Theorem iscyg2 17566

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 17563	. 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 2700	. . . 4 |- ( E =/= (/) <-> { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B } =/= (/) )
6		rabn0 3764	. . . 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 257	. . 3 |- ( E =/= (/) <-> E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B )
8	7	anbi2i 705	. 2 |- ( ( G e. Grp /\ E =/= (/) ) <-> ( G e. Grp /\ E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B ) )
9	3, 8	bitr4i 260	1 |- ( G e. CycGrp <-> ( G e. Grp /\ E =/= (/) ) )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   E.wrex 2750   {crab 2753   (/)c0 3743   |-> cmpt 4475   ran crn 4854   ` cfv 5601  (class class class)co 6315   ZZcz 10966   Basecbs 15170   Grpcgrp 16718  ._gcmg 16721  CycGrpccyg 17561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-cnv 4861  df-dm 4863  df-rn 4864  df-iota 5565  df-fv 5609  df-ov 6318  df-cyg 17562

This theorem is referenced by:  iscygd  17571  iscygodd  17572  cyggex2  17580  cyggexb  17582  cygzn  19190