Theorem iscyggen

Description: The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	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	iscyggen	|- ( X e. E <-> ( X e. B /\ ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )

Proof

Step	Hyp	Ref	Expression
1		iscyg.1	|- B = ( Base ` G )
2		iscyg.2	|- .x. = ( .g ` G )
3		iscyg3.e	|- E = { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B }
4		simpl	|- ( ( x = X /\ n e. ZZ ) -> x = X )
5	4	oveq2d	|- ( ( x = X /\ n e. ZZ ) -> ( n .x. x ) = ( n .x. X ) )
6	5	mpteq2dva	|- ( x = X -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n .x. X ) ) )
7	6	rneqd	|- ( x = X -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n .x. X ) ) )
8	7	eqeq1d	|- ( x = X -> ( ran ( n e. ZZ |-> ( n .x. x ) ) = B <-> ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )
9	8 3	elrab2	|- ( X e. E <-> ( X e. B /\ ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )