Theorem ngpms 24613

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 24609	. 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 2108   ⊆ wss 3951   ∘ ccom 5689  ‘cfv 6561  distcds 17306  Grpcgrp 18951  -_gcsg 18953  MetSpcms 24328  normcnm 24589  NrmGrpcngp 24590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-co 5694  df-iota 6514  df-fv 6569  df-ngp 24596

This theorem is referenced by:  ngpxms  24614  ngptps  24615  ngpmet  24616  isngp4  24625  nmmtri  24635  nmrtri  24637  subgngp  24648  ngptgp  24649  tngngp2  24673  nlmvscnlem2  24706  nlmvscnlem1  24707  nlmvscn  24708  nrginvrcn  24713  nghmcn  24766  nmcn  24866  nmhmcn  25153  ipcnlem2  25278  ipcnlem1  25279  ipcn  25280  nglmle  25336  cssbn  25409  minveclem2  25460  minveclem3b  25462  minveclem3  25463  minveclem4  25466  minveclem7  25469