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Theorem iscyg 15417
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 5669 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscyg.1 . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2438 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5669 . . . . . . . 8  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 iscyg.2 . . . . . . . 8  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2438 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76oveqd 6038 . . . . . 6  |-  ( g  =  G  ->  (
n (.g `  g ) x )  =  ( n 
.x.  x ) )
87mpteq2dv 4238 . . . . 5  |-  ( g  =  G  ->  (
n  e.  ZZ  |->  ( n (.g `  g ) x ) )  =  ( n  e.  ZZ  |->  ( n  .x.  x ) ) )
98rneqd 5038 . . . 4  |-  ( g  =  G  ->  ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) ) )
109, 3eqeq12d 2402 . . 3  |-  ( g  =  G  ->  ( ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
)  <->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
113, 10rexeqbidv 2861 . 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 15416 . 2  |- CycGrp  =  {
g  e.  Grp  |  E. x  e.  ( Base `  g ) ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
) }
1311, 12elrab2 3038 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651    e. cmpt 4208   ran crn 4820   ` cfv 5395  (class class class)co 6021   ZZcz 10215   Basecbs 13397   Grpcgrp 14613  .gcmg 14617  CycGrpccyg 15415
This theorem is referenced by:  iscyg2  15420  iscyg3  15424  cyggrp  15427  cygctb  15429  ghmcyg  15433  ablfac2  15575  zncyg  16753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-cnv 4827  df-dm 4829  df-rn 4830  df-iota 5359  df-fv 5403  df-ov 6024  df-cyg 15416
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