Theorem ngpms 24553

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 24549	. 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 2114   ⊆ wss 3885   ∘ ccom 5624  ‘cfv 6487  distcds 17218  Grpcgrp 18898  -_gcsg 18900  MetSpcms 24271  normcnm 24529  NrmGrpcngp 24530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-co 5629  df-iota 6443  df-fv 6495  df-ngp 24536

This theorem is referenced by:  ngpxms  24554  ngptps  24555  ngpmet  24556  isngp4  24565  nmmtri  24575  nmrtri  24577  subgngp  24588  ngptgp  24589  tngngp2  24605  nlmvscnlem2  24638  nlmvscnlem1  24639  nlmvscn  24640  nrginvrcn  24645  nghmcn  24698  nmcn  24798  nmhmcn  25075  ipcnlem2  25199  ipcnlem1  25200  ipcn  25201  nglmle  25257  cssbn  25330  minveclem2  25381  minveclem3b  25383  minveclem3  25384  minveclem4  25387  minveclem7  25390