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Metamath Proof Explorer

Theorem mulgcl

Description: Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014)

Ref Expression
Hypotheses mulgnncl.b 
|- B = ( Base ` G )
mulgnncl.t 
|- .x. = ( .g ` G )
Assertion mulgcl 
|- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )

Proof

Step Hyp Ref Expression
1 mulgnncl.b 
 |-  B = ( Base ` G )
2 mulgnncl.t 
 |-  .x. = ( .g ` G )
3 eqid 
 |-  ( +g ` G ) = ( +g ` G )
4 id 
 |-  ( G e. Grp -> G e. Grp )
5 ssidd 
 |-  ( G e. Grp -> B C_ B )
6 1 3 grpcl 
 |-  ( ( G e. Grp /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
7 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
8 1 7 grpidcl 
 |-  ( G e. Grp -> ( 0g ` G ) e. B )
9 eqid 
 |-  ( invg ` G ) = ( invg ` G )
10 1 9 grpinvcl 
 |-  ( ( G e. Grp /\ x e. B ) -> ( ( invg ` G ) ` x ) e. B )
11 1 2 3 4 5 6 7 8 9 10 mulgsubcl 
 |-  ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )