Theorem ngpgrp 24107

Description: A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)

Assertion

Ref	Expression
ngpgrp	⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)

Proof of Theorem ngpgrp

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (norm‘𝐺) = (norm‘𝐺)
2		eqid 2732	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
3		eqid 2732	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
4	1, 2, 3	isngp 24104	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)))
5	4	simp1bi 1145	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3948   ∘ ccom 5680  ‘cfv 6543  distcds 17205  Grpcgrp 18818  -_gcsg 18820  MetSpcms 23823  normcnm 24084  NrmGrpcngp 24085

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  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 24091

This theorem is referenced by:  ngpds  24112  ngpds2  24114  ngpds3  24116  ngprcan  24118  isngp4  24120  ngpinvds  24121  ngpsubcan  24122  nmf  24123  nmge0  24125  nmeq0  24126  nminv  24129  nmmtri  24130  nmsub  24131  nmrtri  24132  nm2dif  24133  nmtri  24134  nmtri2  24135  ngpi  24136  nm0  24137  ngptgp  24144  tngngp2  24168  tnggrpr  24171  nrmtngnrm  24174  nlmdsdi  24197  nlmdsdir  24198  nrginvrcnlem  24207  ngpocelbl  24220  nmo0  24251  nmotri  24255  0nghm  24257  nmoid  24258  idnghm  24259  nmods  24260  nmcn  24359  nmoleub2lem2  24631  nmhmcn  24635  cphpyth  24732  cphipval2  24757  4cphipval2  24758  cphipval  24759  ipcnlem2  24760  nglmle  24818  qqhcn  32966