Step |
Hyp |
Ref |
Expression |
1 |
|
djhsumss.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
djhsumss.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
djhsumss.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
djhsumss.p |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
5 |
|
djhsumss.j |
⊢ ∨ = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
djhsumss.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
djhsumss.x |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑉 ) |
8 |
|
djhsumss.y |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑉 ) |
9 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
10 |
1 2 6
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
11 |
3 9 4 7 8 10
|
lsmssspx |
⊢ ( 𝜑 → ( 𝑋 ⊕ 𝑌 ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋 ∪ 𝑌 ) ) ) |
12 |
1 2 3 9 5 6 7 8
|
djhspss |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋 ∪ 𝑌 ) ) ⊆ ( 𝑋 ∨ 𝑌 ) ) |
13 |
11 12
|
sstrd |
⊢ ( 𝜑 → ( 𝑋 ⊕ 𝑌 ) ⊆ ( 𝑋 ∨ 𝑌 ) ) |