Step |
Hyp |
Ref |
Expression |
1 |
|
djhunss.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
djhunss.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
djhunss.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
djhunss.j |
⊢ ∨ = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
djhunss.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
djhunss.x |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑉 ) |
7 |
|
djhunss.y |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑉 ) |
8 |
1 2 5
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
9 |
6 7
|
unssd |
⊢ ( 𝜑 → ( 𝑋 ∪ 𝑌 ) ⊆ 𝑉 ) |
10 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
11 |
3 10
|
lspssid |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑋 ∪ 𝑌 ) ⊆ 𝑉 ) → ( 𝑋 ∪ 𝑌 ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋 ∪ 𝑌 ) ) ) |
12 |
8 9 11
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ∪ 𝑌 ) ⊆ ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋 ∪ 𝑌 ) ) ) |
13 |
1 2 3 10 4 5 6 7
|
djhspss |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ ( 𝑋 ∪ 𝑌 ) ) ⊆ ( 𝑋 ∨ 𝑌 ) ) |
14 |
12 13
|
sstrd |
⊢ ( 𝜑 → ( 𝑋 ∪ 𝑌 ) ⊆ ( 𝑋 ∨ 𝑌 ) ) |