Step |
Hyp |
Ref |
Expression |
1 |
|
2polat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
2polat.p |
⊢ 𝑃 = ( ⊥𝑃 ‘ 𝐾 ) |
3 |
1 2
|
2polssN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → 𝑋 ⊆ ( 𝑃 ‘ ( 𝑃 ‘ 𝑋 ) ) ) |
4 |
3
|
ssrind |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑋 ∩ ( 𝑃 ‘ 𝑋 ) ) ⊆ ( ( 𝑃 ‘ ( 𝑃 ‘ 𝑋 ) ) ∩ ( 𝑃 ‘ 𝑋 ) ) ) |
5 |
|
eqid |
⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 ) |
6 |
|
eqid |
⊢ ( pmap ‘ 𝐾 ) = ( pmap ‘ 𝐾 ) |
7 |
5 1 6 2
|
2polvalN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑃 ‘ ( 𝑃 ‘ 𝑋 ) ) = ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) |
8 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
9 |
5 8 1 6 2
|
polval2N |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑃 ‘ 𝑋 ) = ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) |
10 |
7 9
|
ineq12d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( 𝑃 ‘ ( 𝑃 ‘ 𝑋 ) ) ∩ ( 𝑃 ‘ 𝑋 ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∩ ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) ) |
11 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
12 |
11
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → 𝐾 ∈ OP ) |
13 |
|
hlclat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
15 |
14 1
|
atssbase |
⊢ 𝐴 ⊆ ( Base ‘ 𝐾 ) |
16 |
|
sstr |
⊢ ( ( 𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ ( Base ‘ 𝐾 ) ) → 𝑋 ⊆ ( Base ‘ 𝐾 ) ) |
17 |
15 16
|
mpan2 |
⊢ ( 𝑋 ⊆ 𝐴 → 𝑋 ⊆ ( Base ‘ 𝐾 ) ) |
18 |
14 5
|
clatlubcl |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑋 ⊆ ( Base ‘ 𝐾 ) ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
13 17 18
|
syl2an |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
21 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
22 |
14 8 20 21
|
opnoncon |
⊢ ( ( 𝐾 ∈ OP ∧ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) = ( 0. ‘ 𝐾 ) ) |
23 |
12 19 22
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) = ( 0. ‘ 𝐾 ) ) |
24 |
23
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) = ( ( pmap ‘ 𝐾 ) ‘ ( 0. ‘ 𝐾 ) ) ) |
25 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → 𝐾 ∈ HL ) |
26 |
14 8
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) ) |
27 |
12 19 26
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) ) |
28 |
14 20 1 6
|
pmapmeet |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∩ ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) ) |
29 |
25 19 27 28
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∩ ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) ) |
30 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
31 |
30
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → 𝐾 ∈ AtLat ) |
32 |
21 6
|
pmap0 |
⊢ ( 𝐾 ∈ AtLat → ( ( pmap ‘ 𝐾 ) ‘ ( 0. ‘ 𝐾 ) ) = ∅ ) |
33 |
31 32
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( 0. ‘ 𝐾 ) ) = ∅ ) |
34 |
24 29 33
|
3eqtr3d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( ( pmap ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∩ ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ) = ∅ ) |
35 |
10 34
|
eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( 𝑃 ‘ ( 𝑃 ‘ 𝑋 ) ) ∩ ( 𝑃 ‘ 𝑋 ) ) = ∅ ) |
36 |
4 35
|
sseqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑋 ∩ ( 𝑃 ‘ 𝑋 ) ) ⊆ ∅ ) |
37 |
|
ss0b |
⊢ ( ( 𝑋 ∩ ( 𝑃 ‘ 𝑋 ) ) ⊆ ∅ ↔ ( 𝑋 ∩ ( 𝑃 ‘ 𝑋 ) ) = ∅ ) |
38 |
36 37
|
sylib |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑋 ∩ ( 𝑃 ‘ 𝑋 ) ) = ∅ ) |