Step |
Hyp |
Ref |
Expression |
1 |
|
dochf.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochf.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochf.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochf.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochf.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dochf.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝒫 𝑉 ) → ( 𝐼 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ∈ V ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
9 |
|
eqid |
⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 ) |
10 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
11 |
8 9 10 1 2 3 4 5
|
dochfval |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ⊥ = ( 𝑥 ∈ 𝒫 𝑉 ↦ ( 𝐼 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) ) |
12 |
6 11
|
syl |
⊢ ( 𝜑 → ⊥ = ( 𝑥 ∈ 𝒫 𝑉 ↦ ( 𝐼 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) ) |
13 |
|
elpwi |
⊢ ( 𝑦 ∈ 𝒫 𝑉 → 𝑦 ⊆ 𝑉 ) |
14 |
1 2 3 4 5
|
dochcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑦 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑦 ) ∈ ran 𝐼 ) |
15 |
6 13 14
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑉 ) → ( ⊥ ‘ 𝑦 ) ∈ ran 𝐼 ) |
16 |
7 12 15
|
fmpt2d |
⊢ ( 𝜑 → ⊥ : 𝒫 𝑉 ⟶ ran 𝐼 ) |