Theorem hlipass

Description: Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlipass.1	|- X = ( BaseSet ` U )
	hlipass.4	|- S = ( .sOLD ` U )
	hlipass.7	|- P = ( .iOLD ` U )
Assertion	hlipass	|- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( ( A S B ) P C ) = ( A x. ( B P C ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlipass.1	|- X = ( BaseSet ` U )
2		hlipass.4	|- S = ( .sOLD ` U )
3		hlipass.7	|- P = ( .iOLD ` U )
4		hlph	|- ( U e. CHilOLD -> U e. CPreHilOLD )
5	1 2 3	dipass	|- ( ( U e. CPreHilOLD /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( ( A S B ) P C ) = ( A x. ( B P C ) ) )
6	4 5	sylan	|- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( ( A S B ) P C ) = ( A x. ( B P C ) ) )