Theorem hial2eq

Description: Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	hial2eq	|- ( ( A e. ~H /\ B e. ~H ) -> ( A. x e. ~H ( A .ih x ) = ( B .ih x ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		hvsubcl	|- ( ( A e. ~H /\ B e. ~H ) -> ( A -h B ) e. ~H )
2		oveq2	|- ( x = ( A -h B ) -> ( A .ih x ) = ( A .ih ( A -h B ) ) )
3		oveq2	|- ( x = ( A -h B ) -> ( B .ih x ) = ( B .ih ( A -h B ) ) )
4	2 3	eqeq12d	|- ( x = ( A -h B ) -> ( ( A .ih x ) = ( B .ih x ) <-> ( A .ih ( A -h B ) ) = ( B .ih ( A -h B ) ) ) )
5	4	rspcv	|- ( ( A -h B ) e. ~H -> ( A. x e. ~H ( A .ih x ) = ( B .ih x ) -> ( A .ih ( A -h B ) ) = ( B .ih ( A -h B ) ) ) )
6	1 5	syl	|- ( ( A e. ~H /\ B e. ~H ) -> ( A. x e. ~H ( A .ih x ) = ( B .ih x ) -> ( A .ih ( A -h B ) ) = ( B .ih ( A -h B ) ) ) )
7		hi2eq	|- ( ( A e. ~H /\ B e. ~H ) -> ( ( A .ih ( A -h B ) ) = ( B .ih ( A -h B ) ) <-> A = B ) )
8	6 7	sylibd	|- ( ( A e. ~H /\ B e. ~H ) -> ( A. x e. ~H ( A .ih x ) = ( B .ih x ) -> A = B ) )
9		oveq1	|- ( A = B -> ( A .ih x ) = ( B .ih x ) )
10	9	ralrimivw	|- ( A = B -> A. x e. ~H ( A .ih x ) = ( B .ih x ) )
11	8 10	impbid1	|- ( ( A e. ~H /\ B e. ~H ) -> ( A. x e. ~H ( A .ih x ) = ( B .ih x ) <-> A = B ) )