Proof of Theorem paddasslem13
Step | Hyp | Ref
| Expression |
1 | | simpl1l 1225 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝐾 ∈ HL) |
2 | | simpl21 1252 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑋 ⊆ 𝐴) |
3 | | simpl22 1253 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑌 ⊆ 𝐴) |
4 | | paddasslem.a |
. . . . 5
⊢ 𝐴 = (Atoms‘𝐾) |
5 | | paddasslem.p |
. . . . 5
⊢ + =
(+𝑃‘𝐾) |
6 | 4, 5 | paddssat 37448 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) |
7 | 1, 2, 3, 6 | syl3anc 1372 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑋 + 𝑌) ⊆ 𝐴) |
8 | | simpl23 1254 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑍 ⊆ 𝐴) |
9 | 4, 5 | sspadd1 37449 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍)) |
10 | 1, 7, 8, 9 | syl3anc 1372 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍)) |
11 | 1 | hllatd 36998 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝐾 ∈ Lat) |
12 | | simprll 779 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ∈ 𝑋) |
13 | | simprlr 780 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑦 ∈ 𝑌) |
14 | | simpl3l 1229 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ 𝐴) |
15 | | eqid 2738 |
. . . 4
⊢
(Base‘𝐾) =
(Base‘𝐾) |
16 | | paddasslem.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
17 | 15, 4 | atbase 36923 |
. . . . 5
⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
18 | 14, 17 | syl 17 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ (Base‘𝐾)) |
19 | 2, 12 | sseldd 3879 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ∈ 𝐴) |
20 | 15, 4 | atbase 36923 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (Base‘𝐾)) |
21 | 19, 20 | syl 17 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ∈ (Base‘𝐾)) |
22 | | simpl3r 1230 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑟 ∈ 𝐴) |
23 | 15, 4 | atbase 36923 |
. . . . . 6
⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾)) |
24 | 22, 23 | syl 17 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑟 ∈ (Base‘𝐾)) |
25 | | paddasslem.j |
. . . . . 6
⊢ ∨ =
(join‘𝐾) |
26 | 15, 25 | latjcl 17778 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑥 ∨ 𝑟) ∈ (Base‘𝐾)) |
27 | 11, 21, 24, 26 | syl3anc 1372 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑥 ∨ 𝑟) ∈ (Base‘𝐾)) |
28 | 3, 13 | sseldd 3879 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑦 ∈ 𝐴) |
29 | 15, 4 | atbase 36923 |
. . . . . 6
⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (Base‘𝐾)) |
30 | 28, 29 | syl 17 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑦 ∈ (Base‘𝐾)) |
31 | 15, 25 | latjcl 17778 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥 ∨ 𝑦) ∈ (Base‘𝐾)) |
32 | 11, 21, 30, 31 | syl3anc 1372 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑥 ∨ 𝑦) ∈ (Base‘𝐾)) |
33 | | simprrr 782 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ≤ (𝑥 ∨ 𝑟)) |
34 | 15, 16, 25 | latlej1 17787 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 ≤ (𝑥 ∨ 𝑦)) |
35 | 11, 21, 30, 34 | syl3anc 1372 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑥 ≤ (𝑥 ∨ 𝑦)) |
36 | | simprrl 781 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑟 ≤ (𝑥 ∨ 𝑦)) |
37 | 15, 16, 25 | latjle12 17789 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾) ∧ (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))) → ((𝑥 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑥 ∨ 𝑟) ≤ (𝑥 ∨ 𝑦))) |
38 | 11, 21, 24, 32, 37 | syl13anc 1373 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → ((𝑥 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑥 ∨ 𝑟) ≤ (𝑥 ∨ 𝑦))) |
39 | 35, 36, 38 | mpbi2and 712 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → (𝑥 ∨ 𝑟) ≤ (𝑥 ∨ 𝑦)) |
40 | 15, 16, 11, 18, 27, 32, 33, 39 | lattrd 17785 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ≤ (𝑥 ∨ 𝑦)) |
41 | 16, 25, 4, 5 | elpaddri 37436 |
. . 3
⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑝 ∈ 𝐴 ∧ 𝑝 ≤ (𝑥 ∨ 𝑦))) → 𝑝 ∈ (𝑋 + 𝑌)) |
42 | 11, 2, 3, 12, 13, 14, 40, 41 | syl322anc 1399 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ (𝑋 + 𝑌)) |
43 | 10, 42 | sseldd 3879 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) |