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Theorem iscyg 15166
Description: Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1  |-  B  =  ( Base `  G
)
iscyg.2  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
iscyg  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
Distinct variable groups:    x, n, B    n, G, x    .x. , n, x

Proof of Theorem iscyg
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscyg.1 . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2333 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5525 . . . . . . . 8  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 iscyg.2 . . . . . . . 8  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2333 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76oveqd 5875 . . . . . 6  |-  ( g  =  G  ->  (
n (.g `  g ) x )  =  ( n 
.x.  x ) )
87mpteq2dv 4107 . . . . 5  |-  ( g  =  G  ->  (
n  e.  ZZ  |->  ( n (.g `  g ) x ) )  =  ( n  e.  ZZ  |->  ( n  .x.  x ) ) )
98rneqd 4906 . . . 4  |-  ( g  =  G  ->  ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) ) )
109, 3eqeq12d 2297 . . 3  |-  ( g  =  G  ->  ( ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
)  <->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
113, 10rexeqbidv 2749 . 2  |-  ( g  =  G  ->  ( E. x  e.  ( Base `  g ) ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
)  <->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
12 df-cyg 15165 . 2  |- CycGrp  =  {
g  e.  Grp  |  E. x  e.  ( Base `  g ) ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
) }
1311, 12elrab2 2925 1  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = 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:  iscyg2  15169  iscyg3  15173  cyggrp  15176  cygctb  15178  ghmcyg  15182  ablfac2  15324  zncyg  16502
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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