Step | Hyp | Ref
| Expression |
1 | | hlpos 36992 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
2 | | pmaple.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝐾) |
3 | | eqid 2738 |
. . . . . . . . . 10
⊢
(Atoms‘𝐾) =
(Atoms‘𝐾) |
4 | 2, 3 | atbase 36915 |
. . . . . . . . 9
⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵) |
5 | | pmaple.l |
. . . . . . . . . . . . . . 15
⊢ ≤ =
(le‘𝐾) |
6 | 2, 5 | postr 17672 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Poset ∧ (𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑝 ≤ 𝑌)) |
7 | 6 | exp4b 434 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ Poset → ((𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌)))) |
8 | 7 | 3expd 1354 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ Poset → (𝑝 ∈ 𝐵 → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌)))))) |
9 | 8 | com23 86 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ Poset → (𝑋 ∈ 𝐵 → (𝑝 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌)))))) |
10 | 9 | com34 91 |
. . . . . . . . . 10
⊢ (𝐾 ∈ Poset → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑝 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌)))))) |
11 | 10 | 3imp 1112 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ 𝐵 → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌)))) |
12 | 4, 11 | syl5 34 |
. . . . . . . 8
⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 ≤ 𝑋 → (𝑋 ≤ 𝑌 → 𝑝 ≤ 𝑌)))) |
13 | 12 | com34 91 |
. . . . . . 7
⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) → (𝑋 ≤ 𝑌 → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))) |
14 | 13 | com23 86 |
. . . . . 6
⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑝 ∈ (Atoms‘𝐾) → (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)))) |
15 | 14 | ralrimdv 3100 |
. . . . 5
⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))) |
16 | 1, 15 | syl3an1 1164 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))) |
17 | | ss2rab 3958 |
. . . 4
⊢ ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ↔ ∀𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌)) |
18 | 16, 17 | syl6ibr 255 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})) |
19 | | hlclat 36984 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) |
20 | | ssrab2 3967 |
. . . . . . . . 9
⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ (Atoms‘𝐾) |
21 | 2, 3 | atssbase 36916 |
. . . . . . . . 9
⊢
(Atoms‘𝐾)
⊆ 𝐵 |
22 | 20, 21 | sstri 3884 |
. . . . . . . 8
⊢ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ 𝐵 |
23 | | eqid 2738 |
. . . . . . . . 9
⊢
(lub‘𝐾) =
(lub‘𝐾) |
24 | 2, 5, 23 | lubss 17840 |
. . . . . . . 8
⊢ ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} ⊆ 𝐵 ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})) |
25 | 22, 24 | mp3an2 1450 |
. . . . . . 7
⊢ ((𝐾 ∈ CLat ∧ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})) |
26 | 25 | ex 416 |
. . . . . 6
⊢ (𝐾 ∈ CLat → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))) |
27 | 19, 26 | syl 17 |
. . . . 5
⊢ (𝐾 ∈ HL → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))) |
28 | 27 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}))) |
29 | | hlomcmat 36991 |
. . . . . . 7
⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
30 | 29 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
31 | | simp2 1138 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
32 | 2, 5, 23, 3 | atlatmstc 36945 |
. . . . . 6
⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) = 𝑋) |
33 | 30, 31, 32 | syl2anc 587 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) = 𝑋) |
34 | | simp3 1139 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) |
35 | 2, 5, 23, 3 | atlatmstc 36945 |
. . . . . 6
⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑌 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) = 𝑌) |
36 | 30, 34, 35 | syl2anc 587 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) = 𝑌) |
37 | 33, 36 | breq12d 5040 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) ≤ ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) ↔ 𝑋 ≤ 𝑌)) |
38 | 28, 37 | sylibd 242 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ({𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌} → 𝑋 ≤ 𝑌)) |
39 | 18, 38 | impbid 215 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})) |
40 | | pmaple.m |
. . . . 5
⊢ 𝑀 = (pmap‘𝐾) |
41 | 2, 5, 3, 40 | pmapval 37383 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) |
42 | 41 | 3adant3 1133 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋}) |
43 | 2, 5, 3, 40 | pmapval 37383 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) |
44 | 43 | 3adant2 1132 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘𝑌) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌}) |
45 | 42, 44 | sseq12d 3908 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑀‘𝑋) ⊆ (𝑀‘𝑌) ↔ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑋} ⊆ {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝 ≤ 𝑌})) |
46 | 39, 45 | bitr4d 285 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑀‘𝑋) ⊆ (𝑀‘𝑌))) |