Proof of Theorem dalem21
| Step | Hyp | Ref
| Expression |
| 1 | | dalem.ph |
. . . 4
⊢ (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))))) |
| 2 | 1 | dalemkehl 39625 |
. . 3
⊢ (𝜑 → 𝐾 ∈ HL) |
| 3 | 2 | 3ad2ant1 1134 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
| 4 | | dalem.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
| 5 | | dalem.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
| 6 | | dalem.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
| 7 | | dalem.ps |
. . . 4
⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| 8 | 1, 4, 5, 6, 7 | dalemcjden 39694 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾)) |
| 9 | 8 | 3adant2 1132 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾)) |
| 10 | | dalem21.o |
. . . 4
⊢ 𝑂 = (LPlanes‘𝐾) |
| 11 | | dalem21.y |
. . . 4
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 12 | 1, 4, 5, 6, 10, 11 | dalempjsen 39655 |
. . 3
⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) |
| 13 | 12 | 3ad2ant1 1134 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) |
| 14 | 1, 4, 5, 6, 10, 11 | dalemply 39656 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ≤ 𝑌) |
| 15 | 14 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑃 ≤ 𝑌) |
| 16 | | dalem21.z |
. . . . . . 7
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| 17 | 1, 4, 5, 6, 16 | dalemsly 39657 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌) |
| 18 | 1 | dalemkelat 39626 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 19 | 1, 6 | dalempeb 39641 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
| 20 | 1, 6 | dalemseb 39644 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
| 21 | 1, 10 | dalemyeb 39651 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
| 22 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 23 | 22, 4, 5 | latjle12 18495 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌)) |
| 24 | 18, 19, 20, 21, 23 | syl13anc 1374 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌)) |
| 25 | 24 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌)) |
| 26 | 15, 17, 25 | mpbi2and 712 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑃 ∨ 𝑆) ≤ 𝑌) |
| 27 | 26 | 3adant3 1133 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≤ 𝑌) |
| 28 | 7 | dalem-ccly 39687 |
. . . . . . 7
⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌) |
| 29 | 28 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌) |
| 30 | 18 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Lat) |
| 31 | 7, 6 | dalemcceb 39691 |
. . . . . . . . 9
⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾)) |
| 32 | 31 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾)) |
| 33 | 7 | dalemddea 39686 |
. . . . . . . . . 10
⊢ (𝜓 → 𝑑 ∈ 𝐴) |
| 34 | 22, 6 | atbase 39290 |
. . . . . . . . . 10
⊢ (𝑑 ∈ 𝐴 → 𝑑 ∈ (Base‘𝐾)) |
| 35 | 33, 34 | syl 17 |
. . . . . . . . 9
⊢ (𝜓 → 𝑑 ∈ (Base‘𝐾)) |
| 36 | 35 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝑑 ∈ (Base‘𝐾)) |
| 37 | 22, 4, 5 | latlej1 18493 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 ≤ (𝑐 ∨ 𝑑)) |
| 38 | 30, 32, 36, 37 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → 𝑐 ≤ (𝑐 ∨ 𝑑)) |
| 39 | | eqid 2737 |
. . . . . . . . . 10
⊢
(LLines‘𝐾) =
(LLines‘𝐾) |
| 40 | 22, 39 | llnbase 39511 |
. . . . . . . . 9
⊢ ((𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾)) |
| 41 | 8, 40 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾)) |
| 42 | 21 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾)) |
| 43 | 22, 4 | lattr 18489 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑐 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ 𝑌) → 𝑐 ≤ 𝑌)) |
| 44 | 30, 32, 41, 42, 43 | syl13anc 1374 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ 𝑌) → 𝑐 ≤ 𝑌)) |
| 45 | 38, 44 | mpand 695 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ≤ 𝑌 → 𝑐 ≤ 𝑌)) |
| 46 | 29, 45 | mtod 198 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) |
| 47 | 46 | 3adant2 1132 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) |
| 48 | | nbrne2 5163 |
. . . 4
⊢ (((𝑃 ∨ 𝑆) ≤ 𝑌 ∧ ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑)) |
| 49 | 27, 47, 48 | syl2anc 584 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑)) |
| 50 | 49 | necomd 2996 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆)) |
| 51 | | hlatl 39361 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 52 | 2, 51 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ AtLat) |
| 53 | 52 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ AtLat) |
| 54 | 1 | dalempea 39628 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
| 55 | 1 | dalemsea 39631 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ 𝐴) |
| 56 | 22, 5, 6 | hlatjcl 39368 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 57 | 2, 54, 55, 56 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 58 | 57 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 59 | | dalem21.m |
. . . . . 6
⊢ ∧ =
(meet‘𝐾) |
| 60 | 22, 59 | latmcl 18485 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾)) |
| 61 | 30, 41, 58, 60 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾)) |
| 62 | 1, 4, 5, 6, 10, 11 | dalemcea 39662 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ 𝐴) |
| 63 | 62 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝐴) |
| 64 | 7 | dalemclccjdd 39690 |
. . . . . 6
⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
| 65 | 64 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
| 66 | 1 | dalemclpjs 39636 |
. . . . . 6
⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆)) |
| 67 | 66 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑃 ∨ 𝑆)) |
| 68 | 1, 6 | dalemceb 39640 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
| 69 | 68 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾)) |
| 70 | 22, 4, 59 | latlem12 18511 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)) ↔ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)))) |
| 71 | 30, 69, 41, 58, 70 | syl13anc 1374 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)) ↔ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)))) |
| 72 | 65, 67, 71 | mpbi2and 712 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))) |
| 73 | | eqid 2737 |
. . . . 5
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 74 | 22, 4, 73, 6 | atlen0 39311 |
. . . 4
⊢ (((𝐾 ∈ AtLat ∧ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾) ∧ 𝐶 ∈ 𝐴) ∧ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾)) |
| 75 | 53, 61, 63, 72, 74 | syl31anc 1375 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾)) |
| 76 | 75 | 3adant2 1132 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾)) |
| 77 | 59, 73, 6, 39 | 2llnmat 39526 |
. 2
⊢ (((𝐾 ∈ HL ∧ (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) ∧ ((𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆) ∧ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴) |
| 78 | 3, 9, 13, 50, 76, 77 | syl32anc 1380 |
1
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴) |