Theorem ngpms 23662

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 2738	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2738	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2738	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 23658	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1144	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3883   ∘ ccom 5584  ‘cfv 6418  distcds 16897  Grpcgrp 18492  -_gcsg 18494  MetSpcms 23379  normcnm 23638  NrmGrpcngp 23639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-co 5589  df-iota 6376  df-fv 6426  df-ngp 23645

This theorem is referenced by:  ngpxms  23663  ngptps  23664  ngpmet  23665  isngp4  23674  nmmtri  23684  nmrtri  23686  subgngp  23697  ngptgp  23698  tngngp2  23722  nlmvscnlem2  23755  nlmvscnlem1  23756  nlmvscn  23757  nrginvrcn  23762  nghmcn  23815  nmcn  23913  nmhmcn  24189  ipcnlem2  24313  ipcnlem1  24314  ipcn  24315  nglmle  24371  cssbn  24444  minveclem2  24495  minveclem3b  24497  minveclem3  24498  minveclem4  24501  minveclem7  24504