Metamath Proof Explorer


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 ) ) )