Theorem ngpms 24546

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 2735	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2735	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2735	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24542	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1147	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3900   ∘ ccom 5627  ‘cfv 6491  distcds 17188  Grpcgrp 18865  -_gcsg 18867  MetSpcms 24264  normcnm 24522  NrmGrpcngp 24523

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-co 5632  df-iota 6447  df-fv 6499  df-ngp 24529

This theorem is referenced by:  ngpxms  24547  ngptps  24548  ngpmet  24549  isngp4  24558  nmmtri  24568  nmrtri  24570  subgngp  24581  ngptgp  24582  tngngp2  24598  nlmvscnlem2  24631  nlmvscnlem1  24632  nlmvscn  24633  nrginvrcn  24638  nghmcn  24691  nmcn  24791  nmhmcn  25078  ipcnlem2  25202  ipcnlem1  25203  ipcn  25204  nglmle  25260  cssbn  25333  minveclem2  25384  minveclem3b  25386  minveclem3  25387  minveclem4  25390  minveclem7  25393