Step |
Hyp |
Ref |
Expression |
1 |
|
dochcl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochcl.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochcl.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochcl.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochcl.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 ) |
8 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
9 |
6 7 8 1 2 3 4 5
|
dochval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑋 ) = ( 𝐼 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) |
10 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
11 |
10
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → 𝐾 ∈ OP ) |
12 |
|
hlclat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat ) |
13 |
12
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → 𝐾 ∈ CLat ) |
14 |
|
ssrab2 |
⊢ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ⊆ ( Base ‘ 𝐾 ) |
15 |
6 7
|
clatglbcl |
⊢ ( ( 𝐾 ∈ CLat ∧ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ⊆ ( Base ‘ 𝐾 ) ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ∈ ( Base ‘ 𝐾 ) ) |
16 |
13 14 15
|
sylancl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ∈ ( Base ‘ 𝐾 ) ) |
17 |
6 8
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
11 16 17
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
6 1 2
|
dihcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ∈ ran 𝐼 ) |
20 |
18 19
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( 𝐼 ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑦 ∈ ( Base ‘ 𝐾 ) ∣ 𝑋 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ∈ ran 𝐼 ) |
21 |
9 20
|
eqeltrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑋 ) ∈ ran 𝐼 ) |