Step |
Hyp |
Ref |
Expression |
1 |
|
csslss.c |
|- C = ( ClSubSp ` W ) |
2 |
|
csslss.l |
|- L = ( LSubSp ` W ) |
3 |
|
eqid |
|- ( ocv ` W ) = ( ocv ` W ) |
4 |
3 1
|
cssi |
|- ( S e. C -> S = ( ( ocv ` W ) ` ( ( ocv ` W ) ` S ) ) ) |
5 |
4
|
adantl |
|- ( ( W e. PreHil /\ S e. C ) -> S = ( ( ocv ` W ) ` ( ( ocv ` W ) ` S ) ) ) |
6 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
7 |
6 3
|
ocvss |
|- ( ( ocv ` W ) ` S ) C_ ( Base ` W ) |
8 |
7
|
a1i |
|- ( S e. C -> ( ( ocv ` W ) ` S ) C_ ( Base ` W ) ) |
9 |
6 3 2
|
ocvlss |
|- ( ( W e. PreHil /\ ( ( ocv ` W ) ` S ) C_ ( Base ` W ) ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` S ) ) e. L ) |
10 |
8 9
|
sylan2 |
|- ( ( W e. PreHil /\ S e. C ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` S ) ) e. L ) |
11 |
5 10
|
eqeltrd |
|- ( ( W e. PreHil /\ S e. C ) -> S e. L ) |