Proof of Theorem dia2dimlem2
| Step | Hyp | Ref
| Expression |
| 1 | | dia2dimlem2.k |
. . . . . . . . 9
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 2 | 1 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ HL) |
| 3 | 2 | hllatd 39365 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 4 | | dia2dimlem2.p |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
| 5 | 4 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
| 6 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 7 | | dia2dimlem2.a |
. . . . . . . . 9
⊢ 𝐴 = (Atoms‘𝐾) |
| 8 | 6, 7 | atbase 39290 |
. . . . . . . 8
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
| 9 | 5, 8 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
| 10 | | dia2dimlem2.u |
. . . . . . . . 9
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
| 11 | 10 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
| 12 | 6, 7 | atbase 39290 |
. . . . . . . 8
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) |
| 13 | 11, 12 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
| 14 | | dia2dimlem2.l |
. . . . . . . 8
⊢ ≤ =
(le‘𝐾) |
| 15 | | dia2dimlem2.j |
. . . . . . . 8
⊢ ∨ =
(join‘𝐾) |
| 16 | 6, 14, 15 | latlej2 18494 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑈 ≤ (𝑃 ∨ 𝑈)) |
| 17 | 3, 9, 13, 16 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑈)) |
| 18 | 6, 15, 7 | hlatjcl 39368 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
| 19 | 2, 5, 11, 18 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
| 20 | | dia2dimlem2.m |
. . . . . . . 8
⊢ ∧ =
(meet‘𝐾) |
| 21 | 6, 14, 20 | latleeqm2 18513 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑈 ≤ (𝑃 ∨ 𝑈) ↔ ((𝑃 ∨ 𝑈) ∧ 𝑈) = 𝑈)) |
| 22 | 3, 13, 19, 21 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝑈 ≤ (𝑃 ∨ 𝑈) ↔ ((𝑃 ∨ 𝑈) ∧ 𝑈) = 𝑈)) |
| 23 | 17, 22 | mpbid 232 |
. . . . 5
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑈) = 𝑈) |
| 24 | | dia2dimlem2.rf |
. . . . . . . 8
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
| 25 | | dia2dimlem2.f |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) |
| 26 | | dia2dimlem2.h |
. . . . . . . . . . 11
⊢ 𝐻 = (LHyp‘𝐾) |
| 27 | | dia2dimlem2.t |
. . . . . . . . . . 11
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| 28 | | dia2dimlem2.r |
. . . . . . . . . . 11
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
| 29 | 14, 7, 26, 27, 28 | trlat 40171 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
| 30 | 1, 4, 25, 29 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) |
| 31 | | dia2dimlem2.v |
. . . . . . . . . 10
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
| 32 | 31 | simpld 494 |
. . . . . . . . 9
⊢ (𝜑 → 𝑉 ∈ 𝐴) |
| 33 | | dia2dimlem2.rv |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) |
| 34 | 14, 15, 7 | hlatexch2 39398 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑉) → ((𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉) → 𝑈 ≤ ((𝑅‘𝐹) ∨ 𝑉))) |
| 35 | 2, 30, 11, 32, 33, 34 | syl131anc 1385 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉) → 𝑈 ≤ ((𝑅‘𝐹) ∨ 𝑉))) |
| 36 | 24, 35 | mpd 15 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ≤ ((𝑅‘𝐹) ∨ 𝑉)) |
| 37 | 25 | simpld 494 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ 𝑇) |
| 38 | 14, 15, 20, 7, 26, 27, 28 | trlval2 40165 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
| 39 | 1, 37, 4, 38 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
| 40 | 39 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑉) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉)) |
| 41 | 14, 7, 26, 27 | ltrnel 40141 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
| 42 | 1, 37, 4, 41 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
| 43 | 42 | simpld 494 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
| 44 | 6, 15, 7 | hlatjcl 39368 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
| 45 | 2, 5, 43, 44 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
| 46 | 1 | simprd 495 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
| 47 | 6, 26 | lhpbase 40000 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
| 48 | 46, 47 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
| 49 | 31 | simprd 495 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ≤ 𝑊) |
| 50 | 6, 14, 15, 20, 7 | atmod4i1 39868 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 ≤ 𝑊) → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) ∧ 𝑊)) |
| 51 | 2, 32, 45, 48, 49, 50 | syl131anc 1385 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) ∧ 𝑊)) |
| 52 | 15, 7 | hlatjass 39371 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) = (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) |
| 53 | 2, 5, 43, 32, 52 | syl13anc 1374 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) = (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) |
| 54 | 53 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑉) ∧ 𝑊) = ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
| 55 | 51, 54 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑉) = ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
| 56 | 40, 55 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑉) = ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
| 57 | 36, 56 | breqtrd 5169 |
. . . . . 6
⊢ (𝜑 → 𝑈 ≤ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
| 58 | 6, 15, 7 | hlatjcl 39368 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
| 59 | 2, 43, 32, 58 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
| 60 | 6, 15 | latjcl 18484 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾)) |
| 61 | 3, 9, 59, 60 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾)) |
| 62 | 6, 20 | latmcl 18485 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) ∈ (Base‘𝐾)) |
| 63 | 3, 61, 48, 62 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) ∈ (Base‘𝐾)) |
