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Mirrors > Home > MPE Home > Th. List > ngpmet | Structured version Visualization version GIF version |
Description: The (induced) metric of a normed group is a metric. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 14-Oct-2021.) |
Ref | Expression |
---|---|
ngpmet.x | ⊢ 𝑋 = (Base‘𝐺) |
ngpmet.d | ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
ngpmet | ⊢ (𝐺 ∈ NrmGrp → 𝐷 ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpms 23212 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
2 | ngpmet.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | ngpmet.d | . . 3 ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) | |
4 | 2, 3 | msmet 23070 | . 2 ⊢ (𝐺 ∈ MetSp → 𝐷 ∈ (Met‘𝑋)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 × cxp 5556 ↾ cres 5560 ‘cfv 6358 Basecbs 16486 distcds 16577 Metcmet 20534 MetSpcms 22931 NrmGrpcngp 23190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-xms 22933 df-ms 22934 df-ngp 23196 |
This theorem is referenced by: nmf 23227 nrmtngnrm 23270 ngpocelbl 23316 cssbn 23981 |
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