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 475 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ HL) |
4 | | dia2dimlem1.p |
. . . . 5
⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
5 | 4 | simpld 475 |
. . . 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 35456 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
13 | 2, 4, 6, 12 | syl3anc 1326 |
. . . 4
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) |
14 | | dia2dimlem1.u |
. . . . 5
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
15 | 14 | simpld 475 |
. . . 4
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
16 | 6 | simpld 475 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝑇) |
17 | 7, 8, 9, 10 | ltrnel 35425 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
18 | 2, 16, 4, 17 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
19 | 18 | simpld 475 |
. . . 4
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
20 | | dia2dimlem1.v |
. . . . 5
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
21 | 20 | simpld 475 |
. . . 4
⊢ (𝜑 → 𝑉 ∈ 𝐴) |
22 | 4 | simprd 479 |
. . . . . 6
⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊) |
23 | 7, 9, 10, 11 | trlle 35471 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊) |
24 | 2, 16, 23 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) ≤ 𝑊) |
25 | 14 | simprd 479 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ≤ 𝑊) |
26 | | hllat 34650 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
27 | 3, 26 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Lat) |
28 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(Base‘𝐾) =
(Base‘𝐾) |
29 | 28, 8 | atbase 34576 |
. . . . . . . . . 10
⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾)) |
30 | 13, 29 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾)) |
31 | 28, 8 | atbase 34576 |
. . . . . . . . . 10
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) |
32 | 15, 31 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
33 | 2 | simprd 479 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
34 | 28, 9 | lhpbase 35284 |
. . . . . . . . . 10
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
35 | 33, 34 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
36 | | dia2dimlem1.j |
. . . . . . . . . 10
⊢ ∨ =
(join‘𝐾) |
37 | 28, 7, 36 | latjle12 17062 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝐹) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊)) |
38 | 27, 30, 32, 35, 37 | syl13anc 1328 |
. . . . . . . 8
⊢ (𝜑 → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊)) |
39 | 24, 25, 38 | mpbi2and 956 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) |
40 | 28, 8 | atbase 34576 |
. . . . . . . . 9
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
41 | 5, 40 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
42 | 28, 36, 8 | hlatjcl 34653 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
43 | 3, 13, 15, 42 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
44 | 28, 7 | lattr 17056 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊)) |
45 | 27, 41, 43, 35, 44 | syl13anc 1328 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊)) |
46 | 39, 45 | mpan2d 710 |
. . . . . 6
⊢ (𝜑 → (𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) → 𝑃 ≤ 𝑊)) |
47 | 22, 46 | mtod 189 |
. . . . 5
⊢ (𝜑 → ¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈)) |
48 | 20 | simprd 479 |
. . . . . . 7
⊢ (𝜑 → 𝑉 ≤ 𝑊) |
49 | 18 | simprd 479 |
. . . . . . 7
⊢ (𝜑 → ¬ (𝐹‘𝑃) ≤ 𝑊) |
50 | | nbrne2 4673 |
. . . . . . 7
⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃)) |
51 | 48, 49, 50 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃)) |
52 | 51 | necomd 2849 |
. . . . 5
⊢ (𝜑 → (𝐹‘𝑃) ≠ 𝑉) |
53 | 47, 52 | jca 554 |
. . . 4
⊢ (𝜑 → (¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉)) |
54 | 27 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝐾 ∈ Lat) |
55 | 41 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ∈ (Base‘𝐾)) |
56 | 28, 36, 8 | hlatjcl 34653 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) |
57 | 3, 21, 15, 56 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) |
58 | 57 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) |
59 | 35 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑊 ∈ (Base‘𝐾)) |
60 | 7, 36, 8 | hlatlej2 34662 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
61 | 3, 19, 21, 60 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
62 | 61 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
63 | | simpr 477 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) |
64 | 62, 63 | breqtrrd 4681 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈)) |
65 | | dia2dimlem1.uv |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑈 ≠ 𝑉) |
66 | 65 | necomd 2849 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ≠ 𝑈) |
67 | 7, 36, 8 | hlatexch2 34682 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ 𝑉 ≠ 𝑈) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈))) |
