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Theorem cyggrp 15176
Description: A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
cyggrp  |-  ( G  e. CycGrp  ->  G  e.  Grp )

Proof of Theorem cyggrp
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2283 . . 3  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 15166 . 2  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  (
Base `  G ) ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  ( Base `  G
) ) )
43simplbi 446 1  |-  ( G  e. CycGrp  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   ZZcz 10024   Basecbs 13148   Grpcgrp 14362  .gcmg 14366  CycGrpccyg 15164
This theorem is referenced by:  cygznlem1  16520  cygznlem2a  16521  cygznlem3  16523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263  df-ov 5861  df-cyg 15165
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