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Theorem iscyg 17008
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 5872 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscyg.1 . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2516 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5872 . . . . . . . 8  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 iscyg.2 . . . . . . . 8  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2516 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76oveqd 6313 . . . . . 6  |-  ( g  =  G  ->  (
n (.g `  g ) x )  =  ( n 
.x.  x ) )
87mpteq2dv 4544 . . . . 5  |-  ( g  =  G  ->  (
n  e.  ZZ  |->  ( n (.g `  g ) x ) )  =  ( n  e.  ZZ  |->  ( n  .x.  x ) ) )
98rneqd 5240 . . . 4  |-  ( g  =  G  ->  ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) ) )
109, 3eqeq12d 2479 . . 3  |-  ( g  =  G  ->  ( ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
)  <->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
113, 10rexeqbidv 3069 . 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 17007 . 2  |- CycGrp  =  {
g  e.  Grp  |  E. x  e.  ( Base `  g ) ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
) }
1311, 12elrab2 3259 1  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    |-> cmpt 4515   ran crn 5009   ` cfv 5594  (class class class)co 6296   ZZcz 10885   Basecbs 14643   Grpcgrp 16179  .gcmg 16182  CycGrpccyg 17006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-cnv 5016  df-dm 5018  df-rn 5019  df-iota 5557  df-fv 5602  df-ov 6299  df-cyg 17007
This theorem is referenced by:  iscyg2  17011  iscyg3  17015  cyggrp  17018  cygctb  17020  ghmcyg  17024  ablfac2  17266  zncyg  18713
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