Proof of Theorem 4atlem11a
Step | Hyp | Ref
| Expression |
1 | | simp11 1084 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝐾 ∈ HL) |
2 | | simp13 1086 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝑄 ∈ 𝐴) |
3 | | simp21 1087 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝑈 ∈ 𝐴) |
4 | | hllat 33668 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
5 | 1, 4 | syl 17 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝐾 ∈ Lat) |
6 | | simp12 1085 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝑃 ∈ 𝐴) |
7 | | simp22 1088 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝑉 ∈ 𝐴) |
8 | | eqid 2610 |
. . . . . 6
⊢
(Base‘𝐾) =
(Base‘𝐾) |
9 | | 4at.j |
. . . . . 6
⊢ ∨ =
(join‘𝐾) |
10 | | 4at.a |
. . . . . 6
⊢ 𝐴 = (Atoms‘𝐾) |
11 | 8, 9, 10 | hlatjcl 33671 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
12 | 1, 6, 7, 11 | syl3anc 1318 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
13 | | simp23 1089 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝑊 ∈ 𝐴) |
14 | 8, 10 | atbase 33594 |
. . . . 5
⊢ (𝑊 ∈ 𝐴 → 𝑊 ∈ (Base‘𝐾)) |
15 | 13, 14 | syl 17 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
16 | 8, 9 | latjcl 16874 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑉) ∨ 𝑊) ∈ (Base‘𝐾)) |
17 | 5, 12, 15, 16 | syl3anc 1318 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → ((𝑃 ∨ 𝑉) ∨ 𝑊) ∈ (Base‘𝐾)) |
18 | | simp3 1056 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) |
19 | | 4at.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
20 | 8, 19, 9, 10 | hlexchb2 33689 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑉) ∨ 𝑊) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → (𝑄 ≤ (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)) ↔ (𝑄 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)) = (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)))) |
21 | 1, 2, 3, 17, 18, 20 | syl131anc 1331 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → (𝑄 ≤ (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)) ↔ (𝑄 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)) = (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)))) |
22 | 19, 9, 10 | 4atlem4b 33904 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) → ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) = (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊))) |
23 | 1, 6, 3, 7, 13, 22 | syl32anc 1326 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) = (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊))) |
24 | 23 | breq2d 4595 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → (𝑄 ≤ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ↔ 𝑄 ≤ (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)))) |
25 | 19, 9, 10 | 4atlem4b 33904 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊))) |
26 | 1, 6, 2, 7, 13, 25 | syl32anc 1326 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊))) |
27 | 26, 23 | eqeq12d 2625 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → (((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) = ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ↔ (𝑄 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)) = (𝑈 ∨ ((𝑃 ∨ 𝑉) ∨ 𝑊)))) |
28 | 21, 24, 27 | 3bitr4d 299 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑊 ∈ 𝐴) ∧ ¬ 𝑄 ≤ ((𝑃 ∨ 𝑉) ∨ 𝑊)) → (𝑄 ≤ ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)) ↔ ((𝑃 ∨ 𝑄) ∨ (𝑉 ∨ 𝑊)) = ((𝑃 ∨ 𝑈) ∨ (𝑉 ∨ 𝑊)))) |