Theorem hlass

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

	Ref	Expression
Hypotheses	hladdf.1	|- X = ( BaseSet ` U )
	hladdf.2	|- G = ( +v ` U )
Assertion	hlass	|- ( ( U e. CHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) )

Proof

Step	Hyp	Ref	Expression
1		hladdf.1	|- X = ( BaseSet ` U )
2		hladdf.2	|- G = ( +v ` U )
3		hlnv	|- ( U e. CHilOLD -> U e. NrmCVec )
4	1 2	nvass	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) )
5	3 4	sylan	|- ( ( U e. CHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) )