Proof of Theorem dia2dimlem2
Step | Hyp | Ref
| Expression |
1 | | dia2dimlem2.k |
. . . . . . . . 9
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | 1 | simpld 474 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ HL) |
3 | | hllat 33668 |
. . . . . . . 8
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
4 | 2, 3 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ Lat) |
5 | | dia2dimlem2.p |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
6 | 5 | simpld 474 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
7 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
8 | | dia2dimlem2.a |
. . . . . . . . 9
⊢ 𝐴 = (Atoms‘𝐾) |
9 | 7, 8 | atbase 33594 |
. . . . . . . 8
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
10 | 6, 9 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
11 | | dia2dimlem2.u |
. . . . . . . . 9
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
12 | 11 | simpld 474 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
13 | 7, 8 | atbase 33594 |
. . . . . . . 8
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) |
14 | 12, 13 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
15 | | dia2dimlem2.l |
. . . . . . . 8
⊢ ≤ =
(le‘𝐾) |
16 | | dia2dimlem2.j |
. . . . . . . 8
⊢ ∨ =
(join‘𝐾) |
17 | 7, 15, 16 | latlej2 16884 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ≤ (𝑃 ∨ 𝑈)) |
18 | 4, 10, 14, 17 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑈)) |
19 | 7, 16, 8 | hlatjcl 33671 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
20 | 2, 6, 12, 19 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
21 | | dia2dimlem2.m |
. . . . . . . 8
⊢ ∧ =
(meet‘𝐾) |
22 | 7, 15, 21 | latleeqm2 16903 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑈 ≤ (𝑃 ∨ 𝑈) ↔ ((𝑃 ∨ 𝑈) ∧ 𝑈) = 𝑈)) |
23 | 4, 14, 20, 22 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (𝑈 ≤ (𝑃 ∨ 𝑈) ↔ ((𝑃 ∨ 𝑈) ∧ 𝑈) = 𝑈)) |
24 | 18, 23 | mpbid 221 |
. . . . 5
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑈) = 𝑈) |
25 | | dia2dimlem2.rf |
. . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
26 | | dia2dimlem2.f |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) |
27 | | dia2dimlem2.h |
. . . . . . . . . . 11
⊢ 𝐻 = (LHyp‘𝐾) |
28 | | dia2dimlem2.t |
. . . . . . . . . . 11
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
29 | | dia2dimlem2.r |
. . . . . . . . . . 11
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
30 | 15, 8, 27, 28, 29 | trlat 34474 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
31 | 1, 5, 26, 30 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) |
32 | | dia2dimlem2.v |
. . . . . . . . . 10
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
33 | 32 | simpld 474 |
. . . . . . . . 9
⊢ (𝜑 → 𝑉 ∈ 𝐴) |
34 | | dia2dimlem2.rv |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) |
35 | 15, 16, 8 | hlatexch2 33700 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑉) → ((𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉) → 𝑈 ≤ ((𝑅‘𝐹) ∨ 𝑉))) |
36 | 2, 31, 12, 33, 34, 35 | syl131anc 1331 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉) → 𝑈 ≤ ((𝑅‘𝐹) ∨ 𝑉))) |
37 | 25, 36 | mpd 15 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ≤ ((𝑅‘𝐹) ∨ 𝑉)) |
38 | 26 | simpld 474 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ 𝑇) |
39 | 15, 16, 21, 8, 27, 28, 29 | trlval2 34468 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
40 | 1, 38, 5, 39 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
41 | 40 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑉) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉)) |
42 | 15, 8, 27, 28 | ltrnel 34443 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
43 | 1, 38, 5, 42 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
44 | 43 | simpld 474 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
45 | 7, 16, 8 | hlatjcl 33671 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
46 | 2, 6, 44, 45 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
47 | 1 | simprd 478 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
48 | 7, 27 | lhpbase 34302 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
49 | 47, 48 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
50 | 32 | simprd 478 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ≤ 𝑊) |
51 | 7, 15, 16, 21, 8 | atmod4i1 34170 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 ≤ 𝑊) → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) ∧ 𝑊)) |
52 | 2, 33, 46, 49, 50, 51 | syl131anc 1331 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) ∧ 𝑊)) |
53 | 16, 8 | hlatjass 33674 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) = (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) |
54 | 2, 6, 44, 33, 53 | syl13anc 1320 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) = (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) |
55 | 54 | oveq1d 6564 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) ∧ 𝑊) = ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
56 | 52, 55 | eqtrd 2644 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉) = ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
57 | 41, 56 | eqtrd 2644 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑉) = ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
58 | 37, 57 | breqtrd 4609 |
. . . . . 6
⊢ (𝜑 → 𝑈 ≤ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
59 | 7, 16, 8 | hlatjcl 33671 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
60 | 2, 44, 33, 59 | syl3anc 1318 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
61 | 7, 16 | latjcl 16874 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾)) |
62 | 4, 10, 60, 61 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾)) |
63 | 7, 21 | latmcl 16875 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) ∈ (Base‘𝐾)) |
