Theorem ipfval

Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Hypotheses	ipffval.1	|- V = ( Base ` W )
	ipffval.2	|- ., = ( .i ` W )
	ipffval.3	|- .x. = ( .if ` W )
Assertion	ipfval	|- ( ( X e. V /\ Y e. V ) -> ( X .x. Y ) = ( X ., Y ) )

Proof

Step	Hyp	Ref	Expression
1		ipffval.1	|- V = ( Base ` W )
2		ipffval.2	|- ., = ( .i ` W )
3		ipffval.3	|- .x. = ( .if ` W )
4		oveq12	|- ( ( x = X /\ y = Y ) -> ( x ., y ) = ( X ., Y ) )
5	1 2 3	ipffval	|- .x. = ( x e. V , y e. V |-> ( x ., y ) )
6		ovex	|- ( X ., Y ) e. _V
7	4 5 6	ovmpoa	|- ( ( X e. V /\ Y e. V ) -> ( X .x. Y ) = ( X ., Y ) )