Proof of Theorem dalem38
| Step | Hyp | Ref
| Expression |
| 1 | | dalem38.y |
. 2
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 2 | | dalem.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))))) |
| 3 | | dalem.l |
. . . . . . 7
⊢ ≤ =
(le‘𝐾) |
| 4 | | dalem.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
| 5 | | dalem.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
| 6 | | dalem.ps |
. . . . . . 7
⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| 7 | | dalem38.m |
. . . . . . 7
⊢ ∧ =
(meet‘𝐾) |
| 8 | | dalem38.o |
. . . . . . 7
⊢ 𝑂 = (LPlanes‘𝐾) |
| 9 | | dalem38.z |
. . . . . . 7
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| 10 | | dalem38.g |
. . . . . . 7
⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
| 11 | 2, 3, 4, 5, 6, 7, 8, 1, 9, 10 | dalem28 39702 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝐺 ∨ 𝑐)) |
| 12 | | dalem38.h |
. . . . . . 7
⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) |
| 13 | 2, 3, 4, 5, 6, 7, 8, 1, 9, 12 | dalem33 39707 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐)) |
| 14 | 2 | dalemkelat 39626 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 15 | 14 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) |
| 16 | 2, 5 | dalempeb 39641 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
| 17 | 16 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾)) |
| 18 | 2 | dalemkehl 39625 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ HL) |
| 19 | 18 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
| 20 | 2, 3, 4, 5, 6, 7, 8, 1, 9, 10 | dalem23 39698 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
| 21 | 6 | dalemccea 39685 |
. . . . . . . . 9
⊢ (𝜓 → 𝑐 ∈ 𝐴) |
| 22 | 21 | 3ad2ant3 1136 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴) |
| 23 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 24 | 23, 4, 5 | hlatjcl 39368 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾)) |
| 25 | 19, 20, 22, 24 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾)) |
| 26 | 2, 5 | dalemqeb 39642 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
| 27 | 26 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ (Base‘𝐾)) |
| 28 | 2, 3, 4, 5, 6, 7, 8, 1, 9, 12 | dalem29 39703 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴) |
| 29 | 23, 4, 5 | hlatjcl 39368 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝐻 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾)) |
| 30 | 19, 28, 22, 29 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾)) |
| 31 | 23, 3, 4 | latjlej12 18500 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝑐) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝐺 ∨ 𝑐) ∧ 𝑄 ≤ (𝐻 ∨ 𝑐)) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))) |
| 32 | 15, 17, 25, 27, 30, 31 | syl122anc 1381 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ≤ (𝐺 ∨ 𝑐) ∧ 𝑄 ≤ (𝐻 ∨ 𝑐)) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))) |
| 33 | 11, 13, 32 | mp2and 699 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))) |
| 34 | 23, 5 | atbase 39290 |
. . . . . . 7
⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾)) |
| 35 | 20, 34 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾)) |
| 36 | 23, 5 | atbase 39290 |
. . . . . . 7
⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾)) |
| 37 | 28, 36 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾)) |
| 38 | 6, 5 | dalemcceb 39691 |
. . . . . . 7
⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾)) |
| 39 | 38 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾)) |
| 40 | 23, 4 | latjjdir 18537 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → ((𝐺 ∨ 𝐻) ∨ 𝑐) = ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))) |
| 41 | 15, 35, 37, 39, 40 | syl13anc 1374 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) = ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))) |
| 42 | 33, 41 | breqtrrd 5171 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐)) |
| 43 | | dalem38.i |
. . . . 5
⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) |
| 44 | 2, 3, 4, 5, 6, 7, 8, 1, 9, 43 | dalem37 39711 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐)) |
| 45 | 2, 4, 5 | dalempjqeb 39647 |
. . . . . 6
⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 46 | 45 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 47 | 23, 4, 5 | hlatjcl 39368 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 48 | 19, 20, 28, 47 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 49 | 23, 4 | latjcl 18484 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾)) |
| 50 | 15, 48, 39, 49 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾)) |
| 51 | 2, 5 | dalemreb 39643 |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾)) |
| 52 | 51 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾)) |
| 53 | 2, 3, 4, 5, 6, 7, 8, 1, 9, 43 | dalem34 39708 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴) |
| 54 | 23, 4, 5 | hlatjcl 39368 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾)) |
| 55 | 19, 53, 22, 54 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾)) |
| 56 | 23, 3, 4 | latjlej12 18500 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∧ 𝑅 ≤ (𝐼 ∨ 𝑐)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))) |
| 57 | 15, 46, 50, 52, 55, 56 | syl122anc 1381 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∧ 𝑅 ≤ (𝐼 ∨ 𝑐)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))) |
| 58 | 42, 44, 57 | mp2and 699 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐))) |
| 59 | 23, 5 | atbase 39290 |
. . . . 5
⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾)) |
| 60 | 53, 59 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾)) |
| 61 | 23, 4 | latjjdir 18537 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐))) |
| 62 | 15, 48, 60, 39, 61 | syl13anc 1374 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐))) |
| 63 | 58, 62 | breqtrrd 5171 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) |
| 64 | 1, 63 | eqbrtrid 5178 |
1
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) |