Theorem ngpms 24634

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 2740	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2740	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2740	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24630	. 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 2108   ⊆ wss 3976   ∘ ccom 5704  ‘cfv 6573  distcds 17320  Grpcgrp 18973  -_gcsg 18975  MetSpcms 24349  normcnm 24610  NrmGrpcngp 24611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-co 5709  df-iota 6525  df-fv 6581  df-ngp 24617

This theorem is referenced by:  ngpxms  24635  ngptps  24636  ngpmet  24637  isngp4  24646  nmmtri  24656  nmrtri  24658  subgngp  24669  ngptgp  24670  tngngp2  24694  nlmvscnlem2  24727  nlmvscnlem1  24728  nlmvscn  24729  nrginvrcn  24734  nghmcn  24787  nmcn  24885  nmhmcn  25172  ipcnlem2  25297  ipcnlem1  25298  ipcn  25299  nglmle  25355  cssbn  25428  minveclem2  25479  minveclem3b  25481  minveclem3  25482  minveclem4  25485  minveclem7  25488