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Theorem iscyg 17449
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 5881 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscyg.1 . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2488 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5881 . . . . . . . 8  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 iscyg.2 . . . . . . . 8  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2488 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76oveqd 6322 . . . . . 6  |-  ( g  =  G  ->  (
n (.g `  g ) x )  =  ( n 
.x.  x ) )
87mpteq2dv 4513 . . . . 5  |-  ( g  =  G  ->  (
n  e.  ZZ  |->  ( n (.g `  g ) x ) )  =  ( n  e.  ZZ  |->  ( n  .x.  x ) ) )
98rneqd 5082 . . . 4  |-  ( g  =  G  ->  ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) ) )
109, 3eqeq12d 2451 . . 3  |-  ( g  =  G  ->  ( ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
)  <->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
113, 10rexeqbidv 3047 . 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 17448 . 2  |- CycGrp  =  {
g  e.  Grp  |  E. x  e.  ( Base `  g ) ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
) }
1311, 12elrab2 3237 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 187    /\ wa 370    = 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:  iscyg2  17452  iscyg3  17456  cyggrp  17459  cygctb  17461  ghmcyg  17465  ablfac2  17657  zncyg  19050
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