Description: Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)
Ref | Expression | ||
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Hypotheses | hlipass.1 | |- X = ( BaseSet ` U ) |
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hlipass.4 | |- S = ( .sOLD ` U ) |
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hlipass.7 | |- P = ( .iOLD ` U ) |
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Assertion | hlipass | |- ( ( U e. CHilOLD /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( ( A S B ) P C ) = ( A x. ( B P C ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlipass.1 | |- X = ( BaseSet ` U ) |
|
2 | hlipass.4 | |- S = ( .sOLD ` U ) |
|
3 | hlipass.7 | |- P = ( .iOLD ` U ) |
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4 | hlph | |- ( U e. CHilOLD -> U e. CPreHilOLD ) |
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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 ) ) ) |