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Mirrors > Home > MPE Home > Th. List > tngngpd | Structured version Visualization version GIF version |
Description: Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tngngp.t | β’ π = (πΊ toNrmGrp π) |
tngngp.x | β’ π = (BaseβπΊ) |
tngngp.m | β’ β = (-gβπΊ) |
tngngp.z | β’ 0 = (0gβπΊ) |
tngngpd.1 | β’ (π β πΊ β Grp) |
tngngpd.2 | β’ (π β π:πβΆβ) |
tngngpd.3 | β’ ((π β§ π₯ β π) β ((πβπ₯) = 0 β π₯ = 0 )) |
tngngpd.4 | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (πβ(π₯ β π¦)) β€ ((πβπ₯) + (πβπ¦))) |
Ref | Expression |
---|---|
tngngpd | β’ (π β π β NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngngpd.1 | . 2 β’ (π β πΊ β Grp) | |
2 | tngngpd.2 | . . . 4 β’ (π β π:πβΆβ) | |
3 | tngngp.x | . . . . . 6 β’ π = (BaseβπΊ) | |
4 | 3 | fvexi 6914 | . . . . 5 β’ π β V |
5 | reex 11235 | . . . . 5 β’ β β V | |
6 | fex2 7945 | . . . . 5 β’ ((π:πβΆβ β§ π β V β§ β β V) β π β V) | |
7 | 4, 5, 6 | mp3an23 1449 | . . . 4 β’ (π:πβΆβ β π β V) |
8 | tngngp.t | . . . . 5 β’ π = (πΊ toNrmGrp π) | |
9 | tngngp.m | . . . . 5 β’ β = (-gβπΊ) | |
10 | 8, 9 | tngds 24582 | . . . 4 β’ (π β V β (π β β ) = (distβπ)) |
11 | 2, 7, 10 | 3syl 18 | . . 3 β’ (π β (π β β ) = (distβπ)) |
12 | tngngp.z | . . . 4 β’ 0 = (0gβπΊ) | |
13 | tngngpd.3 | . . . 4 β’ ((π β§ π₯ β π) β ((πβπ₯) = 0 β π₯ = 0 )) | |
14 | tngngpd.4 | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (πβ(π₯ β π¦)) β€ ((πβπ₯) + (πβπ¦))) | |
15 | 3, 9, 12, 1, 2, 13, 14 | nrmmetd 24501 | . . 3 β’ (π β (π β β ) β (Metβπ)) |
16 | 11, 15 | eqeltrrd 2829 | . 2 β’ (π β (distβπ) β (Metβπ)) |
17 | eqid 2727 | . . . 4 β’ (distβπ) = (distβπ) | |
18 | 8, 3, 17 | tngngp2 24587 | . . 3 β’ (π:πβΆβ β (π β NrmGrp β (πΊ β Grp β§ (distβπ) β (Metβπ)))) |
19 | 2, 18 | syl 17 | . 2 β’ (π β (π β NrmGrp β (πΊ β Grp β§ (distβπ) β (Metβπ)))) |
20 | 1, 16, 19 | mpbir2and 711 | 1 β’ (π β π β NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3471 class class class wbr 5150 β ccom 5684 βΆwf 6547 βcfv 6551 (class class class)co 7424 βcr 11143 0cc0 11144 + caddc 11147 β€ cle 11285 Basecbs 17185 distcds 17247 0gc0g 17426 Grpcgrp 18895 -gcsg 18897 Metcmet 21270 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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-plusg 17251 df-tset 17257 df-ds 17260 df-rest 17409 df-topn 17410 df-0g 17428 df-topgen 17430 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18898 df-minusg 18899 df-sbg 18900 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 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: tngngp 24589 tngngp3 24591 tcphcph 25183 |
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