Proof of Theorem dia2dimlem3
Step | Hyp | Ref
| Expression |
1 | | dia2dimlem3.k |
. . . . . . 7
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | 1 | simpld 475 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ HL) |
3 | | dia2dimlem3.f |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) |
4 | 3 | simpld 475 |
. . . . . . . 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 35425 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
11 | 1, 4, 5, 10 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
12 | 11 | simpld 475 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
13 | | dia2dimlem3.v |
. . . . . . 7
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
14 | 13 | simpld 475 |
. . . . . 6
⊢ (𝜑 → 𝑉 ∈ 𝐴) |
15 | | dia2dimlem3.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
16 | 6, 15, 7 | hlatlej2 34662 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
17 | 2, 12, 14, 16 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
18 | | hllat 34650 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
19 | 2, 18 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ Lat) |
20 | | eqid 2622 |
. . . . . . . 8
⊢
(Base‘𝐾) =
(Base‘𝐾) |
21 | 20, 7 | atbase 34576 |
. . . . . . 7
⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾)) |
22 | 14, 21 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾)) |
23 | 20, 15, 7 | hlatjcl 34653 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
24 | 2, 12, 14, 23 | syl3anc 1326 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
25 | | dia2dimlem3.r |
. . . . . . . . 9
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
26 | 6, 7, 8, 9, 25 | trlat 35456 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
27 | 1, 5, 3, 26 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) |
28 | | dia2dimlem3.u |
. . . . . . . 8
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
29 | 28 | simpld 475 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
30 | 20, 15, 7 | hlatjcl 34653 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
31 | 2, 27, 29, 30 | syl3anc 1326 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
32 | | dia2dimlem3.m |
. . . . . . 7
⊢ ∧ =
(meet‘𝐾) |
33 | 20, 6, 32 | latmlem2 17082 |
. . . . . 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)))) |
34 | 19, 22, 24, 31, 33 | syl13anc 1328 |
. . . . 5
⊢ (𝜑 → (𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉) → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))) |
35 | 17, 34 | mpd 15 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
36 | | dia2dimlem3.rf |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
37 | 15, 7 | hlatjcom 34654 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
38 | 2, 29, 14, 37 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
39 | 36, 38 | breqtrd 4679 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈)) |
40 | | dia2dimlem3.ru |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) |
41 | 6, 15, 7 | hlatexch2 34682 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑈) → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
42 | 2, 27, 14, 29, 40, 41 | syl131anc 1339 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
43 | 39, 42 | mpd 15 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)) |
44 | 20, 6, 32 | latleeqm2 17080 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑉 ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉)) |
45 | 19, 22, 31, 44 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉)) |
46 | 43, 45 | mpbid 222 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉) |
47 | | dia2dimlem3.d |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ 𝑇) |
48 | | dia2dimlem3.q |
. . . . . . 7
⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) |
49 | | dia2dimlem3.uv |
. . . . . . 7
⊢ (𝜑 → 𝑈 ≠ 𝑉) |
50 | 6, 15, 32, 7, 8, 9,
25, 48, 1, 28, 13, 5, 3, 36, 49, 40 | dia2dimlem1 36353 |
. . . . . 6
⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
51 | 6, 15, 32, 7, 8, 9,
25 | trlval2 35450 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊)) |
52 | 1, 47, 50, 51 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊)) |
53 | 48 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
54 | | dia2dimlem3.dv |
. . . . . . . . 9
⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) |
55 | 53, 54 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃))) |
56 | 5 | simpld 475 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
57 | 20, 15, 7 | hlatjcl 34653 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
58 | 2, 56, 29, 57 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
59 | 6, 15, 7 | hlatlej1 34661 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
60 | 2, 12, 14, 59 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
61 | 20, 6, 15, 32, 7 | atmod4i1 35152 |
. . . . . . . . 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))) |
62 | 2, 12, 58, 24, 60, 61 | syl131anc 1339 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
63 | 15, 7 | hlatj32 34658 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴)) → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈)) |
