Theorem bra0

Description: The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006) (Revised by Mario Carneiro, 23-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	bra0	|- ( bra ` 0h ) = ( ~H X. { 0 } )

Proof

Step	Hyp	Ref	Expression
1		ax-hv0cl	|- 0h e. ~H
2		brafval	|- ( 0h e. ~H -> ( bra ` 0h ) = ( x e. ~H |-> ( x .ih 0h ) ) )
3		hi02	|- ( x e. ~H -> ( x .ih 0h ) = 0 )
4	3	mpteq2ia	|- ( x e. ~H |-> ( x .ih 0h ) ) = ( x e. ~H |-> 0 )
5	2 4	eqtrdi	|- ( 0h e. ~H -> ( bra ` 0h ) = ( x e. ~H |-> 0 ) )
6	1 5	ax-mp	|- ( bra ` 0h ) = ( x e. ~H |-> 0 )
7		fconstmpt	|- ( ~H X. { 0 } ) = ( x e. ~H |-> 0 )
8	6 7	eqtr4i	|- ( bra ` 0h ) = ( ~H X. { 0 } )