Theorem ngpms 24429

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 2731	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2731	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2731	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24425	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp2bi 1145	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Syntax hints:   → wi 4   ∈ wcel 2105   ⊆ wss 3948   ∘ ccom 5680  ‘cfv 6543  distcds 17213  Grpcgrp 18861  -_gcsg 18863  MetSpcms 24144  normcnm 24405  NrmGrpcngp 24406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-co 5685  df-iota 6495  df-fv 6551  df-ngp 24412

This theorem is referenced by:  ngpxms  24430  ngptps  24431  ngpmet  24432  isngp4  24441  nmmtri  24451  nmrtri  24453  subgngp  24464  ngptgp  24465  tngngp2  24489  nlmvscnlem2  24522  nlmvscnlem1  24523  nlmvscn  24524  nrginvrcn  24529  nghmcn  24582  nmcn  24680  nmhmcn  24967  ipcnlem2  25092  ipcnlem1  25093  ipcn  25094  nglmle  25150  cssbn  25223  minveclem2  25274  minveclem3b  25276  minveclem3  25277  minveclem4  25280  minveclem7  25283