Step |
Hyp |
Ref |
Expression |
1 |
|
lmimlbs.j |
|- J = ( LBasis ` S ) |
2 |
|
lmimlbs.k |
|- K = ( LBasis ` T ) |
3 |
|
lmimlmhm |
|- ( F e. ( S LMIso T ) -> F e. ( S LMHom T ) ) |
4 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
5 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
6 |
4 5
|
lmimf1o |
|- ( F e. ( S LMIso T ) -> F : ( Base ` S ) -1-1-onto-> ( Base ` T ) ) |
7 |
|
f1of1 |
|- ( F : ( Base ` S ) -1-1-onto-> ( Base ` T ) -> F : ( Base ` S ) -1-1-> ( Base ` T ) ) |
8 |
6 7
|
syl |
|- ( F e. ( S LMIso T ) -> F : ( Base ` S ) -1-1-> ( Base ` T ) ) |
9 |
1
|
lbslinds |
|- J C_ ( LIndS ` S ) |
10 |
9
|
sseli |
|- ( B e. J -> B e. ( LIndS ` S ) ) |
11 |
4 5
|
lindsmm2 |
|- ( ( F e. ( S LMHom T ) /\ F : ( Base ` S ) -1-1-> ( Base ` T ) /\ B e. ( LIndS ` S ) ) -> ( F " B ) e. ( LIndS ` T ) ) |
12 |
3 8 10 11
|
syl2an3an |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> ( F " B ) e. ( LIndS ` T ) ) |
13 |
|
eqid |
|- ( LSpan ` S ) = ( LSpan ` S ) |
14 |
4 1 13
|
lbssp |
|- ( B e. J -> ( ( LSpan ` S ) ` B ) = ( Base ` S ) ) |
15 |
14
|
adantl |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> ( ( LSpan ` S ) ` B ) = ( Base ` S ) ) |
16 |
15
|
imaeq2d |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> ( F " ( ( LSpan ` S ) ` B ) ) = ( F " ( Base ` S ) ) ) |
17 |
4 1
|
lbsss |
|- ( B e. J -> B C_ ( Base ` S ) ) |
18 |
|
eqid |
|- ( LSpan ` T ) = ( LSpan ` T ) |
19 |
4 13 18
|
lmhmlsp |
|- ( ( F e. ( S LMHom T ) /\ B C_ ( Base ` S ) ) -> ( F " ( ( LSpan ` S ) ` B ) ) = ( ( LSpan ` T ) ` ( F " B ) ) ) |
20 |
3 17 19
|
syl2an |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> ( F " ( ( LSpan ` S ) ` B ) ) = ( ( LSpan ` T ) ` ( F " B ) ) ) |
21 |
6
|
adantr |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> F : ( Base ` S ) -1-1-onto-> ( Base ` T ) ) |
22 |
|
f1ofo |
|- ( F : ( Base ` S ) -1-1-onto-> ( Base ` T ) -> F : ( Base ` S ) -onto-> ( Base ` T ) ) |
23 |
|
foima |
|- ( F : ( Base ` S ) -onto-> ( Base ` T ) -> ( F " ( Base ` S ) ) = ( Base ` T ) ) |
24 |
21 22 23
|
3syl |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> ( F " ( Base ` S ) ) = ( Base ` T ) ) |
25 |
16 20 24
|
3eqtr3d |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> ( ( LSpan ` T ) ` ( F " B ) ) = ( Base ` T ) ) |
26 |
5 2 18
|
islbs4 |
|- ( ( F " B ) e. K <-> ( ( F " B ) e. ( LIndS ` T ) /\ ( ( LSpan ` T ) ` ( F " B ) ) = ( Base ` T ) ) ) |
27 |
12 25 26
|
sylanbrc |
|- ( ( F e. ( S LMIso T ) /\ B e. J ) -> ( F " B ) e. K ) |