64 | 2, 56, 29, 12, 63 | syl13anc 1328 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈)) |
65 | 64 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
66 | 55, 62, 65 | 3eqtrd 2660 |
. . . . . . 7
⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
67 | 66 | oveq1d 6665 |
. . . . . 6
⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
68 | | hlol 34648 |
. . . . . . . 8
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
69 | 2, 68 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ OL) |
70 | 20, 15, 7 | hlatjcl 34653 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
71 | 2, 56, 12, 70 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
72 | 20, 7 | atbase 34576 |
. . . . . . . . 9
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) |
73 | 29, 72 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
74 | 20, 15 | latjcl 17051 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾)) |
75 | 19, 71, 73, 74 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾)) |
76 | 1 | simprd 479 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
77 | 20, 8 | lhpbase 35284 |
. . . . . . . 8
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
78 | 76, 77 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
79 | 20, 32 | latm32 34518 |
. . . . . . 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))) |
80 | 69, 75, 24, 78, 79 | syl13anc 1328 |
. . . . . 6
⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
81 | 6, 15, 32, 7, 8, 9,
25 | trlval2 35450 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
82 | 1, 4, 5, 81 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
83 | 82 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈)) |
84 | 28 | simprd 479 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ≤ 𝑊) |
85 | 20, 6, 15, 32, 7 | atmod4i1 35152 |
. . . . . . . . 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)) |
86 | 2, 29, 71, 78, 84, 85 | syl131anc 1339 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊)) |
87 | 83, 86 | eqtr2d 2657 |
. . . . . . 7
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈)) |
88 | 87 | oveq1d 6665 |
. . . . . 6
⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
89 | 67, 80, 88 | 3eqtrd 2660 |
. . . . 5
⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
90 | 52, 89 | eqtr2d 2657 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷)) |
91 | 35, 46, 90 | 3brtr3d 4684 |
. . 3
⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷)) |
92 | | hlatl 34647 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
93 | 2, 92 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ AtLat) |
94 | | hlop 34649 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
95 | 2, 94 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ OP) |
96 | | eqid 2622 |
. . . . . . . . . 10
⊢
(0.‘𝐾) =
(0.‘𝐾) |
97 | | eqid 2622 |
. . . . . . . . . 10
⊢
(lt‘𝐾) =
(lt‘𝐾) |
98 | 96, 97, 7 | 0ltat 34578 |
. . . . . . . . 9
⊢ ((𝐾 ∈ OP ∧ 𝑉 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉) |
99 | 95, 14, 98 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉) |
100 | | hlpos 34652 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
101 | 2, 100 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Poset) |
102 | 20, 96 | op0cl 34471 |
. . . . . . . . . 10
⊢ (𝐾 ∈ OP →
(0.‘𝐾) ∈
(Base‘𝐾)) |
103 | 95, 102 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾)) |
104 | 20, 8, 9, 25 | trlcl 35451 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → (𝑅‘𝐷) ∈ (Base‘𝐾)) |
105 | 1, 47, 104 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐷) ∈ (Base‘𝐾)) |
106 | 20, 6, 97 | pltletr 16971 |
. . . . . . . . 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))) |
107 | 101, 103,
22, 105, 106 | syl13anc 1328 |
. . . . . . . 8
⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))) |
108 | 99, 91, 107 | mp2and 715 |
. . . . . . 7
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷)) |
109 | 20, 97, 96 | opltn0 34477 |
. . . . . . . 8
⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾))) |
110 | 95, 105, 109 | syl2anc 693 |
. . . . . . 7
⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾))) |
111 | 108, 110 | mpbid 222 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐷) ≠ (0.‘𝐾)) |
112 | 111 | neneqd 2799 |
. . . . 5
⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾)) |
113 | 96, 7, 8, 9, 25 | trlator0 35458 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾))) |
114 | 1, 47, 113 | syl2anc 693 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾))) |
115 | 114 | orcomd 403 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐷) = (0.‘𝐾) ∨ (𝑅‘𝐷) ∈ 𝐴)) |
116 | 115 | ord 392 |
. . . . 5
⊢ (𝜑 → (¬ (𝑅‘𝐷) = (0.‘𝐾) → (𝑅‘𝐷) ∈ 𝐴)) |
117 | 112, 116 | mpd 15 |
. . . 4
⊢ (𝜑 → (𝑅‘𝐷) ∈ 𝐴) |
118 | 6, 7 | atcmp 34598 |
. . . 4
⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴 ∧ (𝑅‘𝐷) ∈ 𝐴) → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷))) |
119 | 93, 14, 117, 118 | syl3anc 1326 |
. . 3
⊢ (𝜑 → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷))) |
120 | 91, 119 | mpbid 222 |
. 2
⊢ (𝜑 → 𝑉 = (𝑅‘𝐷)) |
121 | 120 | eqcomd 2628 |
1
⊢ (𝜑 → (𝑅‘𝐷) = 𝑉) |