Theorem ngpms 22624

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 2771	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2771	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2771	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 22620	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1140	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2145   ⊆ wss 3723   ∘ ccom 5253  ‘cfv 6031  distcds 16158  Grpcgrp 17630  -_gcsg 17632  MetSpcmt 22343  normcnm 22601  NrmGrpcngp 22602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-co 5258  df-iota 5994  df-fv 6039  df-ngp 22608

This theorem is referenced by:  ngpxms  22625  ngptps  22626  ngpmet  22627  isngp4  22636  nmf  22639  nmmtri  22646  nmrtri  22648  subgngp  22659  ngptgp  22660  tngngp2  22676  nlmvscnlem2  22709  nlmvscnlem1  22710  nlmvscn  22711  nrginvrcn  22716  nghmcn  22769  nmcn  22867  nmhmcn  23139  ipcnlem2  23262  ipcnlem1  23263  ipcn  23264  nglmle  23319  minveclem2  23416  minveclem3b  23418  minveclem3  23419  minveclem4  23422  minveclem7  23425