Theorem ngpms 23756

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 2738	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2738	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2738	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 23752	. 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 2106   ⊆ wss 3887   ∘ ccom 5593  ‘cfv 6433  distcds 16971  Grpcgrp 18577  -_gcsg 18579  MetSpcms 23471  normcnm 23732  NrmGrpcngp 23733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-co 5598  df-iota 6391  df-fv 6441  df-ngp 23739

This theorem is referenced by:  ngpxms  23757  ngptps  23758  ngpmet  23759  isngp4  23768  nmmtri  23778  nmrtri  23780  subgngp  23791  ngptgp  23792  tngngp2  23816  nlmvscnlem2  23849  nlmvscnlem1  23850  nlmvscn  23851  nrginvrcn  23856  nghmcn  23909  nmcn  24007  nmhmcn  24283  ipcnlem2  24408  ipcnlem1  24409  ipcn  24410  nglmle  24466  cssbn  24539  minveclem2  24590  minveclem3b  24592  minveclem3  24593  minveclem4  24596  minveclem7  24599