Theorem ngpms 22404

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( norm ` G ) = ( norm ` G )
2		eqid 2622	. . 3 |- ( -g ` G ) = ( -g ` G )
3		eqid 2622	. . 3 |- ( dist ` G ) = ( dist ` G )
4	1, 2, 3	isngp 22400	. 2 |- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( ( norm ` G ) o. ( -g ` G ) ) C_ ( dist ` G ) ) )
5	4	simp2bi 1077	1 |- ( G e. NrmGrp -> G e. MetSp )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   o. ccom 5118   ` cfv 5888   distcds 15950   Grpcgrp 17422   -gcsg 17424   MetSpcmt 22123   normcnm 22381  NrmGrpcngp 22382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-co 5123  df-iota 5851  df-fv 5896  df-ngp 22388

This theorem is referenced by:  ngpxms  22405  ngptps  22406  ngpmet  22407  isngp4  22416  nmf  22419  nmmtri  22426  nmrtri  22428  subgngp  22439  ngptgp  22440  tngngp2  22456  nlmvscnlem2  22489  nlmvscnlem1  22490  nlmvscn  22491  nrginvrcn  22496  nghmcn  22549  nmcn  22647  nmhmcn  22920  ipcnlem2  23043  ipcnlem1  23044  ipcn  23045  nglmle  23100  minveclem2  23197  minveclem3b  23199  minveclem3  23200  minveclem4  23203  minveclem7  23206