Step |
Hyp |
Ref |
Expression |
1 |
|
lsslindf.u |
⊢ 𝑈 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lsslindf.x |
⊢ 𝑋 = ( 𝑊 ↾s 𝑆 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
4 |
3 1
|
lssss |
⊢ ( 𝑆 ∈ 𝑈 → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
5 |
2 3
|
ressbas2 |
⊢ ( 𝑆 ⊆ ( Base ‘ 𝑊 ) → 𝑆 = ( Base ‘ 𝑋 ) ) |
6 |
4 5
|
syl |
⊢ ( 𝑆 ∈ 𝑈 → 𝑆 = ( Base ‘ 𝑋 ) ) |
7 |
6
|
3ad2ant2 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → 𝑆 = ( Base ‘ 𝑋 ) ) |
8 |
7
|
sseq2d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( 𝐹 ⊆ 𝑆 ↔ 𝐹 ⊆ ( Base ‘ 𝑋 ) ) ) |
9 |
4
|
3ad2ant2 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → 𝑆 ⊆ ( Base ‘ 𝑊 ) ) |
10 |
|
sstr2 |
⊢ ( 𝐹 ⊆ 𝑆 → ( 𝑆 ⊆ ( Base ‘ 𝑊 ) → 𝐹 ⊆ ( Base ‘ 𝑊 ) ) ) |
11 |
9 10
|
mpan9 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ⊆ 𝑆 ) → 𝐹 ⊆ ( Base ‘ 𝑊 ) ) |
12 |
|
simpl3 |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) ∧ 𝐹 ⊆ ( Base ‘ 𝑊 ) ) → 𝐹 ⊆ 𝑆 ) |
13 |
11 12
|
impbida |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( 𝐹 ⊆ 𝑆 ↔ 𝐹 ⊆ ( Base ‘ 𝑊 ) ) ) |
14 |
8 13
|
bitr3d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( 𝐹 ⊆ ( Base ‘ 𝑋 ) ↔ 𝐹 ⊆ ( Base ‘ 𝑊 ) ) ) |
15 |
|
rnresi |
⊢ ran ( I ↾ 𝐹 ) = 𝐹 |
16 |
15
|
sseq1i |
⊢ ( ran ( I ↾ 𝐹 ) ⊆ 𝑆 ↔ 𝐹 ⊆ 𝑆 ) |
17 |
1 2
|
lsslindf |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran ( I ↾ 𝐹 ) ⊆ 𝑆 ) → ( ( I ↾ 𝐹 ) LIndF 𝑋 ↔ ( I ↾ 𝐹 ) LIndF 𝑊 ) ) |
18 |
16 17
|
syl3an3br |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( ( I ↾ 𝐹 ) LIndF 𝑋 ↔ ( I ↾ 𝐹 ) LIndF 𝑊 ) ) |
19 |
14 18
|
anbi12d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( ( 𝐹 ⊆ ( Base ‘ 𝑋 ) ∧ ( I ↾ 𝐹 ) LIndF 𝑋 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ( I ↾ 𝐹 ) LIndF 𝑊 ) ) ) |
20 |
2
|
ovexi |
⊢ 𝑋 ∈ V |
21 |
|
eqid |
⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 ) |
22 |
21
|
islinds |
⊢ ( 𝑋 ∈ V → ( 𝐹 ∈ ( LIndS ‘ 𝑋 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑋 ) ∧ ( I ↾ 𝐹 ) LIndF 𝑋 ) ) ) |
23 |
20 22
|
mp1i |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑋 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑋 ) ∧ ( I ↾ 𝐹 ) LIndF 𝑋 ) ) ) |
24 |
3
|
islinds |
⊢ ( 𝑊 ∈ LMod → ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ( I ↾ 𝐹 ) LIndF 𝑊 ) ) ) |
25 |
24
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑊 ) ↔ ( 𝐹 ⊆ ( Base ‘ 𝑊 ) ∧ ( I ↾ 𝐹 ) LIndF 𝑊 ) ) ) |
26 |
19 23 25
|
3bitr4d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑋 ) ↔ 𝐹 ∈ ( LIndS ‘ 𝑊 ) ) ) |