64 | 4, 62, 49, 63 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) ∈ (Base‘𝐾)) |
65 | 7, 15, 21 | latmlem2 16905 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))) → (𝑈 ≤ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑈) ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)))) |
66 | 4, 14, 64, 20, 65 | syl13anc 1320 |
. . . . . 6
⊢ (𝜑 → (𝑈 ≤ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑈) ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)))) |
67 | 58, 66 | mpd 15 |
. . . . 5
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑈) ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
68 | 24, 67 | eqbrtrrd 4607 |
. . . 4
⊢ (𝜑 → 𝑈 ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
69 | | dia2dimlem2.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ 𝑇) |
70 | 15, 16, 21, 8, 27, 28, 29 | trlval2 34468 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) |
71 | 1, 69, 5, 70 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) |
72 | | dia2dimlem2.gv |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺‘𝑃) = 𝑄) |
73 | | dia2dimlem2.q |
. . . . . . . . . 10
⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) |
74 | 72, 73 | syl6eq 2660 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘𝑃) = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
75 | 74 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ (𝐺‘𝑃)) = (𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))) |
76 | 75 | oveq1d 6564 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊)) |
77 | 15, 16, 8 | hlatlej1 33679 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑈)) |
78 | 2, 6, 12, 77 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑈)) |
79 | 7, 15, 16, 21, 8 | atmod3i1 34168 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ 𝑈)) → (𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) = ((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)))) |
80 | 2, 6, 20, 60, 78, 79 | syl131anc 1331 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) = ((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)))) |
81 | 80 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = (((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊)) |
82 | | hlol 33666 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
83 | 2, 82 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ OL) |
84 | 7, 21 | latmassOLD 33534 |
. . . . . . . . 9
⊢ ((𝐾 ∈ OL ∧ ((𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
85 | 83, 20, 62, 49, 84 | syl13anc 1320 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
86 | 81, 85 | eqtrd 2644 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
87 | 76, 86 | eqtrd 2644 |
. . . . . 6
⊢ (𝜑 → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
88 | 71, 87 | eqtrd 2644 |
. . . . 5
⊢ (𝜑 → (𝑅‘𝐺) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
89 | 88 | eqcomd 2616 |
. . . 4
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) = (𝑅‘𝐺)) |
90 | 68, 89 | breqtrd 4609 |
. . 3
⊢ (𝜑 → 𝑈 ≤ (𝑅‘𝐺)) |
91 | | hlatl 33665 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
92 | 2, 91 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ AtLat) |
93 | | hlop 33667 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
94 | 2, 93 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ OP) |
95 | | eqid 2610 |
. . . . . . . . . 10
⊢
(0.‘𝐾) =
(0.‘𝐾) |
96 | | eqid 2610 |
. . . . . . . . . 10
⊢
(lt‘𝐾) =
(lt‘𝐾) |
97 | 95, 96, 8 | 0ltat 33596 |
. . . . . . . . 9
⊢ ((𝐾 ∈ OP ∧ 𝑈 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑈) |
98 | 94, 12, 97 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑈) |
99 | | hlpos 33670 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
100 | 2, 99 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Poset) |
101 | 7, 95 | op0cl 33489 |
. . . . . . . . . 10
⊢ (𝐾 ∈ OP →
(0.‘𝐾) ∈
(Base‘𝐾)) |
102 | 94, 101 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾)) |
103 | 7, 27, 28, 29 | trlcl 34469 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾)) |
104 | 1, 69, 103 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐺) ∈ (Base‘𝐾)) |
105 | 7, 15, 96 | pltletr 16794 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Poset ∧
((0.‘𝐾) ∈
(Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝑈 ∧ 𝑈 ≤ (𝑅‘𝐺)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺))) |
106 | 100, 102,
14, 104, 105 | syl13anc 1320 |
. . . . . . . 8
⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑈 ∧ 𝑈 ≤ (𝑅‘𝐺)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺))) |
107 | 98, 90, 106 | mp2and 711 |
. . . . . . 7
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺)) |
108 | 7, 96, 95 | opltn0 33495 |
. . . . . . . 8
⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺) ↔ (𝑅‘𝐺) ≠ (0.‘𝐾))) |
109 | 94, 104, 108 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺) ↔ (𝑅‘𝐺) ≠ (0.‘𝐾))) |
110 | 107, 109 | mpbid 221 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐺) ≠ (0.‘𝐾)) |
111 | 110 | neneqd 2787 |
. . . . 5
⊢ (𝜑 → ¬ (𝑅‘𝐺) = (0.‘𝐾)) |
112 | 95, 8, 27, 28, 29 | trlator0 34476 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐺) ∈ 𝐴 ∨ (𝑅‘𝐺) = (0.‘𝐾))) |
113 | 1, 69, 112 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐺) ∈ 𝐴 ∨ (𝑅‘𝐺) = (0.‘𝐾))) |
114 | 113 | orcomd 402 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐺) = (0.‘𝐾) ∨ (𝑅‘𝐺) ∈ 𝐴)) |
115 | 114 | ord 391 |
. . . . 5
⊢ (𝜑 → (¬ (𝑅‘𝐺) = (0.‘𝐾) → (𝑅‘𝐺) ∈ 𝐴)) |
116 | 111, 115 | mpd 15 |
. . . 4
⊢ (𝜑 → (𝑅‘𝐺) ∈ 𝐴) |
117 | 15, 8 | atcmp 33616 |
. . . 4
⊢ ((𝐾 ∈ AtLat ∧ 𝑈 ∈ 𝐴 ∧ (𝑅‘𝐺) ∈ 𝐴) → (𝑈 ≤ (𝑅‘𝐺) ↔ 𝑈 = (𝑅‘𝐺))) |
118 | 92, 12, 116, 117 | syl3anc 1318 |
. . 3
⊢ (𝜑 → (𝑈 ≤ (𝑅‘𝐺) ↔ 𝑈 = (𝑅‘𝐺))) |
119 | 90, 118 | mpbid 221 |
. 2
⊢ (𝜑 → 𝑈 = (𝑅‘𝐺)) |
120 | 119 | eqcomd 2616 |
1
⊢ (𝜑 → (𝑅‘𝐺) = 𝑈) |