68 | 3, 21, 5, 15, 66, 67 | syl131anc 1339 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈))) |
69 | 68 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈))) |
70 | 64, 69 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ (𝑉 ∨ 𝑈)) |
71 | 28, 8 | atbase 34576 |
. . . . . . . . . . . 12
⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾)) |
72 | 21, 71 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾)) |
73 | 28, 7, 36 | latjle12 17062 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊)) |
74 | 27, 72, 32, 35, 73 | syl13anc 1328 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊)) |
75 | 48, 25, 74 | mpbi2and 956 |
. . . . . . . . 9
⊢ (𝜑 → (𝑉 ∨ 𝑈) ≤ 𝑊) |
76 | 75 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ≤ 𝑊) |
77 | 28, 7, 54, 55, 58, 59, 70, 76 | lattrd 17058 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ 𝑊) |
78 | 77 | ex 450 |
. . . . . 6
⊢ (𝜑 → ((𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉) → 𝑃 ≤ 𝑊)) |
79 | 78 | necon3bd 2808 |
. . . . 5
⊢ (𝜑 → (¬ 𝑃 ≤ 𝑊 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉))) |
80 | 22, 79 | mpd 15 |
. . . 4
⊢ (𝜑 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉)) |
81 | 7, 36, 8 | hlatlej2 34662 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃))) |
82 | 3, 5, 19, 81 | syl3anc 1326 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃))) |
83 | | dia2dimlem1.m |
. . . . . . . . . 10
⊢ ∧ =
(meet‘𝐾) |
84 | 7, 36, 83, 8, 9, 10, 11 | trlval2 35450 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
85 | 2, 16, 4, 84 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
86 | 85 | oveq2d 6666 |
. . . . . . 7
⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))) |
87 | 28, 36, 8 | hlatjcl 34653 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
88 | 3, 5, 19, 87 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
89 | 7, 36, 8 | hlatlej1 34661 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) |
90 | 3, 5, 19, 89 | syl3anc 1326 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) |
91 | 28, 7, 36, 83, 8 | atmod3i1 35150 |
. . . . . . . . 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))) |
92 | 3, 5, 88, 35, 90, 91 | syl131anc 1339 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊))) |
93 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(1.‘𝐾) =
(1.‘𝐾) |
94 | 7, 36, 93, 8, 9 | lhpjat2 35307 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾)) |
95 | 2, 4, 94 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ 𝑊) = (1.‘𝐾)) |
96 | 95 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾))) |
97 | | hlol 34648 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
98 | 3, 97 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ OL) |
99 | 28, 83, 93 | olm11 34514 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) |
100 | 98, 88, 99 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) |
101 | 96, 100 | eqtrd 2656 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃))) |
102 | 92, 101 | eqtrd 2656 |
. . . . . . 7
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃))) |
103 | 86, 102 | eqtrd 2656 |
. . . . . 6
⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |
104 | 82, 103 | breqtrrd 4681 |
. . . . 5
⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹))) |
105 | | dia2dimlem1.rf |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
106 | 36, 8 | hlatjcom 34654 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
107 | 3, 15, 21, 106 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
108 | 105, 107 | breqtrd 4679 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈)) |
109 | | dia2dimlem1.ru |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) |
110 | 7, 36, 8 | hlatexch2 34682 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑈) → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
111 | 3, 13, 21, 15, 109, 110 | syl131anc 1339 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
112 | 108, 111 | mpd 15 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)) |
113 | 104, 112 | jca 554 |
. . . 4
⊢ (𝜑 → ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
114 | 7, 36, 83, 8 | ps-2c 34814 |
. . . 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) |
115 | 3, 5, 13, 15, 19, 21, 53, 80, 113, 114 | syl333anc 1358 |
. . 3
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴) |
116 | 1, 115 | syl5eqel 2705 |
. 2
⊢ (𝜑 → 𝑄 ∈ 𝐴) |
117 | 28, 36, 8 | hlatjcl 34653 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
118 | 3, 5, 15, 117 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
119 | 28, 36, 8 | hlatjcl 34653 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
120 | 3, 19, 21, 119 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
121 | 28, 7, 83 | latmle1 17076 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈)) |
122 | 27, 118, 120, 121 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈)) |
123 | 1, 122 | syl5eqbr 4688 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈)) |
