Proof of Theorem dia2dimlem3
| Step | Hyp | Ref
| Expression |
| 1 | | dia2dimlem3.k |
. . . . . . 7
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 2 | 1 | simpld 494 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ HL) |
| 3 | | dia2dimlem3.f |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) |
| 4 | 3 | simpld 494 |
. . . . . . . 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 40141 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
| 11 | 1, 4, 5, 10 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) |
| 12 | 11 | simpld 494 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
| 13 | | dia2dimlem3.v |
. . . . . . 7
⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
| 14 | 13 | simpld 494 |
. . . . . 6
⊢ (𝜑 → 𝑉 ∈ 𝐴) |
| 15 | | dia2dimlem3.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
| 16 | 6, 15, 7 | hlatlej2 39377 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
| 17 | 2, 12, 14, 16 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
| 18 | 2 | hllatd 39365 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 19 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 20 | 19, 7 | atbase 39290 |
. . . . . . 7
⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾)) |
| 21 | 14, 20 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾)) |
| 22 | 19, 15, 7 | hlatjcl 39368 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
| 23 | 2, 12, 14, 22 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) |
| 24 | | dia2dimlem3.r |
. . . . . . . . 9
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
| 25 | 6, 7, 8, 9, 24 | trlat 40171 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
| 26 | 1, 5, 3, 25 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴) |
| 27 | | dia2dimlem3.u |
. . . . . . . 8
⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
| 28 | 27 | simpld 494 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
| 29 | 19, 15, 7 | hlatjcl 39368 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
| 30 | 2, 26, 28, 29 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) |
| 31 | | dia2dimlem3.m |
. . . . . . 7
⊢ ∧ =
(meet‘𝐾) |
| 32 | 19, 6, 31 | latmlem2 18515 |
. . . . . 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)))) |
| 33 | 18, 21, 23, 30, 32 | syl13anc 1374 |
. . . . 5
⊢ (𝜑 → (𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉) → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))) |
| 34 | 17, 33 | mpd 15 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 35 | | dia2dimlem3.rf |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) |
| 36 | 15, 7 | hlatjcom 39369 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
| 37 | 2, 28, 14, 36 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈)) |
| 38 | 35, 37 | breqtrd 5169 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈)) |
| 39 | | dia2dimlem3.ru |
. . . . . . 7
⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) |
| 40 | 6, 15, 7 | hlatexch2 39398 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑈) → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
| 41 | 2, 26, 14, 28, 39, 40 | syl131anc 1385 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))) |
| 42 | 38, 41 | mpd 15 |
. . . . 5
⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)) |
| 43 | 19, 6, 31 | latleeqm2 18513 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑉 ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉)) |
| 44 | 18, 21, 30, 43 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉)) |
| 45 | 42, 44 | mpbid 232 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉) |
| 46 | | dia2dimlem3.d |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ 𝑇) |
| 47 | | dia2dimlem3.q |
. . . . . . 7
⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) |
| 48 | | dia2dimlem3.uv |
. . . . . . 7
⊢ (𝜑 → 𝑈 ≠ 𝑉) |
| 49 | 6, 15, 31, 7, 8, 9,
24, 47, 1, 27, 13, 5, 3, 35, 48, 39 | dia2dimlem1 41066 |
. . . . . 6
⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
| 50 | 6, 15, 31, 7, 8, 9,
24 | trlval2 40165 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊)) |
| 51 | 1, 46, 49, 50 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊)) |
| 52 | 47 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 53 | | dia2dimlem3.dv |
. . . . . . . . 9
⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) |
| 54 | 52, 53 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃))) |
| 55 | 5 | simpld 494 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
| 56 | 19, 15, 7 | hlatjcl 39368 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
| 57 | 2, 55, 28, 56 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) |
| 58 | 6, 15, 7 | hlatlej1 39376 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
| 59 | 2, 12, 14, 58 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) |
| 60 | 19, 6, 15, 31, 7 | atmod4i1 39868 |
. . . . . . . . 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))) |
| 61 | 2, 12, 57, 23, 59, 60 | syl131anc 1385 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 62 | 15, 7 | hlatj32 39373 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴)) → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈)) |
