Theorem ngpms 24581

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 2737	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2737	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2737	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24577	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1147	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3890   ∘ ccom 5632  ‘cfv 6496  distcds 17226  Grpcgrp 18906  -_gcsg 18908  MetSpcms 24299  normcnm 24557  NrmGrpcngp 24558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-co 5637  df-iota 6452  df-fv 6504  df-ngp 24564

This theorem is referenced by:  ngpxms  24582  ngptps  24583  ngpmet  24584  isngp4  24593  nmmtri  24603  nmrtri  24605  subgngp  24616  ngptgp  24617  tngngp2  24633  nlmvscnlem2  24666  nlmvscnlem1  24667  nlmvscn  24668  nrginvrcn  24673  nghmcn  24726  nmcn  24826  nmhmcn  25103  ipcnlem2  25227  ipcnlem1  25228  ipcn  25229  nglmle  25285  cssbn  25358  minveclem2  25409  minveclem3b  25411  minveclem3  25412  minveclem4  25415  minveclem7  25418