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 37633 |
. . 3
⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 2 | 3ad2ant1 1132 |
. 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 37702 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾)) |
9 | 8 | 3adant2 1130 |
. 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 37663 |
. . 3
⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) |
13 | 12 | 3ad2ant1 1132 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) |
14 | 1, 4, 5, 6, 10, 11 | dalemply 37664 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ≤ 𝑌) |
15 | 14 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑃 ≤ 𝑌) |
16 | | dalem21.z |
. . . . . . 7
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
17 | 1, 4, 5, 6, 16 | dalemsly 37665 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌) |
18 | 1 | dalemkelat 37634 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ Lat) |
19 | 1, 6 | dalempeb 37649 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
20 | 1, 6 | dalemseb 37652 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
21 | 1, 10 | dalemyeb 37659 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
22 | | eqid 2740 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
23 | 22, 4, 5 | latjle12 18166 |
. . . . . . . 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 1371 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌)) |
25 | 24 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌)) |
26 | 15, 17, 25 | mpbi2and 709 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑃 ∨ 𝑆) ≤ 𝑌) |
27 | 26 | 3adant3 1131 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≤ 𝑌) |
28 | 7 | dalem-ccly 37695 |
. . . . . . 7
⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌) |
29 | 28 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌) |
30 | 18 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Lat) |
31 | 7, 6 | dalemcceb 37699 |
. . . . . . . . 9
⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾)) |
32 | 31 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾)) |
33 | 7 | dalemddea 37694 |
. . . . . . . . . 10
⊢ (𝜓 → 𝑑 ∈ 𝐴) |
34 | 22, 6 | atbase 37299 |
. . . . . . . . . 10
⊢ (𝑑 ∈ 𝐴 → 𝑑 ∈ (Base‘𝐾)) |
35 | 33, 34 | syl 17 |
. . . . . . . . 9
⊢ (𝜓 → 𝑑 ∈ (Base‘𝐾)) |
36 | 35 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝑑 ∈ (Base‘𝐾)) |
37 | 22, 4, 5 | latlej1 18164 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 ≤ (𝑐 ∨ 𝑑)) |
38 | 30, 32, 36, 37 | syl3anc 1370 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → 𝑐 ≤ (𝑐 ∨ 𝑑)) |
39 | | eqid 2740 |
. . . . . . . . . 10
⊢
(LLines‘𝐾) =
(LLines‘𝐾) |
40 | 22, 39 | llnbase 37519 |
. . . . . . . . 9
⊢ ((𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾)) |
41 | 8, 40 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾)) |
42 | 21 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾)) |
43 | 22, 4 | lattr 18160 |
. . . . . . . 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 1371 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ 𝑌) → 𝑐 ≤ 𝑌)) |
45 | 38, 44 | mpand 692 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ≤ 𝑌 → 𝑐 ≤ 𝑌)) |
46 | 29, 45 | mtod 197 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) |
47 | 46 | 3adant2 1130 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) |
48 | | nbrne2 5099 |
. . . 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 3001 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆)) |
51 | | hlatl 37370 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
52 | 2, 51 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ AtLat) |
53 | 52 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ AtLat) |
54 | 1 | dalempea 37636 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
55 | 1 | dalemsea 37639 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ 𝐴) |
56 | 22, 5, 6 | hlatjcl 37377 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
57 | 2, 54, 55, 56 | syl3anc 1370 |
. . . . . 6
⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
58 | 57 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
59 | | dalem21.m |
. . . . . 6
⊢ ∧ =
(meet‘𝐾) |
60 | 22, 59 | latmcl 18156 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾)) |
61 | 30, 41, 58, 60 | syl3anc 1370 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾)) |
62 | 1, 4, 5, 6, 10, 11 | dalemcea 37670 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ 𝐴) |
63 | 62 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝐴) |
64 | 7 | dalemclccjdd 37698 |
. . . . . 6
⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
65 | 64 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
66 | 1 | dalemclpjs 37644 |
. . . . . 6
⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆)) |
67 | 66 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑃 ∨ 𝑆)) |
68 | 1, 6 | dalemceb 37648 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
69 | 68 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾)) |
70 | 22, 4, 59 | latlem12 18182 |
. . . . . 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 1371 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)) ↔ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)))) |
72 | 65, 67, 71 | mpbi2and 709 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))) |
73 | | eqid 2740 |
. . . . 5
⊢
(0.‘𝐾) =
(0.‘𝐾) |
74 | 22, 4, 73, 6 | atlen0 37320 |
. . . 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 1372 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾)) |
76 | 75 | 3adant2 1130 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾)) |
77 | 59, 73, 6, 39 | 2llnmat 37534 |
. 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 1377 |
1
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴) |