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Theorem iscyg2 15185
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 15182 . 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 2469 . . . 4  |-  ( E  =/=  (/)  <->  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }  =/=  (/) )
6 rabn0 3487 . . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   (/)c0 3468    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   ZZcz 10040   Basecbs 13164   Grpcgrp 14378  .gcmg 14382  CycGrpccyg 15180
This theorem is referenced by:  iscygd  15190  iscygodd  15191  cyggex2  15199  cyggexb  15201  cygzn  16540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279  df-ov 5877  df-cyg 15181
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