e. LinOp /\ ( normop ` T ) e. RR ) )
35	34	biimpri 218	. . . . . . . . 9 |- ( ( T e. LinOp /\ ( normop ` T ) e. RR ) -> T e. BndLinOp )
36	33, 35	sylan 488	. . . . . . . 8 |- ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) -> T e. BndLinOp )
37		nmbdoplb 28884	. . . . . . . 8 |- ( ( T e. BndLinOp /\ x e. ~H ) -> ( normh ` ( T ` x ) ) <_ ( ( normop ` T ) x. ( normh ` x ) ) )
38	36, 37	sylan 488	. . . . . . 7 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( normh ` ( T ` x ) ) <_ ( ( normop ` T ) x. ( normh ` x ) ) )
39		lemul1a 10877	. . . . . . 7 |- ( ( ( ( normh ` ( T ` x ) ) e. RR /\ ( ( normop ` T ) x. ( normh ` x ) ) e. RR /\ ( ( normh ` x ) e. RR /\ 0 <_ ( normh ` x ) ) ) /\ ( normh ` ( T ` x ) ) <_ ( ( normop ` T ) x. ( normh ` x ) ) ) -> ( ( normh ` ( T ` x ) ) x. ( normh ` x ) ) <_ ( ( ( normop ` T ) x. ( normh ` x ) ) x. ( normh ` x ) ) )
40	19, 29, 32, 38, 39	syl31anc 1329	. . . . . 6 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( normh ` ( T ` x ) ) x. ( normh ` x ) ) <_ ( ( ( normop ` T ) x. ( normh ` x ) ) x. ( normh ` x ) ) )
41		recn 10026	. . . . . . . . . 10 |- ( ( normop ` T ) e. RR -> ( normop ` T ) e. CC )
42	41	ad2antlr 763	. . . . . . . . 9 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( normop ` T ) e. CC )
43	21	recnd 10068	. . . . . . . . 9 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( normh ` x ) e. CC )
44	42, 43, 43	mulassd 10063	. . . . . . . 8 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( normop ` T ) x. ( normh ` x ) ) x. ( normh ` x ) ) = ( ( normop ` T ) x. ( ( normh ` x ) x. ( normh ` x ) ) ) )
45		simpr 477	. . . . . . . . . 10 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> x e. ~H )
46		ax-his3 27941	. . . . . . . . . 10 |- ( ( ( normop ` T ) e. CC /\ x e. ~H /\ x e. ~H ) -> ( ( ( normop ` T ) .h x ) .ih x ) = ( ( normop ` T ) x. ( x .ih x ) ) )
47	42, 45, 45, 46	syl3anc 1326	. . . . . . . . 9 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( normop ` T ) .h x ) .ih x ) = ( ( normop ` T ) x. ( x .ih x ) ) )
48	20	recnd 10068	. . . . . . . . . . . . 13 |- ( x e. ~H -> ( normh ` x ) e. CC )
49	48	sqvald 13005	. . . . . . . . . . . 12 |- ( x e. ~H -> ( ( normh ` x ) ^ 2 ) = ( ( normh ` x ) x. ( normh ` x ) ) )
50		normsq 27991	. . . . . . . . . . . 12 |- ( x e. ~H -> ( ( normh ` x ) ^ 2 ) = ( x .ih x ) )
51	49, 50	eqtr3d 2658	. . . . . . . . . . 11 |- ( x e. ~H -> ( ( normh ` x ) x. ( normh ` x ) ) = ( x .ih x ) )
52	51	oveq2d 6666	. . . . . . . . . 10 |- ( x e. ~H -> ( ( normop ` T ) x. ( ( normh ` x ) x. ( normh ` x ) ) ) = ( ( normop ` T ) x. ( x .ih x ) ) )
53	52	adantl 482	. . . . . . . . 9 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( normop ` T ) x. ( ( normh ` x ) x. ( normh ` x ) ) ) = ( ( normop ` T ) x. ( x .ih x ) ) )
54	47, 53	eqtr4d 2659	. . . . . . . 8 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( normop ` T ) .h x ) .ih x ) = ( ( normop ` T ) x. ( ( normh ` x ) x. ( normh ` x ) ) ) )
55	44, 54	eqtr4d 2659	. . . . . . 7 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( normop ` T ) x. ( normh ` x ) ) x. ( normh ` x ) ) = ( ( ( normop ` T ) .h x ) .ih x ) )
56		hoif 28613	. . . . . . . . . . 11 |- Iop : ~H -1-1-onto-> ~H
57		f1of 6137	. . . . . . . . . . 11 |- ( Iop : ~H -1-1-onto-> ~H -> Iop : ~H --> ~H )
58	56, 57	mp1i 13	. . . . . . . . . 10 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> Iop : ~H --> ~H )
59		homval 28600	. . . . . . . . . 10 |- ( ( ( normop ` T ) e. CC /\ Iop : ~H --> ~H /\ x e. ~H ) -> ( ( ( normop ` T ) .op Iop ) ` x ) = ( ( normop ` T ) .h ( Iop ` x ) ) )
60	42, 58, 45, 59	syl3anc 1326	. . . . . . . . 9 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( normop ` T ) .op Iop ) ` x ) = ( ( normop ` T ) .h ( Iop ` x ) ) )
61		hoival 28614	. . . . . . . . . . 11 |- ( x e. ~H -> ( Iop ` x ) = x )
62	61	oveq2d 6666	. . . . . . . . . 10 |- ( x e. ~H -> ( ( normop ` T ) .h ( Iop ` x ) ) = ( ( normop ` T ) .h x ) )
63	62	adantl 482	. . . . . . . . 9 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( normop ` T ) .h ( Iop ` x ) ) = ( ( normop ` T ) .h x ) )
64	60, 63	eqtrd 2656	. . . . . . . 8 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( normop ` T ) .op Iop ) ` x ) = ( ( normop ` T ) .h x ) )
65	64	oveq1d 6665	. . . . . . 7 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) = ( ( ( normop ` T ) .h x ) .ih x ) )
66	55, 65	eqtr4d 2659	. . . . . 6 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( ( normop ` T ) x. ( normh ` x ) ) x. ( normh ` x ) ) = ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) )
67	40, 66	breqtrd 4679	. . . . 5 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( normh ` ( T ` x ) ) x. ( normh ` x ) ) <_ ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) )
68	5, 22, 11, 26, 67	letrd 10194	. . . 4 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( abs ` ( ( T ` x ) .ih x ) ) <_ ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) )
69	2, 5, 11, 13, 68	letrd 10194	. . 3 |- ( ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) /\ x e. ~H ) -> ( ( T ` x ) .ih x ) <_ ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) )
70	69	ralrimiva 2966	. 2 |- ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) -> A. x e. ~H ( ( T ` x ) .ih x ) <_ ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) )
71		leop2 28983	. . 3 |- ( ( T e. HrmOp /\ ( ( normop ` T ) .op Iop ) e. HrmOp ) -> ( T <_op ( ( normop ` T ) .op Iop ) <-> A. x e. ~H ( ( T ` x ) .ih x ) <_ ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) ) )
72	8, 71	sylan2 491	. 2 |- ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) -> ( T <_op ( ( normop ` T ) .op Iop ) <-> A. x e. ~H ( ( T ` x ) .ih x ) <_ ( ( ( ( normop ` T ) .op Iop ) ` x ) .ih x ) ) )
73	70, 72	mpbird 247	1 |- ( ( T e. HrmOp /\ ( normop ` T ) e. RR ) -> T <_op ( ( normop ` T ) .op Iop ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   <_ cle 10075   2c2 11070   ^cexp 12860   abscabs 13974   ~Hchil 27776   .h csm 27778   .ih csp 27779   normhcno 27780   .op chot 27796   Iop chio 27801   normopcnop 27802   LinOpclo 27804   BndLinOpcbo 27805   HrmOpcho 27807   <_op cleo 27815

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  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-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942  ax-hcompl 28059

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-lm 21033  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cfil 23053  df-cau 23054  df-cmet 23055  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-dip 27556  df-ssp 27577  df-ph 27668  df-cbn 27719  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-hcau 27830  df-sh 28064  df-ch 28078  df-oc 28109  df-ch0 28110  df-shs 28167  df-pjh 28254  df-hosum 28589  df-homul 28590  df-hodif 28591  df-h0op 28607  df-iop 28608  df-nmop 28698  df-lnop 28700  df-bdop 28701  df-hmop 28703  df-leop 28711

This theorem is referenced by:  nmopleid  28998