Theorem iscyg2 17012

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 17009	. 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 2742	. . . 4 |- ( E =/= (/) <-> { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B } =/= (/) )
6		rabn0 3814	. . . 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 1395   e. wcel 1819   =/= wne 2652   E.wrex 2808   {crab 2811   (/)c0 3793   |-> cmpt 4515   ran crn 5009   ` cfv 5594  (class class class)co 6296   ZZcz 10885   Basecbs 14644   Grpcgrp 16180  ._gcmg 16183  CycGrpccyg 17007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-cnv 5016  df-dm 5018  df-rn 5019  df-iota 5557  df-fv 5602  df-ov 6299  df-cyg 17008

This theorem is referenced by:  iscygd  17017  iscygodd  17018  cyggex2  17026  cyggexb  17028  cygzn  18736