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Mirrors > Home > MPE Home > Th. List > nmcn | Structured version Visualization version GIF version |
Description: The norm of a normed group is a continuous function. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nmcn.n | โข ๐ = (normโ๐บ) |
nmcn.j | โข ๐ฝ = (TopOpenโ๐บ) |
nmcn.k | โข ๐พ = (topGenโran (,)) |
Ref | Expression |
---|---|
nmcn | โข (๐บ โ NrmGrp โ ๐ โ (๐ฝ Cn ๐พ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmcn.n | . . 3 โข ๐ = (normโ๐บ) | |
2 | eqid 2724 | . . 3 โข (Baseโ๐บ) = (Baseโ๐บ) | |
3 | eqid 2724 | . . 3 โข (0gโ๐บ) = (0gโ๐บ) | |
4 | eqid 2724 | . . 3 โข (distโ๐บ) = (distโ๐บ) | |
5 | 1, 2, 3, 4 | nmfval 24421 | . 2 โข ๐ = (๐ฅ โ (Baseโ๐บ) โฆ (๐ฅ(distโ๐บ)(0gโ๐บ))) |
6 | nmcn.j | . . 3 โข ๐ฝ = (TopOpenโ๐บ) | |
7 | nmcn.k | . . 3 โข ๐พ = (topGenโran (,)) | |
8 | ngpms 24433 | . . 3 โข (๐บ โ NrmGrp โ ๐บ โ MetSp) | |
9 | ngptps 24435 | . . . 4 โข (๐บ โ NrmGrp โ ๐บ โ TopSp) | |
10 | 2, 6 | istps 22760 | . . . 4 โข (๐บ โ TopSp โ ๐ฝ โ (TopOnโ(Baseโ๐บ))) |
11 | 9, 10 | sylib 217 | . . 3 โข (๐บ โ NrmGrp โ ๐ฝ โ (TopOnโ(Baseโ๐บ))) |
12 | 11 | cnmptid 23489 | . . 3 โข (๐บ โ NrmGrp โ (๐ฅ โ (Baseโ๐บ) โฆ ๐ฅ) โ (๐ฝ Cn ๐ฝ)) |
13 | ngpgrp 24432 | . . . . 5 โข (๐บ โ NrmGrp โ ๐บ โ Grp) | |
14 | 2, 3 | grpidcl 18887 | . . . . 5 โข (๐บ โ Grp โ (0gโ๐บ) โ (Baseโ๐บ)) |
15 | 13, 14 | syl 17 | . . . 4 โข (๐บ โ NrmGrp โ (0gโ๐บ) โ (Baseโ๐บ)) |
16 | 11, 11, 15 | cnmptc 23490 | . . 3 โข (๐บ โ NrmGrp โ (๐ฅ โ (Baseโ๐บ) โฆ (0gโ๐บ)) โ (๐ฝ Cn ๐ฝ)) |
17 | 4, 6, 7, 8, 11, 12, 16 | cnmpt1ds 24682 | . 2 โข (๐บ โ NrmGrp โ (๐ฅ โ (Baseโ๐บ) โฆ (๐ฅ(distโ๐บ)(0gโ๐บ))) โ (๐ฝ Cn ๐พ)) |
18 | 5, 17 | eqeltrid 2829 | 1 โข (๐บ โ NrmGrp โ ๐ โ (๐ฝ Cn ๐พ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โฆ cmpt 5222 ran crn 5668 โcfv 6534 (class class class)co 7402 (,)cioo 13322 Basecbs 17145 distcds 17207 TopOpenctopn 17368 topGenctg 17384 0gc0g 17386 Grpcgrp 18855 TopOnctopon 22736 TopSpctps 22758 Cn ccn 23052 normcnm 24409 NrmGrpcngp 24410 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-ec 8702 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-ioo 13326 df-ioc 13327 df-ico 13328 df-icc 13329 df-fz 13483 df-fzo 13626 df-seq 13965 df-exp 14026 df-hash 14289 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-hom 17222 df-cco 17223 df-rest 17369 df-topn 17370 df-0g 17388 df-gsum 17389 df-topgen 17390 df-pt 17391 df-prds 17394 df-ordt 17448 df-xrs 17449 df-qtop 17454 df-imas 17455 df-xps 17457 df-mre 17531 df-mrc 17532 df-acs 17534 df-ps 18523 df-tsr 18524 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-mulg 18988 df-cntz 19225 df-cmn 19694 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-top 22720 df-topon 22737 df-topsp 22759 df-bases 22773 df-cn 23055 df-cnp 23056 df-tx 23390 df-hmeo 23583 df-xms 24150 df-ms 24151 df-tms 24152 df-nm 24415 df-ngp 24416 |
This theorem is referenced by: ngnmcncn 24685 abscn 24686 |
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