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Theorem iscyg2 16449
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 16446 . 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 2730 . . . 4  |-  ( E  =/=  (/)  <->  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }  =/=  (/) )
6 rabn0 3741 . . . 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 1370    e. wcel 1757    =/= wne 2641   E.wrex 2793   {crab 2796   (/)c0 3721    |-> cmpt 4434   ran crn 4925   ` cfv 5502  (class class class)co 6176   ZZcz 10733   Basecbs 14262   Grpcgrp 15498  .gcmg 15502  CycGrpccyg 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-cnv 4932  df-dm 4934  df-rn 4935  df-iota 5465  df-fv 5510  df-ov 6179  df-cyg 16445
This theorem is referenced by:  iscygd  16454  iscygodd  16455  cyggex2  16463  cyggexb  16465  cygzn  18098
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