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 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑆 ∈ LMod ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑆 ∈ LMod ) |
11 |
|
simpl1 |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑇 ∈ LMod ) |
12 |
|
simpl2 |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑋 ∈ 𝐿 ) |
13 |
1 2
|
lsslmod |
⊢ ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ) → 𝑈 ∈ LMod ) |
14 |
11 12 13
|
syl2anc |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑈 ∈ LMod ) |
15 |
|
eqid |
⊢ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑇 ) |
16 |
1 15
|
resssca |
⊢ ( 𝑋 ∈ 𝐿 → ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑈 ) ) |
17 |
16
|
3ad2ant2 |
⊢ ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) → ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑈 ) ) |
18 |
6 15
|
lmhmsca |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑆 ) ) |
19 |
17 18
|
sylan9req |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑆 ) ) |
20 |
|
lmghm |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
21 |
2
|
lsssubg |
⊢ ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ) → 𝑋 ∈ ( SubGrp ‘ 𝑇 ) ) |
22 |
1
|
resghm2b |
⊢ ( ( 𝑋 ∈ ( SubGrp ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) → ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ↔ 𝐹 ∈ ( 𝑆 GrpHom 𝑈 ) ) ) |
23 |
21 22
|
stoic3 |
⊢ ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) → ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ↔ 𝐹 ∈ ( 𝑆 GrpHom 𝑈 ) ) ) |
24 |
23
|
biimpa |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑈 ) ) |
25 |
20 24
|
sylan2 |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑈 ) ) |
26 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑇 ) = ( ·𝑠 ‘ 𝑇 ) |
27 |
6 8 3 4 26
|
lmhmlin |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
28 |
27
|
3expb |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
29 |
28
|
adantll |
⊢ ( ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( ·𝑠 ‘ 𝑆 ) 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
30 |
|
simpll2 |
⊢ ( ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → 𝑋 ∈ 𝐿 ) |
31 |
1 26
|
ressvsca |
⊢ ( 𝑋 ∈ 𝐿 → ( ·𝑠 ‘ 𝑇 ) = ( ·𝑠 ‘ 𝑈 ) ) |
32 |
31
|
oveqd |
⊢ ( 𝑋 ∈ 𝐿 → ( 𝑥 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
33 |
30 32
|
syl |
⊢ ( ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
34 |
29 33
|
eqtrd |
⊢ ( ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( 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 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ) |
36 |
|
simpr |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ) → 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ) |
37 |
|
simpl1 |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ) → 𝑇 ∈ LMod ) |
38 |
|
simpl2 |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ) → 𝑋 ∈ 𝐿 ) |
39 |
1 2
|
reslmhm2 |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
40 |
36 37 38 39
|
syl3anc |
⊢ ( ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
41 |
35 40
|
impbida |
⊢ ( ( 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋 ) → ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ↔ 𝐹 ∈ ( 𝑆 LMHom 𝑈 ) ) ) |