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 494 | . . . 4
⊢ (𝜑 → 𝐾 ∈ HL) | 
| 4 |  | dia2dimlem1.p | . . . . 5
⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | 
| 5 | 4 | simpld 494 | . . . 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 40172 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) | 
| 13 | 2, 4, 6, 12 | syl3anc 1372 | . . . 4
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) | 
| 14 |  | dia2dimlem1.u | . . . . 5
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) | 
| 15 | 14 | simpld 494 | . . . 4
⊢ (𝜑 → 𝑈 ∈ 𝐴) | 
| 16 | 6 | simpld 494 | . . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝑇) | 
| 17 | 7, 8, 9, 10 | ltrnel 40142 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) | 
| 18 | 2, 16, 4, 17 | syl3anc 1372 | . . . . 5
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) | 
| 19 | 18 | simpld 494 | . . . 4
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) | 
| 20 |  | dia2dimlem1.v | . . . . 5
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) | 
| 21 | 20 | simpld 494 | . . . 4
⊢ (𝜑 → 𝑉 ∈ 𝐴) | 
| 22 | 4 | simprd 495 | . . . . . 6
⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊) | 
| 23 | 7, 9, 10, 11 | trlle 40187 | . . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊) | 
| 24 | 2, 16, 23 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) ≤ 𝑊) | 
| 25 | 14 | simprd 495 | . . . . . . . 8
⊢ (𝜑 → 𝑈 ≤ 𝑊) | 
| 26 | 3 | hllatd 39366 | . . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Lat) | 
| 27 |  | eqid 2736 | . . . . . . . . . . 11
⊢
(Base‘𝐾) =
(Base‘𝐾) | 
| 28 | 27, 8 | atbase 39291 | . . . . . . . . . 10
⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾)) | 
| 29 | 13, 28 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾)) | 
| 30 | 27, 8 | atbase 39291 | . . . . . . . . . 10
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) | 
| 31 | 15, 30 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) | 
| 32 | 2 | simprd 495 | . . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ 𝐻) | 
| 33 | 27, 9 | lhpbase 40001 | . . . . . . . . . 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 18496 | . . . . . . . . 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 1373 | . . . . . . . 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 39291 | . . . . . . . . 9
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) | 
| 40 | 5, 39 | syl 17 | . . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) | 
| 41 | 27, 35, 8 | hlatjcl 39369 | . . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) | 
| 42 | 3, 13, 15, 41 | syl3anc 1372 | . . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) | 
| 43 | 27, 7 | lattr 18490 | . . . . . . . 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 1373 | . . . . . . 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 198 | . . . . 5
⊢ (𝜑 → ¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈)) | 
| 47 | 20 | simprd 495 | . . . . . . 7
⊢ (𝜑 → 𝑉 ≤ 𝑊) | 
| 48 | 18 | simprd 495 | . . . . . . 7
⊢ (𝜑 → ¬ (𝐹‘𝑃) ≤ 𝑊) | 
| 49 |  | nbrne2 5162 | . . . . . . 7
⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃)) | 
| 50 | 47, 48, 49 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃)) | 
| 51 | 50 | necomd 2995 | . . . . 5
⊢ (𝜑 → (𝐹‘𝑃) ≠ 𝑉) | 
| 52 | 46, 51 | jca 511 | . . . 4
⊢ (𝜑 → (¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉)) | 
| 53 | 26 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝐾 ∈ Lat) | 
| 54 | 40 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ∈ (Base‘𝐾)) | 
| 55 | 27, 35, 8 | hlatjcl 39369 | . . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) | 
| 56 | 3, 21, 15, 55 | syl3anc 1372 | . . . . . . . . 9
⊢ (𝜑 → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) | 
| 57 | 56 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾)) | 
| 58 | 34 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑊 ∈ (Base‘𝐾)) | 
| 59 | 7, 35, 8 | hlatlej2 39378 | . . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) | 
| 60 | 3, 19, 21, 59 | syl3anc 1372 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) | 
| 61 | 60 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) | 
| 62 |  | simpr 484 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) | 
| 63 | 61, 62 | breqtrrd 5170 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈)) | 
| 64 |  | dia2dimlem1.uv | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑈 ≠ 𝑉) | 
| 65 | 64 | necomd 2995 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ≠ 𝑈) | 
| 66 | 7, 35, 8 | hlatexch2 39399 | . . . . . . . . . . 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 1384 | . . . . . . . . . 10
⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈))) | 
