Step |
Hyp |
Ref |
Expression |
1 |
|
atlatle.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atlatle.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
atlatle.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
simpl13 |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ AtLat ) |
5 |
|
atlpos |
⊢ ( 𝐾 ∈ AtLat → 𝐾 ∈ Poset ) |
6 |
4 5
|
syl |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ Poset ) |
7 |
1 3
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 ) |
8 |
7
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑝 ∈ 𝐵 ) |
9 |
|
simpl2 |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 ) |
10 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 ) |
11 |
1 2
|
postr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) → 𝑝 ≤ 𝑌 ) ) |
12 |
6 8 9 10 11
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) → 𝑝 ≤ 𝑌 ) ) |
13 |
12
|
expcomd |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑌 → ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) ) ) |
14 |
13
|
ralrimdva |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) ) ) |
15 |
|
ss2rab |
⊢ ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ↔ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) ) |
16 |
|
simpl12 |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) → 𝐾 ∈ CLat ) |
17 |
|
ssrab2 |
⊢ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ⊆ 𝐴 |
18 |
1 3
|
atssbase |
⊢ 𝐴 ⊆ 𝐵 |
19 |
17 18
|
sstri |
⊢ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ⊆ 𝐵 |
20 |
|
eqid |
⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 ) |
21 |
1 2 20
|
lubss |
⊢ ( ( 𝐾 ∈ CLat ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ⊆ 𝐵 ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ) ≤ ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) ) |
22 |
19 21
|
mp3an2 |
⊢ ( ( 𝐾 ∈ CLat ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ) ≤ ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) ) |
23 |
16 22
|
sylancom |
⊢ ( ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ) ≤ ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) ) |
24 |
23
|
ex |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ) ≤ ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) ) ) |
25 |
1 2 20 3
|
atlatmstc |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ) = 𝑋 ) |
26 |
25
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ) = 𝑋 ) |
27 |
1 2 20 3
|
atlatmstc |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑌 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) = 𝑌 ) |
28 |
27
|
3adant2 |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) = 𝑌 ) |
29 |
26 28
|
breq12d |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ) ≤ ( ( lub ‘ 𝐾 ) ‘ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } ) ↔ 𝑋 ≤ 𝑌 ) ) |
30 |
24 29
|
sylibd |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑋 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ 𝑌 } → 𝑋 ≤ 𝑌 ) ) |
31 |
15 30
|
syl5bir |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) → 𝑋 ≤ 𝑌 ) ) |
32 |
14 31
|
impbid |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) ) ) |