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 ` T ) = ( .s ` T ) |
6 |
|
eqid |
|- ( Scalar ` S ) = ( Scalar ` S ) |
7 |
|
eqid |
|- ( Scalar ` T ) = ( Scalar ` T ) |
8 |
|
eqid |
|- ( Base ` ( Scalar ` S ) ) = ( Base ` ( Scalar ` S ) ) |
9 |
|
lmhmlmod1 |
|- ( F e. ( S LMHom U ) -> S e. LMod ) |
10 |
9
|
3ad2ant1 |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> S e. LMod ) |
11 |
|
simp2 |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> T e. LMod ) |
12 |
1 7
|
resssca |
|- ( X e. L -> ( Scalar ` T ) = ( Scalar ` U ) ) |
13 |
12
|
3ad2ant3 |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> ( Scalar ` T ) = ( Scalar ` U ) ) |
14 |
|
eqid |
|- ( Scalar ` U ) = ( Scalar ` U ) |
15 |
6 14
|
lmhmsca |
|- ( F e. ( S LMHom U ) -> ( Scalar ` U ) = ( Scalar ` S ) ) |
16 |
15
|
3ad2ant1 |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> ( Scalar ` U ) = ( Scalar ` S ) ) |
17 |
13 16
|
eqtrd |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> ( Scalar ` T ) = ( Scalar ` S ) ) |
18 |
|
lmghm |
|- ( F e. ( S LMHom U ) -> F e. ( S GrpHom U ) ) |
19 |
18
|
3ad2ant1 |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> F e. ( S GrpHom U ) ) |
20 |
2
|
lsssubg |
|- ( ( T e. LMod /\ X e. L ) -> X e. ( SubGrp ` T ) ) |
21 |
20
|
3adant1 |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> X e. ( SubGrp ` T ) ) |
22 |
1
|
resghm2 |
|- ( ( F e. ( S GrpHom U ) /\ X e. ( SubGrp ` T ) ) -> F e. ( S GrpHom T ) ) |
23 |
19 21 22
|
syl2anc |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> F e. ( S GrpHom T ) ) |
24 |
|
eqid |
|- ( .s ` U ) = ( .s ` U ) |
25 |
6 8 3 4 24
|
lmhmlin |
|- ( ( F e. ( S LMHom U ) /\ x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
26 |
25
|
3expb |
|- ( ( F e. ( S LMHom U ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
27 |
26
|
3ad2antl1 |
|- ( ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
28 |
|
simpl3 |
|- ( ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> X e. L ) |
29 |
1 5
|
ressvsca |
|- ( X e. L -> ( .s ` T ) = ( .s ` U ) ) |
30 |
29
|
oveqd |
|- ( X e. L -> ( x ( .s ` T ) ( F ` y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
31 |
28 30
|
syl |
|- ( ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( x ( .s ` T ) ( F ` y ) ) = ( x ( .s ` U ) ( F ` y ) ) ) |
32 |
27 31
|
eqtr4d |
|- ( ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) /\ ( x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` T ) ( F ` y ) ) ) |
33 |
3 4 5 6 7 8 10 11 17 23 32
|
islmhmd |
|- ( ( F e. ( S LMHom U ) /\ T e. LMod /\ X e. L ) -> F e. ( S LMHom T ) ) |