Step |
Hyp |
Ref |
Expression |
1 |
|
lspprabs.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspprabs.p |
⊢ + = ( +g ‘ 𝑊 ) |
3 |
|
lspprabs.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lspprabs.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
5 |
|
lspprabs.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
6 |
|
lspprabs.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
7 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
8 |
7
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
9 |
4 8
|
syl |
⊢ ( 𝜑 → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
10 |
1 7 3
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
11 |
4 5 10
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
12 |
9 11
|
sseldd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) |
13 |
1 7 3
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
14 |
4 6 13
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
15 |
9 14
|
sseldd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝑊 ) ) |
16 |
|
eqid |
⊢ ( LSSum ‘ 𝑊 ) = ( LSSum ‘ 𝑊 ) |
17 |
16
|
lsmub1 |
⊢ ( ( ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
18 |
12 15 17
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
19 |
7 16
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) ) → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
20 |
4 11 14 19
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
21 |
1 3
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
22 |
4 5 21
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
23 |
1 3
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑌 } ) ) |
24 |
4 6 23
|
syl2anc |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑁 ‘ { 𝑌 } ) ) |
25 |
2 16
|
lsmelvali |
⊢ ( ( ( ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑌 } ) ) ) → ( 𝑋 + 𝑌 ) ∈ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
26 |
12 15 22 24 25
|
syl22anc |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
27 |
7 3 4 20 26
|
lspsnel5a |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
28 |
1 2
|
lmodvacl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 ) |
29 |
4 5 6 28
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑉 ) |
30 |
1 7 3
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 + 𝑌 ) ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
31 |
4 29 30
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
32 |
9 31
|
sseldd |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( SubGrp ‘ 𝑊 ) ) |
33 |
9 20
|
sseldd |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( SubGrp ‘ 𝑊 ) ) |
34 |
16
|
lsmlub |
⊢ ( ( ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ∧ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) ↔ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
35 |
12 32 33 34
|
syl3anc |
⊢ ( 𝜑 → ( ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ∧ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) ↔ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
36 |
18 27 35
|
mpbi2and |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
37 |
16
|
lsmub1 |
⊢ ( ( ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
38 |
12 32 37
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
39 |
7 16
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑊 ) ) → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
40 |
4 11 31 39
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
41 |
|
eqid |
⊢ ( -g ‘ 𝑊 ) = ( -g ‘ 𝑊 ) |
42 |
1 3
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 + 𝑌 ) ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) |
43 |
4 29 42
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) |
44 |
41 16 32 12 43 22
|
lsmelvalmi |
⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) ( -g ‘ 𝑊 ) 𝑋 ) ∈ ( ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑋 } ) ) ) |
45 |
|
lmodabl |
⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Abel ) |
46 |
4 45
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ Abel ) |
47 |
1 2 41
|
ablpncan2 |
⊢ ( ( 𝑊 ∈ Abel ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( ( 𝑋 + 𝑌 ) ( -g ‘ 𝑊 ) 𝑋 ) = 𝑌 ) |
48 |
46 5 6 47
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) ( -g ‘ 𝑊 ) 𝑋 ) = 𝑌 ) |
49 |
16
|
lsmcom |
⊢ ( ( 𝑊 ∈ Abel ∧ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑋 } ) ) = ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
50 |
46 32 12 49
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑋 } ) ) = ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
51 |
44 48 50
|
3eltr3d |
⊢ ( 𝜑 → 𝑌 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
52 |
7 3 4 40 51
|
lspsnel5a |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
53 |
9 40
|
sseldd |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ ( SubGrp ‘ 𝑊 ) ) |
54 |
16
|
lsmlub |
⊢ ( ( ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ∧ ( 𝑁 ‘ { 𝑌 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) ↔ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) ) |
55 |
12 15 53 54
|
syl3anc |
⊢ ( 𝜑 → ( ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ∧ ( 𝑁 ‘ { 𝑌 } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) ↔ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) ) |
56 |
38 52 55
|
mpbi2and |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
57 |
36 56
|
eqssd |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
58 |
1 3 16 4 5 29
|
lsmpr |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , ( 𝑋 + 𝑌 ) } ) = ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
59 |
1 3 16 4 5 6
|
lsmpr |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑊 ) ( 𝑁 ‘ { 𝑌 } ) ) ) |
60 |
57 58 59
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , ( 𝑋 + 𝑌 ) } ) = ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |