Step |
Hyp |
Ref |
Expression |
1 |
|
lsatexch.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lsatexch.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
3 |
|
lsatexch.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
4 |
|
lsatexch.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
5 |
|
lsatexch.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
6 |
|
lsatexch.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
7 |
|
lsatexch.q |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
8 |
|
lsatexch.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
9 |
|
lsatexch.l |
⊢ ( 𝜑 → 𝑄 ⊆ ( 𝑈 ⊕ 𝑅 ) ) |
10 |
|
lsatexch.z |
⊢ ( 𝜑 → ( 𝑈 ∩ 𝑄 ) = { 0 } ) |
11 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
12 |
5 11
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
13 |
1
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
15 |
14 6
|
sseldd |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ) |
16 |
1 4 12 8
|
lsatlssel |
⊢ ( 𝜑 → 𝑅 ∈ 𝑆 ) |
17 |
14 16
|
sseldd |
⊢ ( 𝜑 → 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ) |
18 |
2
|
lsmub2 |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ) → 𝑅 ⊆ ( 𝑈 ⊕ 𝑅 ) ) |
19 |
15 17 18
|
syl2anc |
⊢ ( 𝜑 → 𝑅 ⊆ ( 𝑈 ⊕ 𝑅 ) ) |
20 |
|
eqid |
⊢ ( ⋖L ‘ 𝑊 ) = ( ⋖L ‘ 𝑊 ) |
21 |
1 2
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆 ) → ( 𝑈 ⊕ 𝑅 ) ∈ 𝑆 ) |
22 |
12 6 16 21
|
syl3anc |
⊢ ( 𝜑 → ( 𝑈 ⊕ 𝑅 ) ∈ 𝑆 ) |
23 |
1 4 12 7
|
lsatlssel |
⊢ ( 𝜑 → 𝑄 ∈ 𝑆 ) |
24 |
1 2
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆 ) → ( 𝑈 ⊕ 𝑄 ) ∈ 𝑆 ) |
25 |
12 6 23 24
|
syl3anc |
⊢ ( 𝜑 → ( 𝑈 ⊕ 𝑄 ) ∈ 𝑆 ) |
26 |
1 2 3 4 20 5 6 7
|
lcvp |
⊢ ( 𝜑 → ( ( 𝑈 ∩ 𝑄 ) = { 0 } ↔ 𝑈 ( ⋖L ‘ 𝑊 ) ( 𝑈 ⊕ 𝑄 ) ) ) |
27 |
10 26
|
mpbid |
⊢ ( 𝜑 → 𝑈 ( ⋖L ‘ 𝑊 ) ( 𝑈 ⊕ 𝑄 ) ) |
28 |
1 20 5 6 25 27
|
lcvpss |
⊢ ( 𝜑 → 𝑈 ⊊ ( 𝑈 ⊕ 𝑄 ) ) |
29 |
2
|
lsmub1 |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ) → 𝑈 ⊆ ( 𝑈 ⊕ 𝑅 ) ) |
30 |
15 17 29
|
syl2anc |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑈 ⊕ 𝑅 ) ) |
31 |
14 23
|
sseldd |
⊢ ( 𝜑 → 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ) |
32 |
14 22
|
sseldd |
⊢ ( 𝜑 → ( 𝑈 ⊕ 𝑅 ) ∈ ( SubGrp ‘ 𝑊 ) ) |
33 |
2
|
lsmlub |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑈 ⊕ 𝑅 ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( ( 𝑈 ⊆ ( 𝑈 ⊕ 𝑅 ) ∧ 𝑄 ⊆ ( 𝑈 ⊕ 𝑅 ) ) ↔ ( 𝑈 ⊕ 𝑄 ) ⊆ ( 𝑈 ⊕ 𝑅 ) ) ) |
34 |
15 31 32 33
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑈 ⊆ ( 𝑈 ⊕ 𝑅 ) ∧ 𝑄 ⊆ ( 𝑈 ⊕ 𝑅 ) ) ↔ ( 𝑈 ⊕ 𝑄 ) ⊆ ( 𝑈 ⊕ 𝑅 ) ) ) |
35 |
30 9 34
|
mpbi2and |
⊢ ( 𝜑 → ( 𝑈 ⊕ 𝑄 ) ⊆ ( 𝑈 ⊕ 𝑅 ) ) |
36 |
28 35
|
psssstrd |
⊢ ( 𝜑 → 𝑈 ⊊ ( 𝑈 ⊕ 𝑅 ) ) |
37 |
1 2 4 20 5 6 8
|
lcv2 |
⊢ ( 𝜑 → ( 𝑈 ⊊ ( 𝑈 ⊕ 𝑅 ) ↔ 𝑈 ( ⋖L ‘ 𝑊 ) ( 𝑈 ⊕ 𝑅 ) ) ) |
38 |
36 37
|
mpbid |
⊢ ( 𝜑 → 𝑈 ( ⋖L ‘ 𝑊 ) ( 𝑈 ⊕ 𝑅 ) ) |
39 |
1 20 5 6 22 25 38 28 35
|
lcvnbtwn2 |
⊢ ( 𝜑 → ( 𝑈 ⊕ 𝑄 ) = ( 𝑈 ⊕ 𝑅 ) ) |
40 |
19 39
|
sseqtrrd |
⊢ ( 𝜑 → 𝑅 ⊆ ( 𝑈 ⊕ 𝑄 ) ) |