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Mirrors > Home > MPE Home > Th. List > ngpms | Structured version Visualization version GIF version |
Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpms | ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | eqid 2798 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2798 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 1, 2, 3 | isngp 23202 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
5 | 4 | simp2bi 1143 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3881 ∘ ccom 5523 ‘cfv 6324 distcds 16566 Grpcgrp 18095 -gcsg 18097 MetSpcms 22925 normcnm 23183 NrmGrpcngp 23184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-co 5528 df-iota 6283 df-fv 6332 df-ngp 23190 |
This theorem is referenced by: ngpxms 23207 ngptps 23208 ngpmet 23209 isngp4 23218 nmmtri 23228 nmrtri 23230 subgngp 23241 ngptgp 23242 tngngp2 23258 nlmvscnlem2 23291 nlmvscnlem1 23292 nlmvscn 23293 nrginvrcn 23298 nghmcn 23351 nmcn 23449 nmhmcn 23725 ipcnlem2 23848 ipcnlem1 23849 ipcn 23850 nglmle 23906 cssbn 23979 minveclem2 24030 minveclem3b 24032 minveclem3 24033 minveclem4 24036 minveclem7 24039 |
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