![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nmeq0 | Structured version Visualization version GIF version |
Description: The identity is the only element of the group with zero norm. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (Revised by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nmf.x | ⊢ 𝑋 = (Base‘𝐺) |
nmf.n | ⊢ 𝑁 = (norm‘𝐺) |
nmeq0.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
nmeq0 | ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmf.n | . . . . 5 ⊢ 𝑁 = (norm‘𝐺) | |
2 | nmf.x | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
3 | nmeq0.z | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | eqid 2778 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
5 | 1, 2, 3, 4 | nmval 22813 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴(dist‘𝐺) 0 )) |
6 | 5 | adantl 475 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴(dist‘𝐺) 0 )) |
7 | 6 | eqeq1d 2780 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ (𝐴(dist‘𝐺) 0 ) = 0)) |
8 | ngpgrp 22822 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ Grp) |
10 | 2, 3 | grpidcl 17848 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋) |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ∈ 𝑋) |
12 | ngpxms 22824 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) | |
13 | 2, 4 | xmseq0 22688 | . . . 4 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋) → ((𝐴(dist‘𝐺) 0 ) = 0 ↔ 𝐴 = 0 )) |
14 | 12, 13 | syl3an1 1163 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋) → ((𝐴(dist‘𝐺) 0 ) = 0 ↔ 𝐴 = 0 )) |
15 | 11, 14 | mpd3an3 1535 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝐴(dist‘𝐺) 0 ) = 0 ↔ 𝐴 = 0 )) |
16 | 7, 15 | bitrd 271 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 0cc0 10274 Basecbs 16266 distcds 16358 0gc0g 16497 Grpcgrp 17820 ∞MetSpcxms 22541 normcnm 22800 NrmGrpcngp 22801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-n0 11648 df-z 11734 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-0g 16499 df-topgen 16501 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-psmet 20145 df-xmet 20146 df-bl 20148 df-mopn 20149 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-xms 22544 df-ms 22545 df-nm 22806 df-ngp 22807 |
This theorem is referenced by: nmne0 22842 ngpi 22851 nm0 22852 nmgt0 22853 tngngp 22877 tngngp3 22879 nlmmul0or 22906 nmoeq0 22959 ncvs1 23375 |
Copyright terms: Public domain | W3C validator |