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

Theorem ngpgrp 23208
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 2824 . . 3 (norm‘𝐺) = (norm‘𝐺)
2 eqid 2824 . . 3 (-g𝐺) = (-g𝐺)
3 eqid 2824 . . 3 (dist‘𝐺) = (dist‘𝐺)
41, 2, 3isngp 23205 . 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 2115  wss 3919  ccom 5546  cfv 6343  distcds 16574  Grpcgrp 18103  -gcsg 18105  MetSpcms 22928  normcnm 23186  NrmGrpcngp 23187
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-co 5551  df-iota 6302  df-fv 6351  df-ngp 23193
This theorem is referenced by:  ngpds  23213  ngpds2  23215  ngpds3  23217  ngprcan  23219  isngp4  23221  ngpinvds  23222  ngpsubcan  23223  nmf  23224  nmge0  23226  nmeq0  23227  nminv  23230  nmmtri  23231  nmsub  23232  nmrtri  23233  nm2dif  23234  nmtri  23235  nmtri2  23236  ngpi  23237  nm0  23238  ngptgp  23245  tngngp2  23261  tnggrpr  23264  nrmtngnrm  23267  nlmdsdi  23290  nlmdsdir  23291  nrginvrcnlem  23300  ngpocelbl  23313  nmo0  23344  nmotri  23348  0nghm  23350  nmoid  23351  idnghm  23352  nmods  23353  nmcn  23452  nmoleub2lem2  23724  nmhmcn  23728  cphipval2  23848  4cphipval2  23849  cphipval  23850  ipcnlem2  23851  nglmle  23909  qqhcn  31289
  Copyright terms: Public domain W3C validator