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Mirrors > Home > MPE Home > Th. List > nrmtngdist | Structured version Visualization version GIF version |
Description: The augmentation of a normed group by its own norm has the same distance function as the normed group (restricted to the base set). (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
nrmtngdist.t | โข ๐ = (๐บ toNrmGrp (normโ๐บ)) |
nrmtngdist.x | โข ๐ = (Baseโ๐บ) |
Ref | Expression |
---|---|
nrmtngdist | โข (๐บ โ NrmGrp โ (distโ๐) = ((distโ๐บ) โพ (๐ ร ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6905 | . . 3 โข (normโ๐บ) โ V | |
2 | nrmtngdist.t | . . . 4 โข ๐ = (๐บ toNrmGrp (normโ๐บ)) | |
3 | eqid 2725 | . . . 4 โข (-gโ๐บ) = (-gโ๐บ) | |
4 | 2, 3 | tngds 24582 | . . 3 โข ((normโ๐บ) โ V โ ((normโ๐บ) โ (-gโ๐บ)) = (distโ๐)) |
5 | 1, 4 | ax-mp 5 | . 2 โข ((normโ๐บ) โ (-gโ๐บ)) = (distโ๐) |
6 | eqid 2725 | . . . 4 โข (normโ๐บ) = (normโ๐บ) | |
7 | eqid 2725 | . . . 4 โข (distโ๐บ) = (distโ๐บ) | |
8 | nrmtngdist.x | . . . 4 โข ๐ = (Baseโ๐บ) | |
9 | eqid 2725 | . . . 4 โข ((distโ๐บ) โพ (๐ ร ๐)) = ((distโ๐บ) โพ (๐ ร ๐)) | |
10 | 6, 3, 7, 8, 9 | isngp2 24524 | . . 3 โข (๐บ โ NrmGrp โ (๐บ โ Grp โง ๐บ โ MetSp โง ((normโ๐บ) โ (-gโ๐บ)) = ((distโ๐บ) โพ (๐ ร ๐)))) |
11 | 10 | simp3bi 1144 | . 2 โข (๐บ โ NrmGrp โ ((normโ๐บ) โ (-gโ๐บ)) = ((distโ๐บ) โพ (๐ ร ๐))) |
12 | 5, 11 | eqtr3id 2779 | 1 โข (๐บ โ NrmGrp โ (distโ๐) = ((distโ๐บ) โพ (๐ ร ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 Vcvv 3463 ร cxp 5670 โพ cres 5674 โ ccom 5676 โcfv 6543 (class class class)co 7416 Basecbs 17179 distcds 17241 Grpcgrp 18894 -gcsg 18896 MetSpcms 24242 normcnm 24503 NrmGrpcngp 24504 toNrmGrp ctng 24505 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-sets 17132 df-slot 17150 df-ndx 17162 df-tset 17251 df-ds 17254 df-0g 17422 df-topgen 17424 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-xms 24244 df-ms 24245 df-nm 24509 df-ngp 24510 df-tng 24511 |
This theorem is referenced by: nrmtngnrm 24593 |
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