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Metamath Proof Explorer

Theorem dipdi

Description: Distributive law for inner product. (Contributed by NM, 20-Nov-2007) (New usage is discouraged.)

Ref Expression
Hypotheses dipdir.1 
|- X = ( BaseSet ` U )
dipdir.2 
|- G = ( +v ` U )
dipdir.7 
|- P = ( .iOLD ` U )
Assertion dipdi 
|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A P ( B G C ) ) = ( ( A P B ) + ( A P C ) ) )

Proof

Step Hyp Ref Expression
1 dipdir.1 
 |-  X = ( BaseSet ` U )
2 dipdir.2 
 |-  G = ( +v ` U )
3 dipdir.7 
 |-  P = ( .iOLD ` U )
4 id 
 |-  ( ( C e. X /\ B e. X /\ A e. X ) -> ( C e. X /\ B e. X /\ A e. X ) )
5 4 3com13 
 |-  ( ( A e. X /\ B e. X /\ C e. X ) -> ( C e. X /\ B e. X /\ A e. X ) )
6 id 
 |-  ( ( B e. X /\ C e. X /\ A e. X ) -> ( B e. X /\ C e. X /\ A e. X ) )
7 6 3com12 
 |-  ( ( C e. X /\ B e. X /\ A e. X ) -> ( B e. X /\ C e. X /\ A e. X ) )
8 1 2 3 dipdir 
 |-  ( ( U e. CPreHilOLD /\ ( B e. X /\ C e. X /\ A e. X ) ) -> ( ( B G C ) P A ) = ( ( B P A ) + ( C P A ) ) )
9 7 8 sylan2 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( ( B G C ) P A ) = ( ( B P A ) + ( C P A ) ) )
10 9 fveq2d 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B G C ) P A ) ) = ( * ` ( ( B P A ) + ( C P A ) ) ) )
11 phnv 
 |-  ( U e. CPreHilOLD -> U e. NrmCVec )
12 simpl 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> U e. NrmCVec )
13 1 2 nvgcl 
 |-  ( ( U e. NrmCVec /\ B e. X /\ C e. X ) -> ( B G C ) e. X )
14 13 3com23 
 |-  ( ( U e. NrmCVec /\ C e. X /\ B e. X ) -> ( B G C ) e. X )
15 14 3adant3r3 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( B G C ) e. X )
16 simpr3 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> A e. X )
17 1 3 dipcj 
 |-  ( ( U e. NrmCVec /\ ( B G C ) e. X /\ A e. X ) -> ( * ` ( ( B G C ) P A ) ) = ( A P ( B G C ) ) )
18 12 15 16 17 syl3anc 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B G C ) P A ) ) = ( A P ( B G C ) ) )
19 11 18 sylan 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B G C ) P A ) ) = ( A P ( B G C ) ) )
20 1 3 dipcl 
 |-  ( ( U e. NrmCVec /\ B e. X /\ A e. X ) -> ( B P A ) e. CC )
21 20 3adant3r1 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( B P A ) e. CC )
22 1 3 dipcl 
 |-  ( ( U e. NrmCVec /\ C e. X /\ A e. X ) -> ( C P A ) e. CC )
23 22 3adant3r2 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( C P A ) e. CC )
24 21 23 cjaddd 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B P A ) + ( C P A ) ) ) = ( ( * ` ( B P A ) ) + ( * ` ( C P A ) ) ) )
25 1 3 dipcj 
 |-  ( ( U e. NrmCVec /\ B e. X /\ A e. X ) -> ( * ` ( B P A ) ) = ( A P B ) )
26 25 3adant3r1 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( B P A ) ) = ( A P B ) )
27 1 3 dipcj 
 |-  ( ( U e. NrmCVec /\ C e. X /\ A e. X ) -> ( * ` ( C P A ) ) = ( A P C ) )
28 27 3adant3r2 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( C P A ) ) = ( A P C ) )
29 26 28 oveq12d 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( ( * ` ( B P A ) ) + ( * ` ( C P A ) ) ) = ( ( A P B ) + ( A P C ) ) )
30 24 29 eqtrd 
 |-  ( ( U e. NrmCVec /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B P A ) + ( C P A ) ) ) = ( ( A P B ) + ( A P C ) ) )
31 11 30 sylan 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( * ` ( ( B P A ) + ( C P A ) ) ) = ( ( A P B ) + ( A P C ) ) )
32 10 19 31 3eqtr3d 
 |-  ( ( U e. CPreHilOLD /\ ( C e. X /\ B e. X /\ A e. X ) ) -> ( A P ( B G C ) ) = ( ( A P B ) + ( A P C ) ) )
33 5 32 sylan2 
 |-  ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A P ( B G C ) ) = ( ( A P B ) + ( A P C ) ) )