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Theorem iscyg2 15169
Description: A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1  |-  B  =  ( Base `  G
)
iscyg.2  |-  .x.  =  (.g
`  G )
iscyg3.e  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
Assertion
Ref Expression
iscyg2  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\  E  =/=  (/) ) )
Distinct variable groups:    x, n, B    n, G, x    .x. , n, x
Allowed substitution hints:    E( x, n)

Proof of Theorem iscyg2
StepHypRef Expression
1 iscyg.1 . . 3  |-  B  =  ( Base `  G
)
2 iscyg.2 . . 3  |-  .x.  =  (.g
`  G )
31, 2iscyg 15166 . 2  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
4 iscyg3.e . . . . 5  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
54neeq1i 2456 . . . 4  |-  ( E  =/=  (/)  <->  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }  =/=  (/) )
6 rabn0 3474 . . . 4  |-  ( { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }  =/=  (/)  <->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B )
75, 6bitri 240 . . 3  |-  ( E  =/=  (/)  <->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B )
87anbi2i 675 . 2  |-  ( ( G  e.  Grp  /\  E  =/=  (/) )  <->  ( G  e.  Grp  /\  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) )  =  B ) )
93, 8bitr4i 243 1  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\  E  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   (/)c0 3455    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:  iscygd  15174  iscygodd  15175  cyggex2  15183  cyggexb  15185  cygzn  16524
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-ne 2448  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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