MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ngpgrp Structured version   Visualization version   GIF version

Theorem ngpgrp 24552
Description: A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
ngpgrp (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)

Proof of Theorem ngpgrp
StepHypRef Expression
1 eqid 2725 . . 3 (norm‘𝐺) = (norm‘𝐺)
2 eqid 2725 . . 3 (-g𝐺) = (-g𝐺)
3 eqid 2725 . . 3 (dist‘𝐺) = (dist‘𝐺)
41, 2, 3isngp 24549 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g𝐺)) ⊆ (dist‘𝐺)))
54simp1bi 1142 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3944  ccom 5682  cfv 6549  distcds 17245  Grpcgrp 18898  -gcsg 18900  MetSpcms 24268  normcnm 24529  NrmGrpcngp 24530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-co 5687  df-iota 6501  df-fv 6557  df-ngp 24536
This theorem is referenced by:  ngpds  24557  ngpds2  24559  ngpds3  24561  ngprcan  24563  isngp4  24565  ngpinvds  24566  ngpsubcan  24567  nmf  24568  nmge0  24570  nmeq0  24571  nminv  24574  nmmtri  24575  nmsub  24576  nmrtri  24577  nm2dif  24578  nmtri  24579  nmtri2  24580  ngpi  24581  nm0  24582  ngptgp  24589  tngngp2  24613  tnggrpr  24616  nrmtngnrm  24619  nlmdsdi  24642  nlmdsdir  24643  nrginvrcnlem  24652  ngpocelbl  24665  nmo0  24696  nmotri  24700  0nghm  24702  nmoid  24703  idnghm  24704  nmods  24705  nmcn  24804  nmoleub2lem2  25087  nmhmcn  25091  cphpyth  25188  cphipval2  25213  4cphipval2  25214  cphipval  25215  ipcnlem2  25216  nglmle  25274  qqhcn  33723
  Copyright terms: Public domain W3C validator