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Theorem ngpms 23143
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
StepHypRef Expression
1 eqid 2826 . . 3 (norm‘𝐺) = (norm‘𝐺)
2 eqid 2826 . . 3 (-g𝐺) = (-g𝐺)
3 eqid 2826 . . 3 (dist‘𝐺) = (dist‘𝐺)
41, 2, 3isngp 23139 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g𝐺)) ⊆ (dist‘𝐺)))
54simp2bi 1140 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3940  ccom 5558  cfv 6354  distcds 16569  Grpcgrp 18048  -gcsg 18050  MetSpcms 22862  normcnm 23120  NrmGrpcngp 23121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-co 5563  df-iota 6313  df-fv 6362  df-ngp 23127
This theorem is referenced by:  ngpxms  23144  ngptps  23145  ngpmet  23146  isngp4  23155  nmmtri  23165  nmrtri  23167  subgngp  23178  ngptgp  23179  tngngp2  23195  nlmvscnlem2  23228  nlmvscnlem1  23229  nlmvscn  23230  nrginvrcn  23235  nghmcn  23288  nmcn  23386  nmhmcn  23658  ipcnlem2  23781  ipcnlem1  23782  ipcn  23783  nglmle  23839  cssbn  23912  minveclem2  23963  minveclem3b  23965  minveclem3  23966  minveclem4  23969  minveclem7  23972
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