| 68 | 67 | adantr 480 | . . . . . . . . 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 39291 | . . . . . . . . . . . 12
⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾)) | 
| 71 | 21, 70 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾)) | 
| 72 | 27, 7, 35 | latjle12 18496 | . . . . . . . . . . 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 1373 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊)) | 
| 74 | 47, 25, 73 | mpbi2and 712 | . . . . . . . . 9
⊢ (𝜑 → (𝑉 ∨ 𝑈) ≤ 𝑊) | 
| 75 | 74 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ≤ 𝑊) | 
| 76 | 27, 7, 53, 54, 57, 58, 69, 75 | lattrd 18492 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ 𝑊) | 
| 77 | 76 | ex 412 | . . . . . 6
⊢ (𝜑 → ((𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉) → 𝑃 ≤ 𝑊)) | 
| 78 | 77 | necon3bd 2953 | . . . . 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 39378 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃))) | 
| 81 | 3, 5, 19, 80 | syl3anc 1372 | . . . . . 6
⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃))) | 
| 82 |  | dia2dimlem1.m | . . . . . . . . . 10
⊢  ∧ =
(meet‘𝐾) | 
| 83 | 7, 35, 82, 8, 9, 10, 11 | trlval2 40166 | . . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) | 
| 84 | 2, 16, 4, 83 | syl3anc 1372 | . . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) | 
| 85 | 84 | oveq2d 7448 | . . . . . . 7
⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))) | 
| 86 | 27, 35, 8 | hlatjcl 39369 | . . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) | 
| 87 | 3, 5, 19, 86 | syl3anc 1372 | . . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) | 
| 88 | 7, 35, 8 | hlatlej1 39377 | . . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) | 
| 89 | 3, 5, 19, 88 | syl3anc 1372 | . . . . . . . . 9
⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) | 
| 90 | 27, 7, 35, 82, 8 | atmod3i1 39867 | . . . . . . . . 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 1384 | . . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊))) | 
| 92 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(1.‘𝐾) =
(1.‘𝐾) | 
| 93 | 7, 35, 92, 8, 9 | lhpjat2 40024 | . . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾)) | 
| 94 | 2, 4, 93 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ 𝑊) = (1.‘𝐾)) | 
| 95 | 94 | oveq2d 7448 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾))) | 
| 96 |  | hlol 39363 | . . . . . . . . . . 11
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | 
| 97 | 3, 96 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ OL) | 
| 98 | 27, 82, 92 | olm11 39229 | . . . . . . . . . 10
⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) | 
| 99 | 97, 87, 98 | syl2anc 584 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) | 
| 100 | 95, 99 | eqtrd 2776 | . . . . . . . 8
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃))) | 
| 101 | 91, 100 | eqtrd 2776 | . . . . . . 7
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃))) | 
| 102 | 85, 101 | eqtrd 2776 | . . . . . 6
⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) | 
| 103 | 81, 102 | breqtrrd 5170 | . . . . 5
⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹))) | 
| 104 |  | dia2dimlem1.rf | . . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) | 
| 105 | 35, 8 | hlatjcom 39370 | . . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) | 
| 106 | 3, 15, 21, 105 | syl3anc 1372 | . . . . . . 7
⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) | 
| 107 | 104, 106 | breqtrd 5168 | . . . . . 6
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈)) | 
| 108 |  | dia2dimlem1.ru | . . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) | 
| 109 | 7, 35, 8 | hlatexch2 39399 | . . . . . . 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 1384 | . . . . . 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 511 | . . . 4
⊢ (𝜑 → ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) | 
| 113 | 7, 35, 82, 8 | ps-2c 39531 | . . . 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 1403 | . . 3
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴) | 
| 115 | 1, 114 | eqeltrid 2844 | . 2
⊢ (𝜑 → 𝑄 ∈ 𝐴) | 
| 116 | 27, 35, 8 | hlatjcl 39369 | . . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) | 
| 117 | 3, 5, 15, 116 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) | 
| 118 | 27, 35, 8 | hlatjcl 39369 | . . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) | 
| 119 | 3, 19, 21, 118 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) | 
| 120 | 27, 7, 82 | latmle1 18510 | . . . . . . . . . . . 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 1372 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈)) | 
| 122 | 1, 121 | eqbrtrid 5177 | . . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈)) | 
