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

Step	Hyp	Ref	Expression
1		iscyg.1	. . 3 |- B = ( Base ` G )
2		iscyg.2	. . 3 |- .x. = (.g ` G )
3	1, 2	iscyg 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 }
5	4	neeq1i 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 )
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 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