Step |
Hyp |
Ref |
Expression |
1 |
|
dochsnshp.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochsnshp.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochsnshp.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochsnshp.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochsnshp.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
6 |
|
dochsnshp.y |
⊢ 𝑌 = ( LSHyp ‘ 𝑈 ) |
7 |
|
dochsnshp.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
dochsnshp.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
9 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
10 |
8
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
11 |
10
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 ) |
12 |
1 3 2 4 9 7 11
|
dochocsp |
⊢ ( 𝜑 → ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) = ( ⊥ ‘ { 𝑋 } ) ) |
13 |
|
eqid |
⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 ) |
14 |
1 3 7
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
15 |
4 9 5 13 14 8
|
lsatlspsn |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∈ ( LSAtoms ‘ 𝑈 ) ) |
16 |
1 3 2 13 6 7 15
|
dochsatshp |
⊢ ( 𝜑 → ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) ∈ 𝑌 ) |
17 |
12 16
|
eqeltrrd |
⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 } ) ∈ 𝑌 ) |