Theorem pwsabl

Description: The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015)

	Ref	Expression
Hypothesis	pwscmn.y	|- Y = ( R ^s I )
Assertion	pwsabl	|- ( ( R e. Abel /\ I e. V ) -> Y e. Abel )

Proof

Step	Hyp	Ref	Expression
1		pwscmn.y	|- Y = ( R ^s I )
2		eqid	|- ( Scalar ` R ) = ( Scalar ` R )
3	1 2	pwsval	|- ( ( R e. Abel /\ I e. V ) -> Y = ( ( Scalar ` R ) Xs_ ( I X. { R } ) ) )
4		eqid	|- ( ( Scalar ` R ) Xs_ ( I X. { R } ) ) = ( ( Scalar ` R ) Xs_ ( I X. { R } ) )
5		simpr	|- ( ( R e. Abel /\ I e. V ) -> I e. V )
6		fvexd	|- ( ( R e. Abel /\ I e. V ) -> ( Scalar ` R ) e. _V )
7		fconst6g	|- ( R e. Abel -> ( I X. { R } ) : I --> Abel )
8	7	adantr	|- ( ( R e. Abel /\ I e. V ) -> ( I X. { R } ) : I --> Abel )
9	4 5 6 8	prdsabld	|- ( ( R e. Abel /\ I e. V ) -> ( ( Scalar ` R ) Xs_ ( I X. { R } ) ) e. Abel )
10	3 9	eqeltrd	|- ( ( R e. Abel /\ I e. V ) -> Y e. Abel )