Theorem nglmle 23100

Description: If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by AV, 16-Oct-2021.)

Hypotheses

Ref	Expression
nglmle.1	|- X = ( Base ` G )
nglmle.2	|- D = ( ( dist ` G ) |` ( X X. X ) )
nglmle.3	|- J = ( MetOpen ` D )
nglmle.5	|- N = ( norm ` G )
nglmle.6	|- ( ph -> G e. NrmGrp )
nglmle.7	|- ( ph -> F : NN --> X )
nglmle.8	|- ( ph -> F ( ~~> t ` J ) P )
nglmle.9	|- ( ph -> R e. RR* )
nglmle.10	|- ( ( ph /\ k e. NN ) -> ( N ` ( F ` k ) ) <_ R )

Assertion

Ref	Expression
nglmle	|- ( ph -> ( N ` P ) <_ R )

Distinct variable groups:   k, F   D, k   k, G   k, J   P, k   R, k   k, X   ph, k

Allowed substitution hint:   N( k)

Proof of Theorem nglmle

Step	Hyp	Ref	Expression
1		nglmle.6	. . . . 5 |- ( ph -> G e. NrmGrp )
2		ngpgrp 22403	. . . . 5 |- ( G e. NrmGrp -> G e. Grp )
3	1, 2	syl 17	. . . 4 |- ( ph -> G e. Grp )
4		ngpms 22404	. . . . . . . . 9 |- ( G e. NrmGrp -> G e. MetSp )
5	1, 4	syl 17	. . . . . . . 8 |- ( ph -> G e. MetSp )
6		msxms 22259	. . . . . . . 8 |- ( G e. MetSp -> G e. *MetSp )
7	5, 6	syl 17	. . . . . . 7 |- ( ph -> G e. *MetSp )
8		nglmle.1	. . . . . . . 8 |- X = ( Base ` G )
9		nglmle.2	. . . . . . . 8 |- D = ( ( dist ` G ) |` ( X X. X ) )
10	8, 9	xmsxmet 22261	. . . . . . 7 |- ( G e. *MetSp -> D e. ( *Met ` X ) )
11	7, 10	syl 17	. . . . . 6 |- ( ph -> D e. ( *Met ` X ) )
12		nglmle.3	. . . . . . 7 |- J = ( MetOpen ` D )
13	12	mopntopon 22244	. . . . . 6 |- ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) )
14	11, 13	syl 17	. . . . 5 |- ( ph -> J e. (TopOn ` X ) )
15		nglmle.8	. . . . 5 |- ( ph -> F ( ~~> t ` J ) P )
16		lmcl 21101	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> P e. X )
17	14, 15, 16	syl2anc 693	. . . 4 |- ( ph -> P e. X )
18		nglmle.5	. . . . 5 |- N = ( norm ` G )
19		eqid 2622	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
20		eqid 2622	. . . . 5 |- ( dist ` G ) = ( dist ` G )
21	18, 8, 19, 20, 9	nmval2 22396	. . . 4 |- ( ( G e. Grp /\ P e. X ) -> ( N ` P ) = ( P D ( 0g ` G ) ) )
22	3, 17, 21	syl2anc 693	. . 3 |- ( ph -> ( N ` P ) = ( P D ( 0g ` G ) ) )
23	8, 19	grpidcl 17450	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. X )
24	3, 23	syl 17	. . . 4 |- ( ph -> ( 0g ` G ) e. X )
25		xmetsym 22152	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ ( 0g ` G ) e. X ) -> ( P D ( 0g ` G ) ) = ( ( 0g ` G ) D P ) )
26	11, 17, 24, 25	syl3anc 1326	. . 3 |- ( ph -> ( P D ( 0g ` G ) ) = ( ( 0g ` G ) D P ) )
27	22, 26	eqtrd 2656	. 2 |- ( ph -> ( N ` P ) = ( ( 0g ` G ) D P ) )
28		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
29		1zzd 11408	. . 3 |- ( ph -> 1 e. ZZ )
30		nglmle.9	. . 3 |- ( ph -> R e. RR* )
31	3	adantr 481	. . . . . 6 |- ( ( ph /\ k e. NN ) -> G e. Grp )
32		nglmle.7	. . . . . . 7 |- ( ph -> F : NN --> X )
33	32	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. X )
34	18, 8, 19, 20, 9	nmval2 22396	. . . . . 6 |- ( ( G e. Grp /\ ( F ` k ) e. X ) -> ( N ` ( F ` k ) ) = ( ( F ` k ) D ( 0g ` G ) ) )
35	31, 33, 34	syl2anc 693	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( N ` ( F ` k ) ) = ( ( F ` k ) D ( 0g ` G ) ) )
36	11	adantr 481	. . . . . 6 |- ( ( ph /\ k e. NN ) -> D e. ( *Met ` X ) )
37	24	adantr 481	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( 0g ` G ) e. X )
38		xmetsym 22152	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ ( F ` k ) e. X /\ ( 0g ` G ) e. X ) -> ( ( F ` k ) D ( 0g ` G ) ) = ( ( 0g ` G ) D ( F ` k ) ) )
39	36, 33, 37, 38	syl3anc 1326	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( F ` k ) D ( 0g ` G ) ) = ( ( 0g ` G ) D ( F ` k ) ) )
40	35, 39	eqtrd 2656	. . . 4 |- ( ( ph /\ k e. NN ) -> ( N ` ( F ` k ) ) = ( ( 0g ` G ) D ( F ` k ) ) )
41		nglmle.10	. . . 4 |- ( ( ph /\ k e. NN ) -> ( N ` ( F ` k ) ) <_ R )
42	40, 41	eqbrtrrd 4677	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( 0g ` G ) D ( F ` k ) ) <_ R )
43	28, 12, 11, 29, 15, 24, 30, 42	lmle 23099	. 2 |- ( ph -> ( ( 0g ` G ) D P ) <_ R )
44	27, 43	eqbrtrd 4675	1 |- ( ph -> ( N ` P ) <_ R )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   X. cxp 5112   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   1c1 9937   RR*cxr 10073   <_ cle 10075   NNcn 11020   Basecbs 15857   distcds 15950   0gc0g 16100   Grpcgrp 17422   *Metcxmt 19731   MetOpencmopn 19736  TopOnctopon 20715   ~~> tclm 21030   *MetSpcxme 22122   MetSpcmt 22123   normcnm 22381  NrmGrpcngp 22382

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-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

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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388

This theorem is referenced by: (None)