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 ) |