Step |
Hyp |
Ref |
Expression |
1 |
|
ipcl.1 |
|- X = ( BaseSet ` U ) |
2 |
|
ipcl.7 |
|- P = ( .iOLD ` U ) |
3 |
|
fveq2 |
|- ( ( A P B ) = 0 -> ( * ` ( A P B ) ) = ( * ` 0 ) ) |
4 |
|
cj0 |
|- ( * ` 0 ) = 0 |
5 |
3 4
|
eqtrdi |
|- ( ( A P B ) = 0 -> ( * ` ( A P B ) ) = 0 ) |
6 |
1 2
|
dipcj |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( * ` ( A P B ) ) = ( B P A ) ) |
7 |
6
|
eqeq1d |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( * ` ( A P B ) ) = 0 <-> ( B P A ) = 0 ) ) |
8 |
5 7
|
syl5ib |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( A P B ) = 0 -> ( B P A ) = 0 ) ) |
9 |
|
fveq2 |
|- ( ( B P A ) = 0 -> ( * ` ( B P A ) ) = ( * ` 0 ) ) |
10 |
9 4
|
eqtrdi |
|- ( ( B P A ) = 0 -> ( * ` ( B P A ) ) = 0 ) |
11 |
1 2
|
dipcj |
|- ( ( U e. NrmCVec /\ B e. X /\ A e. X ) -> ( * ` ( B P A ) ) = ( A P B ) ) |
12 |
11
|
3com23 |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( * ` ( B P A ) ) = ( A P B ) ) |
13 |
12
|
eqeq1d |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( * ` ( B P A ) ) = 0 <-> ( A P B ) = 0 ) ) |
14 |
10 13
|
syl5ib |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( B P A ) = 0 -> ( A P B ) = 0 ) ) |
15 |
8 14
|
impbid |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( A P B ) = 0 <-> ( B P A ) = 0 ) ) |