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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 24107 | . . . . 5 β’ (πΊ β NrmGrp β πΊ β Grp) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β πΊ β Grp) |
4 | ngpms 24108 | . . . . . . . . 9 β’ (πΊ β NrmGrp β πΊ β MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 β’ (π β πΊ β MetSp) |
6 | msxms 23959 | . . . . . . . 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 23961 | . . . . . . 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 23944 | . . . . . 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 22800 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΉ(βπ‘βπ½)π) β π β π) | |
17 | 14, 15, 16 | syl2anc 584 | . . . 4 β’ (π β π β π) |
18 | nglmle.5 | . . . . 5 β’ π = (normβπΊ) | |
19 | eqid 2732 | . . . . 5 β’ (0gβπΊ) = (0gβπΊ) | |
20 | eqid 2732 | . . . . 5 β’ (distβπΊ) = (distβπΊ) | |
21 | 18, 8, 19, 20, 9 | nmval2 24100 | . . . 4 β’ ((πΊ β Grp β§ π β π) β (πβπ) = (ππ·(0gβπΊ))) |
22 | 3, 17, 21 | syl2anc 584 | . . 3 β’ (π β (πβπ) = (ππ·(0gβπΊ))) |
23 | 8, 19 | grpidcl 18849 | . . . . 5 β’ (πΊ β Grp β (0gβπΊ) β π) |
24 | 3, 23 | syl 17 | . . . 4 β’ (π β (0gβπΊ) β π) |
25 | xmetsym 23852 | . . . 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 1371 | . . 3 β’ (π β (ππ·(0gβπΊ)) = ((0gβπΊ)π·π)) |
27 | 22, 26 | eqtrd 2772 | . 2 β’ (π β (πβπ) = ((0gβπΊ)π·π)) |
28 | nnuz 12864 | . . 3 β’ β = (β€β₯β1) | |
29 | 1zzd 12592 | . . 3 β’ (π β 1 β β€) | |
30 | nglmle.9 | . . 3 β’ (π β π β β*) | |
31 | 3 | adantr 481 | . . . . . 6 β’ ((π β§ π β β) β πΊ β Grp) |
32 | nglmle.7 | . . . . . . 7 β’ (π β πΉ:ββΆπ) | |
33 | 32 | ffvelcdmda 7086 | . . . . . 6 β’ ((π β§ π β β) β (πΉβπ) β π) |
34 | 18, 8, 19, 20, 9 | nmval2 24100 | . . . . . 6 β’ ((πΊ β Grp β§ (πΉβπ) β π) β (πβ(πΉβπ)) = ((πΉβπ)π·(0gβπΊ))) |
35 | 31, 33, 34 | syl2anc 584 | . . . . 5 β’ ((π β§ π β β) β (πβ(πΉβπ)) = ((πΉβπ)π·(0gβπΊ))) |
36 | 11 | adantr 481 | . . . . . 6 β’ ((π β§ π β β) β π· β (βMetβπ)) |
37 | 24 | adantr 481 | . . . . . 6 β’ ((π β§ π β β) β (0gβπΊ) β π) |
38 | xmetsym 23852 | . . . . . 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 1371 | . . . . 5 β’ ((π β§ π β β) β ((πΉβπ)π·(0gβπΊ)) = ((0gβπΊ)π·(πΉβπ))) |
40 | 35, 39 | eqtrd 2772 | . . . 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 5172 | . . 3 β’ ((π β§ π β β) β ((0gβπΊ)π·(πΉβπ)) β€ π ) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 24817 | . 2 β’ (π β ((0gβπΊ)π·π) β€ π ) |
44 | 27, 43 | eqbrtrd 5170 | 1 β’ (π β (πβπ) β€ π ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 class class class wbr 5148 Γ cxp 5674 βΎ cres 5678 βΆwf 6539 βcfv 6543 (class class class)co 7408 1c1 11110 β*cxr 11246 β€ cle 11248 βcn 12211 Basecbs 17143 distcds 17205 0gc0g 17384 Grpcgrp 18818 βMetcxmet 20928 MetOpencmopn 20933 TopOnctopon 22411 βπ‘clm 22729 βMetSpcxms 23822 MetSpcms 23823 normcnm 24084 NrmGrpcngp 24085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-0g 17386 df-topgen 17388 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-psmet 20935 df-xmet 20936 df-bl 20938 df-mopn 20939 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-lm 22732 df-xms 23825 df-ms 23826 df-nm 24090 df-ngp 24091 |
This theorem is referenced by: (None) |
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