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Mirrors > Home > MPE Home > Th. List > ngpgrp | Structured version Visualization version GIF version |
Description: A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpgrp | ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | eqid 2818 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
3 | eqid 2818 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 1, 2, 3 | isngp 23132 | . 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
5 | 4 | simp1bi 1137 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3933 ∘ ccom 5552 ‘cfv 6348 distcds 16562 Grpcgrp 18041 -gcsg 18043 MetSpcms 22855 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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-co 5557 df-iota 6307 df-fv 6356 df-ngp 23120 |
This theorem is referenced by: ngpds 23140 ngpds2 23142 ngpds3 23144 ngprcan 23146 isngp4 23148 ngpinvds 23149 ngpsubcan 23150 nmf 23151 nmge0 23153 nmeq0 23154 nminv 23157 nmmtri 23158 nmsub 23159 nmrtri 23160 nm2dif 23161 nmtri 23162 nmtri2 23163 ngpi 23164 nm0 23165 ngptgp 23172 tngngp2 23188 tnggrpr 23191 nrmtngnrm 23194 nlmdsdi 23217 nlmdsdir 23218 nrginvrcnlem 23227 ngpocelbl 23240 nmo0 23271 nmotri 23275 0nghm 23277 nmoid 23278 idnghm 23279 nmods 23280 nmcn 23379 nmoleub2lem2 23647 nmhmcn 23651 cphipval2 23771 4cphipval2 23772 cphipval 23773 ipcnlem2 23774 nglmle 23832 qqhcn 31131 |
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