Step | Hyp | Ref
| Expression |
1 | | cdleme24.b |
. . 3
⊢ 𝐵 = (Base‘𝐾) |
2 | | cdleme24.l |
. . 3
⊢ ≤ =
(le‘𝐾) |
3 | | cdleme24.j |
. . 3
⊢ ∨ =
(join‘𝐾) |
4 | | cdleme24.m |
. . 3
⊢ ∧ =
(meet‘𝐾) |
5 | | cdleme24.a |
. . 3
⊢ 𝐴 = (Atoms‘𝐾) |
6 | | cdleme24.h |
. . 3
⊢ 𝐻 = (LHyp‘𝐾) |
7 | | cdleme24.u |
. . 3
⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
8 | | cdleme24.f |
. . 3
⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) |
9 | | cdleme24.n |
. . 3
⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | cdleme25a 38104 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵)) |
11 | | eqid 2737 |
. . 3
⊢ ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
12 | | eqid 2737 |
. . 3
⊢ ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11,
12 | cdleme24 38103 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))))) |
14 | | breq1 5056 |
. . . . . 6
⊢ (𝑠 = 𝑡 → (𝑠 ≤ 𝑊 ↔ 𝑡 ≤ 𝑊)) |
15 | 14 | notbid 321 |
. . . . 5
⊢ (𝑠 = 𝑡 → (¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑡 ≤ 𝑊)) |
16 | | breq1 5056 |
. . . . . 6
⊢ (𝑠 = 𝑡 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑡 ≤ (𝑃 ∨ 𝑄))) |
17 | 16 | notbid 321 |
. . . . 5
⊢ (𝑠 = 𝑡 → (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) |
18 | 15, 17 | anbi12d 634 |
. . . 4
⊢ (𝑠 = 𝑡 → ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ↔ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))) |
19 | | oveq1 7220 |
. . . . . . . . 9
⊢ (𝑠 = 𝑡 → (𝑠 ∨ 𝑈) = (𝑡 ∨ 𝑈)) |
20 | | oveq2 7221 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑡 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑡)) |
21 | 20 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑡) ∧ 𝑊)) |
22 | 21 | oveq2d 7229 |
. . . . . . . . 9
⊢ (𝑠 = 𝑡 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
23 | 19, 22 | oveq12d 7231 |
. . . . . . . 8
⊢ (𝑠 = 𝑡 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))) |
24 | 8, 23 | syl5eq 2790 |
. . . . . . 7
⊢ (𝑠 = 𝑡 → 𝐹 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))) |
25 | | oveq2 7221 |
. . . . . . . 8
⊢ (𝑠 = 𝑡 → (𝑅 ∨ 𝑠) = (𝑅 ∨ 𝑡)) |
26 | 25 | oveq1d 7228 |
. . . . . . 7
⊢ (𝑠 = 𝑡 → ((𝑅 ∨ 𝑠) ∧ 𝑊) = ((𝑅 ∨ 𝑡) ∧ 𝑊)) |
27 | 24, 26 | oveq12d 7231 |
. . . . . 6
⊢ (𝑠 = 𝑡 → (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊)) = (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) |
28 | 27 | oveq2d 7229 |
. . . . 5
⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))) |
29 | 9, 28 | syl5eq 2790 |
. . . 4
⊢ (𝑠 = 𝑡 → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))) |
30 | 18, 29 | reusv3 5298 |
. . 3
⊢
(∃𝑠 ∈
𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵) → (∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))) ↔ ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))) |
31 | 30 | biimpd 232 |
. 2
⊢
(∃𝑠 ∈
𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵) → (∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))) → ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))) |
32 | 10, 13, 31 | sylc 65 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) |