Theorem mulvfn

Description: Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012)

	Ref	Expression
Assertion	mulvfn	|- ( ( A e. C /\ B e. D ) -> ( A .v B ) Fn RR )

Proof

Step	Hyp	Ref	Expression
1		ovex	|- ( A x. ( B ` x ) ) e. _V
2		eqid	|- ( x e. RR |-> ( A x. ( B ` x ) ) ) = ( x e. RR |-> ( A x. ( B ` x ) ) )
3	1 2	fnmpti	|- ( x e. RR |-> ( A x. ( B ` x ) ) ) Fn RR
4		mulvval	|- ( ( A e. C /\ B e. D ) -> ( A .v B ) = ( x e. RR |-> ( A x. ( B ` x ) ) ) )
5	4	fneq1d	|- ( ( A e. C /\ B e. D ) -> ( ( A .v B ) Fn RR <-> ( x e. RR |-> ( A x. ( B ` x ) ) ) Fn RR ) )
6	3 5	mpbiri	|- ( ( A e. C /\ B e. D ) -> ( A .v B ) Fn RR )