Description: Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)
Ref | Expression | ||
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Hypotheses | hlipdir.1 | |- X = ( BaseSet ` U ) |
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hlipdir.2 | |- G = ( +v ` U ) |
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hlipdir.7 | |- P = ( .iOLD ` U ) |
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Assertion | hlipdir | |- ( ( U e. CHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) P C ) = ( ( A P C ) + ( B P C ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlipdir.1 | |- X = ( BaseSet ` U ) |
|
2 | hlipdir.2 | |- G = ( +v ` U ) |
|
3 | hlipdir.7 | |- P = ( .iOLD ` U ) |
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4 | hlph | |- ( U e. CHilOLD -> U e. CPreHilOLD ) |
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5 | 1 2 3 | dipdir | |- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) P C ) = ( ( A P C ) + ( B P C ) ) ) |
6 | 4 5 | sylan | |- ( ( U e. CHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) P C ) = ( ( A P C ) + ( B P C ) ) ) |