Step |
Hyp |
Ref |
Expression |
1 |
|
islmodfg.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
2 |
|
islmodfg.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
df-lfig |
⊢ LFinGen = { 𝑎 ∈ LMod ∣ ( Base ‘ 𝑎 ) ∈ ( ( LSpan ‘ 𝑎 ) “ ( 𝒫 ( Base ‘ 𝑎 ) ∩ Fin ) ) } |
4 |
3
|
eleq2i |
⊢ ( 𝑊 ∈ LFinGen ↔ 𝑊 ∈ { 𝑎 ∈ LMod ∣ ( Base ‘ 𝑎 ) ∈ ( ( LSpan ‘ 𝑎 ) “ ( 𝒫 ( Base ‘ 𝑎 ) ∩ Fin ) ) } ) |
5 |
|
fveq2 |
⊢ ( 𝑎 = 𝑊 → ( Base ‘ 𝑎 ) = ( Base ‘ 𝑊 ) ) |
6 |
|
fveq2 |
⊢ ( 𝑎 = 𝑊 → ( LSpan ‘ 𝑎 ) = ( LSpan ‘ 𝑊 ) ) |
7 |
6 2
|
eqtr4di |
⊢ ( 𝑎 = 𝑊 → ( LSpan ‘ 𝑎 ) = 𝑁 ) |
8 |
5
|
pweqd |
⊢ ( 𝑎 = 𝑊 → 𝒫 ( Base ‘ 𝑎 ) = 𝒫 ( Base ‘ 𝑊 ) ) |
9 |
8
|
ineq1d |
⊢ ( 𝑎 = 𝑊 → ( 𝒫 ( Base ‘ 𝑎 ) ∩ Fin ) = ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) |
10 |
7 9
|
imaeq12d |
⊢ ( 𝑎 = 𝑊 → ( ( LSpan ‘ 𝑎 ) “ ( 𝒫 ( Base ‘ 𝑎 ) ∩ Fin ) ) = ( 𝑁 “ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) ) |
11 |
5 10
|
eleq12d |
⊢ ( 𝑎 = 𝑊 → ( ( Base ‘ 𝑎 ) ∈ ( ( LSpan ‘ 𝑎 ) “ ( 𝒫 ( Base ‘ 𝑎 ) ∩ Fin ) ) ↔ ( Base ‘ 𝑊 ) ∈ ( 𝑁 “ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) ) ) |
12 |
11
|
elrab3 |
⊢ ( 𝑊 ∈ LMod → ( 𝑊 ∈ { 𝑎 ∈ LMod ∣ ( Base ‘ 𝑎 ) ∈ ( ( LSpan ‘ 𝑎 ) “ ( 𝒫 ( Base ‘ 𝑎 ) ∩ Fin ) ) } ↔ ( Base ‘ 𝑊 ) ∈ ( 𝑁 “ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) ) ) |
13 |
4 12
|
syl5bb |
⊢ ( 𝑊 ∈ LMod → ( 𝑊 ∈ LFinGen ↔ ( Base ‘ 𝑊 ) ∈ ( 𝑁 “ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
15 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
16 |
14 15 2
|
lspf |
⊢ ( 𝑊 ∈ LMod → 𝑁 : 𝒫 ( Base ‘ 𝑊 ) ⟶ ( LSubSp ‘ 𝑊 ) ) |
17 |
16
|
ffnd |
⊢ ( 𝑊 ∈ LMod → 𝑁 Fn 𝒫 ( Base ‘ 𝑊 ) ) |
18 |
|
inss1 |
⊢ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ⊆ 𝒫 ( Base ‘ 𝑊 ) |
19 |
|
fvelimab |
⊢ ( ( 𝑁 Fn 𝒫 ( Base ‘ 𝑊 ) ∧ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ⊆ 𝒫 ( Base ‘ 𝑊 ) ) → ( ( Base ‘ 𝑊 ) ∈ ( 𝑁 “ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) ↔ ∃ 𝑏 ∈ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ( 𝑁 ‘ 𝑏 ) = ( Base ‘ 𝑊 ) ) ) |
20 |
17 18 19
|
sylancl |
⊢ ( 𝑊 ∈ LMod → ( ( Base ‘ 𝑊 ) ∈ ( 𝑁 “ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) ↔ ∃ 𝑏 ∈ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ( 𝑁 ‘ 𝑏 ) = ( Base ‘ 𝑊 ) ) ) |
21 |
|
elin |
⊢ ( 𝑏 ∈ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ↔ ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑊 ) ∧ 𝑏 ∈ Fin ) ) |
22 |
1
|
eqcomi |
⊢ ( Base ‘ 𝑊 ) = 𝐵 |
23 |
22
|
pweqi |
⊢ 𝒫 ( Base ‘ 𝑊 ) = 𝒫 𝐵 |
24 |
23
|
eleq2i |
⊢ ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑊 ) ↔ 𝑏 ∈ 𝒫 𝐵 ) |
25 |
24
|
anbi1i |
⊢ ( ( 𝑏 ∈ 𝒫 ( Base ‘ 𝑊 ) ∧ 𝑏 ∈ Fin ) ↔ ( 𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin ) ) |
26 |
21 25
|
bitri |
⊢ ( 𝑏 ∈ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ↔ ( 𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin ) ) |
27 |
22
|
eqeq2i |
⊢ ( ( 𝑁 ‘ 𝑏 ) = ( Base ‘ 𝑊 ) ↔ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) |
28 |
26 27
|
anbi12i |
⊢ ( ( 𝑏 ∈ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ∧ ( 𝑁 ‘ 𝑏 ) = ( Base ‘ 𝑊 ) ) ↔ ( ( 𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin ) ∧ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) ) |
29 |
|
anass |
⊢ ( ( ( 𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin ) ∧ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) ↔ ( 𝑏 ∈ 𝒫 𝐵 ∧ ( 𝑏 ∈ Fin ∧ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) ) ) |
30 |
28 29
|
bitri |
⊢ ( ( 𝑏 ∈ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ∧ ( 𝑁 ‘ 𝑏 ) = ( Base ‘ 𝑊 ) ) ↔ ( 𝑏 ∈ 𝒫 𝐵 ∧ ( 𝑏 ∈ Fin ∧ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) ) ) |
31 |
30
|
rexbii2 |
⊢ ( ∃ 𝑏 ∈ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ( 𝑁 ‘ 𝑏 ) = ( Base ‘ 𝑊 ) ↔ ∃ 𝑏 ∈ 𝒫 𝐵 ( 𝑏 ∈ Fin ∧ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) ) |
32 |
20 31
|
syl6bb |
⊢ ( 𝑊 ∈ LMod → ( ( Base ‘ 𝑊 ) ∈ ( 𝑁 “ ( 𝒫 ( Base ‘ 𝑊 ) ∩ Fin ) ) ↔ ∃ 𝑏 ∈ 𝒫 𝐵 ( 𝑏 ∈ Fin ∧ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) ) ) |
33 |
13 32
|
bitrd |
⊢ ( 𝑊 ∈ LMod → ( 𝑊 ∈ LFinGen ↔ ∃ 𝑏 ∈ 𝒫 𝐵 ( 𝑏 ∈ Fin ∧ ( 𝑁 ‘ 𝑏 ) = 𝐵 ) ) ) |