Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> M e. LMod ) |
2 |
|
eqid |
|- ( Base ` M ) = ( Base ` M ) |
3 |
|
eqid |
|- ( LSubSp ` M ) = ( LSubSp ` M ) |
4 |
2 3
|
lssss |
|- ( S e. ( LSubSp ` M ) -> S C_ ( Base ` M ) ) |
5 |
4
|
3ad2ant2 |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> S C_ ( Base ` M ) ) |
6 |
|
sstr |
|- ( ( V C_ S /\ S C_ ( Base ` M ) ) -> V C_ ( Base ` M ) ) |
7 |
|
fvex |
|- ( Base ` M ) e. _V |
8 |
7
|
ssex |
|- ( V C_ ( Base ` M ) -> V e. _V ) |
9 |
|
elpwg |
|- ( V e. _V -> ( V e. ~P ( Base ` M ) <-> V C_ ( Base ` M ) ) ) |
10 |
9
|
biimprd |
|- ( V e. _V -> ( V C_ ( Base ` M ) -> V e. ~P ( Base ` M ) ) ) |
11 |
8 10
|
mpcom |
|- ( V C_ ( Base ` M ) -> V e. ~P ( Base ` M ) ) |
12 |
6 11
|
syl |
|- ( ( V C_ S /\ S C_ ( Base ` M ) ) -> V e. ~P ( Base ` M ) ) |
13 |
12
|
ex |
|- ( V C_ S -> ( S C_ ( Base ` M ) -> V e. ~P ( Base ` M ) ) ) |
14 |
13
|
3ad2ant3 |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( S C_ ( Base ` M ) -> V e. ~P ( Base ` M ) ) ) |
15 |
5 14
|
mpd |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> V e. ~P ( Base ` M ) ) |
16 |
|
eqid |
|- ( Scalar ` M ) = ( Scalar ` M ) |
17 |
|
eqid |
|- ( Base ` ( Scalar ` M ) ) = ( Base ` ( Scalar ` M ) ) |
18 |
2 16 17
|
lcoval |
|- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( x e. ( M LinCo V ) <-> ( x e. ( Base ` M ) /\ E. f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` ( Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) ) ) |
19 |
1 15 18
|
syl2anc |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( x e. ( M LinCo V ) <-> ( x e. ( Base ` M ) /\ E. f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` ( Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) ) ) |
20 |
|
lincellss |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` ( Scalar ` M ) ) ) -> ( f ( linC ` M ) V ) e. S ) ) |
21 |
20
|
imp |
|- ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` ( Scalar ` M ) ) ) ) -> ( f ( linC ` M ) V ) e. S ) |
22 |
|
eleq1 |
|- ( x = ( f ( linC ` M ) V ) -> ( x e. S <-> ( f ( linC ` M ) V ) e. S ) ) |
23 |
21 22
|
syl5ibr |
|- ( x = ( f ( linC ` M ) V ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` ( Scalar ` M ) ) ) ) -> x e. S ) ) |
24 |
23
|
expd |
|- ( x = ( f ( linC ` M ) V ) -> ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` ( Scalar ` M ) ) ) -> x e. S ) ) ) |
25 |
24
|
com12 |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( x = ( f ( linC ` M ) V ) -> ( ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` ( Scalar ` M ) ) ) -> x e. S ) ) ) |
26 |
25
|
adantr |
|- ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> ( x = ( f ( linC ` M ) V ) -> ( ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` ( Scalar ` M ) ) ) -> x e. S ) ) ) |
27 |
26
|
com13 |
|- ( ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` ( Scalar ` M ) ) ) -> ( x = ( f ( linC ` M ) V ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> x e. S ) ) ) |
28 |
27
|
impr |
|- ( ( f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) /\ ( f finSupp ( 0g ` ( Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> x e. S ) ) |
29 |
28
|
rexlimiva |
|- ( E. f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` ( Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> x e. S ) ) |
30 |
29
|
com12 |
|- ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> ( E. f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` ( Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) -> x e. S ) ) |
31 |
30
|
expimpd |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( ( x e. ( Base ` M ) /\ E. f e. ( ( Base ` ( Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` ( Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) -> x e. S ) ) |
32 |
19 31
|
sylbid |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( x e. ( M LinCo V ) -> x e. S ) ) |
33 |
32
|
ralrimiv |
|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> A. x e. ( M LinCo V ) x e. S ) |