Theorem iscyg2 18284

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 18281	. 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 2858	. . . 4 |- ( E =/= (/) <-> { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B } =/= (/) )
6		rabn0 3958	. . . 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 264	. . 3 |- ( E =/= (/) <-> E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B )
8	7	anbi2i 730	. 2 |- ( ( G e. Grp /\ E =/= (/) ) <-> ( G e. Grp /\ E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B ) )
9	3, 8	bitr4i 267	1 |- ( G e. CycGrp <-> ( G e. Grp /\ E =/= (/) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   (/)c0 3915   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   ZZcz 11377   Basecbs 15857   Grpcgrp 17422  ._gcmg 17540  CycGrpccyg 18279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-cyg 18280

This theorem is referenced by:  iscygd  18289  iscygodd  18290  cyggex2  18298  cyggexb  18300  cygzn  19919