Theorem ngpms 24488

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 2729	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2729	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2729	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24484	. 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 2109   ⊆ wss 3914   ∘ ccom 5642  ‘cfv 6511  distcds 17229  Grpcgrp 18865  -_gcsg 18867  MetSpcms 24206  normcnm 24464  NrmGrpcngp 24465

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-co 5647  df-iota 6464  df-fv 6519  df-ngp 24471

This theorem is referenced by:  ngpxms  24489  ngptps  24490  ngpmet  24491  isngp4  24500  nmmtri  24510  nmrtri  24512  subgngp  24523  ngptgp  24524  tngngp2  24540  nlmvscnlem2  24573  nlmvscnlem1  24574  nlmvscn  24575  nrginvrcn  24580  nghmcn  24633  nmcn  24733  nmhmcn  25020  ipcnlem2  25144  ipcnlem1  25145  ipcn  25146  nglmle  25202  cssbn  25275  minveclem2  25326  minveclem3b  25328  minveclem3  25329  minveclem4  25332  minveclem7  25335