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Theorem cyggrp 15501
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 2438 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2438 . . 3  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 15491 . 2  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  (
Base `  G ) ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  ( Base `  G
) ) )
43simplbi 448 1  |-  ( G  e. CycGrp  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   E.wrex 2708    e. cmpt 4268   ran crn 4881   ` cfv 5456  (class class class)co 6083   ZZcz 10284   Basecbs 13471   Grpcgrp 14687  .gcmg 14691  CycGrpccyg 15489
This theorem is referenced by:  cygznlem1  16849  cygznlem2a  16850  cygznlem3  16852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891  df-iota 5420  df-fv 5464  df-ov 6086  df-cyg 15490
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