124 | 28, 8 | atbase 34576 |
. . . . . . . . . . . . 13
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
125 | 116, 124 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
126 | 28, 7, 83 | latlem12 17078 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
127 | 27, 125, 118, 35, 126 | syl13anc 1328 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
128 | 127 | biimpd 219 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
129 | 123, 128 | mpand 711 |
. . . . . . . . 9
⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) |
130 | 129 | imp 445 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)) |
131 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢
(0.‘𝐾) =
(0.‘𝐾) |
132 | 7, 83, 131, 8, 9 | lhpmat 35316 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = (0.‘𝐾)) |
133 | 2, 4, 132 | syl2anc 693 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃 ∧ 𝑊) = (0.‘𝐾)) |
134 | 133 | oveq1d 6665 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈)) |
135 | 28, 7, 36, 83, 8 | atmod4i1 35152 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 ≤ 𝑊) → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊)) |
136 | 3, 15, 41, 35, 25, 135 | syl131anc 1339 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊)) |
137 | 28, 36, 131 | olj02 34513 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑈) = 𝑈) |
138 | 98, 32, 137 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑈) = 𝑈) |
139 | 134, 136,
138 | 3eqtr3d 2664 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈) |
140 | 139 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈) |
141 | 130, 140 | breqtrd 4679 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈) |
142 | | hlatl 34647 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
143 | 3, 142 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ AtLat) |
144 | 143 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝐾 ∈ AtLat) |
145 | 116 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ∈ 𝐴) |
146 | 15 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 ∈ 𝐴) |
147 | 7, 8 | atcmp 34598 |
. . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈)) |
148 | 144, 145,
146, 147 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈)) |
149 | 141, 148 | mpbid 222 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑈) |
150 | 28, 7, 83 | latmle2 17077 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
151 | 27, 118, 120, 150 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
152 | 1, 151 | syl5eqbr 4688 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
153 | 28, 7, 83 | latlem12 17078 |
. . . . . . . . . . . 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))) |
154 | 27, 125, 120, 35, 153 | syl13anc 1328 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) |
155 | 154 | biimpd 219 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) |
156 | 152, 155 | mpand 711 |
. . . . . . . . 9
⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) |
157 | 156 | imp 445 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) |
158 | 7, 83, 131, 8, 9 | lhpmat 35316 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾)) |
159 | 2, 18, 158 | syl2anc 693 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾)) |
160 | 159 | oveq1d 6665 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉)) |
161 | 28, 8 | atbase 34576 |
. . . . . . . . . . . 12
⊢ ((𝐹‘𝑃) ∈ 𝐴 → (𝐹‘𝑃) ∈ (Base‘𝐾)) |
162 | 19, 161 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹‘𝑃) ∈ (Base‘𝐾)) |
163 | 28, 7, 36, 83, 8 | atmod4i1 35152 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 ≤ 𝑊) → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) |
164 | 3, 21, 162, 35, 48, 163 | syl131anc 1339 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) |
165 | 28, 36, 131 | olj02 34513 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑉) = 𝑉) |
166 | 98, 72, 165 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑉) = 𝑉) |
167 | 160, 164,
166 | 3eqtr3d 2664 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉) |
168 | 167 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉) |
169 | 157, 168 | breqtrd 4679 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉) |
170 | 21 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴) |
171 | 7, 8 | atcmp 34598 |
. . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉)) |
172 | 144, 145,
170, 171 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉)) |
173 | 169, 172 | mpbid 222 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑉) |
174 | 149, 173 | eqtr3d 2658 |
. . . . 5
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉) |
175 | 174 | ex 450 |
. . . 4
⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉)) |
176 | 175 | necon3ad 2807 |
. . 3
⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ 𝑄 ≤ 𝑊)) |
177 | 65, 176 | mpd 15 |
. 2
⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊) |
178 | 116, 177 | jca 554 |
1
⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |