Proof of Theorem dia2dimlem1
Step | Hyp | Ref
| Expression |
1 | | dia2dimlem1.q |
. . 3
⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) |
2 | | dia2dimlem1.k |
. . . . 5
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
3 | 2 | simpld 498 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ HL) |
4 | | dia2dimlem1.p |
. . . . 5
⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
5 | 4 | simpld 498 |
. . . 4
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
6 | | dia2dimlem1.f |
. . . . 5
⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) |
7 | | dia2dimlem1.l |
. . . . . 6
⊢ ≤ =
(le‘𝐾) |
8 | | dia2dimlem1.a |
. . . . . 6
⊢ 𝐴 = (Atoms‘𝐾) |
9 | | dia2dimlem1.h |
. . . . . 6
⊢ 𝐻 = (LHyp‘𝐾) |
10 | | dia2dimlem1.t |
. . . . . 6
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
11 | | dia2dimlem1.r |
. . . . . 6
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
12 | 7, 8, 9, 10, 11 | trlat 37920 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
13 | 2, 4, 6, 12 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) |
14 | | dia2dimlem1.u |
. . . . 5
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
15 | 14 | simpld 498 |
. . . 4
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
16 | 6 | simpld 498 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝑇) |
17 | 7, 8, 9, 10 | ltrnel 37890 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
18 | 2, 16, 4, 17 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
19 | 18 | simpld 498 |
. . . 4
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
20 | | dia2dimlem1.v |
. . . . 5
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
21 | 20 | simpld 498 |
. . . 4
⊢ (𝜑 → 𝑉 ∈ 𝐴) |
22 | 4 | simprd 499 |
. . . . . 6
⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊) |
23 | 7, 9, 10, 11 | trlle 37935 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊) |
24 | 2, 16, 23 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) ≤ 𝑊) |
25 | 14 | simprd 499 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ≤ 𝑊) |
26 | 3 | hllatd 37115 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Lat) |
27 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘𝐾) =
(Base‘𝐾) |
28 | 27, 8 | atbase 37040 |
. . . . . . . . . 10
⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾)) |
29 | 13, 28 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾)) |
30 | 27, 8 | atbase 37040 |
. . . . . . . . . 10
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) |
31 | 15, 30 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
32 | 2 | simprd 499 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
33 | 27, 9 | lhpbase 37749 |
. . . . . . . . . 10
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
34 | 32, 33 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
35 | | dia2dimlem1.j |
. . . . . . . . . 10
⊢ ∨ =
(join‘𝐾) |
36 | 27, 7, 35 | latjle12 17956 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝐹) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊)) |
37 | 26, 29, 31, 34, 36 | syl13anc 1374 |
. . . . . . . 8
⊢ (𝜑 → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊)) |
38 | 24, 25, 37 | mpbi2and 712 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) |
39 | 27, 8 | atbase 37040 |
. . . . . . . . 9
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
40 | 5, 39 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
41 | 27, 35, 8 | hlatjcl 37118 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
42 | 3, 13, 15, 41 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
43 | 27, 7 | lattr 17950 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊)) |
44 | 26, 40, 42, 34, 43 | syl13anc 1374 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊)) |
45 | 38, 44 | mpan2d 694 |
. . . . . 6
⊢ (𝜑 → (𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) → 𝑃 ≤ 𝑊)) |
46 | 22, 45 | mtod 201 |
. . . . 5
⊢ (𝜑 → ¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈)) |
47 | 20 | simprd 499 |
. . . . . . 7
⊢ (𝜑 → 𝑉 ≤ 𝑊) |
48 | 18 | simprd 499 |
. . . . . . 7
⊢ (𝜑 → ¬ (𝐹‘𝑃) ≤ 𝑊) |
49 | | nbrne2 5073 |
. . . . . . 7
⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃)) |
50 | 47, 48, 49 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃)) |
51 | 50 | necomd 2996 |
. . . . 5
⊢ (𝜑 → (𝐹‘𝑃) ≠ 𝑉) |
52 | 46, 51 | jca 515 |
. . . 4
⊢ (𝜑 → (¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉)) |
53 | 26 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝐾 ∈ Lat) |
54 | 40 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ∈ (Base‘𝐾)) |
55 | 27, 35, 8 | hlatjcl 37118 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) |
56 | 3, 21, 15, 55 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) |
57 | 56 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) |
58 | 34 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑊 ∈ (Base‘𝐾)) |
59 | 7, 35, 8 | hlatlej2 37127 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
60 | 3, 19, 21, 59 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
61 | 60 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
62 | | simpr 488 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) |
