Step |
Hyp |
Ref |
Expression |
1 |
|
polssat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
polssat.p |
⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 ) |
3 |
|
ssid |
⊢ 𝐴 ⊆ 𝐴 |
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 ∧ 𝐴 ⊆ 𝐴 ) → ( ⊥ ‘ 𝐴 ) = ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝐴 ) ) ) ) |
8 |
3 7
|
mpan2 |
⊢ ( 𝐾 ∈ HL → ( ⊥ ‘ 𝐴 ) = ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝐴 ) ) ) ) |
9 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
11 |
10 1
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ ( Base ‘ 𝐾 ) ) |
12 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
13 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
14 |
10 12 13
|
ople1 |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑝 ∈ ( Base ‘ 𝐾 ) ) → 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
15 |
9 11 14
|
syl2an |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑝 ∈ 𝐴 ) → 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
16 |
15
|
ralrimiva |
⊢ ( 𝐾 ∈ HL → ∀ 𝑝 ∈ 𝐴 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
17 |
|
rabid2 |
⊢ ( 𝐴 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) } ↔ ∀ 𝑝 ∈ 𝐴 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
18 |
16 17
|
sylibr |
⊢ ( 𝐾 ∈ HL → 𝐴 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) } ) |
19 |
18
|
fveq2d |
⊢ ( 𝐾 ∈ HL → ( ( lub ‘ 𝐾 ) ‘ 𝐴 ) = ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) } ) ) |
20 |
|
hlomcmat |
⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ) |
21 |
10 13
|
op1cl |
⊢ ( 𝐾 ∈ OP → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
22 |
9 21
|
syl |
⊢ ( 𝐾 ∈ HL → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
10 12 4 1
|
atlatmstc |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) } ) = ( 1. ‘ 𝐾 ) ) |
24 |
20 22 23
|
syl2anc |
⊢ ( 𝐾 ∈ HL → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ( le ‘ 𝐾 ) ( 1. ‘ 𝐾 ) } ) = ( 1. ‘ 𝐾 ) ) |
25 |
19 24
|
eqtr2d |
⊢ ( 𝐾 ∈ HL → ( 1. ‘ 𝐾 ) = ( ( lub ‘ 𝐾 ) ‘ 𝐴 ) ) |
26 |
25
|
fveq2d |
⊢ ( 𝐾 ∈ HL → ( ( oc ‘ 𝐾 ) ‘ ( 1. ‘ 𝐾 ) ) = ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝐴 ) ) ) |
27 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
28 |
27 13 5
|
opoc1 |
⊢ ( 𝐾 ∈ OP → ( ( oc ‘ 𝐾 ) ‘ ( 1. ‘ 𝐾 ) ) = ( 0. ‘ 𝐾 ) ) |
29 |
9 28
|
syl |
⊢ ( 𝐾 ∈ HL → ( ( oc ‘ 𝐾 ) ‘ ( 1. ‘ 𝐾 ) ) = ( 0. ‘ 𝐾 ) ) |
30 |
26 29
|
eqtr3d |
⊢ ( 𝐾 ∈ HL → ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝐴 ) ) = ( 0. ‘ 𝐾 ) ) |
31 |
30
|
fveq2d |
⊢ ( 𝐾 ∈ HL → ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝐴 ) ) ) = ( ( pmap ‘ 𝐾 ) ‘ ( 0. ‘ 𝐾 ) ) ) |
32 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
33 |
27 6
|
pmap0 |
⊢ ( 𝐾 ∈ AtLat → ( ( pmap ‘ 𝐾 ) ‘ ( 0. ‘ 𝐾 ) ) = ∅ ) |
34 |
32 33
|
syl |
⊢ ( 𝐾 ∈ HL → ( ( pmap ‘ 𝐾 ) ‘ ( 0. ‘ 𝐾 ) ) = ∅ ) |
35 |
8 31 34
|
3eqtrd |
⊢ ( 𝐾 ∈ HL → ( ⊥ ‘ 𝐴 ) = ∅ ) |