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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 | ⊢ 𝑁 = (norm‘𝐺) |
nmcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
nmcn.k | ⊢ 𝐾 = (topGen‘ran (,)) |
Ref | Expression |
---|---|
nmcn | ⊢ (𝐺 ∈ NrmGrp → 𝑁 ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmcn.n | . . 3 ⊢ 𝑁 = (norm‘𝐺) | |
2 | eqid 2737 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | eqid 2737 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2737 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
5 | 1, 2, 3, 4 | nmfval 23486 | . 2 ⊢ 𝑁 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) |
6 | nmcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
7 | nmcn.k | . . 3 ⊢ 𝐾 = (topGen‘ran (,)) | |
8 | ngpms 23498 | . . 3 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
9 | ngptps 23500 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp) | |
10 | 2, 6 | istps 21831 | . . . 4 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
11 | 9, 10 | sylib 221 | . . 3 ⊢ (𝐺 ∈ NrmGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
12 | 11 | cnmptid 22558 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
13 | ngpgrp 23497 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
14 | 2, 3 | grpidcl 18395 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺)) |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (0g‘𝐺) ∈ (Base‘𝐺)) |
16 | 11, 11, 15 | cnmptc 22559 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (𝑥 ∈ (Base‘𝐺) ↦ (0g‘𝐺)) ∈ (𝐽 Cn 𝐽)) |
17 | 4, 6, 7, 8, 11, 12, 16 | cnmpt1ds 23739 | . 2 ⊢ (𝐺 ∈ NrmGrp → (𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) ∈ (𝐽 Cn 𝐾)) |
18 | 5, 17 | eqeltrid 2842 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝑁 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5135 ran crn 5552 ‘cfv 6380 (class class class)co 7213 (,)cioo 12935 Basecbs 16760 distcds 16811 TopOpenctopn 16926 topGenctg 16942 0gc0g 16944 Grpcgrp 18365 TopOnctopon 21807 TopSpctps 21829 Cn ccn 22121 normcnm 23474 NrmGrpcngp 23475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-ec 8393 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-ordt 17006 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-ps 18072 df-tsr 18073 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cn 22124 df-cnp 22125 df-tx 22459 df-hmeo 22652 df-xms 23218 df-ms 23219 df-tms 23220 df-nm 23480 df-ngp 23481 |
This theorem is referenced by: ngnmcncn 23742 abscn 23743 |
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