Step |
Hyp |
Ref |
Expression |
1 |
|
lmhmima.x |
⊢ 𝑋 = ( LSubSp ‘ 𝑆 ) |
2 |
|
lmhmima.y |
⊢ 𝑌 = ( LSubSp ‘ 𝑇 ) |
3 |
|
lmghm |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
4 |
|
lmhmlmod1 |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑆 ∈ LMod ) |
5 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → 𝑈 ∈ 𝑋 ) |
6 |
1
|
lsssubg |
⊢ ( ( 𝑆 ∈ LMod ∧ 𝑈 ∈ 𝑋 ) → 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ) |
7 |
4 5 6
|
syl2an2r |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ) |
8 |
|
ghmima |
⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ) → ( 𝐹 “ 𝑈 ) ∈ ( SubGrp ‘ 𝑇 ) ) |
9 |
3 7 8
|
syl2an2r |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ( 𝐹 “ 𝑈 ) ∈ ( SubGrp ‘ 𝑇 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
12 |
10 11
|
lmhmf |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
13 |
12
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
14 |
|
ffn |
⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → 𝐹 Fn ( Base ‘ 𝑆 ) ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → 𝐹 Fn ( Base ‘ 𝑆 ) ) |
16 |
10 1
|
lssss |
⊢ ( 𝑈 ∈ 𝑋 → 𝑈 ⊆ ( Base ‘ 𝑆 ) ) |
17 |
5 16
|
syl |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → 𝑈 ⊆ ( Base ‘ 𝑆 ) ) |
18 |
15 17
|
fvelimabd |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ( 𝑏 ∈ ( 𝐹 “ 𝑈 ) ↔ ∃ 𝑐 ∈ 𝑈 ( 𝐹 ‘ 𝑐 ) = 𝑏 ) ) |
19 |
18
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ) → ( 𝑏 ∈ ( 𝐹 “ 𝑈 ) ↔ ∃ 𝑐 ∈ 𝑈 ( 𝐹 ‘ 𝑐 ) = 𝑏 ) ) |
20 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
21 |
|
eqid |
⊢ ( Scalar ‘ 𝑆 ) = ( Scalar ‘ 𝑆 ) |
22 |
|
eqid |
⊢ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑇 ) |
23 |
21 22
|
lmhmsca |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑆 ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑆 ) ) |
25 |
24
|
fveq2d |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ( Base ‘ ( Scalar ‘ 𝑇 ) ) = ( Base ‘ ( Scalar ‘ 𝑆 ) ) ) |
26 |
25
|
eleq2d |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ↔ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ) ) |
27 |
26
|
biimpa |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ) → 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ) |
28 |
27
|
adantrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ) |
29 |
17
|
sselda |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ 𝑐 ∈ 𝑈 ) → 𝑐 ∈ ( Base ‘ 𝑆 ) ) |
30 |
29
|
adantrl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝑐 ∈ ( Base ‘ 𝑆 ) ) |
31 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑆 ) ) = ( Base ‘ ( Scalar ‘ 𝑆 ) ) |
32 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 ) |
33 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑇 ) = ( ·𝑠 ‘ 𝑇 ) |
34 |
21 31 10 32 33
|
lmhmlin |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑐 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑐 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑐 ) ) ) |
35 |
20 28 30 34
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑐 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑐 ) ) ) |
36 |
20 12 14
|
3syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝐹 Fn ( Base ‘ 𝑆 ) ) |
37 |
|
simplr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝑈 ∈ 𝑋 ) |
38 |
37 16
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝑈 ⊆ ( Base ‘ 𝑆 ) ) |
39 |
4
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → 𝑆 ∈ LMod ) |
40 |
39
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝑆 ∈ LMod ) |
41 |
|
simprr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → 𝑐 ∈ 𝑈 ) |
42 |
21 32 31 1
|
lssvscl |
⊢ ( ( ( 𝑆 ∈ LMod ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑐 ) ∈ 𝑈 ) |
43 |
40 37 28 41 42
|
syl22anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑐 ) ∈ 𝑈 ) |
44 |
|
fnfvima |
⊢ ( ( 𝐹 Fn ( Base ‘ 𝑆 ) ∧ 𝑈 ⊆ ( Base ‘ 𝑆 ) ∧ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑐 ) ∈ 𝑈 ) → ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑐 ) ) ∈ ( 𝐹 “ 𝑈 ) ) |
45 |
36 38 43 44
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → ( 𝐹 ‘ ( 𝑎 ( ·𝑠 ‘ 𝑆 ) 𝑐 ) ) ∈ ( 𝐹 “ 𝑈 ) ) |
46 |
35 45
|
eqeltrrd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑐 ∈ 𝑈 ) ) → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑐 ) ) ∈ ( 𝐹 “ 𝑈 ) ) |
47 |
46
|
anassrs |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ) ∧ 𝑐 ∈ 𝑈 ) → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑐 ) ) ∈ ( 𝐹 “ 𝑈 ) ) |
48 |
|
oveq2 |
⊢ ( ( 𝐹 ‘ 𝑐 ) = 𝑏 → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑐 ) ) = ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ) |
49 |
48
|
eleq1d |
⊢ ( ( 𝐹 ‘ 𝑐 ) = 𝑏 → ( ( 𝑎 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑐 ) ) ∈ ( 𝐹 “ 𝑈 ) ↔ ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) ) |
50 |
47 49
|
syl5ibcom |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ) ∧ 𝑐 ∈ 𝑈 ) → ( ( 𝐹 ‘ 𝑐 ) = 𝑏 → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) ) |
51 |
50
|
rexlimdva |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ) → ( ∃ 𝑐 ∈ 𝑈 ( 𝐹 ‘ 𝑐 ) = 𝑏 → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) ) |
52 |
19 51
|
sylbid |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ) → ( 𝑏 ∈ ( 𝐹 “ 𝑈 ) → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) ) |
53 |
52
|
impr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) ∧ ( 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝑈 ) ) ) → ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) |
54 |
53
|
ralrimivva |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∀ 𝑏 ∈ ( 𝐹 “ 𝑈 ) ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) |
55 |
|
lmhmlmod2 |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑇 ∈ LMod ) |
56 |
55
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → 𝑇 ∈ LMod ) |
57 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑇 ) ) = ( Base ‘ ( Scalar ‘ 𝑇 ) ) |
58 |
22 57 11 33 2
|
islss4 |
⊢ ( 𝑇 ∈ LMod → ( ( 𝐹 “ 𝑈 ) ∈ 𝑌 ↔ ( ( 𝐹 “ 𝑈 ) ∈ ( SubGrp ‘ 𝑇 ) ∧ ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∀ 𝑏 ∈ ( 𝐹 “ 𝑈 ) ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) ) ) |
59 |
56 58
|
syl |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ( ( 𝐹 “ 𝑈 ) ∈ 𝑌 ↔ ( ( 𝐹 “ 𝑈 ) ∈ ( SubGrp ‘ 𝑇 ) ∧ ∀ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑇 ) ) ∀ 𝑏 ∈ ( 𝐹 “ 𝑈 ) ( 𝑎 ( ·𝑠 ‘ 𝑇 ) 𝑏 ) ∈ ( 𝐹 “ 𝑈 ) ) ) ) |
60 |
9 54 59
|
mpbir2and |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑈 ∈ 𝑋 ) → ( 𝐹 “ 𝑈 ) ∈ 𝑌 ) |