Step |
Hyp |
Ref |
Expression |
1 |
|
reslmhm2.u |
|- U = ( T |`s X ) |
2 |
|
reslmhm2.l |
|- L = ( LSubSp ` T ) |
3 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
4 |
|
eqid |
|- ( .s ` S ) = ( .s ` S ) |
5 |
|
eqid |
|- ( .s ` U ) = ( .s ` U ) |
6 |
|
eqid |
|- ( Scalar ` S ) = ( Scalar ` S ) |
7 |
|
eqid |
|- ( Scalar ` U ) = ( Scalar ` U ) |
8 |
|
eqid |
|- ( Base ` ( Scalar ` S ) ) = ( Base ` ( Scalar ` S ) ) |
9 |
|
lmhmlmod1 |
|- ( F e. ( S LMHom T ) -> S e. LMod ) |
10 |
9
|
adantl |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) -> S e. LMod ) |
11 |
|
simpl1 |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) -> T e. LMod ) |
12 |
|
simpl2 |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) -> X e. L ) |
13 |
1 2
|
lsslmod |
|- ( ( T e. LMod /\ X e. L ) -> U e. LMod ) |
14 |
11 12 13
|
syl2anc |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) -> U e. LMod ) |
15 |
|
eqid |
|- ( Scalar ` T ) = ( Scalar ` T ) |
16 |
1 15
|
resssca |
|- ( X e. L -> ( Scalar ` T ) = ( Scalar ` U ) ) |
17 |
16
|
3ad2ant2 |
|- ( ( T e. LMod /\ X e. L /\ ran F C_ X ) -> ( Scalar ` T ) = ( Scalar ` U ) ) |
18 |
6 15
|
lmhmsca |
|- ( F e. ( S LMHom T ) -> ( Scalar ` T ) = ( Scalar ` S ) ) |
19 |
17 18
|
sylan9req |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) -> ( Scalar ` U ) = ( Scalar ` S ) ) |
20 |
|
lmghm |
|- ( F e. ( S LMHom T ) -> F e. ( S GrpHom T ) ) |
21 |
2
|
lsssubg |
|- ( ( T e. LMod /\ X e. L ) -> X e. ( SubGrp ` T ) ) |
22 |
1
|
resghm2b |
|- ( ( X e. ( SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) ) |
23 |
21 22
|
stoic3 |
|- ( ( T e. LMod /\ X e. L /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) ) |
24 |
23
|
biimpa |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S GrpHom T ) ) -> F e. ( S GrpHom U ) ) |
25 |
20 24
|
sylan2 |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) -> F e. ( S GrpHom U ) ) |
26 |
|
eqid |
|- ( .s ` T ) = ( .s ` T ) |
27 |
6 8 3 4 26
|
lmhmlin |
|- ( ( F e. ( S LMHom T ) /\ x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` T ) ( F ` y ) ) ) |
28 |
27
|
3expb |
|- ( ( F e. ( S LMHom T ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` T ) ( F ` y ) ) ) |
29 |
28
|
adantll |
|- ( ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` T ) ( F ` y ) ) ) |
30 |
|
simpll2 |
|- ( ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> X e. L ) |
31 |
1 26
|
ressvsca |
|- ( X e. L -> ( .s ` T ) = ( .s ` U ) ) |
32 |
31
|
oveqd |
|- ( X e. L -> ( x ( .s ` T ) ( F ` y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
33 |
30 32
|
syl |
|- ( ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( x ( .s ` T ) ( F ` y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
34 |
29 33
|
eqtrd |
|- ( ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
35 |
3 4 5 6 7 8 10 14 19 25 34
|
islmhmd |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom T ) ) -> F e. ( S LMHom U ) ) |
36 |
|
simpr |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom U ) ) -> F e. ( S LMHom U ) ) |
37 |
|
simpl1 |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom U ) ) -> T e. LMod ) |
38 |
|
simpl2 |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom U ) ) -> X e. L ) |
39 |
1 2
|
reslmhm2 |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> F e. ( S LMHom T ) ) |
40 |
36 37 38 39
|
syl3anc |
|- ( ( ( T e. LMod /\ X e. L /\ ran F C_ X ) /\ F e. ( S LMHom U ) ) -> F e. ( S LMHom T ) ) |
41 |
35 40
|
impbida |
|- ( ( T e. LMod /\ X e. L /\ ran F C_ X ) -> ( F e. ( S LMHom T ) <-> F e. ( S LMHom U ) ) ) |