Proof of Theorem dalem-cly
| Step | Hyp | Ref
| Expression |
| 1 | | dalema.ph |
. . . . . . 7
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) |
| 2 | 1 | dalemkelat 39626 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 3 | | dalemc.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
| 4 | 1, 3 | dalemceb 39640 |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
| 5 | | dalem-cly.o |
. . . . . . 7
⊢ 𝑂 = (LPlanes‘𝐾) |
| 6 | 1, 5 | dalemyeb 39651 |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
| 7 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 8 | | dalemc.l |
. . . . . . 7
⊢ ≤ =
(le‘𝐾) |
| 9 | | dalemc.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
| 10 | 7, 8, 9 | latleeqj1 18496 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌)) |
| 11 | 2, 4, 6, 10 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌)) |
| 12 | 1 | dalemclpjs 39636 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆)) |
| 13 | 1 | dalemkehl 39625 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐾 ∈ HL) |
| 14 | | dalem-cly.y |
. . . . . . . . . . . . . . 15
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 15 | 1, 8, 9, 3, 5, 14 | dalemcea 39662 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ∈ 𝐴) |
| 16 | 1 | dalemsea 39631 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑆 ∈ 𝐴) |
| 17 | 1 | dalempea 39628 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
| 18 | 1 | dalemqea 39629 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑄 ∈ 𝐴) |
| 19 | 1 | dalem-clpjq 39639 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) |
| 20 | 8, 9, 3 | atnlej1 39381 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) → 𝐶 ≠ 𝑃) |
| 21 | 13, 15, 17, 18, 19, 20 | syl131anc 1385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ≠ 𝑃) |
| 22 | 8, 9, 3 | hlatexch1 39397 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝐶 ≠ 𝑃) → (𝐶 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝐶))) |
| 23 | 13, 15, 16, 17, 21, 22 | syl131anc 1385 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐶 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝐶))) |
| 24 | 12, 23 | mpd 15 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ≤ (𝑃 ∨ 𝐶)) |
| 25 | 9, 3 | hlatjcom 39369 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶)) |
| 26 | 13, 15, 17, 25 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶)) |
| 27 | 24, 26 | breqtrrd 5171 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆 ≤ (𝐶 ∨ 𝑃)) |
| 28 | 1 | dalemclqjt 39637 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇)) |
| 29 | 1 | dalemtea 39632 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ∈ 𝐴) |
| 30 | 1 | dalemrea 39630 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑅 ∈ 𝐴) |
| 31 | | simp312 1322 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))) → ¬ 𝐶 ≤ (𝑄 ∨ 𝑅)) |
| 32 | 1, 31 | sylbi 217 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ¬ 𝐶 ≤ (𝑄 ∨ 𝑅)) |
| 33 | 8, 9, 3 | atnlej1 39381 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅)) → 𝐶 ≠ 𝑄) |
| 34 | 13, 15, 18, 30, 32, 33 | syl131anc 1385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ≠ 𝑄) |
| 35 | 8, 9, 3 | hlatexch1 39397 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝐶 ≠ 𝑄) → (𝐶 ≤ (𝑄 ∨ 𝑇) → 𝑇 ≤ (𝑄 ∨ 𝐶))) |
| 36 | 13, 15, 29, 18, 34, 35 | syl131anc 1385 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐶 ≤ (𝑄 ∨ 𝑇) → 𝑇 ≤ (𝑄 ∨ 𝐶))) |
| 37 | 28, 36 | mpd 15 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 ≤ (𝑄 ∨ 𝐶)) |
| 38 | 9, 3 | hlatjcom 39369 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶)) |
| 39 | 13, 15, 18, 38 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶)) |
| 40 | 37, 39 | breqtrrd 5171 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ≤ (𝐶 ∨ 𝑄)) |
| 41 | 1, 3 | dalemseb 39644 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
| 42 | 7, 9, 3 | hlatjcl 39368 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾)) |
| 43 | 13, 15, 17, 42 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾)) |
| 44 | 1, 3 | dalemteb 39645 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
| 45 | 7, 9, 3 | hlatjcl 39368 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 46 | 13, 15, 18, 45 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 47 | 7, 8, 9 | latjlej12 18500 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝐶 ∨ 𝑃) ∈ (Base‘𝐾)) ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑆 ≤ (𝐶 ∨ 𝑃) ∧ 𝑇 ≤ (𝐶 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))) |
| 48 | 2, 41, 43, 44, 46, 47 | syl122anc 1381 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑆 ≤ (𝐶 ∨ 𝑃) ∧ 𝑇 ≤ (𝐶 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))) |
