Theorem iscyggen 18282

Description: The property of being a cyclic generator for a group. (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
iscyggen	|- ( X e. E <-> ( X e. B /\ ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )

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

Allowed substitution hints:   E( x, n)

Proof of Theorem iscyggen

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( x = X /\ n e. ZZ ) -> x = X )
2	1	oveq2d 6666	. . . . 5 |- ( ( x = X /\ n e. ZZ ) -> ( n .x. x ) = ( n .x. X ) )
3	2	mpteq2dva 4744	. . . 4 |- ( x = X -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n .x. X ) ) )
4	3	rneqd 5353	. . 3 |- ( x = X -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n .x. X ) ) )
5	4	eqeq1d 2624	. 2 |- ( x = X -> ( ran ( n e. ZZ |-> ( n .x. x ) ) = B <-> ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )
6		iscyg3.e	. 2 |- E = { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B }
7	5, 6	elrab2 3366	1 |- ( X e. E <-> ( X e. B /\ ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   ZZcz 11377   Basecbs 15857  ._gcmg 17540

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

This theorem is referenced by:  iscyggen2  18283  cyggenod  18286  cyggenod2  18287  cygznlem1  19915  cygznlem3  19918