Proof of Theorem dalem24
| Step | Hyp | Ref
| Expression |
| 1 | | dalem23.g |
. . . . 5
⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
| 2 | 1 | oveq1i 7441 |
. . . 4
⊢ (𝐺 ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∧ 𝑌) |
| 3 | | dalem.ph |
. . . . . . . 8
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) |
| 4 | 3 | dalemkehl 39625 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ HL) |
| 5 | | hlol 39362 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
| 6 | 4, 5 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ OL) |
| 7 | 6 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ OL) |
| 8 | 4 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
| 9 | | dalem.ps |
. . . . . . . 8
⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| 10 | 9 | dalemccea 39685 |
. . . . . . 7
⊢ (𝜓 → 𝑐 ∈ 𝐴) |
| 11 | 10 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴) |
| 12 | 3 | dalempea 39628 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
| 13 | 12 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴) |
| 14 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 15 | | dalem.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
| 16 | | dalem.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
| 17 | 14, 15, 16 | hlatjcl 39368 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾)) |
| 18 | 8, 11, 13, 17 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾)) |
| 19 | 9 | dalemddea 39686 |
. . . . . . 7
⊢ (𝜓 → 𝑑 ∈ 𝐴) |
| 20 | 19 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴) |
| 21 | 3 | dalemsea 39631 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ 𝐴) |
| 22 | 21 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴) |
| 23 | 14, 15, 16 | hlatjcl 39368 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 24 | 8, 20, 22, 23 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 25 | | dalem23.o |
. . . . . . 7
⊢ 𝑂 = (LPlanes‘𝐾) |
| 26 | 3, 25 | dalemyeb 39651 |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
| 27 | 26 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾)) |
| 28 | | dalem23.m |
. . . . . 6
⊢ ∧ =
(meet‘𝐾) |
| 29 | 14, 28 | latmmdir 39236 |
. . . . 5
⊢ ((𝐾 ∈ OL ∧ ((𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌))) |
| 30 | 7, 18, 24, 27, 29 | syl13anc 1374 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌))) |
| 31 | 2, 30 | eqtrid 2789 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌))) |
| 32 | 15, 16 | hlatjcom 39369 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) = (𝑃 ∨ 𝑐)) |
| 33 | 8, 11, 13, 32 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) = (𝑃 ∨ 𝑐)) |
| 34 | 33 | oveq1d 7446 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ 𝑌) = ((𝑃 ∨ 𝑐) ∧ 𝑌)) |
| 35 | | dalem.l |
. . . . . . . 8
⊢ ≤ =
(le‘𝐾) |
| 36 | | dalem23.y |
. . . . . . . 8
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 37 | 3, 35, 15, 16, 25, 36 | dalemply 39656 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ≤ 𝑌) |
| 38 | 37 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ 𝑌) |
| 39 | 9 | dalem-ccly 39687 |
. . . . . . 7
⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌) |
| 40 | 39 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌) |
| 41 | 14, 35, 15, 28, 16 | 2atjm 39447 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑃 ≤ 𝑌 ∧ ¬ 𝑐 ≤ 𝑌)) → ((𝑃 ∨ 𝑐) ∧ 𝑌) = 𝑃) |
| 42 | 8, 13, 11, 27, 38, 40, 41 | syl132anc 1390 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑐) ∧ 𝑌) = 𝑃) |
| 43 | 34, 42 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ 𝑌) = 𝑃) |
| 44 | 15, 16 | hlatjcom 39369 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑)) |
| 45 | 8, 20, 22, 44 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑)) |
| 46 | 45 | oveq1d 7446 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∨ 𝑆) ∧ 𝑌) = ((𝑆 ∨ 𝑑) ∧ 𝑌)) |
| 47 | | dalem23.z |
. . . . . . . 8
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| 48 | 3, 35, 15, 16, 47 | dalemsly 39657 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌) |
| 49 | 48 | 3adant3 1133 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ≤ 𝑌) |
| 50 | 9 | dalem-ddly 39688 |
. . . . . . 7
⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌) |
| 51 | 50 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑌) |
| 52 | 14, 35, 15, 28, 16 | 2atjm 39447 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 ≤ 𝑌 ∧ ¬ 𝑑 ≤ 𝑌)) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆) |
| 53 | 8, 22, 20, 27, 49, 51, 52 | syl132anc 1390 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆) |
| 54 | 46, 53 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∨ 𝑆) ∧ 𝑌) = 𝑆) |
| 55 | 43, 54 | oveq12d 7449 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌)) = (𝑃 ∧ 𝑆)) |
| 56 | 3, 35, 15, 16, 25, 36 | dalempnes 39653 |
. . . . 5
⊢ (𝜑 → 𝑃 ≠ 𝑆) |
| 57 | | hlatl 39361 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 58 | 4, 57 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ AtLat) |
| 59 | | eqid 2737 |
. . . . . . 7
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 60 | 28, 59, 16 | atnem0 39319 |
. . . . . 6
⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ≠ 𝑆 ↔ (𝑃 ∧ 𝑆) = (0.‘𝐾))) |
| 61 | 58, 12, 21, 60 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → (𝑃 ≠ 𝑆 ↔ (𝑃 ∧ 𝑆) = (0.‘𝐾))) |
| 62 | 56, 61 | mpbid 232 |
. . . 4
⊢ (𝜑 → (𝑃 ∧ 𝑆) = (0.‘𝐾)) |
| 63 | 62 | 3ad2ant1 1134 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∧ 𝑆) = (0.‘𝐾)) |
| 64 | 31, 55, 63 | 3eqtrd 2781 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∧ 𝑌) = (0.‘𝐾)) |
| 65 | 58 | 3ad2ant1 1134 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ AtLat) |
| 66 | 3, 35, 15, 16, 9, 28, 25, 36, 47, 1 | dalem23 39698 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
| 67 | 14, 35, 28, 59, 16 | atnle 39318 |
. . 3
⊢ ((𝐾 ∈ AtLat ∧ 𝐺 ∈ 𝐴 ∧ 𝑌 ∈ (Base‘𝐾)) → (¬ 𝐺 ≤ 𝑌 ↔ (𝐺 ∧ 𝑌) = (0.‘𝐾))) |
| 68 | 65, 66, 27, 67 | syl3anc 1373 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (¬ 𝐺 ≤ 𝑌 ↔ (𝐺 ∧ 𝑌) = (0.‘𝐾))) |
| 69 | 64, 68 | mpbird 257 |
1
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌) |