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Theorem ngpgrp 24612
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
StepHypRef Expression
1 eqid 2737 . . 3 (norm‘𝐺) = (norm‘𝐺)
2 eqid 2737 . . 3 (-g𝐺) = (-g𝐺)
3 eqid 2737 . . 3 (dist‘𝐺) = (dist‘𝐺)
41, 2, 3isngp 24609 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g𝐺)) ⊆ (dist‘𝐺)))
54simp1bi 1146 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951  ccom 5689  cfv 6561  distcds 17306  Grpcgrp 18951  -gcsg 18953  MetSpcms 24328  normcnm 24589  NrmGrpcngp 24590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-co 5694  df-iota 6514  df-fv 6569  df-ngp 24596
This theorem is referenced by:  ngpds  24617  ngpds2  24619  ngpds3  24621  ngprcan  24623  isngp4  24625  ngpinvds  24626  ngpsubcan  24627  nmf  24628  nmge0  24630  nmeq0  24631  nminv  24634  nmmtri  24635  nmsub  24636  nmrtri  24637  nm2dif  24638  nmtri  24639  nmtri2  24640  ngpi  24641  nm0  24642  ngptgp  24649  tngngp2  24673  tnggrpr  24676  nrmtngnrm  24679  nlmdsdi  24702  nlmdsdir  24703  nrginvrcnlem  24712  ngpocelbl  24725  nmo0  24756  nmotri  24760  0nghm  24762  nmoid  24763  idnghm  24764  nmods  24765  nmcn  24866  nmoleub2lem2  25149  nmhmcn  25153  cphpyth  25250  cphipval2  25275  4cphipval2  25276  cphipval  25277  ipcnlem2  25278  nglmle  25336  qqhcn  33992
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