| 63 | 2, 55, 28, 12, 62 | syl13anc 1374 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈)) |
| 64 | 63 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 65 | 54, 61, 64 | 3eqtrd 2781 |
. . . . . . 7
⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 66 | 65 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊)) |
| 67 | | hlol 39362 |
. . . . . . . 8
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
| 68 | 2, 67 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ OL) |
| 69 | 19, 15, 7 | hlatjcl 39368 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
| 70 | 2, 55, 12, 69 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
| 71 | 19, 7 | atbase 39290 |
. . . . . . . . 9
⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾)) |
| 72 | 28, 71 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
| 73 | 19, 15 | latjcl 18484 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾)) |
| 74 | 18, 70, 72, 73 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾)) |
| 75 | 1 | simprd 495 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
| 76 | 19, 8 | lhpbase 40000 |
. . . . . . . 8
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
| 77 | 75, 76 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
| 78 | 19, 31 | latm32 39232 |
. . . . . . 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))) |
| 79 | 68, 74, 23, 77, 78 | syl13anc 1374 |
. . . . . 6
⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 80 | 6, 15, 31, 7, 8, 9,
24 | trlval2 40165 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
| 81 | 1, 4, 5, 80 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) |
| 82 | 81 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈)) |
| 83 | 27 | simprd 495 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ≤ 𝑊) |
| 84 | 19, 6, 15, 31, 7 | atmod4i1 39868 |
. . . . . . . . 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)) |
| 85 | 2, 28, 70, 77, 83, 84 | syl131anc 1385 |
. . . . . . . 8
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊)) |
| 86 | 82, 85 | eqtr2d 2778 |
. . . . . . 7
⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈)) |
| 87 | 86 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 88 | 66, 79, 87 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))) |
| 89 | 51, 88 | eqtr2d 2778 |
. . . 4
⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷)) |
| 90 | 34, 45, 89 | 3brtr3d 5174 |
. . 3
⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷)) |
| 91 | | hlatl 39361 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 92 | 2, 91 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ AtLat) |
| 93 | | hlop 39363 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
| 94 | 2, 93 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ OP) |
| 95 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 96 | | eqid 2737 |
. . . . . . . . . 10
⊢
(lt‘𝐾) =
(lt‘𝐾) |
| 97 | 95, 96, 7 | 0ltat 39292 |
. . . . . . . . 9
⊢ ((𝐾 ∈ OP ∧ 𝑉 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉) |
| 98 | 94, 14, 97 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉) |
| 99 | | hlpos 39367 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
| 100 | 2, 99 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ Poset) |
| 101 | 19, 95 | op0cl 39185 |
. . . . . . . . . 10
⊢ (𝐾 ∈ OP →
(0.‘𝐾) ∈
(Base‘𝐾)) |
| 102 | 94, 101 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾)) |
| 103 | 19, 8, 9, 24 | trlcl 40166 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → (𝑅‘𝐷) ∈ (Base‘𝐾)) |
| 104 | 1, 46, 103 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (𝑅‘𝐷) ∈ (Base‘𝐾)) |
| 105 | 19, 6, 96 | pltletr 18388 |
. . . . . . . . 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))) |
| 106 | 100, 102,
21, 104, 105 | syl13anc 1374 |
. . . . . . . 8
⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))) |
| 107 | 98, 90, 106 | mp2and 699 |
. . . . . . 7
⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷)) |
| 108 | 19, 96, 95 | opltn0 39191 |
. . . . . . . 8
⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾))) |
| 109 | 94, 104, 108 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾))) |
| 110 | 107, 109 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (𝑅‘𝐷) ≠ (0.‘𝐾)) |
| 111 | 110 | neneqd 2945 |
. . . . 5
⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾)) |
| 112 | 95, 7, 8, 9, 24 | trlator0 40173 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾))) |
| 113 | 1, 46, 112 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾))) |
| 114 | 113 | orcomd 872 |
. . . . . 6
⊢ (𝜑 → ((𝑅‘𝐷) = (0.‘𝐾) ∨ (𝑅‘𝐷) ∈ 𝐴)) |
| 115 | 114 | ord 865 |
. . . . 5
⊢ (𝜑 → (¬ (𝑅‘𝐷) = (0.‘𝐾) → (𝑅‘𝐷) ∈ 𝐴)) |
| 116 | 111, 115 | mpd 15 |
. . . 4
⊢ (𝜑 → (𝑅‘𝐷) ∈ 𝐴) |
| 117 | 6, 7 | atcmp 39312 |
. . . 4
⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴 ∧ (𝑅‘𝐷) ∈ 𝐴) → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷))) |
| 118 | 92, 14, 116, 117 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷))) |
| 119 | 90, 118 | mpbid 232 |
. 2
⊢ (𝜑 → 𝑉 = (𝑅‘𝐷)) |
| 120 | 119 | eqcomd 2743 |
1
⊢ (𝜑 → (𝑅‘𝐷) = 𝑉) |