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

Theorem ngpgrp 22213
 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 2610 . . 3 (norm‘𝐺) = (norm‘𝐺)
2 eqid 2610 . . 3 (-g𝐺) = (-g𝐺)
3 eqid 2610 . . 3 (dist‘𝐺) = (dist‘𝐺)
41, 2, 3isngp 22210 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g𝐺)) ⊆ (dist‘𝐺)))
54simp1bi 1069 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540   ∘ ccom 5042  ‘cfv 5804  distcds 15777  Grpcgrp 17245  -gcsg 17247  MetSpcmt 21933  normcnm 22191  NrmGrpcngp 22192 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-co 5047  df-iota 5768  df-fv 5812  df-ngp 22198 This theorem is referenced by:  ngpds  22218  ngpds2  22220  ngpds3  22222  ngprcan  22224  isngp4  22226  ngpinvds  22227  ngpsubcan  22228  nmf  22229  nmge0  22231  nmeq0  22232  nminv  22235  nmmtri  22236  nmsub  22237  nmrtri  22238  nm2dif  22239  nmtri  22240  nmtri2  22241  ngpi  22242  nm0  22243  ngptgp  22250  tngngp2  22266  tnggrpr  22269  nrmtngnrm  22272  nlmdsdi  22295  nlmdsdir  22296  nrginvrcnlem  22305  ngpocelbl  22318  nmo0  22349  nmotri  22353  0nghm  22355  nmoid  22356  idnghm  22357  nmods  22358  nmcn  22455  nmoleub2lem2  22724  nmhmcn  22728  cphipval2  22848  4cphipval2  22849  cphipval  22850  ipcnlem2  22851  nglmle  22908  qqhcn  29363
 Copyright terms: Public domain W3C validator