Theorem ngpms 24629

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 24625	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1145	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3963   ∘ ccom 5693  ‘cfv 6563  distcds 17307  Grpcgrp 18964  -_gcsg 18966  MetSpcms 24344  normcnm 24605  NrmGrpcngp 24606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-co 5698  df-iota 6516  df-fv 6571  df-ngp 24612

This theorem is referenced by:  ngpxms  24630  ngptps  24631  ngpmet  24632  isngp4  24641  nmmtri  24651  nmrtri  24653  subgngp  24664  ngptgp  24665  tngngp2  24689  nlmvscnlem2  24722  nlmvscnlem1  24723  nlmvscn  24724  nrginvrcn  24729  nghmcn  24782  nmcn  24880  nmhmcn  25167  ipcnlem2  25292  ipcnlem1  25293  ipcn  25294  nglmle  25350  cssbn  25423  minveclem2  25474  minveclem3b  25476  minveclem3  25477  minveclem4  25480  minveclem7  25483