Step |
Hyp |
Ref |
Expression |
1 |
|
reslmhm.u |
⊢ 𝑈 = ( LSubSp ‘ 𝑆 ) |
2 |
|
reslmhm.r |
⊢ 𝑅 = ( 𝑆 ↾s 𝑋 ) |
3 |
|
lmhmlmod1 |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑆 ∈ LMod ) |
4 |
2 1
|
lsslmod |
⊢ ( ( 𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈 ) → 𝑅 ∈ LMod ) |
5 |
3 4
|
sylan |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → 𝑅 ∈ LMod ) |
6 |
|
lmhmlmod2 |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑇 ∈ LMod ) |
7 |
6
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → 𝑇 ∈ LMod ) |
8 |
|
lmghm |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
9 |
1
|
lsssubg |
⊢ ( ( 𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ ( SubGrp ‘ 𝑆 ) ) |
10 |
3 9
|
sylan |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ∈ ( SubGrp ‘ 𝑆 ) ) |
11 |
2
|
resghm |
⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑋 ∈ ( SubGrp ‘ 𝑆 ) ) → ( 𝐹 ↾ 𝑋 ) ∈ ( 𝑅 GrpHom 𝑇 ) ) |
12 |
8 10 11
|
syl2an2r |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( 𝐹 ↾ 𝑋 ) ∈ ( 𝑅 GrpHom 𝑇 ) ) |
13 |
|
eqid |
⊢ ( Scalar ‘ 𝑆 ) = ( Scalar ‘ 𝑆 ) |
14 |
|
eqid |
⊢ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑇 ) |
15 |
13 14
|
lmhmsca |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑆 ) ) |
16 |
2 13
|
resssca |
⊢ ( 𝑋 ∈ 𝑈 → ( Scalar ‘ 𝑆 ) = ( Scalar ‘ 𝑅 ) ) |
17 |
15 16
|
sylan9eq |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑅 ) ) |
18 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
19 |
|
simprl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ) |
20 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
21 |
20 1
|
lssss |
⊢ ( 𝑋 ∈ 𝑈 → 𝑋 ⊆ ( Base ‘ 𝑆 ) ) |
22 |
21
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → 𝑋 ⊆ ( Base ‘ 𝑆 ) ) |
23 |
22
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑋 ⊆ ( Base ‘ 𝑆 ) ) |
24 |
2 20
|
ressbas2 |
⊢ ( 𝑋 ⊆ ( Base ‘ 𝑆 ) → 𝑋 = ( Base ‘ 𝑅 ) ) |
25 |
22 24
|
syl |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → 𝑋 = ( Base ‘ 𝑅 ) ) |
26 |
25
|
eleq2d |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( 𝑏 ∈ 𝑋 ↔ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) |
27 |
26
|
biimpar |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) → 𝑏 ∈ 𝑋 ) |
28 |
27
|
adantrl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑏 ∈ 𝑋 ) |
29 |
23 28
|
sseldd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑏 ∈ ( Base ‘ 𝑆 ) ) |
30 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑆 ) ) = ( Base ‘ ( Scalar ‘ 𝑆 ) ) |
31 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 ) |
32 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑇 ) = ( ·𝑠 ‘ 𝑇 ) |
33 |
13 30 20 31 32
|
lmhmlin |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑏 ) ) ) |
34 |
18 19 29 33
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑏 ) ) ) |
35 |
3
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → 𝑆 ∈ LMod ) |
36 |
35
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑆 ∈ LMod ) |
37 |
|
simplr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑋 ∈ 𝑈 ) |
38 |
13 31 30 1
|
lssvscl |
⊢ ( ( ( 𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ 𝑋 ) ) → ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ∈ 𝑋 ) |
39 |
36 37 19 28 38
|
syl22anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ∈ 𝑋 ) |
40 |
39
|
fvresd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) ) |
41 |
|
fvres |
⊢ ( 𝑏 ∈ 𝑋 → ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) = ( 𝐹 ‘ 𝑏 ) ) |
42 |
41
|
oveq2d |
⊢ ( 𝑏 ∈ 𝑋 → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑏 ) ) ) |
43 |
28 42
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑏 ) ) ) |
44 |
34 40 43
|
3eqtr4d |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑏 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) |
45 |
44
|
ralrimivva |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) |
46 |
16
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( Scalar ‘ 𝑆 ) = ( Scalar ‘ 𝑅 ) ) |
47 |
46
|
fveq2d |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( Base ‘ ( Scalar ‘ 𝑆 ) ) = ( Base ‘ ( Scalar ‘ 𝑅 ) ) ) |
48 |
2 31
|
ressvsca |
⊢ ( 𝑋 ∈ 𝑈 → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑅 ) ) |
49 |
48
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑅 ) ) |
50 |
49
|
oveqd |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) = ( 𝑎 ( ·𝑠 ‘ 𝑅 ) 𝑏 ) ) |
51 |
50
|
fveqeq2d |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ↔ ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) ) |
52 |
51
|
ralbidv |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ↔ ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) ) |
53 |
47 52
|
raleqbidv |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ↔ ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) ) |
54 |
45 53
|
mpbid |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) |
55 |
12 17 54
|
3jca |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( ( 𝐹 ↾ 𝑋 ) ∈ ( 𝑅 GrpHom 𝑇 ) ∧ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑅 ) ∧ ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) ) |
56 |
|
eqid |
⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 ) |
57 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑅 ) ) = ( Base ‘ ( Scalar ‘ 𝑅 ) ) |
58 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
59 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑅 ) = ( ·𝑠 ‘ 𝑅 ) |
60 |
56 14 57 58 59 32
|
islmhm |
⊢ ( ( 𝐹 ↾ 𝑋 ) ∈ ( 𝑅 LMHom 𝑇 ) ↔ ( ( 𝑅 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( ( 𝐹 ↾ 𝑋 ) ∈ ( 𝑅 GrpHom 𝑇 ) ∧ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑅 ) ∧ ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) ∀ 𝑏 ∈ ( Base ‘ 𝑅 ) ( ( 𝐹 ↾ 𝑋 ) ‘ ( 𝑎 ( ·𝑠 ‘ 𝑅 ) 𝑏 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( ( 𝐹 ↾ 𝑋 ) ‘ 𝑏 ) ) ) ) ) |
61 |
5 7 55 60
|
syl21anbrc |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑋 ∈ 𝑈 ) → ( 𝐹 ↾ 𝑋 ) ∈ ( 𝑅 LMHom 𝑇 ) ) |