Theorem ipfeq

Description: If the inner product operation is already a function, the functionalization of it is equal to the original operation. (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	ipfeq	|- ( ., Fn ( V X. V ) -> .x. = ., )

Proof

Step	Hyp	Ref	Expression
1		ipffval.1	|- V = ( Base ` W )
2		ipffval.2	|- ., = ( .i ` W )
3		ipffval.3	|- .x. = ( .if ` W )
4	1 2 3	ipffval	|- .x. = ( x e. V , y e. V |-> ( x ., y ) )
5		fnov	|- ( ., Fn ( V X. V ) <-> ., = ( x e. V , y e. V |-> ( x ., y ) ) )
6	5	biimpi	|- ( ., Fn ( V X. V ) -> ., = ( x e. V , y e. V |-> ( x ., y ) ) )
7	4 6	eqtr4id	|- ( ., Fn ( V X. V ) -> .x. = ., )