Proof of Theorem dia2dimlem3
Step | Hyp | Ref
| Expression |
1 | | dia2dimlem3.k |
. . . . . . 7
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | 1 | simpld 498 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ HL) |
3 | | dia2dimlem3.f |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) |
4 | 3 | simpld 498 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ 𝑇) |
5 | | dia2dimlem3.p |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
6 | | dia2dimlem3.l |
. . . . . . . . 9
⊢ ≤ =
(le‘𝐾) |
7 | | dia2dimlem3.a |
. . . . . . . . 9
⊢ 𝐴 = (Atoms‘𝐾) |
8 | | dia2dimlem3.h |
. . . . . . . . 9
⊢ 𝐻 = (LHyp‘𝐾) |
9 | | dia2dimlem3.t |
. . . . . . . . 9
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
10 | 6, 7, 8, 9 | ltrnel 37796 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
11 | 1, 4, 5, 10 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
12 | 11 | simpld 498 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
13 | | dia2dimlem3.v |
. . . . . . 7
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
14 | 13 | simpld 498 |
. . . . . 6
⊢ (𝜑 → 𝑉 ∈ 𝐴) |
15 | | dia2dimlem3.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
16 | 6, 15, 7 | hlatlej2 37033 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
17 | 2, 12, 14, 16 | syl3anc 1372 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
18 | 2 | hllatd 37021 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ Lat) |
19 | | eqid 2738 |
. . . . . . . 8
⊢
(Base‘𝐾) =
(Base‘𝐾) |
20 | 19, 7 | atbase 36946 |
. . . . . . 7
⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾)) |
21 | 14, 20 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾)) |
22 | 19, 15, 7 | hlatjcl 37024 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
23 | 2, 12, 14, 22 | syl3anc 1372 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
24 | | dia2dimlem3.r |
. . . . . . . . 9
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
25 | 6, 7, 8, 9, 24 | trlat 37826 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
26 | 1, 5, 3, 25 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) |
27 | | dia2dimlem3.u |
. . . . . . . 8
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
28 | 27 | simpld 498 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
29 | 19, 15, 7 | hlatjcl 37024 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
30 | 2, 26, 28, 29 | syl3anc 1372 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
31 | | dia2dimlem3.m |
. . . . . . 7
⊢ ∧ =
(meet‘𝐾) |
32 | 19, 6, 31 | latmlem2 17808 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))) → (𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉) → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))) |
33 | 18, 21, 23, 30, 32 | syl13anc 1373 |
. . . . 5
⊢ (𝜑 → (𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉) → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))) |
34 | 17, 33 | mpd 15 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
35 | | dia2dimlem3.rf |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
36 | 15, 7 | hlatjcom 37025 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
37 | 2, 28, 14, 36 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
38 | 35, 37 | breqtrd 5056 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈)) |
39 | | dia2dimlem3.ru |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) |
40 | 6, 15, 7 | hlatexch2 37053 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑈) → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
41 | 2, 26, 14, 28, 39, 40 | syl131anc 1384 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
42 | 38, 41 | mpd 15 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)) |
43 | 19, 6, 31 | latleeqm2 17806 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑉 ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉)) |
44 | 18, 21, 30, 43 | syl3anc 1372 |
. . . . 5
⊢ (𝜑 → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉)) |
45 | 42, 44 | mpbid 235 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉) |
46 | | dia2dimlem3.d |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ 𝑇) |
47 | | dia2dimlem3.q |
. . . . . . 7
⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) |
48 | | dia2dimlem3.uv |
. . . . . . 7
⊢ (𝜑 → 𝑈 ≠ 𝑉) |
49 | 6, 15, 31, 7, 8, 9,
24, 47, 1, 27, 13, 5, 3, 35, 48, 39 | dia2dimlem1 38721 |
. . . . . 6
⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
50 | 6, 15, 31, 7, 8, 9,
24 | trlval2 37820 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊)) |
51 | 1, 46, 49, 50 | syl3anc 1372 |
. . . . 5
⊢ (𝜑 → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊)) |
52 | 47 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
53 | | dia2dimlem3.dv |
. . . . . . . . 9
⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) |
54 | 52, 53 | oveq12d 7188 |
. . . . . . . 8
⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃))) |
55 | 5 | simpld 498 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
56 | 19, 15, 7 | hlatjcl 37024 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
57 | 2, 55, 28, 56 | syl3anc 1372 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
58 | 6, 15, 7 | hlatlej1 37032 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
59 | 2, 12, 14, 58 | syl3anc 1372 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
60 | 19, 6, 15, 31, 7 | atmod4i1 37523 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) ∧ (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
61 | 2, 12, 57, 23, 59, 60 | syl131anc 1384 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
62 | 15, 7 | hlatj32 37029 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴)) → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈)) |
63 | 2, 55, 28, 12, 62 | syl13anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈)) |
64 | 63 | oveq1d 7185 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
65 | 54, 61, 64 | 3eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
66 | 65 | oveq1d 7185 |
. . . . . 6
⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
67 | | hlol 37018 |
. . . . . . . 8
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
68 | 2, 67 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ OL) |
69 | 19, 15, 7 | hlatjcl 37024 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
70 | 2, 55, 12, 69 | syl3anc 1372 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
71 | 19, 7 | atbase 36946 |
. . . . . . . . 9
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) |
72 | 28, 71 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
73 | 19, 15 | latjcl 17777 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾)) |
74 | 18, 70, 72, 73 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾)) |
75 | 1 | simprd 499 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
76 | 19, 8 | lhpbase 37655 |
. . . . . . . 8
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
77 | 75, 76 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
78 | 19, 31 | latm32 36888 |
. . . . . . 7
⊢ ((𝐾 ∈ OL ∧ (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
79 | 68, 74, 23, 77, 78 | syl13anc 1373 |
. . . . . 6
⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
80 | 6, 15, 31, 7, 8, 9,
24 | trlval2 37820 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
81 | 1, 4, 5, 80 | syl3anc 1372 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
82 | 81 | oveq1d 7185 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈)) |
83 | 27 | simprd 499 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ≤ 𝑊) |
84 | 19, 6, 15, 31, 7 | atmod4i1 37523 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 ≤ 𝑊) → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊)) |
85 | 2, 28, 70, 77, 83, 84 | syl131anc 1384 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊)) |
86 | 82, 85 | eqtr2d 2774 |
. . . . . . 7
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈)) |
87 | 86 | oveq1d 7185 |
. . . . . 6
⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
88 | 66, 79, 87 | 3eqtrd 2777 |
. . . . 5
⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
89 | 51, 88 | eqtr2d 2774 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷)) |
90 | 34, 45, 89 | 3brtr3d 5061 |
. . 3
⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷)) |
91 | | hlatl 37017 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
92 | 2, 91 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ AtLat) |
93 | | hlop 37019 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
94 | 2, 93 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ OP) |
95 | | eqid 2738 |
. . . . . . . . . 10
⊢
(0.‘𝐾) =
(0.‘𝐾) |
96 | | eqid 2738 |
. . . . . . . . . 10
⊢
(lt‘𝐾) =
(lt‘𝐾) |
97 | 95, 96, 7 | 0ltat 36948 |
. . . . . . . . 9
⊢ ((𝐾 ∈ OP ∧ 𝑉 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉) |
98 | 94, 14, 97 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉) |
99 | | hlpos 37023 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
100 | 2, 99 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Poset) |
101 | 19, 95 | op0cl 36841 |
. . . . . . . . . 10
⊢ (𝐾 ∈ OP →
(0.‘𝐾) ∈
(Base‘𝐾)) |
102 | 94, 101 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾)) |
103 | 19, 8, 9, 24 | trlcl 37821 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → (𝑅‘𝐷) ∈ (Base‘𝐾)) |
104 | 1, 46, 103 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐷) ∈ (Base‘𝐾)) |
105 | 19, 6, 96 | pltletr 17697 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Poset ∧
((0.‘𝐾) ∈
(Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑅‘𝐷) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))) |
106 | 100, 102,
21, 104, 105 | syl13anc 1373 |
. . . . . . . 8
⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))) |
107 | 98, 90, 106 | mp2and 699 |
. . . . . . 7
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷)) |
108 | 19, 96, 95 | opltn0 36847 |
. . . . . . . 8
⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾))) |
109 | 94, 104, 108 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾))) |
110 | 107, 109 | mpbid 235 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐷) ≠ (0.‘𝐾)) |
111 | 110 | neneqd 2939 |
. . . . 5
⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾)) |
112 | 95, 7, 8, 9, 24 | trlator0 37828 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾))) |
113 | 1, 46, 112 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾))) |
114 | 113 | orcomd 870 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐷) = (0.‘𝐾) ∨ (𝑅‘𝐷) ∈ 𝐴)) |
115 | 114 | ord 863 |
. . . . 5
⊢ (𝜑 → (¬ (𝑅‘𝐷) = (0.‘𝐾) → (𝑅‘𝐷) ∈ 𝐴)) |
116 | 111, 115 | mpd 15 |
. . . 4
⊢ (𝜑 → (𝑅‘𝐷) ∈ 𝐴) |
117 | 6, 7 | atcmp 36968 |
. . . 4
⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴 ∧ (𝑅‘𝐷) ∈ 𝐴) → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷))) |
118 | 92, 14, 116, 117 | syl3anc 1372 |
. . 3
⊢ (𝜑 → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷))) |
119 | 90, 118 | mpbid 235 |
. 2
⊢ (𝜑 → 𝑉 = (𝑅‘𝐷)) |
120 | 119 | eqcomd 2744 |
1
⊢ (𝜑 → (𝑅‘𝐷) = 𝑉) |