Step |
Hyp |
Ref |
Expression |
1 |
|
lnmepi.b |
⊢ 𝐵 = ( Base ‘ 𝑇 ) |
2 |
|
lmhmlmod2 |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝑇 ∈ LMod ) |
3 |
2
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) → 𝑇 ∈ LMod ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
5 |
4 1
|
lmhmf |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝐵 ) |
6 |
5
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝐵 ) |
7 |
|
simp3 |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) → ran 𝐹 = 𝐵 ) |
8 |
|
dffo2 |
⊢ ( 𝐹 : ( Base ‘ 𝑆 ) –onto→ 𝐵 ↔ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝐵 ∧ ran 𝐹 = 𝐵 ) ) |
9 |
6 7 8
|
sylanbrc |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) → 𝐹 : ( Base ‘ 𝑆 ) –onto→ 𝐵 ) |
10 |
|
eqid |
⊢ ( LSubSp ‘ 𝑇 ) = ( LSubSp ‘ 𝑇 ) |
11 |
1 10
|
lssss |
⊢ ( 𝑎 ∈ ( LSubSp ‘ 𝑇 ) → 𝑎 ⊆ 𝐵 ) |
12 |
|
foimacnv |
⊢ ( ( 𝐹 : ( Base ‘ 𝑆 ) –onto→ 𝐵 ∧ 𝑎 ⊆ 𝐵 ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑎 ) ) = 𝑎 ) |
13 |
9 11 12
|
syl2an |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑎 ) ) = 𝑎 ) |
14 |
13
|
oveq2d |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → ( 𝑇 ↾s ( 𝐹 “ ( ◡ 𝐹 “ 𝑎 ) ) ) = ( 𝑇 ↾s 𝑎 ) ) |
15 |
|
eqid |
⊢ ( 𝑇 ↾s ( 𝐹 “ ( ◡ 𝐹 “ 𝑎 ) ) ) = ( 𝑇 ↾s ( 𝐹 “ ( ◡ 𝐹 “ 𝑎 ) ) ) |
16 |
|
eqid |
⊢ ( 𝑆 ↾s ( ◡ 𝐹 “ 𝑎 ) ) = ( 𝑆 ↾s ( ◡ 𝐹 “ 𝑎 ) ) |
17 |
|
eqid |
⊢ ( LSubSp ‘ 𝑆 ) = ( LSubSp ‘ 𝑆 ) |
18 |
|
simpl2 |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → 𝑆 ∈ LNoeM ) |
19 |
17 10
|
lmhmpreima |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → ( ◡ 𝐹 “ 𝑎 ) ∈ ( LSubSp ‘ 𝑆 ) ) |
20 |
19
|
3ad2antl1 |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → ( ◡ 𝐹 “ 𝑎 ) ∈ ( LSubSp ‘ 𝑆 ) ) |
21 |
17 16
|
lnmlssfg |
⊢ ( ( 𝑆 ∈ LNoeM ∧ ( ◡ 𝐹 “ 𝑎 ) ∈ ( LSubSp ‘ 𝑆 ) ) → ( 𝑆 ↾s ( ◡ 𝐹 “ 𝑎 ) ) ∈ LFinGen ) |
22 |
18 20 21
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → ( 𝑆 ↾s ( ◡ 𝐹 “ 𝑎 ) ) ∈ LFinGen ) |
23 |
|
simpl1 |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
24 |
15 16 17 22 20 23
|
lmhmfgima |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → ( 𝑇 ↾s ( 𝐹 “ ( ◡ 𝐹 “ 𝑎 ) ) ) ∈ LFinGen ) |
25 |
14 24
|
eqeltrrd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) ∧ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ) → ( 𝑇 ↾s 𝑎 ) ∈ LFinGen ) |
26 |
25
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) → ∀ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ( 𝑇 ↾s 𝑎 ) ∈ LFinGen ) |
27 |
10
|
islnm |
⊢ ( 𝑇 ∈ LNoeM ↔ ( 𝑇 ∈ LMod ∧ ∀ 𝑎 ∈ ( LSubSp ‘ 𝑇 ) ( 𝑇 ↾s 𝑎 ) ∈ LFinGen ) ) |
28 |
3 26 27
|
sylanbrc |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵 ) → 𝑇 ∈ LNoeM ) |