Theorem ngpms 24522

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 24518	. 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 17206  Grpcgrp 18848  -_gcsg 18850  MetSpcms 24240  normcnm 24498  NrmGrpcngp 24499

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 24505

This theorem is referenced by:  ngpxms  24523  ngptps  24524  ngpmet  24525  isngp4  24534  nmmtri  24544  nmrtri  24546  subgngp  24557  ngptgp  24558  tngngp2  24574  nlmvscnlem2  24607  nlmvscnlem1  24608  nlmvscn  24609  nrginvrcn  24614  nghmcn  24667  nmcn  24767  nmhmcn  25054  ipcnlem2  25178  ipcnlem1  25179  ipcn  25180  nglmle  25236  cssbn  25309  minveclem2  25360  minveclem3b  25362  minveclem3  25363  minveclem4  25366  minveclem7  25369