Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
|- ( ( A .ih B ) = 0 -> ( * ` ( A .ih B ) ) = ( * ` 0 ) ) |
2 |
|
cj0 |
|- ( * ` 0 ) = 0 |
3 |
1 2
|
eqtrdi |
|- ( ( A .ih B ) = 0 -> ( * ` ( A .ih B ) ) = 0 ) |
4 |
|
ax-his1 |
|- ( ( B e. ~H /\ A e. ~H ) -> ( B .ih A ) = ( * ` ( A .ih B ) ) ) |
5 |
4
|
ancoms |
|- ( ( A e. ~H /\ B e. ~H ) -> ( B .ih A ) = ( * ` ( A .ih B ) ) ) |
6 |
5
|
eqeq1d |
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( B .ih A ) = 0 <-> ( * ` ( A .ih B ) ) = 0 ) ) |
7 |
3 6
|
syl5ibr |
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( A .ih B ) = 0 -> ( B .ih A ) = 0 ) ) |
8 |
|
fveq2 |
|- ( ( B .ih A ) = 0 -> ( * ` ( B .ih A ) ) = ( * ` 0 ) ) |
9 |
8 2
|
eqtrdi |
|- ( ( B .ih A ) = 0 -> ( * ` ( B .ih A ) ) = 0 ) |
10 |
|
ax-his1 |
|- ( ( A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) ) |
11 |
10
|
eqeq1d |
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( A .ih B ) = 0 <-> ( * ` ( B .ih A ) ) = 0 ) ) |
12 |
9 11
|
syl5ibr |
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( B .ih A ) = 0 -> ( A .ih B ) = 0 ) ) |
13 |
7 12
|
impbid |
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( A .ih B ) = 0 <-> ( B .ih A ) = 0 ) ) |