Theorem brafval

Description: The bra of a vector, expressed as <. A | in Dirac notation. See df-bra . (Contributed by NM, 15-May-2006) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	brafval	|- ( A e. ~H -> ( bra ` A ) = ( x e. ~H |-> ( x .ih A ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq2	|- ( y = A -> ( x .ih y ) = ( x .ih A ) )
2	1	mpteq2dv	|- ( y = A -> ( x e. ~H |-> ( x .ih y ) ) = ( x e. ~H |-> ( x .ih A ) ) )
3		df-bra	|- bra = ( y e. ~H |-> ( x e. ~H |-> ( x .ih y ) ) )
4		ax-hilex	|- ~H e. _V
5	4	mptex	|- ( x e. ~H |-> ( x .ih A ) ) e. _V
6	2 3 5	fvmpt	|- ( A e. ~H -> ( bra ` A ) = ( x e. ~H |-> ( x .ih A ) ) )