63 | 61, 62 | breqtrrd 5081 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈)) |
64 | | dia2dimlem1.uv |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑈 ≠ 𝑉) |
65 | 64 | necomd 2996 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ≠ 𝑈) |
66 | 7, 35, 8 | hlatexch2 37147 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ 𝑉 ≠ 𝑈) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈))) |
67 | 3, 21, 5, 15, 65, 66 | syl131anc 1385 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈))) |
68 | 67 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈))) |
69 | 63, 68 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ (𝑉 ∨ 𝑈)) |
70 | 27, 8 | atbase 37040 |
. . . . . . . . . . . 12
⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾)) |
71 | 21, 70 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾)) |
72 | 27, 7, 35 | latjle12 17956 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊)) |
73 | 26, 71, 31, 34, 72 | syl13anc 1374 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊)) |
74 | 47, 25, 73 | mpbi2and 712 |
. . . . . . . . 9
⊢ (𝜑 → (𝑉 ∨ 𝑈) ≤ 𝑊) |
75 | 74 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ≤ 𝑊) |
76 | 27, 7, 53, 54, 57, 58, 69, 75 | lattrd 17952 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ 𝑊) |
77 | 76 | ex 416 |
. . . . . 6
⊢ (𝜑 → ((𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉) → 𝑃 ≤ 𝑊)) |
78 | 77 | necon3bd 2954 |
. . . . 5
⊢ (𝜑 → (¬ 𝑃 ≤ 𝑊 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉))) |
79 | 22, 78 | mpd 15 |
. . . 4
⊢ (𝜑 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉)) |
80 | 7, 35, 8 | hlatlej2 37127 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃))) |
81 | 3, 5, 19, 80 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃))) |
82 | | dia2dimlem1.m |
. . . . . . . . . 10
⊢ ∧ =
(meet‘𝐾) |
83 | 7, 35, 82, 8, 9, 10, 11 | trlval2 37914 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
84 | 2, 16, 4, 83 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
85 | 84 | oveq2d 7229 |
. . . . . . 7
⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))) |
86 | 27, 35, 8 | hlatjcl 37118 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
87 | 3, 5, 19, 86 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
88 | 7, 35, 8 | hlatlej1 37126 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) |
89 | 3, 5, 19, 88 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) |
90 | 27, 7, 35, 82, 8 | atmod3i1 37615 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊))) |
91 | 3, 5, 87, 34, 89, 90 | syl131anc 1385 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊))) |
92 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(1.‘𝐾) =
(1.‘𝐾) |
93 | 7, 35, 92, 8, 9 | lhpjat2 37772 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾)) |
94 | 2, 4, 93 | syl2anc 587 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ 𝑊) = (1.‘𝐾)) |
95 | 94 | oveq2d 7229 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾))) |
96 | | hlol 37112 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
97 | 3, 96 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ OL) |
98 | 27, 82, 92 | olm11 36978 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) |
99 | 97, 87, 98 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) |
100 | 95, 99 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃))) |
101 | 91, 100 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃))) |
102 | 85, 101 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |
103 | 81, 102 | breqtrrd 5081 |
. . . . 5
⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹))) |
104 | | dia2dimlem1.rf |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
105 | 35, 8 | hlatjcom 37119 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
106 | 3, 15, 21, 105 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
107 | 104, 106 | breqtrd 5079 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈)) |
108 | | dia2dimlem1.ru |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) |
109 | 7, 35, 8 | hlatexch2 37147 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑈) → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
110 | 3, 13, 21, 15, 108, 109 | syl131anc 1385 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
111 | 107, 110 | mpd 15 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)) |
112 | 103, 111 | jca 515 |
. . . 4
⊢ (𝜑 → ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
113 | 7, 35, 82, 8 | ps-2c 37279 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) ∧ ((¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉) ∧ (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉) ∧ ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴) |
114 | 3, 5, 13, 15, 19, 21, 52, 79, 112, 113 | syl333anc 1404 |
. . 3
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴) |
115 | 1, 114 | eqeltrid 2842 |
. 2
⊢ (𝜑 → 𝑄 ∈ 𝐴) |
116 | 27, 35, 8 | hlatjcl 37118 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
117 | 3, 5, 15, 116 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
118 | 27, 35, 8 | hlatjcl 37118 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
119 | 3, 19, 21, 118 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
120 | 27, 7, 82 | latmle1 17970 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈)) |
121 | 26, 117, 119, 120 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈)) |
122 | 1, 121 | eqbrtrid 5088 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈)) |
123 | 27, 8 | atbase 37040 |
. . . . . . . . . . . . 13
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
124 | 115, 123 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
125 | 27, 7, 82 | latlem12 17972 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
126 | 26, 124, 117, 34, 125 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
127 | 126 | biimpd 232 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
128 | 122, 127 | mpand 695 |
. . . . . . . . 9
⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
129 | 128 | imp 410 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)) |
130 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(0.‘𝐾) =
(0.‘𝐾) |
131 | 7, 82, 130, 8, 9 | lhpmat 37781 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = (0.‘𝐾)) |
132 | 2, 4, 131 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃 ∧ 𝑊) = (0.‘𝐾)) |
133 | 132 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈)) |
134 | 27, 7, 35, 82, 8 | atmod4i1 37617 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 ≤ 𝑊) → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊)) |
135 | 3, 15, 40, 34, 25, 134 | syl131anc 1385 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊)) |
136 | 27, 35, 130 | olj02 36977 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑈) = 𝑈) |
137 | 97, 31, 136 | syl2anc 587 |
. . . . . . . . . 10
⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑈) = 𝑈) |
138 | 133, 135,
137 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈) |
139 | 138 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈) |
140 | 129, 139 | breqtrd 5079 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈) |
141 | | hlatl 37111 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
142 | 3, 141 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ AtLat) |
143 | 142 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝐾 ∈ AtLat) |
144 | 115 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ∈ 𝐴) |
145 | 15 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 ∈ 𝐴) |
146 | 7, 8 | atcmp 37062 |
. . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈)) |
147 | 143, 144,
145, 146 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈)) |
148 | 140, 147 | mpbid 235 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑈) |
149 | 27, 7, 82 | latmle2 17971 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
150 | 26, 117, 119, 149 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
151 | 1, 150 | eqbrtrid 5088 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
152 | 27, 7, 82 | latlem12 17972 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) |
153 | 26, 124, 119, 34, 152 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) |
154 | 153 | biimpd 232 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) |
155 | 151, 154 | mpand 695 |
. . . . . . . . 9
⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) |
156 | 155 | imp 410 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) |
157 | 7, 82, 130, 8, 9 | lhpmat 37781 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾)) |
158 | 2, 18, 157 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾)) |
159 | 158 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉)) |
160 | 27, 8 | atbase 37040 |
. . . . . . . . . . . 12
⊢ ((𝐹‘𝑃) ∈ 𝐴 → (𝐹‘𝑃) ∈ (Base‘𝐾)) |
161 | 19, 160 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹‘𝑃) ∈ (Base‘𝐾)) |
162 | 27, 7, 35, 82, 8 | atmod4i1 37617 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 ≤ 𝑊) → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) |
163 | 3, 21, 161, 34, 47, 162 | syl131anc 1385 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) |
164 | 27, 35, 130 | olj02 36977 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑉) = 𝑉) |
165 | 97, 71, 164 | syl2anc 587 |
. . . . . . . . . 10
⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑉) = 𝑉) |
166 | 159, 163,
165 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉) |
167 | 166 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉) |
168 | 156, 167 | breqtrd 5079 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉) |
169 | 21 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴) |
170 | 7, 8 | atcmp 37062 |
. . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉)) |
171 | 143, 144,
169, 170 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉)) |
172 | 168, 171 | mpbid 235 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑉) |
173 | 148, 172 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉) |
174 | 173 | ex 416 |
. . . 4
⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉)) |
175 | 174 | necon3ad 2953 |
. . 3
⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ 𝑄 ≤ 𝑊)) |
176 | 64, 175 | mpd 15 |
. 2
⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊) |
177 | 115, 176 | jca 515 |
1
⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |