Step |
Hyp |
Ref |
Expression |
1 |
|
islindf3.l |
|- L = ( Scalar ` W ) |
2 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
3 |
2 1
|
lindff1 |
|- ( ( W e. LMod /\ L e. NzRing /\ F LIndF W ) -> F : dom F -1-1-> ( Base ` W ) ) |
4 |
3
|
3expa |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ F LIndF W ) -> F : dom F -1-1-> ( Base ` W ) ) |
5 |
|
ssv |
|- ( Base ` W ) C_ _V |
6 |
|
f1ss |
|- ( ( F : dom F -1-1-> ( Base ` W ) /\ ( Base ` W ) C_ _V ) -> F : dom F -1-1-> _V ) |
7 |
4 5 6
|
sylancl |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ F LIndF W ) -> F : dom F -1-1-> _V ) |
8 |
|
lindfrn |
|- ( ( W e. LMod /\ F LIndF W ) -> ran F e. ( LIndS ` W ) ) |
9 |
8
|
adantlr |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ F LIndF W ) -> ran F e. ( LIndS ` W ) ) |
10 |
7 9
|
jca |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ F LIndF W ) -> ( F : dom F -1-1-> _V /\ ran F e. ( LIndS ` W ) ) ) |
11 |
|
simpll |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ ( F : dom F -1-1-> _V /\ ran F e. ( LIndS ` W ) ) ) -> W e. LMod ) |
12 |
|
simprr |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ ( F : dom F -1-1-> _V /\ ran F e. ( LIndS ` W ) ) ) -> ran F e. ( LIndS ` W ) ) |
13 |
|
f1f1orn |
|- ( F : dom F -1-1-> _V -> F : dom F -1-1-onto-> ran F ) |
14 |
|
f1of1 |
|- ( F : dom F -1-1-onto-> ran F -> F : dom F -1-1-> ran F ) |
15 |
13 14
|
syl |
|- ( F : dom F -1-1-> _V -> F : dom F -1-1-> ran F ) |
16 |
15
|
ad2antrl |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ ( F : dom F -1-1-> _V /\ ran F e. ( LIndS ` W ) ) ) -> F : dom F -1-1-> ran F ) |
17 |
|
f1linds |
|- ( ( W e. LMod /\ ran F e. ( LIndS ` W ) /\ F : dom F -1-1-> ran F ) -> F LIndF W ) |
18 |
11 12 16 17
|
syl3anc |
|- ( ( ( W e. LMod /\ L e. NzRing ) /\ ( F : dom F -1-1-> _V /\ ran F e. ( LIndS ` W ) ) ) -> F LIndF W ) |
19 |
10 18
|
impbida |
|- ( ( W e. LMod /\ L e. NzRing ) -> ( F LIndF W <-> ( F : dom F -1-1-> _V /\ ran F e. ( LIndS ` W ) ) ) ) |