Step |
Hyp |
Ref |
Expression |
1 |
|
polsubcl.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
polsubcl.p |
⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 ) |
3 |
|
polsubcl.c |
⊢ 𝐶 = ( PSubCl ‘ 𝐾 ) |
4 |
|
eqid |
⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 ) |
5 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
6 |
|
eqid |
⊢ ( pmap ‘ 𝐾 ) = ( pmap ‘ 𝐾 ) |
7 |
4 5 1 6 2
|
polval2N |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) = ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) |
8 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
9 |
8
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → 𝐾 ∈ OP ) |
10 |
|
hlclat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
12 |
11 1
|
atssbase |
⊢ 𝐴 ⊆ ( Base ‘ 𝐾 ) |
13 |
|
sstr |
⊢ ( ( 𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ ( Base ‘ 𝐾 ) ) → 𝑋 ⊆ ( Base ‘ 𝐾 ) ) |
14 |
12 13
|
mpan2 |
⊢ ( 𝑋 ⊆ 𝐴 → 𝑋 ⊆ ( Base ‘ 𝐾 ) ) |
15 |
11 4
|
clatlubcl |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑋 ⊆ ( Base ‘ 𝐾 ) ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
16 |
10 14 15
|
syl2an |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
17 |
11 5
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
9 16 17
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
11 6 3
|
pmapsubclN |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ∈ 𝐶 ) |
20 |
18 19
|
syldan |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ∈ 𝐶 ) |
21 |
7 20
|
eqeltrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐶 ) |