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

Theorem ngpms 22344
Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
ngpms (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Proof of Theorem ngpms
StepHypRef Expression
1 eqid 2621 . . 3 (norm‘𝐺) = (norm‘𝐺)
2 eqid 2621 . . 3 (-g𝐺) = (-g𝐺)
3 eqid 2621 . . 3 (dist‘𝐺) = (dist‘𝐺)
41, 2, 3isngp 22340 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g𝐺)) ⊆ (dist‘𝐺)))
54simp2bi 1075 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3560  ccom 5088  cfv 5857  distcds 15890  Grpcgrp 17362  -gcsg 17364  MetSpcmt 22063  normcnm 22321  NrmGrpcngp 22322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-co 5093  df-iota 5820  df-fv 5865  df-ngp 22328
This theorem is referenced by:  ngpxms  22345  ngptps  22346  ngpmet  22347  isngp4  22356  nmf  22359  nmmtri  22366  nmrtri  22368  subgngp  22379  ngptgp  22380  tngngp2  22396  nlmvscnlem2  22429  nlmvscnlem1  22430  nlmvscn  22431  nrginvrcn  22436  nghmcn  22489  nmcn  22587  nmhmcn  22860  ipcnlem2  22983  ipcnlem1  22984  ipcn  22985  nglmle  23040  minveclem2  23137  minveclem3b  23139  minveclem3  23140  minveclem4  23143  minveclem7  23146
  Copyright terms: Public domain W3C validator