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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 24430 | . . . . 5 β’ (πΊ β NrmGrp β πΊ β Grp) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β πΊ β Grp) |
4 | ngpms 24431 | . . . . . . . . 9 β’ (πΊ β NrmGrp β πΊ β MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 β’ (π β πΊ β MetSp) |
6 | msxms 24282 | . . . . . . . 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 24284 | . . . . . . 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 24267 | . . . . . 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 23123 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΉ(βπ‘βπ½)π) β π β π) | |
17 | 14, 15, 16 | syl2anc 583 | . . . 4 β’ (π β π β π) |
18 | nglmle.5 | . . . . 5 β’ π = (normβπΊ) | |
19 | eqid 2724 | . . . . 5 β’ (0gβπΊ) = (0gβπΊ) | |
20 | eqid 2724 | . . . . 5 β’ (distβπΊ) = (distβπΊ) | |
21 | 18, 8, 19, 20, 9 | nmval2 24423 | . . . 4 β’ ((πΊ β Grp β§ π β π) β (πβπ) = (ππ·(0gβπΊ))) |
22 | 3, 17, 21 | syl2anc 583 | . . 3 β’ (π β (πβπ) = (ππ·(0gβπΊ))) |
23 | 8, 19 | grpidcl 18885 | . . . . 5 β’ (πΊ β Grp β (0gβπΊ) β π) |
24 | 3, 23 | syl 17 | . . . 4 β’ (π β (0gβπΊ) β π) |
25 | xmetsym 24175 | . . . 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 1368 | . . 3 β’ (π β (ππ·(0gβπΊ)) = ((0gβπΊ)π·π)) |
27 | 22, 26 | eqtrd 2764 | . 2 β’ (π β (πβπ) = ((0gβπΊ)π·π)) |
28 | nnuz 12862 | . . 3 β’ β = (β€β₯β1) | |
29 | 1zzd 12590 | . . 3 β’ (π β 1 β β€) | |
30 | nglmle.9 | . . 3 β’ (π β π β β*) | |
31 | 3 | adantr 480 | . . . . . 6 β’ ((π β§ π β β) β πΊ β Grp) |
32 | nglmle.7 | . . . . . . 7 β’ (π β πΉ:ββΆπ) | |
33 | 32 | ffvelcdmda 7076 | . . . . . 6 β’ ((π β§ π β β) β (πΉβπ) β π) |
34 | 18, 8, 19, 20, 9 | nmval2 24423 | . . . . . 6 β’ ((πΊ β Grp β§ (πΉβπ) β π) β (πβ(πΉβπ)) = ((πΉβπ)π·(0gβπΊ))) |
35 | 31, 33, 34 | syl2anc 583 | . . . . 5 β’ ((π β§ π β β) β (πβ(πΉβπ)) = ((πΉβπ)π·(0gβπΊ))) |
36 | 11 | adantr 480 | . . . . . 6 β’ ((π β§ π β β) β π· β (βMetβπ)) |
37 | 24 | adantr 480 | . . . . . 6 β’ ((π β§ π β β) β (0gβπΊ) β π) |
38 | xmetsym 24175 | . . . . . 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 1368 | . . . . 5 β’ ((π β§ π β β) β ((πΉβπ)π·(0gβπΊ)) = ((0gβπΊ)π·(πΉβπ))) |
40 | 35, 39 | eqtrd 2764 | . . . 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 5162 | . . 3 β’ ((π β§ π β β) β ((0gβπΊ)π·(πΉβπ)) β€ π ) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 25151 | . 2 β’ (π β ((0gβπΊ)π·π) β€ π ) |
44 | 27, 43 | eqbrtrd 5160 | 1 β’ (π β (πβπ) β€ π ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5138 Γ cxp 5664 βΎ cres 5668 βΆwf 6529 βcfv 6533 (class class class)co 7401 1c1 11107 β*cxr 11244 β€ cle 11246 βcn 12209 Basecbs 17143 distcds 17205 0gc0g 17384 Grpcgrp 18853 βMetcxmet 21213 MetOpencmopn 21218 TopOnctopon 22734 βπ‘clm 23052 βMetSpcxms 24145 MetSpcms 24146 normcnm 24407 NrmGrpcngp 24408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-0g 17386 df-topgen 17388 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-psmet 21220 df-xmet 21221 df-bl 21223 df-mopn 21224 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-lm 23055 df-xms 24148 df-ms 24149 df-nm 24413 df-ngp 24414 |
This theorem is referenced by: (None) |
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