Step |
Hyp |
Ref |
Expression |
1 |
|
lmisfree.j |
|- J = ( LBasis ` W ) |
2 |
|
lmisfree.f |
|- F = ( Scalar ` W ) |
3 |
|
n0 |
|- ( J =/= (/) <-> E. j j e. J ) |
4 |
|
vex |
|- j e. _V |
5 |
4
|
enref |
|- j ~~ j |
6 |
2 1
|
lbslcic |
|- ( ( W e. LMod /\ j e. J /\ j ~~ j ) -> W ~=m ( F freeLMod j ) ) |
7 |
5 6
|
mp3an3 |
|- ( ( W e. LMod /\ j e. J ) -> W ~=m ( F freeLMod j ) ) |
8 |
|
oveq2 |
|- ( k = j -> ( F freeLMod k ) = ( F freeLMod j ) ) |
9 |
8
|
breq2d |
|- ( k = j -> ( W ~=m ( F freeLMod k ) <-> W ~=m ( F freeLMod j ) ) ) |
10 |
4 9
|
spcev |
|- ( W ~=m ( F freeLMod j ) -> E. k W ~=m ( F freeLMod k ) ) |
11 |
7 10
|
syl |
|- ( ( W e. LMod /\ j e. J ) -> E. k W ~=m ( F freeLMod k ) ) |
12 |
11
|
ex |
|- ( W e. LMod -> ( j e. J -> E. k W ~=m ( F freeLMod k ) ) ) |
13 |
12
|
exlimdv |
|- ( W e. LMod -> ( E. j j e. J -> E. k W ~=m ( F freeLMod k ) ) ) |
14 |
3 13
|
syl5bi |
|- ( W e. LMod -> ( J =/= (/) -> E. k W ~=m ( F freeLMod k ) ) ) |
15 |
|
lmicsym |
|- ( W ~=m ( F freeLMod k ) -> ( F freeLMod k ) ~=m W ) |
16 |
|
lmiclcl |
|- ( W ~=m ( F freeLMod k ) -> W e. LMod ) |
17 |
2
|
lmodring |
|- ( W e. LMod -> F e. Ring ) |
18 |
|
vex |
|- k e. _V |
19 |
|
eqid |
|- ( F freeLMod k ) = ( F freeLMod k ) |
20 |
|
eqid |
|- ( F unitVec k ) = ( F unitVec k ) |
21 |
|
eqid |
|- ( LBasis ` ( F freeLMod k ) ) = ( LBasis ` ( F freeLMod k ) ) |
22 |
19 20 21
|
frlmlbs |
|- ( ( F e. Ring /\ k e. _V ) -> ran ( F unitVec k ) e. ( LBasis ` ( F freeLMod k ) ) ) |
23 |
17 18 22
|
sylancl |
|- ( W e. LMod -> ran ( F unitVec k ) e. ( LBasis ` ( F freeLMod k ) ) ) |
24 |
23
|
ne0d |
|- ( W e. LMod -> ( LBasis ` ( F freeLMod k ) ) =/= (/) ) |
25 |
16 24
|
syl |
|- ( W ~=m ( F freeLMod k ) -> ( LBasis ` ( F freeLMod k ) ) =/= (/) ) |
26 |
21 1
|
lmiclbs |
|- ( ( F freeLMod k ) ~=m W -> ( ( LBasis ` ( F freeLMod k ) ) =/= (/) -> J =/= (/) ) ) |
27 |
15 25 26
|
sylc |
|- ( W ~=m ( F freeLMod k ) -> J =/= (/) ) |
28 |
27
|
exlimiv |
|- ( E. k W ~=m ( F freeLMod k ) -> J =/= (/) ) |
29 |
14 28
|
impbid1 |
|- ( W e. LMod -> ( J =/= (/) <-> E. k W ~=m ( F freeLMod k ) ) ) |