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Metamath Proof Explorer

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 ) )