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

Step	Hyp	Ref	Expression
1		iscyg.1	. . 3 |- B = ( Base ` G )
2		iscyg.2	. . 3 |- .x. = (.g ` G )
3	1, 2	iscyg 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 }
5	4	neeq1i 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 )
7	5, 6	bitri 249	. . 3 |- ( E =/= (/) <-> E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B )
8	7	anbi2i 694	. 2 |- ( ( G e. Grp /\ E =/= (/) ) <-> ( G e. Grp /\ E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B ) )
9	3, 8	bitr4i 252	1 |- ( G e. CycGrp <-> ( G e. Grp /\ E =/= (/) ) )

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