Theorem ngpms 24586

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 2741	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2741	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2741	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24582	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1153	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2121   ⊆ wss 3884   ∘ ccom 5624  ‘cfv 6488  distcds 17224  Grpcgrp 18904  -_gcsg 18906  MetSpcms 24304  normcnm 24562  NrmGrpcngp 24563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-co 5629  df-iota 6444  df-fv 6496  df-ngp 24569

This theorem is referenced by:  ngpxms  24587  ngptps  24588  ngpmet  24589  isngp4  24598  nmmtri  24608  nmrtri  24610  subgngp  24621  ngptgp  24622  tngngp2  24638  nlmvscnlem2  24671  nlmvscnlem1  24672  nlmvscn  24673  nrginvrcn  24678  nghmcn  24731  nmcn  24831  nmhmcn  25108  ipcnlem2  25232  ipcnlem1  25233  ipcn  25234  nglmle  25290  cssbn  25363  minveclem2  25414  minveclem3b  25416  minveclem3  25417  minveclem4  25420  minveclem7  25423