Theorem scafval

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	scaffval.b	|- B = ( Base ` W )
	scaffval.f	|- F = ( Scalar ` W )
	scaffval.k	|- K = ( Base ` F )
	scaffval.a	|- .xb = ( .sf ` W )
	scaffval.s	|- .x. = ( .s ` W )
Assertion	scafval	|- ( ( X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )

Proof

Step	Hyp	Ref	Expression
1		scaffval.b	|- B = ( Base ` W )
2		scaffval.f	|- F = ( Scalar ` W )
3		scaffval.k	|- K = ( Base ` F )
4		scaffval.a	|- .xb = ( .sf ` W )
5		scaffval.s	|- .x. = ( .s ` W )
6		oveq12	|- ( ( x = X /\ y = Y ) -> ( x .x. y ) = ( X .x. Y ) )
7	1 2 3 4 5	scaffval	|- .xb = ( x e. K , y e. B |-> ( x .x. y ) )
8		ovex	|- ( X .x. Y ) e. _V
9	6 7 8	ovmpoa	|- ( ( X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )