Theorem scaffn

Description: The scalar multiplication operation is 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 )
Assertion	scaffn	|- .xb Fn ( K X. B )

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		eqid	|- ( .s ` W ) = ( .s ` W )
6	1 2 3 4 5	scaffval	|- .xb = ( x e. K , y e. B |-> ( x ( .s ` W ) y ) )
7		ovex	|- ( x ( .s ` W ) y ) e. _V
8	6 7	fnmpoi	|- .xb Fn ( K X. B )