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Mirrors > Home > MPE Home > Th. List > nmge0 | Structured version Visualization version GIF version |
Description: The norm of a normed group is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nmf.x | ⊢ 𝑋 = (Base‘𝐺) |
nmf.n | ⊢ 𝑁 = (norm‘𝐺) |
Ref | Expression |
---|---|
nmge0 | ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpgrp 22774 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
2 | nmf.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
3 | eqid 2826 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 2, 3 | grpidcl 17805 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (0g‘𝐺) ∈ 𝑋) |
6 | 5 | adantr 474 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑋) |
7 | ngpxms 22776 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) | |
8 | eqid 2826 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
9 | 2, 8 | xmsge0 22639 | . . . 4 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
10 | 7, 9 | syl3an1 1208 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
11 | 6, 10 | mpd3an3 1592 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
12 | nmf.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
13 | 12, 2, 3, 8 | nmval 22765 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴(dist‘𝐺)(0g‘𝐺))) |
14 | 13 | adantl 475 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴(dist‘𝐺)(0g‘𝐺))) |
15 | 11, 14 | breqtrrd 4902 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 0cc0 10253 ≤ cle 10393 Basecbs 16223 distcds 16315 0gc0g 16454 Grpcgrp 17777 ∞MetSpcxms 22493 normcnm 22752 NrmGrpcngp 22753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-n0 11620 df-z 11706 df-uz 11970 df-q 12073 df-rp 12114 df-xneg 12233 df-xadd 12234 df-xmul 12235 df-0g 16456 df-topgen 16458 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-psmet 20099 df-xmet 20100 df-bl 20102 df-mopn 20103 df-top 21070 df-topon 21087 df-topsp 21109 df-bases 21122 df-xms 22496 df-ms 22497 df-nm 22758 df-ngp 22759 |
This theorem is referenced by: nmrpcl 22795 nmgt0 22805 nlmvscnlem2 22860 nlmvscnlem1 22861 nmoeq0 22911 nmoleub2lem3 23285 ipcnlem2 23413 ipcnlem1 23414 minveclem1 23593 minveclem6 23603 pjthlem1 23606 |
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