Theorem ngpms 24515

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 2731	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2731	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2731	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24511	. 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 2111   ⊆ wss 3897   ∘ ccom 5618  ‘cfv 6481  distcds 17170  Grpcgrp 18846  -_gcsg 18848  MetSpcms 24233  normcnm 24491  NrmGrpcngp 24492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-co 5623  df-iota 6437  df-fv 6489  df-ngp 24498

This theorem is referenced by:  ngpxms  24516  ngptps  24517  ngpmet  24518  isngp4  24527  nmmtri  24537  nmrtri  24539  subgngp  24550  ngptgp  24551  tngngp2  24567  nlmvscnlem2  24600  nlmvscnlem1  24601  nlmvscn  24602  nrginvrcn  24607  nghmcn  24660  nmcn  24760  nmhmcn  25047  ipcnlem2  25171  ipcnlem1  25172  ipcn  25173  nglmle  25229  cssbn  25302  minveclem2  25353  minveclem3b  25355  minveclem3  25356  minveclem4  25359  minveclem7  25362