Theorem ngpms 24539

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 24535	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1146	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3926   ∘ ccom 5658  ‘cfv 6531  distcds 17280  Grpcgrp 18916  -_gcsg 18918  MetSpcms 24257  normcnm 24515  NrmGrpcngp 24516

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-co 5663  df-iota 6484  df-fv 6539  df-ngp 24522

This theorem is referenced by:  ngpxms  24540  ngptps  24541  ngpmet  24542  isngp4  24551  nmmtri  24561  nmrtri  24563  subgngp  24574  ngptgp  24575  tngngp2  24591  nlmvscnlem2  24624  nlmvscnlem1  24625  nlmvscn  24626  nrginvrcn  24631  nghmcn  24684  nmcn  24784  nmhmcn  25071  ipcnlem2  25196  ipcnlem1  25197  ipcn  25198  nglmle  25254  cssbn  25327  minveclem2  25378  minveclem3b  25380  minveclem3  25381  minveclem4  25384  minveclem7  25387