Step |
Hyp |
Ref |
Expression |
1 |
|
lmhmeql.u |
|- U = ( LSubSp ` S ) |
2 |
|
lmghm |
|- ( F e. ( S LMHom T ) -> F e. ( S GrpHom T ) ) |
3 |
|
lmghm |
|- ( G e. ( S LMHom T ) -> G e. ( S GrpHom T ) ) |
4 |
|
ghmeql |
|- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> dom ( F i^i G ) e. ( SubGrp ` S ) ) |
5 |
2 3 4
|
syl2an |
|- ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) -> dom ( F i^i G ) e. ( SubGrp ` S ) ) |
6 |
|
fveq2 |
|- ( z = ( x ( .s ` S ) y ) -> ( F ` z ) = ( F ` ( x ( .s ` S ) y ) ) ) |
7 |
|
fveq2 |
|- ( z = ( x ( .s ` S ) y ) -> ( G ` z ) = ( G ` ( x ( .s ` S ) y ) ) ) |
8 |
6 7
|
eqeq12d |
|- ( z = ( x ( .s ` S ) y ) -> ( ( F ` z ) = ( G ` z ) <-> ( F ` ( x ( .s ` S ) y ) ) = ( G ` ( x ( .s ` S ) y ) ) ) ) |
9 |
|
lmhmlmod1 |
|- ( F e. ( S LMHom T ) -> S e. LMod ) |
10 |
9
|
adantr |
|- ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) -> S e. LMod ) |
11 |
10
|
ad2antrr |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> S e. LMod ) |
12 |
|
simplr |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> x e. ( Base ` ( Scalar ` S ) ) ) |
13 |
|
simprl |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> y e. ( Base ` S ) ) |
14 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
15 |
|
eqid |
|- ( Scalar ` S ) = ( Scalar ` S ) |
16 |
|
eqid |
|- ( .s ` S ) = ( .s ` S ) |
17 |
|
eqid |
|- ( Base ` ( Scalar ` S ) ) = ( Base ` ( Scalar ` S ) ) |
18 |
14 15 16 17
|
lmodvscl |
|- ( ( S e. LMod /\ x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) -> ( x ( .s ` S ) y ) e. ( Base ` S ) ) |
19 |
11 12 13 18
|
syl3anc |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> ( x ( .s ` S ) y ) e. ( Base ` S ) ) |
20 |
|
oveq2 |
|- ( ( F ` y ) = ( G ` y ) -> ( x ( .s ` T ) ( F ` y ) ) = ( x ( .s ` T ) ( G ` y ) ) ) |
21 |
20
|
ad2antll |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> ( x ( .s ` T ) ( F ` y ) ) = ( x ( .s ` T ) ( G ` y ) ) ) |
22 |
|
simplll |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> F e. ( S LMHom T ) ) |
23 |
|
eqid |
|- ( .s ` T ) = ( .s ` T ) |
24 |
15 17 14 16 23
|
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 ) ) ) |
25 |
22 12 13 24
|
syl3anc |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( x ( .s ` T ) ( F ` y ) ) ) |
26 |
|
simpllr |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> G e. ( S LMHom T ) ) |
27 |
15 17 14 16 23
|
lmhmlin |
|- ( ( G e. ( S LMHom T ) /\ x e. ( Base ` ( Scalar ` S ) ) /\ y e. ( Base ` S ) ) -> ( G ` ( x ( .s ` S ) y ) ) = ( x ( .s ` T ) ( G ` y ) ) ) |
28 |
26 12 13 27
|
syl3anc |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> ( G ` ( x ( .s ` S ) y ) ) = ( x ( .s ` T ) ( G ` y ) ) ) |
29 |
21 25 28
|
3eqtr4d |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> ( F ` ( x ( .s ` S ) y ) ) = ( G ` ( x ( .s ` S ) y ) ) ) |
30 |
8 19 29
|
elrabd |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ ( y e. ( Base ` S ) /\ ( F ` y ) = ( G ` y ) ) ) -> ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) |
31 |
30
|
expr |
|- ( ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) /\ y e. ( Base ` S ) ) -> ( ( F ` y ) = ( G ` y ) -> ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) ) |
32 |
31
|
ralrimiva |
|- ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) -> A. y e. ( Base ` S ) ( ( F ` y ) = ( G ` y ) -> ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) ) |
33 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
34 |
14 33
|
lmhmf |
|- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) ) |
35 |
34
|
ffnd |
|- ( F e. ( S LMHom T ) -> F Fn ( Base ` S ) ) |
36 |
14 33
|
lmhmf |
|- ( G e. ( S LMHom T ) -> G : ( Base ` S ) --> ( Base ` T ) ) |
37 |
36
|
ffnd |
|- ( G e. ( S LMHom T ) -> G Fn ( Base ` S ) ) |
38 |
|
fndmin |
|- ( ( F Fn ( Base ` S ) /\ G Fn ( Base ` S ) ) -> dom ( F i^i G ) = { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) |
39 |
35 37 38
|
syl2an |
|- ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) -> dom ( F i^i G ) = { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) |
40 |
39
|
adantr |
|- ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) -> dom ( F i^i G ) = { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) |
41 |
|
eleq2 |
|- ( dom ( F i^i G ) = { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } -> ( ( x ( .s ` S ) y ) e. dom ( F i^i G ) <-> ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) ) |
42 |
41
|
raleqbi1dv |
|- ( dom ( F i^i G ) = { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } -> ( A. y e. dom ( F i^i G ) ( x ( .s ` S ) y ) e. dom ( F i^i G ) <-> A. y e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) ) |
43 |
|
fveq2 |
|- ( z = y -> ( F ` z ) = ( F ` y ) ) |
44 |
|
fveq2 |
|- ( z = y -> ( G ` z ) = ( G ` y ) ) |
45 |
43 44
|
eqeq12d |
|- ( z = y -> ( ( F ` z ) = ( G ` z ) <-> ( F ` y ) = ( G ` y ) ) ) |
46 |
45
|
ralrab |
|- ( A. y e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } <-> A. y e. ( Base ` S ) ( ( F ` y ) = ( G ` y ) -> ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) ) |
47 |
42 46
|
bitrdi |
|- ( dom ( F i^i G ) = { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } -> ( A. y e. dom ( F i^i G ) ( x ( .s ` S ) y ) e. dom ( F i^i G ) <-> A. y e. ( Base ` S ) ( ( F ` y ) = ( G ` y ) -> ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) ) ) |
48 |
40 47
|
syl |
|- ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) -> ( A. y e. dom ( F i^i G ) ( x ( .s ` S ) y ) e. dom ( F i^i G ) <-> A. y e. ( Base ` S ) ( ( F ` y ) = ( G ` y ) -> ( x ( .s ` S ) y ) e. { z e. ( Base ` S ) | ( F ` z ) = ( G ` z ) } ) ) ) |
49 |
32 48
|
mpbird |
|- ( ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) /\ x e. ( Base ` ( Scalar ` S ) ) ) -> A. y e. dom ( F i^i G ) ( x ( .s ` S ) y ) e. dom ( F i^i G ) ) |
50 |
49
|
ralrimiva |
|- ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) -> A. x e. ( Base ` ( Scalar ` S ) ) A. y e. dom ( F i^i G ) ( x ( .s ` S ) y ) e. dom ( F i^i G ) ) |
51 |
15 17 14 16 1
|
islss4 |
|- ( S e. LMod -> ( dom ( F i^i G ) e. U <-> ( dom ( F i^i G ) e. ( SubGrp ` S ) /\ A. x e. ( Base ` ( Scalar ` S ) ) A. y e. dom ( F i^i G ) ( x ( .s ` S ) y ) e. dom ( F i^i G ) ) ) ) |
52 |
10 51
|
syl |
|- ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) -> ( dom ( F i^i G ) e. U <-> ( dom ( F i^i G ) e. ( SubGrp ` S ) /\ A. x e. ( Base ` ( Scalar ` S ) ) A. y e. dom ( F i^i G ) ( x ( .s ` S ) y ) e. dom ( F i^i G ) ) ) ) |
53 |
5 50 52
|
mpbir2and |
|- ( ( F e. ( S LMHom T ) /\ G e. ( S LMHom T ) ) -> dom ( F i^i G ) e. U ) |