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 ) ) ) |