Theorem iscyg 18281

Description: Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
iscyg.1	|- B = ( Base ` G )
iscyg.2	|- .x. = (.g ` G )

Assertion

Ref	Expression
iscyg	|- ( G e. CycGrp <-> ( G e. Grp /\ E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B ) )

Distinct variable groups:   x, n, B   n, G, x   .x. , n, x

Proof of Theorem iscyg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		iscyg.1	. . . 4 |- B = ( Base ` G )
3	1, 2	syl6eqr 2674	. . 3 |- ( g = G -> ( Base ` g ) = B )
4		fveq2 6191	. . . . . . . 8 |- ( g = G -> (.g ` g ) = (.g ` G ) )
5		iscyg.2	. . . . . . . 8 |- .x. = (.g ` G )
6	4, 5	syl6eqr 2674	. . . . . . 7 |- ( g = G -> (.g ` g ) = .x. )
7	6	oveqd 6667	. . . . . 6 |- ( g = G -> ( n (.g ` g ) x ) = ( n .x. x ) )
8	7	mpteq2dv 4745	. . . . 5 |- ( g = G -> ( n e. ZZ |-> ( n (.g ` g ) x ) ) = ( n e. ZZ |-> ( n .x. x ) ) )
9	8	rneqd 5353	. . . 4 |- ( g = G -> ran ( n e. ZZ |-> ( n (.g ` g ) x ) ) = ran ( n e. ZZ |-> ( n .x. x ) ) )
10	9, 3	eqeq12d 2637	. . 3 |- ( g = G -> ( ran ( n e. ZZ |-> ( n (.g ` g ) x ) ) = ( Base ` g ) <-> ran ( n e. ZZ |-> ( n .x. x ) ) = B ) )
11	3, 10	rexeqbidv 3153	. 2 |- ( g = G -> ( E. x e. ( Base ` g ) ran ( n e. ZZ |-> ( n (.g ` g ) x ) ) = ( Base ` g ) <-> E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B ) )
12		df-cyg 18280	. 2 |- CycGrp = { g e. Grp | E. x e. ( Base ` g ) ran ( n e. ZZ |-> ( n (.g ` g ) x ) ) = ( Base ` g ) }
13	11, 12	elrab2 3366	1 |- ( G e. CycGrp <-> ( G e. Grp /\ E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   |-> 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-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:  iscyg2  18284  iscyg3  18288  cyggrp  18291  cygctb  18293  ghmcyg  18297  ablfac2  18488  zncyg  19897