Proof of Theorem 4atexlemcnd
Step | Hyp | Ref
| Expression |
1 | | 4thatlem.ph |
. . . 4
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) |
2 | | 4thatlem0.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
3 | | 4thatlem0.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
4 | | 4thatlem0.m |
. . . 4
⊢ ∧ =
(meet‘𝐾) |
5 | | 4thatlem0.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
6 | | 4thatlem0.h |
. . . 4
⊢ 𝐻 = (LHyp‘𝐾) |
7 | | 4thatlem0.u |
. . . 4
⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
8 | | 4thatlem0.v |
. . . 4
⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemtlw 36137 |
. . 3
⊢ (𝜑 → 𝑇 ≤ 𝑊) |
10 | | 4thatlem0.c |
. . . 4
⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 10 | 4atexlemnclw 36140 |
. . 3
⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊) |
12 | | nbrne2 4895 |
. . 3
⊢ ((𝑇 ≤ 𝑊 ∧ ¬ 𝐶 ≤ 𝑊) → 𝑇 ≠ 𝐶) |
13 | 9, 11, 12 | syl2anc 579 |
. 2
⊢ (𝜑 → 𝑇 ≠ 𝐶) |
14 | 1 | 4atexlemk 36117 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ HL) |
15 | 1 | 4atexlemq 36121 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ 𝐴) |
16 | 1 | 4atexlemt 36123 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ 𝐴) |
17 | 3, 5 | hlatjcom 35438 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄)) |
18 | 14, 15, 16, 17 | syl3anc 1494 |
. . . . . . . 8
⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄)) |
19 | | simp221 1417 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴) |
20 | 1, 19 | sylbi 209 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ 𝐴) |
21 | 3, 5 | hlatjcom 35438 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅)) |
22 | 14, 20, 16, 21 | syl3anc 1494 |
. . . . . . . 8
⊢ (𝜑 → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅)) |
23 | 18, 22 | oveq12d 6928 |
. . . . . . 7
⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) = ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅))) |
24 | 1 | 4atexlemkc 36128 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ CvLat) |
25 | 1 | 4atexlemp 36120 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
26 | 1 | 4atexlempnq 36125 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ≠ 𝑄) |
27 | | simp223 1419 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) |
28 | 1, 27 | sylbi 209 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) |
29 | 5, 3 | cvlsupr6 35417 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑄) |
30 | 29 | necomd 3054 |
. . . . . . . . 9
⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑄 ≠ 𝑅) |
31 | 24, 25, 15, 20, 26, 28, 30 | syl132anc 1511 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ≠ 𝑅) |
32 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemntlpq 36138 |
. . . . . . . . 9
⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) |
33 | 5, 3 | cvlsupr7 35418 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) |
34 | 24, 25, 15, 20, 26, 28, 33 | syl132anc 1511 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) |
35 | 3, 5 | hlatjcom 35438 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄)) |
36 | 14, 15, 20, 35 | syl3anc 1494 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄)) |
37 | 34, 36 | eqtr4d 2864 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅)) |
38 | 37 | breq2d 4887 |
. . . . . . . . 9
⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) ↔ 𝑇 ≤ (𝑄 ∨ 𝑅))) |
39 | 32, 38 | mtbid 316 |
. . . . . . . 8
⊢ (𝜑 → ¬ 𝑇 ≤ (𝑄 ∨ 𝑅)) |
40 | 2, 3, 4, 5 | 2llnma2 35859 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ (𝑄 ∨ 𝑅))) → ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇) |
41 | 14, 15, 20, 16, 31, 39, 40 | syl132anc 1511 |
. . . . . . 7
⊢ (𝜑 → ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇) |
42 | 23, 41 | eqtr2d 2862 |
. . . . . 6
⊢ (𝜑 → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))) |
43 | 42 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))) |
44 | 1 | 4atexlemkl 36127 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ Lat) |
45 | 1, 3, 5 | 4atexlemqtb 36131 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) |
46 | 1, 3, 5 | 4atexlempsb 36130 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
47 | | eqid 2825 |
. . . . . . . . . . 11
⊢
(Base‘𝐾) =
(Base‘𝐾) |
48 | 47, 2, 4 | latmle1 17436 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇)) |
49 | 44, 45, 46, 48 | syl3anc 1494 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇)) |
50 | 10, 49 | syl5eqbr 4910 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇)) |
51 | 50 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑄 ∨ 𝑇)) |
52 | | simpr 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷) |
53 | | 4thatlem0.d |
. . . . . . . . . 10
⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) |
54 | 47, 3, 5 | hlatjcl 35437 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾)) |
55 | 14, 20, 16, 54 | syl3anc 1494 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾)) |
56 | 47, 2, 4 | latmle1 17436 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇)) |
57 | 44, 55, 46, 56 | syl3anc 1494 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇)) |
58 | 53, 57 | syl5eqbr 4910 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ≤ (𝑅 ∨ 𝑇)) |
59 | 58 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ≤ (𝑅 ∨ 𝑇)) |
60 | 52, 59 | eqbrtrd 4897 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑅 ∨ 𝑇)) |
61 | 1, 2, 3, 4, 5, 6, 7, 8, 10 | 4atexlemc 36139 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ 𝐴) |
62 | 47, 5 | atbase 35359 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (Base‘𝐾)) |
63 | 61, 62 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
64 | 47, 2, 4 | latlem12 17438 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))) → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))) |
65 | 44, 63, 45, 55, 64 | syl13anc 1495 |
. . . . . . . 8
⊢ (𝜑 → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))) |
66 | 65 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))) |
67 | 51, 60, 66 | mpbi2and 703 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))) |
68 | | hlatl 35430 |
. . . . . . . . 9
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
69 | 14, 68 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ AtLat) |
70 | 42, 16 | eqeltrrd 2907 |
. . . . . . . 8
⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴) |
71 | 2, 5 | atcmp 35381 |
. . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝐶 ∈ 𝐴 ∧ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴) → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))) |
72 | 69, 61, 70, 71 | syl3anc 1494 |
. . . . . . 7
⊢ (𝜑 → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))) |
73 | 72 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))) |
74 | 67, 73 | mpbid 224 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))) |
75 | 43, 74 | eqtr4d 2864 |
. . . 4
⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = 𝐶) |
76 | 75 | ex 403 |
. . 3
⊢ (𝜑 → (𝐶 = 𝐷 → 𝑇 = 𝐶)) |
77 | 76 | necon3d 3020 |
. 2
⊢ (𝜑 → (𝑇 ≠ 𝐶 → 𝐶 ≠ 𝐷)) |
78 | 13, 77 | mpd 15 |
1
⊢ (𝜑 → 𝐶 ≠ 𝐷) |