Theorem hladdid

Description: Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hladdid.1	|- X = ( BaseSet ` U )
	hladdid.2	|- G = ( +v ` U )
	hladdid.5	|- Z = ( 0vec ` U )
Assertion	hladdid	|- ( ( U e. CHilOLD /\ A e. X ) -> ( A G Z ) = A )

Proof

Step	Hyp	Ref	Expression
1		hladdid.1	|- X = ( BaseSet ` U )
2		hladdid.2	|- G = ( +v ` U )
3		hladdid.5	|- Z = ( 0vec ` U )
4		hlnv	|- ( U e. CHilOLD -> U e. NrmCVec )
5	1 2 3	nv0rid	|- ( ( U e. NrmCVec /\ A e. X ) -> ( A G Z ) = A )
6	4 5	sylan	|- ( ( U e. CHilOLD /\ A e. X ) -> ( A G Z ) = A )