Proof of Theorem cdlemn2
Step | Hyp | Ref
| Expression |
1 | | simp1 1061 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | | simp21 1094 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
3 | | simp22 1095 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) |
4 | | cdlemn2.l |
. . . . . . 7
⊢ ≤ =
(le‘𝐾) |
5 | | cdlemn2.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
6 | | cdlemn2.h |
. . . . . . 7
⊢ 𝐻 = (LHyp‘𝐾) |
7 | | cdlemn2.t |
. . . . . . 7
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
8 | | cdlemn2.f |
. . . . . . 7
⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑆) |
9 | 4, 5, 6, 7, 8 | ltrniotacl 35867 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
10 | 1, 2, 3, 9 | syl3anc 1326 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝐹 ∈ 𝑇) |
11 | | cdlemn2.j |
. . . . . 6
⊢ ∨ =
(join‘𝐾) |
12 | | eqid 2622 |
. . . . . 6
⊢
(meet‘𝐾) =
(meet‘𝐾) |
13 | | cdlemn2.r |
. . . . . 6
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
14 | 4, 11, 12, 5, 6, 7,
13 | trlval2 35450 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑄 ∨ (𝐹‘𝑄))(meet‘𝐾)𝑊)) |
15 | 1, 10, 2, 14 | syl3anc 1326 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) = ((𝑄 ∨ (𝐹‘𝑄))(meet‘𝐾)𝑊)) |
16 | 4, 5, 6, 7, 8 | ltrniotaval 35869 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → (𝐹‘𝑄) = 𝑆) |
17 | 1, 2, 3, 16 | syl3anc 1326 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝐹‘𝑄) = 𝑆) |
18 | 17 | oveq2d 6666 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ (𝐹‘𝑄)) = (𝑄 ∨ 𝑆)) |
19 | 18 | oveq1d 6665 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ (𝐹‘𝑄))(meet‘𝐾)𝑊) = ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊)) |
20 | 15, 19 | eqtrd 2656 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) = ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊)) |
21 | | simp1l 1085 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝐾 ∈ HL) |
22 | | hllat 34650 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
23 | 21, 22 | syl 17 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝐾 ∈ Lat) |
24 | | simp21l 1178 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑄 ∈ 𝐴) |
25 | | cdlemn2.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐾) |
26 | 25, 5 | atbase 34576 |
. . . . . . 7
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
27 | 24, 26 | syl 17 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑄 ∈ 𝐵) |
28 | | simp23l 1182 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑋 ∈ 𝐵) |
29 | 25, 4, 11 | latlej1 17060 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑄 ≤ (𝑄 ∨ 𝑋)) |
30 | 23, 27, 28, 29 | syl3anc 1326 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑄 ≤ (𝑄 ∨ 𝑋)) |
31 | | simp3 1063 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑆 ≤ (𝑄 ∨ 𝑋)) |
32 | | simp22l 1180 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑆 ∈ 𝐴) |
33 | 25, 5 | atbase 34576 |
. . . . . . 7
⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐵) |
34 | 32, 33 | syl 17 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑆 ∈ 𝐵) |
35 | 25, 11 | latjcl 17051 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) ∈ 𝐵) |
36 | 23, 27, 28, 35 | syl3anc 1326 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ 𝑋) ∈ 𝐵) |
37 | 25, 4, 11 | latjle12 17062 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵 ∧ (𝑄 ∨ 𝑋) ∈ 𝐵)) → ((𝑄 ≤ (𝑄 ∨ 𝑋) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) ↔ (𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋))) |
38 | 23, 27, 34, 36, 37 | syl13anc 1328 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ≤ (𝑄 ∨ 𝑋) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) ↔ (𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋))) |
39 | 30, 31, 38 | mpbi2and 956 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋)) |
40 | 25, 11, 5 | hlatjcl 34653 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑄 ∨ 𝑆) ∈ 𝐵) |
41 | 21, 24, 32, 40 | syl3anc 1326 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑄 ∨ 𝑆) ∈ 𝐵) |
42 | | simp1r 1086 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑊 ∈ 𝐻) |
43 | 25, 6 | lhpbase 35284 |
. . . . . 6
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
44 | 42, 43 | syl 17 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → 𝑊 ∈ 𝐵) |
45 | 25, 4, 12 | latmlem1 17081 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ ((𝑄 ∨ 𝑆) ∈ 𝐵 ∧ (𝑄 ∨ 𝑋) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋) → ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊))) |
46 | 23, 41, 36, 44, 45 | syl13anc 1328 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ 𝑆) ≤ (𝑄 ∨ 𝑋) → ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊))) |
47 | 39, 46 | mpd 15 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊)) |
48 | 20, 47 | eqbrtrd 4675 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ≤ ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊)) |
49 | | simp23 1096 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) |
50 | 25, 4, 11, 12, 5, 6 | lhple 35328 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊) = 𝑋) |
51 | 1, 2, 49, 50 | syl3anc 1326 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → ((𝑄 ∨ 𝑋)(meet‘𝐾)𝑊) = 𝑋) |
52 | 48, 51 | breqtrd 4679 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ 𝑆 ≤ (𝑄 ∨ 𝑋)) → (𝑅‘𝐹) ≤ 𝑋) |