| 123 | 27, 8 | atbase 39291 | . . . . . . . . . . . . 13
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) | 
| 124 | 115, 123 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) | 
| 125 | 27, 7, 82 | latlem12 18512 | . . . . . . . . . . . 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 1373 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))) | 
| 127 | 126 | biimpd 229 | . . . . . . . . . 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 406 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)) | 
| 130 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢
(0.‘𝐾) =
(0.‘𝐾) | 
| 131 | 7, 82, 130, 8, 9 | lhpmat 40033 | . . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = (0.‘𝐾)) | 
| 132 | 2, 4, 131 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑃 ∧ 𝑊) = (0.‘𝐾)) | 
| 133 | 132 | oveq1d 7447 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈)) | 
| 134 | 27, 7, 35, 82, 8 | atmod4i1 39869 | . . . . . . . . . . 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 1384 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊)) | 
| 136 | 27, 35, 130 | olj02 39228 | . . . . . . . . . . 11
⊢ ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑈) = 𝑈) | 
| 137 | 97, 31, 136 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑈) = 𝑈) | 
| 138 | 133, 135,
137 | 3eqtr3d 2784 | . . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈) | 
| 139 | 138 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈) | 
| 140 | 129, 139 | breqtrd 5168 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈) | 
| 141 |  | hlatl 39362 | . . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | 
| 142 | 3, 141 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ AtLat) | 
| 143 | 142 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝐾 ∈ AtLat) | 
| 144 | 115 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ∈ 𝐴) | 
| 145 | 15 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 ∈ 𝐴) | 
| 146 | 7, 8 | atcmp 39313 | . . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈)) | 
| 147 | 143, 144,
145, 146 | syl3anc 1372 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈)) | 
| 148 | 140, 147 | mpbid 232 | . . . . . 6
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑈) | 
| 149 | 27, 7, 82 | latmle2 18511 | . . . . . . . . . . . 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 1372 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉)) | 
| 151 | 1, 150 | eqbrtrid 5177 | . . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉)) | 
| 152 | 27, 7, 82 | latlem12 18512 | . . . . . . . . . . . 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 1373 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))) | 
| 154 | 153 | biimpd 229 | . . . . . . . . . 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 406 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) | 
| 157 | 7, 82, 130, 8, 9 | lhpmat 40033 | . . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾)) | 
| 158 | 2, 18, 157 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾)) | 
| 159 | 158 | oveq1d 7447 | . . . . . . . . . 10
⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉)) | 
| 160 | 27, 8 | atbase 39291 | . . . . . . . . . . . 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 39869 | . . . . . . . . . . 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 1384 | . . . . . . . . . 10
⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)) | 
| 164 | 27, 35, 130 | olj02 39228 | . . . . . . . . . . 11
⊢ ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑉) = 𝑉) | 
| 165 | 97, 71, 164 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑉) = 𝑉) | 
| 166 | 159, 163,
165 | 3eqtr3d 2784 | . . . . . . . . 9
⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉) | 
| 167 | 166 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉) | 
| 168 | 156, 167 | breqtrd 5168 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉) | 
| 169 | 21 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴) | 
| 170 | 7, 8 | atcmp 39313 | . . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉)) | 
| 171 | 143, 144,
169, 170 | syl3anc 1372 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉)) | 
| 172 | 168, 171 | mpbid 232 | . . . . . 6
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑉) | 
| 173 | 148, 172 | eqtr3d 2778 | . . . . 5
⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉) | 
| 174 | 173 | ex 412 | . . . 4
⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉)) | 
| 175 | 174 | necon3ad 2952 | . . 3
⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ 𝑄 ≤ 𝑊)) | 
| 176 | 64, 175 | mpd 15 | . 2
⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊) | 
| 177 | 115, 176 | jca 511 | 1
⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |