Theorem ngpms 24521

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 24517	. 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 3911   ∘ ccom 5635  ‘cfv 6499  distcds 17205  Grpcgrp 18847  -_gcsg 18849  MetSpcms 24239  normcnm 24497  NrmGrpcngp 24498

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-co 5640  df-iota 6452  df-fv 6507  df-ngp 24504

This theorem is referenced by:  ngpxms  24522  ngptps  24523  ngpmet  24524  isngp4  24533  nmmtri  24543  nmrtri  24545  subgngp  24556  ngptgp  24557  tngngp2  24573  nlmvscnlem2  24606  nlmvscnlem1  24607  nlmvscn  24608  nrginvrcn  24613  nghmcn  24666  nmcn  24766  nmhmcn  25053  ipcnlem2  25177  ipcnlem1  25178  ipcn  25179  nglmle  25235  cssbn  25308  minveclem2  25359  minveclem3b  25361  minveclem3  25362  minveclem4  25365  minveclem7  25368