Theorem ngpms 24495

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 2730	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2730	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2730	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24491	. 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 3917   ∘ ccom 5645  ‘cfv 6514  distcds 17236  Grpcgrp 18872  -_gcsg 18874  MetSpcms 24213  normcnm 24471  NrmGrpcngp 24472

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-co 5650  df-iota 6467  df-fv 6522  df-ngp 24478

This theorem is referenced by:  ngpxms  24496  ngptps  24497  ngpmet  24498  isngp4  24507  nmmtri  24517  nmrtri  24519  subgngp  24530  ngptgp  24531  tngngp2  24547  nlmvscnlem2  24580  nlmvscnlem1  24581  nlmvscn  24582  nrginvrcn  24587  nghmcn  24640  nmcn  24740  nmhmcn  25027  ipcnlem2  25151  ipcnlem1  25152  ipcn  25153  nglmle  25209  cssbn  25282  minveclem2  25333  minveclem3b  25335  minveclem3  25336  minveclem4  25339  minveclem7  25342