| 49 | 27, 40, 48 | mp2and 699 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄))) |
| 50 | 1, 3 | dalempeb 39641 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
| 51 | 1, 3 | dalemqeb 39642 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
| 52 | 7, 9 | latjjdi 18536 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐶 ∨ (𝑃 ∨ 𝑄)) = ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄))) |
| 53 | 2, 4, 50, 51, 52 | syl13anc 1374 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) = ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄))) |
| 54 | 49, 53 | breqtrrd 5171 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄))) |
| 55 | 1 | dalemclrju 39638 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈)) |
| 56 | 1 | dalemuea 39633 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑈 ∈ 𝐴) |
| 57 | | simp313 1323 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))) → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) |
| 58 | 1, 57 | sylbi 217 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) |
| 59 | 8, 9, 3 | atnlej1 39381 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) → 𝐶 ≠ 𝑅) |
| 60 | 13, 15, 30, 17, 58, 59 | syl131anc 1385 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ≠ 𝑅) |
| 61 | 8, 9, 3 | hlatexch1 39397 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝐶 ≠ 𝑅) → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶))) |
| 62 | 13, 15, 56, 30, 60, 61 | syl131anc 1385 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶))) |
| 63 | 55, 62 | mpd 15 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑈 ≤ (𝑅 ∨ 𝐶)) |
| 64 | 9, 3 | hlatjcom 39369 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶)) |
| 65 | 13, 15, 30, 64 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶)) |
| 66 | 63, 65 | breqtrrd 5171 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ≤ (𝐶 ∨ 𝑅)) |
| 67 | 1, 9, 3 | dalemsjteb 39648 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) |
| 68 | 1, 9, 3 | dalempjqeb 39647 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 69 | 7, 9 | latjcl 18484 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾)) |
| 70 | 2, 4, 68, 69 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾)) |
| 71 | 1, 3 | dalemueb 39646 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
| 72 | 7, 9, 3 | hlatjcl 39368 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 73 | 13, 15, 30, 72 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 74 | 7, 8, 9 | latjlej12 18500 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ Lat ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾)) ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))) → (((𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝐶 ∨ 𝑅)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))) |
| 75 | 2, 67, 70, 71, 73, 74 | syl122anc 1381 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝐶 ∨ 𝑅)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))) |
| 76 | 54, 66, 75 | mp2and 699 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅))) |
| 77 | 1, 3 | dalemreb 39643 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾)) |
| 78 | 7, 9 | latjjdi 18536 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)) = ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅))) |
| 79 | 2, 4, 68, 77, 78 | syl13anc 1374 |
. . . . . . . 8
⊢ (𝜑 → (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)) = ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅))) |
| 80 | 76, 79 | breqtrrd 5171 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅))) |
| 81 | | dalem-cly.z |
. . . . . . 7
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| 82 | 14 | oveq2i 7442 |
. . . . . . 7
⊢ (𝐶 ∨ 𝑌) = (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 83 | 80, 81, 82 | 3brtr4g 5177 |
. . . . . 6
⊢ (𝜑 → 𝑍 ≤ (𝐶 ∨ 𝑌)) |
| 84 | | breq2 5147 |
. . . . . 6
⊢ ((𝐶 ∨ 𝑌) = 𝑌 → (𝑍 ≤ (𝐶 ∨ 𝑌) ↔ 𝑍 ≤ 𝑌)) |
| 85 | 83, 84 | syl5ibcom 245 |
. . . . 5
⊢ (𝜑 → ((𝐶 ∨ 𝑌) = 𝑌 → 𝑍 ≤ 𝑌)) |
| 86 | 11, 85 | sylbid 240 |
. . . 4
⊢ (𝜑 → (𝐶 ≤ 𝑌 → 𝑍 ≤ 𝑌)) |
| 87 | 1 | dalemzeo 39635 |
. . . . . 6
⊢ (𝜑 → 𝑍 ∈ 𝑂) |
| 88 | 1 | dalemyeo 39634 |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ 𝑂) |
| 89 | 8, 5 | lplncmp 39564 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌)) |
| 90 | 13, 87, 88, 89 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌)) |
| 91 | | eqcom 2744 |
. . . . 5
⊢ (𝑍 = 𝑌 ↔ 𝑌 = 𝑍) |
| 92 | 90, 91 | bitrdi 287 |
. . . 4
⊢ (𝜑 → (𝑍 ≤ 𝑌 ↔ 𝑌 = 𝑍)) |
| 93 | 86, 92 | sylibd 239 |
. . 3
⊢ (𝜑 → (𝐶 ≤ 𝑌 → 𝑌 = 𝑍)) |
| 94 | 93 | necon3ad 2953 |
. 2
⊢ (𝜑 → (𝑌 ≠ 𝑍 → ¬ 𝐶 ≤ 𝑌)) |
| 95 | 94 | imp 406 |
1
⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ¬ 𝐶 ≤ 𝑌) |