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Theorem cyggrp 17459
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 2429 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2429 . . 3  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg 17449 . 2  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  (
Base `  G ) ran  ( n  e.  ZZ  |->  ( n (.g `  G
) x ) )  =  ( Base `  G
) ) )
43simplbi 461 1  |-  ( G  e. CycGrp  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   E.wrex 2783    |-> cmpt 4484   ran crn 4855   ` cfv 5601  (class class class)co 6305   ZZcz 10937   Basecbs 15084   Grpcgrp 16620  .gcmg 16623  CycGrpccyg 17447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-cnv 4862  df-dm 4864  df-rn 4865  df-iota 5565  df-fv 5609  df-ov 6308  df-cyg 17448
This theorem is referenced by:  cygznlem1  19068  cygznlem2a  19069  cygznlem3  19071
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