Step |
Hyp |
Ref |
Expression |
1 |
|
css0.c |
|- C = ( ClSubSp ` W ) |
2 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
3 |
|
eqid |
|- ( ocv ` W ) = ( ocv ` W ) |
4 |
2 3
|
ocvss |
|- ( ( ocv ` W ) ` A ) C_ ( Base ` W ) |
5 |
2 3
|
ocvss |
|- ( ( ocv ` W ) ` B ) C_ ( Base ` W ) |
6 |
4 5
|
unssi |
|- ( ( ( ocv ` W ) ` A ) u. ( ( ocv ` W ) ` B ) ) C_ ( Base ` W ) |
7 |
2 1 3
|
ocvcss |
|- ( ( W e. PreHil /\ ( ( ( ocv ` W ) ` A ) u. ( ( ocv ` W ) ` B ) ) C_ ( Base ` W ) ) -> ( ( ocv ` W ) ` ( ( ( ocv ` W ) ` A ) u. ( ( ocv ` W ) ` B ) ) ) e. C ) |
8 |
6 7
|
mpan2 |
|- ( W e. PreHil -> ( ( ocv ` W ) ` ( ( ( ocv ` W ) ` A ) u. ( ( ocv ` W ) ` B ) ) ) e. C ) |
9 |
3 1
|
cssi |
|- ( A e. C -> A = ( ( ocv ` W ) ` ( ( ocv ` W ) ` A ) ) ) |
10 |
3 1
|
cssi |
|- ( B e. C -> B = ( ( ocv ` W ) ` ( ( ocv ` W ) ` B ) ) ) |
11 |
9 10
|
ineqan12d |
|- ( ( A e. C /\ B e. C ) -> ( A i^i B ) = ( ( ( ocv ` W ) ` ( ( ocv ` W ) ` A ) ) i^i ( ( ocv ` W ) ` ( ( ocv ` W ) ` B ) ) ) ) |
12 |
3
|
unocv |
|- ( ( ocv ` W ) ` ( ( ( ocv ` W ) ` A ) u. ( ( ocv ` W ) ` B ) ) ) = ( ( ( ocv ` W ) ` ( ( ocv ` W ) ` A ) ) i^i ( ( ocv ` W ) ` ( ( ocv ` W ) ` B ) ) ) |
13 |
11 12
|
eqtr4di |
|- ( ( A e. C /\ B e. C ) -> ( A i^i B ) = ( ( ocv ` W ) ` ( ( ( ocv ` W ) ` A ) u. ( ( ocv ` W ) ` B ) ) ) ) |
14 |
13
|
eleq1d |
|- ( ( A e. C /\ B e. C ) -> ( ( A i^i B ) e. C <-> ( ( ocv ` W ) ` ( ( ( ocv ` W ) ` A ) u. ( ( ocv ` W ) ` B ) ) ) e. C ) ) |
15 |
8 14
|
syl5ibrcom |
|- ( W e. PreHil -> ( ( A e. C /\ B e. C ) -> ( A i^i B ) e. C ) ) |
16 |
15
|
3impib |
|- ( ( W e. PreHil /\ A e. C /\ B e. C ) -> ( A i^i B ) e. C ) |