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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 23135 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
2 | nmf.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
3 | eqid 2818 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 2, 3 | grpidcl 18069 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (0g‘𝐺) ∈ 𝑋) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑋) |
7 | ngpxms 23137 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) | |
8 | eqid 2818 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
9 | 2, 8 | xmsge0 23000 | . . . 4 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
10 | 7, 9 | syl3an1 1155 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
11 | 6, 10 | mpd3an3 1453 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝐴(dist‘𝐺)(0g‘𝐺))) |
12 | nmf.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
13 | 12, 2, 3, 8 | nmval 23126 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴(dist‘𝐺)(0g‘𝐺))) |
14 | 13 | adantl 482 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴(dist‘𝐺)(0g‘𝐺))) |
15 | 11, 14 | breqtrrd 5085 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ≤ cle 10664 Basecbs 16471 distcds 16562 0gc0g 16701 Grpcgrp 18041 ∞MetSpcxms 22854 normcnm 23113 NrmGrpcngp 23114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-0g 16703 df-topgen 16705 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-psmet 20465 df-xmet 20466 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-xms 22857 df-ms 22858 df-nm 23119 df-ngp 23120 |
This theorem is referenced by: nmrpcl 23156 nmgt0 23166 nlmvscnlem2 23221 nlmvscnlem1 23222 nmoeq0 23272 nmoleub2lem3 23646 ipcnlem2 23774 ipcnlem1 23775 minveclem1 23954 minveclem6 23964 pjthlem1 23967 |
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