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Theorem ngpms 18639
Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
ngpms  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )

Proof of Theorem ngpms
StepHypRef Expression
1 eqid 2435 . . 3  |-  ( norm `  G )  =  (
norm `  G )
2 eqid 2435 . . 3  |-  ( -g `  G )  =  (
-g `  G )
3 eqid 2435 . . 3  |-  ( dist `  G )  =  (
dist `  G )
41, 2, 3isngp 18635 . 2  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  (
( norm `  G )  o.  ( -g `  G
) )  C_  ( dist `  G ) ) )
54simp2bi 973 1  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    C_ wss 3312    o. ccom 4874   ` cfv 5446   distcds 13530   Grpcgrp 14677   -gcsg 14680   MetSpcmt 18340   normcnm 18616  NrmGrpcngp 18617
This theorem is referenced by:  ngpxms  18640  ngptps  18641  isngp4  18650  nmf  18653  nmmtri  18660  nmrtri  18662  subgngp  18668  ngptgp  18669  tngngp2  18685  nlmvscnlem2  18713  nlmvscnlem1  18714  nlmvscn  18715  nrginvrcn  18719  nghmcn  18771  nmcn  18867  nmhmcn  19120  ipcnlem2  19190  ipcnlem1  19191  ipcn  19192  minveclem2  19319  minveclem3b  19321  minveclem3  19322  minveclem4  19325  minveclem7  19328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-co 4879  df-iota 5410  df-fv 5454  df-ngp 18623
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