Theorem ngpms 24108

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

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2732	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2732	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24104	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1146	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3948   ∘ ccom 5680  ‘cfv 6543  distcds 17205  Grpcgrp 18818  -_gcsg 18820  MetSpcms 23823  normcnm 24084  NrmGrpcngp 24085

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-co 5685  df-iota 6495  df-fv 6551  df-ngp 24091

This theorem is referenced by:  ngpxms  24109  ngptps  24110  ngpmet  24111  isngp4  24120  nmmtri  24130  nmrtri  24132  subgngp  24143  ngptgp  24144  tngngp2  24168  nlmvscnlem2  24201  nlmvscnlem1  24202  nlmvscn  24203  nrginvrcn  24208  nghmcn  24261  nmcn  24359  nmhmcn  24635  ipcnlem2  24760  ipcnlem1  24761  ipcn  24762  nglmle  24818  cssbn  24891  minveclem2  24942  minveclem3b  24944  minveclem3  24945  minveclem4  24948  minveclem7  24951