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Theorem ngpgrp 21411
Description: A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
ngpgrp  |-  ( G  e. NrmGrp  ->  G  e.  Grp )

Proof of Theorem ngpgrp
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( norm `  G )  =  (
norm `  G )
2 eqid 2402 . . 3  |-  ( -g `  G )  =  (
-g `  G )
3 eqid 2402 . . 3  |-  ( dist `  G )  =  (
dist `  G )
41, 2, 3isngp 21408 . 2  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  (
( norm `  G )  o.  ( -g `  G
) )  C_  ( dist `  G ) ) )
54simp1bi 1012 1  |-  ( G  e. NrmGrp  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842    C_ wss 3414    o. ccom 4827   ` cfv 5569   distcds 14918   Grpcgrp 16377   -gcsg 16379   MetSpcmt 21113   normcnm 21389  NrmGrpcngp 21390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-co 4832  df-iota 5533  df-fv 5577  df-ngp 21396
This theorem is referenced by:  ngpds  21415  ngpds2  21417  ngpds3  21419  ngprcan  21421  isngp4  21423  ngpinvds  21424  ngpsubcan  21425  nmf  21426  nmge0  21428  nmeq0  21429  nminv  21432  nmmtri  21433  nmsub  21434  nmrtri  21435  nm2dif  21436  nmtri  21437  nm0  21438  ngptgp  21442  tngngp2  21458  nlmdsdi  21482  nlmdsdir  21483  nrginvrcnlem  21491  nmo0  21534  nmotri  21538  0nghm  21540  nmoid  21541  idnghm  21542  nmods  21543  nmcn  21641  nmoleub2lem2  21891  nmhmcn  21895  ipcnlem2  21976  qqhcn  28424
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