| 64 | 6, 14, 20 | latmlem2 18515 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))) → (𝑈 ≤ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑈) ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)))) |
| 65 | 3, 13, 63, 19, 64 | syl13anc 1374 |
. . . . . 6
⊢ (𝜑 → (𝑈 ≤ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑈) ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)))) |
| 66 | 57, 65 | mpd 15 |
. . . . 5
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑈) ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
| 67 | 23, 66 | eqbrtrrd 5167 |
. . . 4
⊢ (𝜑 → 𝑈 ≤ ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
| 68 | | dia2dimlem2.g |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ 𝑇) |
| 69 | 14, 15, 20, 7, 26, 27, 28 | trlval2 40165 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) |
| 70 | 1, 68, 4, 69 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) |
| 71 | | dia2dimlem2.gv |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺‘𝑃) = 𝑄) |
| 72 | | dia2dimlem2.q |
. . . . . . . . . 10
⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) |
| 73 | 71, 72 | eqtrdi 2793 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘𝑃) = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 74 | 73 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ (𝐺‘𝑃)) = (𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))) |
| 75 | 74 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊)) |
| 76 | 14, 15, 7 | hlatlej1 39376 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑈)) |
| 77 | 2, 5, 11, 76 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑈)) |
| 78 | 6, 14, 15, 20, 7 | atmod3i1 39866 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ 𝑈)) → (𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) = ((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)))) |
| 79 | 2, 5, 19, 59, 77, 78 | syl131anc 1385 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) = ((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)))) |
| 80 | 79 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = (((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊)) |
| 81 | | hlol 39362 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
| 82 | 2, 81 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ OL) |
| 83 | 6, 20 | latmassOLD 39230 |
. . . . . . . . 9
⊢ ((𝐾 ∈ OL ∧ ((𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
| 84 | 82, 19, 61, 48, 83 | syl13anc 1374 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ (𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
| 85 | 80, 84 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
| 86 | 75, 85 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
| 87 | 70, 86 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (𝑅‘𝐺) = ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))) |
| 88 | 87 | eqcomd 2743 |
. . . 4
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝑃 ∨ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) = (𝑅‘𝐺)) |
| 89 | 67, 88 | breqtrd 5169 |
. . 3
⊢ (𝜑 → 𝑈 ≤ (𝑅‘𝐺)) |
| 90 | | hlatl 39361 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 91 | 2, 90 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ AtLat) |
| 92 | | hlop 39363 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
| 93 | 2, 92 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ OP) |
| 94 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 95 | | eqid 2737 |
. . . . . . . . . 10
⊢
(lt‘𝐾) =
(lt‘𝐾) |
| 96 | 94, 95, 7 | 0ltat 39292 |
. . . . . . . . 9
⊢ ((𝐾 ∈ OP ∧ 𝑈 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑈) |
| 97 | 93, 11, 96 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑈) |
| 98 | | hlpos 39367 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
| 99 | 2, 98 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Poset) |
| 100 | 6, 94 | op0cl 39185 |
. . . . . . . . . 10
⊢ (𝐾 ∈ OP →
(0.‘𝐾) ∈
(Base‘𝐾)) |
| 101 | 93, 100 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾)) |
| 102 | 6, 26, 27, 28 | trlcl 40166 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾)) |
| 103 | 1, 68, 102 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐺) ∈ (Base‘𝐾)) |
| 104 | 6, 14, 95 | pltletr 18388 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Poset ∧
((0.‘𝐾) ∈
(Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝑈 ∧ 𝑈 ≤ (𝑅‘𝐺)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺))) |
| 105 | 99, 101, 13, 103, 104 | syl13anc 1374 |
. . . . . . . 8
⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑈 ∧ 𝑈 ≤ (𝑅‘𝐺)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺))) |
| 106 | 97, 89, 105 | mp2and 699 |
. . . . . . 7
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺)) |
| 107 | 6, 95, 94 | opltn0 39191 |
. . . . . . . 8
⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺) ↔ (𝑅‘𝐺) ≠ (0.‘𝐾))) |
| 108 | 93, 103, 107 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐺) ↔ (𝑅‘𝐺) ≠ (0.‘𝐾))) |
| 109 | 106, 108 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐺) ≠ (0.‘𝐾)) |
| 110 | 109 | neneqd 2945 |
. . . . 5
⊢ (𝜑 → ¬ (𝑅‘𝐺) = (0.‘𝐾)) |
| 111 | 94, 7, 26, 27, 28 | trlator0 40173 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐺) ∈ 𝐴 ∨ (𝑅‘𝐺) = (0.‘𝐾))) |
| 112 | 1, 68, 111 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐺) ∈ 𝐴 ∨ (𝑅‘𝐺) = (0.‘𝐾))) |
| 113 | 112 | orcomd 872 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐺) = (0.‘𝐾) ∨ (𝑅‘𝐺) ∈ 𝐴)) |
| 114 | 113 | ord 865 |
. . . . 5
⊢ (𝜑 → (¬ (𝑅‘𝐺) = (0.‘𝐾) → (𝑅‘𝐺) ∈ 𝐴)) |
| 115 | 110, 114 | mpd 15 |
. . . 4
⊢ (𝜑 → (𝑅‘𝐺) ∈ 𝐴) |
| 116 | 14, 7 | atcmp 39312 |
. . . 4
⊢ ((𝐾 ∈ AtLat ∧ 𝑈 ∈ 𝐴 ∧ (𝑅‘𝐺) ∈ 𝐴) → (𝑈 ≤ (𝑅‘𝐺) ↔ 𝑈 = (𝑅‘𝐺))) |
| 117 | 91, 11, 115, 116 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (𝑈 ≤ (𝑅‘𝐺) ↔ 𝑈 = (𝑅‘𝐺))) |
| 118 | 89, 117 | mpbid 232 |
. 2
⊢ (𝜑 → 𝑈 = (𝑅‘𝐺)) |
| 119 | 118 | eqcomd 2743 |
1
⊢ (𝜑 → (𝑅‘𝐺) = 𝑈) |