![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nglmle | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nglmle.1 | β’ π = (BaseβπΊ) |
nglmle.2 | β’ π· = ((distβπΊ) βΎ (π Γ π)) |
nglmle.3 | β’ π½ = (MetOpenβπ·) |
nglmle.5 | β’ π = (normβπΊ) |
nglmle.6 | β’ (π β πΊ β NrmGrp) |
nglmle.7 | β’ (π β πΉ:ββΆπ) |
nglmle.8 | β’ (π β πΉ(βπ‘βπ½)π) |
nglmle.9 | β’ (π β π β β*) |
nglmle.10 | β’ ((π β§ π β β) β (πβ(πΉβπ)) β€ π ) |
Ref | Expression |
---|---|
nglmle | β’ (π β (πβπ) β€ π ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nglmle.6 | . . . . 5 β’ (π β πΊ β NrmGrp) | |
2 | ngpgrp 23978 | . . . . 5 β’ (πΊ β NrmGrp β πΊ β Grp) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β πΊ β Grp) |
4 | ngpms 23979 | . . . . . . . . 9 β’ (πΊ β NrmGrp β πΊ β MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 β’ (π β πΊ β MetSp) |
6 | msxms 23830 | . . . . . . . 8 β’ (πΊ β MetSp β πΊ β βMetSp) | |
7 | 5, 6 | syl 17 | . . . . . . 7 β’ (π β πΊ β βMetSp) |
8 | nglmle.1 | . . . . . . . 8 β’ π = (BaseβπΊ) | |
9 | nglmle.2 | . . . . . . . 8 β’ π· = ((distβπΊ) βΎ (π Γ π)) | |
10 | 8, 9 | xmsxmet 23832 | . . . . . . 7 β’ (πΊ β βMetSp β π· β (βMetβπ)) |
11 | 7, 10 | syl 17 | . . . . . 6 β’ (π β π· β (βMetβπ)) |
12 | nglmle.3 | . . . . . . 7 β’ π½ = (MetOpenβπ·) | |
13 | 12 | mopntopon 23815 | . . . . . 6 β’ (π· β (βMetβπ) β π½ β (TopOnβπ)) |
14 | 11, 13 | syl 17 | . . . . 5 β’ (π β π½ β (TopOnβπ)) |
15 | nglmle.8 | . . . . 5 β’ (π β πΉ(βπ‘βπ½)π) | |
16 | lmcl 22671 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΉ(βπ‘βπ½)π) β π β π) | |
17 | 14, 15, 16 | syl2anc 585 | . . . 4 β’ (π β π β π) |
18 | nglmle.5 | . . . . 5 β’ π = (normβπΊ) | |
19 | eqid 2733 | . . . . 5 β’ (0gβπΊ) = (0gβπΊ) | |
20 | eqid 2733 | . . . . 5 β’ (distβπΊ) = (distβπΊ) | |
21 | 18, 8, 19, 20, 9 | nmval2 23971 | . . . 4 β’ ((πΊ β Grp β§ π β π) β (πβπ) = (ππ·(0gβπΊ))) |
22 | 3, 17, 21 | syl2anc 585 | . . 3 β’ (π β (πβπ) = (ππ·(0gβπΊ))) |
23 | 8, 19 | grpidcl 18786 | . . . . 5 β’ (πΊ β Grp β (0gβπΊ) β π) |
24 | 3, 23 | syl 17 | . . . 4 β’ (π β (0gβπΊ) β π) |
25 | xmetsym 23723 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π β§ (0gβπΊ) β π) β (ππ·(0gβπΊ)) = ((0gβπΊ)π·π)) | |
26 | 11, 17, 24, 25 | syl3anc 1372 | . . 3 β’ (π β (ππ·(0gβπΊ)) = ((0gβπΊ)π·π)) |
27 | 22, 26 | eqtrd 2773 | . 2 β’ (π β (πβπ) = ((0gβπΊ)π·π)) |
28 | nnuz 12814 | . . 3 β’ β = (β€β₯β1) | |
29 | 1zzd 12542 | . . 3 β’ (π β 1 β β€) | |
30 | nglmle.9 | . . 3 β’ (π β π β β*) | |
31 | 3 | adantr 482 | . . . . . 6 β’ ((π β§ π β β) β πΊ β Grp) |
32 | nglmle.7 | . . . . . . 7 β’ (π β πΉ:ββΆπ) | |
33 | 32 | ffvelcdmda 7039 | . . . . . 6 β’ ((π β§ π β β) β (πΉβπ) β π) |
34 | 18, 8, 19, 20, 9 | nmval2 23971 | . . . . . 6 β’ ((πΊ β Grp β§ (πΉβπ) β π) β (πβ(πΉβπ)) = ((πΉβπ)π·(0gβπΊ))) |
35 | 31, 33, 34 | syl2anc 585 | . . . . 5 β’ ((π β§ π β β) β (πβ(πΉβπ)) = ((πΉβπ)π·(0gβπΊ))) |
36 | 11 | adantr 482 | . . . . . 6 β’ ((π β§ π β β) β π· β (βMetβπ)) |
37 | 24 | adantr 482 | . . . . . 6 β’ ((π β§ π β β) β (0gβπΊ) β π) |
38 | xmetsym 23723 | . . . . . 6 β’ ((π· β (βMetβπ) β§ (πΉβπ) β π β§ (0gβπΊ) β π) β ((πΉβπ)π·(0gβπΊ)) = ((0gβπΊ)π·(πΉβπ))) | |
39 | 36, 33, 37, 38 | syl3anc 1372 | . . . . 5 β’ ((π β§ π β β) β ((πΉβπ)π·(0gβπΊ)) = ((0gβπΊ)π·(πΉβπ))) |
40 | 35, 39 | eqtrd 2773 | . . . 4 β’ ((π β§ π β β) β (πβ(πΉβπ)) = ((0gβπΊ)π·(πΉβπ))) |
41 | nglmle.10 | . . . 4 β’ ((π β§ π β β) β (πβ(πΉβπ)) β€ π ) | |
42 | 40, 41 | eqbrtrrd 5133 | . . 3 β’ ((π β§ π β β) β ((0gβπΊ)π·(πΉβπ)) β€ π ) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 24688 | . 2 β’ (π β ((0gβπΊ)π·π) β€ π ) |
44 | 27, 43 | eqbrtrd 5131 | 1 β’ (π β (πβπ) β€ π ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5109 Γ cxp 5635 βΎ cres 5639 βΆwf 6496 βcfv 6500 (class class class)co 7361 1c1 11060 β*cxr 11196 β€ cle 11198 βcn 12161 Basecbs 17091 distcds 17150 0gc0g 17329 Grpcgrp 18756 βMetcxmet 20804 MetOpencmopn 20809 TopOnctopon 22282 βπ‘clm 22600 βMetSpcxms 23693 MetSpcms 23694 normcnm 23955 NrmGrpcngp 23956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-0g 17331 df-topgen 17333 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-psmet 20811 df-xmet 20812 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-lm 22603 df-xms 23696 df-ms 23697 df-nm 23961 df-ngp 23962 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |