Theorem dipassr

Description: "Associative" law for second argument of inner product (compare dipass ). (Contributed by NM, 22-Nov-2007) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ipass.1	|- X = ( BaseSet ` U )
2		ipass.4	|- S = ( .sOLD ` U )
3		ipass.7	|- P = ( .iOLD ` U )
4		3anrot	|- ( ( A e. X /\ B e. CC /\ C e. X ) <-> ( B e. CC /\ C e. X /\ A e. X ) )
5	1 2 3	dipass	|- ( ( U e. CPreHilOLD /\ ( B e. CC /\ C e. X /\ A e. X ) ) -> ( ( B S C ) P A ) = ( B x. ( C P A ) ) )
6	4 5	sylan2b	|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( ( B S C ) P A ) = ( B x. ( C P A ) ) )
7	6	fveq2d	|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( * ` ( ( B S C ) P A ) ) = ( * ` ( B x. ( C P A ) ) ) )
8		phnv	|- ( U e. CPreHilOLD -> U e. NrmCVec )
9		simpl	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> U e. NrmCVec )
10	1 2	nvscl	|- ( ( U e. NrmCVec /\ B e. CC /\ C e. X ) -> ( B S C ) e. X )
11	10	3adant3r1	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( B S C ) e. X )
12		simpr1	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> A e. X )
13	1 3	dipcj	|- ( ( U e. NrmCVec /\ ( B S C ) e. X /\ A e. X ) -> ( * ` ( ( B S C ) P A ) ) = ( A P ( B S C ) ) )
14	9 11 12 13	syl3anc	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( * ` ( ( B S C ) P A ) ) = ( A P ( B S C ) ) )
15	8 14	sylan	|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( * ` ( ( B S C ) P A ) ) = ( A P ( B S C ) ) )
16		simpr2	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> B e. CC )
17	1 3	dipcl	|- ( ( U e. NrmCVec /\ C e. X /\ A e. X ) -> ( C P A ) e. CC )
18	17	3com23	|- ( ( U e. NrmCVec /\ A e. X /\ C e. X ) -> ( C P A ) e. CC )
19	18	3adant3r2	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( C P A ) e. CC )
20	16 19	cjmuld	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( * ` ( B x. ( C P A ) ) ) = ( ( * ` B ) x. ( * ` ( C P A ) ) ) )
21	1 3	dipcj	|- ( ( U e. NrmCVec /\ C e. X /\ A e. X ) -> ( * ` ( C P A ) ) = ( A P C ) )
22	21	3com23	|- ( ( U e. NrmCVec /\ A e. X /\ C e. X ) -> ( * ` ( C P A ) ) = ( A P C ) )
23	22	3adant3r2	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( * ` ( C P A ) ) = ( A P C ) )
24	23	oveq2d	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( ( * ` B ) x. ( * ` ( C P A ) ) ) = ( ( * ` B ) x. ( A P C ) ) )
25	20 24	eqtrd	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( * ` ( B x. ( C P A ) ) ) = ( ( * ` B ) x. ( A P C ) ) )
26	8 25	sylan	|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( * ` ( B x. ( C P A ) ) ) = ( ( * ` B ) x. ( A P C ) ) )
27	7 15 26	3eqtr3d	|- ( ( U e. CPreHilOLD /\ ( A e. X /\ B e. CC /\ C e. X ) ) -> ( A P ( B S C ) ) = ( ( * ` B ) x. ( A P C ) ) )