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Theorem iscyg2 15492
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 15489 . 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 2611 . . . 4  |-  ( E  =/=  (/)  <->  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }  =/=  (/) )
6 rabn0 3647 . . . 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 241 . . 3  |-  ( E  =/=  (/)  <->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B )
87anbi2i 676 . 2  |-  ( ( G  e.  Grp  /\  E  =/=  (/) )  <->  ( G  e.  Grp  /\  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) )  =  B ) )
93, 8bitr4i 244 1  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\  E  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709   (/)c0 3628    e. cmpt 4266   ran crn 4879   ` cfv 5454  (class class class)co 6081   ZZcz 10282   Basecbs 13469   Grpcgrp 14685  .gcmg 14689  CycGrpccyg 15487
This theorem is referenced by:  iscygd  15497  iscygodd  15498  cyggex2  15506  cyggexb  15508  cygzn  16851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462  df-ov 6084  df-cyg 15488
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