Step |
Hyp |
Ref |
Expression |
1 |
|
eqidd |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( Scalar ` M ) = ( Scalar ` M ) ) |
2 |
|
eqidd |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( Base ` ( Scalar ` M ) ) = ( Base ` ( Scalar ` M ) ) ) |
3 |
|
eqidd |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( Base ` M ) = ( Base ` M ) ) |
4 |
|
eqidd |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( +g ` M ) = ( +g ` M ) ) |
5 |
|
eqidd |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( .s ` M ) = ( .s ` M ) ) |
6 |
|
eqidd |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( LSubSp ` M ) = ( LSubSp ` M ) ) |
7 |
|
eqid |
|- ( Base ` M ) = ( Base ` M ) |
8 |
|
eqid |
|- ( Scalar ` M ) = ( Scalar ` M ) |
9 |
|
eqid |
|- ( Base ` ( Scalar ` M ) ) = ( Base ` ( Scalar ` M ) ) |
10 |
7 8 9
|
lcoval |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( v e. ( M LinCo V ) <-> ( v e. ( Base ` M ) /\ E. s e. ( ( Base ` ( Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` ( Scalar ` M ) ) /\ v = ( s ( linC ` M ) V ) ) ) ) ) |
11 |
|
simpl |
|- ( ( v e. ( Base ` M ) /\ E. s e. ( ( Base ` ( Scalar ` M ) ) ^m V ) ( s finSupp ( 0g ` ( Scalar ` M ) ) /\ v = ( s ( linC ` M ) V ) ) ) -> v e. ( Base ` M ) ) |
12 |
10 11
|
syl6bi |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( v e. ( M LinCo V ) -> v e. ( Base ` M ) ) ) |
13 |
12
|
ssrdv |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( M LinCo V ) C_ ( Base ` M ) ) |
14 |
|
lcoel0 |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( 0g ` M ) e. ( M LinCo V ) ) |
15 |
14
|
ne0d |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( M LinCo V ) =/= (/) ) |
16 |
|
eqid |
|- ( .s ` M ) = ( .s ` M ) |
17 |
|
eqid |
|- ( +g ` M ) = ( +g ` M ) |
18 |
16 9 17
|
lincsumscmcl |
|- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ ( x e. ( Base ` ( Scalar ` M ) ) /\ a e. ( M LinCo V ) /\ b e. ( M LinCo V ) ) ) -> ( ( x ( .s ` M ) a ) ( +g ` M ) b ) e. ( M LinCo V ) ) |
19 |
1 2 3 4 5 6 13 15 18
|
islssd |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( M LinCo V ) e. ( LSubSp ` M ) ) |