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Theorem ngpgrp 22343
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 2621 . . 3 (norm‘𝐺) = (norm‘𝐺)
2 eqid 2621 . . 3 (-g𝐺) = (-g𝐺)
3 eqid 2621 . . 3 (dist‘𝐺) = (dist‘𝐺)
41, 2, 3isngp 22340 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g𝐺)) ⊆ (dist‘𝐺)))
54simp1bi 1074 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3560  ccom 5088  cfv 5857  distcds 15890  Grpcgrp 17362  -gcsg 17364  MetSpcmt 22063  normcnm 22321  NrmGrpcngp 22322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-co 5093  df-iota 5820  df-fv 5865  df-ngp 22328
This theorem is referenced by:  ngpds  22348  ngpds2  22350  ngpds3  22352  ngprcan  22354  isngp4  22356  ngpinvds  22357  ngpsubcan  22358  nmf  22359  nmge0  22361  nmeq0  22362  nminv  22365  nmmtri  22366  nmsub  22367  nmrtri  22368  nm2dif  22369  nmtri  22370  nmtri2  22371  ngpi  22372  nm0  22373  ngptgp  22380  tngngp2  22396  tnggrpr  22399  nrmtngnrm  22402  nlmdsdi  22425  nlmdsdir  22426  nrginvrcnlem  22435  ngpocelbl  22448  nmo0  22479  nmotri  22483  0nghm  22485  nmoid  22486  idnghm  22487  nmods  22488  nmcn  22587  nmoleub2lem2  22856  nmhmcn  22860  cphipval2  22980  4cphipval2  22981  cphipval  22982  ipcnlem2  22983  nglmle  23040  qqhcn  29859
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