Theorem ngpms 24880

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 2760	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2760	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2760	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24876	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1164	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3898   ∘ ccom 5651  ‘cfv 6527  distcds 17398  Grpcgrp 19105  -_gcsg 19107  MetSpcms 24598  normcnm 24856  NrmGrpcngp 24857

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-co 5656  df-iota 6483  df-fv 6535  df-ngp 24863

This theorem is used by:  ngpxms  24881  ngptps  24882  ngpmet  24883  isngp4  24892  nmmtri  24902  nmrtri  24904  subgngp  24915  ngptgp  24916  tngngp2  24932  nlmvscnlem2  24965  nlmvscnlem1  24966  nlmvscn  24967  nrginvrcn  24972  nghmcn  25025  nmcn  25125  nmhmcn  25402  ipcnlem2  25526  ipcnlem1  25527  ipcn  25528  nglmle  25584  cssbn  25657  minveclem2  25708  minveclem3b  25710  minveclem3  25711  minveclem4  25714  minveclem7  25717