Step |
Hyp |
Ref |
Expression |
1 |
|
simp3 |
|- ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) -> C C_ B ) |
2 |
|
eqid |
|- ( 0g ` W ) = ( 0g ` W ) |
3 |
2
|
obsne0 |
|- ( ( B e. ( OBasis ` W ) /\ x e. B ) -> x =/= ( 0g ` W ) ) |
4 |
3
|
3ad2antl1 |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> x =/= ( 0g ` W ) ) |
5 |
|
eqid |
|- ( ocv ` W ) = ( ocv ` W ) |
6 |
5
|
obselocv |
|- ( ( B e. ( OBasis ` W ) /\ C C_ B /\ x e. B ) -> ( x e. ( ( ocv ` W ) ` C ) <-> -. x e. C ) ) |
7 |
6
|
3expa |
|- ( ( ( B e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> ( x e. ( ( ocv ` W ) ` C ) <-> -. x e. C ) ) |
8 |
7
|
3adantl2 |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> ( x e. ( ( ocv ` W ) ` C ) <-> -. x e. C ) ) |
9 |
|
simpl2 |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> C e. ( OBasis ` W ) ) |
10 |
2 5
|
obsocv |
|- ( C e. ( OBasis ` W ) -> ( ( ocv ` W ) ` C ) = { ( 0g ` W ) } ) |
11 |
9 10
|
syl |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> ( ( ocv ` W ) ` C ) = { ( 0g ` W ) } ) |
12 |
11
|
eleq2d |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> ( x e. ( ( ocv ` W ) ` C ) <-> x e. { ( 0g ` W ) } ) ) |
13 |
|
elsni |
|- ( x e. { ( 0g ` W ) } -> x = ( 0g ` W ) ) |
14 |
12 13
|
syl6bi |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> ( x e. ( ( ocv ` W ) ` C ) -> x = ( 0g ` W ) ) ) |
15 |
8 14
|
sylbird |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> ( -. x e. C -> x = ( 0g ` W ) ) ) |
16 |
15
|
necon1ad |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> ( x =/= ( 0g ` W ) -> x e. C ) ) |
17 |
4 16
|
mpd |
|- ( ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) /\ x e. B ) -> x e. C ) |
18 |
1 17
|
eqelssd |
|- ( ( B e. ( OBasis ` W ) /\ C e. ( OBasis ` W ) /\ C C_ B ) -> C = B ) |