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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 23207 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
2 | nmf.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
3 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 2, 3 | grpidcl 18130 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (0g‘𝐺) ∈ 𝑋) |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑋) |
7 | ngpxms 23209 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) | |
8 | eqid 2821 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
9 | 2, 8 | xmsge0 23072 | . . . 4 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
10 | 7, 9 | syl3an1 1159 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
11 | 6, 10 | mpd3an3 1458 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
12 | nmf.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
13 | 12, 2, 3, 8 | nmval 23198 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴(dist‘𝐺)(0g‘𝐺))) |
14 | 13 | adantl 484 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴(dist‘𝐺)(0g‘𝐺))) |
15 | 11, 14 | breqtrrd 5093 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 0cc0 10536 ≤ cle 10675 Basecbs 16482 distcds 16573 0gc0g 16712 Grpcgrp 18102 ∞MetSpcxms 22926 normcnm 23185 NrmGrpcngp 23186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-0g 16714 df-topgen 16716 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-psmet 20536 df-xmet 20537 df-bl 20539 df-mopn 20540 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-xms 22929 df-ms 22930 df-nm 23191 df-ngp 23192 |
This theorem is referenced by: nmrpcl 23228 nmgt0 23238 nlmvscnlem2 23293 nlmvscnlem1 23294 nmoeq0 23344 nmoleub2lem3 23718 ipcnlem2 23846 ipcnlem1 23847 minveclem1 24026 minveclem6 24036 pjthlem1 24039 |
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