Theorem ngpms 24561

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 2737	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2737	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2737	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24557	. 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 3903   ∘ ccom 5638  ‘cfv 6502  distcds 17200  Grpcgrp 18880  -_gcsg 18882  MetSpcms 24279  normcnm 24537  NrmGrpcngp 24538

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-co 5643  df-iota 6458  df-fv 6510  df-ngp 24544

This theorem is referenced by:  ngpxms  24562  ngptps  24563  ngpmet  24564  isngp4  24573  nmmtri  24583  nmrtri  24585  subgngp  24596  ngptgp  24597  tngngp2  24613  nlmvscnlem2  24646  nlmvscnlem1  24647  nlmvscn  24648  nrginvrcn  24653  nghmcn  24706  nmcn  24806  nmhmcn  25093  ipcnlem2  25217  ipcnlem1  25218  ipcn  25219  nglmle  25275  cssbn  25348  minveclem2  25399  minveclem3b  25401  minveclem3  25402  minveclem4  25405  minveclem7  25408