Theorem ngpms 22214

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 2610	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2610	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2610	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 22210	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1070	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540   ∘ ccom 5042  ‘cfv 5804  distcds 15777  Grpcgrp 17245  -_gcsg 17247  MetSpcmt 21933  normcnm 22191  NrmGrpcngp 22192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-co 5047  df-iota 5768  df-fv 5812  df-ngp 22198

This theorem is referenced by:  ngpxms  22215  ngptps  22216  ngpmet  22217  isngp4  22226  nmf  22229  nmmtri  22236  nmrtri  22238  subgngp  22249  ngptgp  22250  tngngp2  22266  nlmvscnlem2  22299  nlmvscnlem1  22300  nlmvscn  22301  nrginvrcn  22306  nghmcn  22359  nmcn  22455  nmhmcn  22728  ipcnlem2  22851  ipcnlem1  22852  ipcn  22853  nglmle  22908  minveclem2  23005  minveclem3b  23007  minveclem3  23008  minveclem4  23011  minveclem7  23014