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Theorem iscyg2 16673
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 16670 . 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 2752 . . . 4  |-  ( E  =/=  (/)  <->  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }  =/=  (/) )
6 rabn0 3805 . . . 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 249 . . 3  |-  ( E  =/=  (/)  <->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B )
87anbi2i 694 . 2  |-  ( ( G  e.  Grp  /\  E  =/=  (/) )  <->  ( G  e.  Grp  /\  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) )  =  B ) )
93, 8bitr4i 252 1  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\  E  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818   (/)c0 3785    |-> cmpt 4505   ran crn 5000   ` cfv 5586  (class class class)co 6282   ZZcz 10860   Basecbs 14483   Grpcgrp 15720  .gcmg 15724  CycGrpccyg 16668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5549  df-fv 5594  df-ov 6285  df-cyg 16669
This theorem is referenced by:  iscygd  16678  iscygodd  16679  cyggex2  16687  cyggexb  16689  cygzn  18373
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