Step |
Hyp |
Ref |
Expression |
1 |
|
lsslsp.x |
⊢ 𝑋 = ( 𝑊 ↾s 𝑈 ) |
2 |
|
lsslsp.m |
⊢ 𝑀 = ( LSpan ‘ 𝑊 ) |
3 |
|
lsslsp.n |
⊢ 𝑁 = ( LSpan ‘ 𝑋 ) |
4 |
|
lsslsp.l |
⊢ 𝐿 = ( LSubSp ‘ 𝑊 ) |
5 |
1 4
|
lsslmod |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ) → 𝑋 ∈ LMod ) |
6 |
5
|
3adant3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝑋 ∈ LMod ) |
7 |
|
simp1 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝑊 ∈ LMod ) |
8 |
|
simp3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝐺 ⊆ 𝑈 ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
10 |
9 4
|
lssss |
⊢ ( 𝑈 ∈ 𝐿 → 𝑈 ⊆ ( Base ‘ 𝑊 ) ) |
11 |
10
|
3ad2ant2 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝑈 ⊆ ( Base ‘ 𝑊 ) ) |
12 |
8 11
|
sstrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝐺 ⊆ ( Base ‘ 𝑊 ) ) |
13 |
9 4 2
|
lspcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ⊆ ( Base ‘ 𝑊 ) ) → ( 𝑀 ‘ 𝐺 ) ∈ 𝐿 ) |
14 |
7 12 13
|
syl2anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑀 ‘ 𝐺 ) ∈ 𝐿 ) |
15 |
4 2
|
lspssp |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑀 ‘ 𝐺 ) ⊆ 𝑈 ) |
16 |
|
eqid |
⊢ ( LSubSp ‘ 𝑋 ) = ( LSubSp ‘ 𝑋 ) |
17 |
1 4 16
|
lsslss |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ) → ( ( 𝑀 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ↔ ( ( 𝑀 ‘ 𝐺 ) ∈ 𝐿 ∧ ( 𝑀 ‘ 𝐺 ) ⊆ 𝑈 ) ) ) |
18 |
17
|
3adant3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( ( 𝑀 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ↔ ( ( 𝑀 ‘ 𝐺 ) ∈ 𝐿 ∧ ( 𝑀 ‘ 𝐺 ) ⊆ 𝑈 ) ) ) |
19 |
14 15 18
|
mpbir2and |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑀 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ) |
20 |
9 2
|
lspssid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ⊆ ( Base ‘ 𝑊 ) ) → 𝐺 ⊆ ( 𝑀 ‘ 𝐺 ) ) |
21 |
7 12 20
|
syl2anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝐺 ⊆ ( 𝑀 ‘ 𝐺 ) ) |
22 |
16 3
|
lspssp |
⊢ ( ( 𝑋 ∈ LMod ∧ ( 𝑀 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ∧ 𝐺 ⊆ ( 𝑀 ‘ 𝐺 ) ) → ( 𝑁 ‘ 𝐺 ) ⊆ ( 𝑀 ‘ 𝐺 ) ) |
23 |
6 19 21 22
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑁 ‘ 𝐺 ) ⊆ ( 𝑀 ‘ 𝐺 ) ) |
24 |
1 9
|
ressbas2 |
⊢ ( 𝑈 ⊆ ( Base ‘ 𝑊 ) → 𝑈 = ( Base ‘ 𝑋 ) ) |
25 |
11 24
|
syl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝑈 = ( Base ‘ 𝑋 ) ) |
26 |
8 25
|
sseqtrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝐺 ⊆ ( Base ‘ 𝑋 ) ) |
27 |
|
eqid |
⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 ) |
28 |
27 16 3
|
lspcl |
⊢ ( ( 𝑋 ∈ LMod ∧ 𝐺 ⊆ ( Base ‘ 𝑋 ) ) → ( 𝑁 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ) |
29 |
6 26 28
|
syl2anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑁 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ) |
30 |
1 4 16
|
lsslss |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ) → ( ( 𝑁 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ↔ ( ( 𝑁 ‘ 𝐺 ) ∈ 𝐿 ∧ ( 𝑁 ‘ 𝐺 ) ⊆ 𝑈 ) ) ) |
31 |
30
|
3adant3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( ( 𝑁 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑋 ) ↔ ( ( 𝑁 ‘ 𝐺 ) ∈ 𝐿 ∧ ( 𝑁 ‘ 𝐺 ) ⊆ 𝑈 ) ) ) |
32 |
29 31
|
mpbid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( ( 𝑁 ‘ 𝐺 ) ∈ 𝐿 ∧ ( 𝑁 ‘ 𝐺 ) ⊆ 𝑈 ) ) |
33 |
32
|
simpld |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑁 ‘ 𝐺 ) ∈ 𝐿 ) |
34 |
27 3
|
lspssid |
⊢ ( ( 𝑋 ∈ LMod ∧ 𝐺 ⊆ ( Base ‘ 𝑋 ) ) → 𝐺 ⊆ ( 𝑁 ‘ 𝐺 ) ) |
35 |
6 26 34
|
syl2anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → 𝐺 ⊆ ( 𝑁 ‘ 𝐺 ) ) |
36 |
4 2
|
lspssp |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ 𝐺 ) ∈ 𝐿 ∧ 𝐺 ⊆ ( 𝑁 ‘ 𝐺 ) ) → ( 𝑀 ‘ 𝐺 ) ⊆ ( 𝑁 ‘ 𝐺 ) ) |
37 |
7 33 35 36
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑀 ‘ 𝐺 ) ⊆ ( 𝑁 ‘ 𝐺 ) ) |
38 |
23 37
|
eqssd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈 ) → ( 𝑁 ‘ 𝐺 ) = ( 𝑀 ‘ 𝐺 ) ) |