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Theorem iscyg 16356
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 5691 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscyg.1 . . . 4  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2493 . . 3  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 fveq2 5691 . . . . . . . 8  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
5 iscyg.2 . . . . . . . 8  |-  .x.  =  (.g
`  G )
64, 5syl6eqr 2493 . . . . . . 7  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
76oveqd 6108 . . . . . 6  |-  ( g  =  G  ->  (
n (.g `  g ) x )  =  ( n 
.x.  x ) )
87mpteq2dv 4379 . . . . 5  |-  ( g  =  G  ->  (
n  e.  ZZ  |->  ( n (.g `  g ) x ) )  =  ( n  e.  ZZ  |->  ( n  .x.  x ) ) )
98rneqd 5067 . . . 4  |-  ( g  =  G  ->  ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) ) )
109, 3eqeq12d 2457 . . 3  |-  ( g  =  G  ->  ( ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
)  <->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
113, 10rexeqbidv 2932 . 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 16355 . 2  |- CycGrp  =  {
g  e.  Grp  |  E. x  e.  ( Base `  g ) ran  ( n  e.  ZZ  |->  ( n (.g `  g
) x ) )  =  ( Base `  g
) }
1311, 12elrab2 3119 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 1369    e. wcel 1756   E.wrex 2716    e. cmpt 4350   ran crn 4841   ` cfv 5418  (class class class)co 6091   ZZcz 10646   Basecbs 14174   Grpcgrp 15410  .gcmg 15414  CycGrpccyg 16354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-cnv 4848  df-dm 4850  df-rn 4851  df-iota 5381  df-fv 5426  df-ov 6094  df-cyg 16355
This theorem is referenced by:  iscyg2  16359  iscyg3  16363  cyggrp  16366  cygctb  16368  ghmcyg  16372  ablfac2  16590  zncyg  17981
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