Step |
Hyp |
Ref |
Expression |
1 |
|
lcvexch.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lcvexch.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
3 |
|
lcvexch.c |
⊢ 𝐶 = ( ⋖L ‘ 𝑊 ) |
4 |
|
lcvexch.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
5 |
|
lcvexch.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
6 |
|
lcvexch.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
7 |
|
lcvexch.q |
⊢ ( 𝜑 → 𝑅 ∈ 𝑆 ) |
8 |
|
lcvexch.d |
⊢ ( 𝜑 → 𝑇 ⊆ 𝑅 ) |
9 |
|
lcvexch.e |
⊢ ( 𝜑 → 𝑅 ⊆ ( 𝑇 ⊕ 𝑈 ) ) |
10 |
1
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
11 |
4 10
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
12 |
11 7
|
sseldd |
⊢ ( 𝜑 → 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ) |
13 |
11 6
|
sseldd |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ) |
14 |
11 5
|
sseldd |
⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝑊 ) ) |
15 |
2
|
lsmmod2 |
⊢ ( ( ( 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝑊 ) ) ∧ 𝑇 ⊆ 𝑅 ) → ( 𝑅 ∩ ( 𝑈 ⊕ 𝑇 ) ) = ( ( 𝑅 ∩ 𝑈 ) ⊕ 𝑇 ) ) |
16 |
12 13 14 8 15
|
syl31anc |
⊢ ( 𝜑 → ( 𝑅 ∩ ( 𝑈 ⊕ 𝑇 ) ) = ( ( 𝑅 ∩ 𝑈 ) ⊕ 𝑇 ) ) |
17 |
|
lmodabl |
⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Abel ) |
18 |
4 17
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ Abel ) |
19 |
2
|
lsmcom |
⊢ ( ( 𝑊 ∈ Abel ∧ 𝑇 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ) → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |
20 |
18 14 13 19
|
syl3anc |
⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |
21 |
9 20
|
sseqtrd |
⊢ ( 𝜑 → 𝑅 ⊆ ( 𝑈 ⊕ 𝑇 ) ) |
22 |
|
df-ss |
⊢ ( 𝑅 ⊆ ( 𝑈 ⊕ 𝑇 ) ↔ ( 𝑅 ∩ ( 𝑈 ⊕ 𝑇 ) ) = 𝑅 ) |
23 |
21 22
|
sylib |
⊢ ( 𝜑 → ( 𝑅 ∩ ( 𝑈 ⊕ 𝑇 ) ) = 𝑅 ) |
24 |
16 23
|
eqtr3d |
⊢ ( 𝜑 → ( ( 𝑅 ∩ 𝑈 ) ⊕ 𝑇 ) = 𝑅 ) |