Proof of Theorem dalem55
| Step | Hyp | Ref
| Expression |
| 1 | | dalem.ph |
. . . . . 6
⊢ (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 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 3 | 2 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) |
| 4 | 1 | dalemkehl 39625 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ HL) |
| 5 | 4 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
| 6 | | dalem.l |
. . . . . 6
⊢ ≤ =
(le‘𝐾) |
| 7 | | dalem.j |
. . . . . 6
⊢ ∨ =
(join‘𝐾) |
| 8 | | dalem.a |
. . . . . 6
⊢ 𝐴 = (Atoms‘𝐾) |
| 9 | | dalem.ps |
. . . . . 6
⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| 10 | | dalem54.m |
. . . . . 6
⊢ ∧ =
(meet‘𝐾) |
| 11 | | dalem54.o |
. . . . . 6
⊢ 𝑂 = (LPlanes‘𝐾) |
| 12 | | dalem54.y |
. . . . . 6
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 13 | | dalem54.z |
. . . . . 6
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| 14 | | dalem54.g |
. . . . . 6
⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
| 15 | 1, 6, 7, 8, 9, 10,
11, 12, 13, 14 | dalem23 39698 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
| 16 | | dalem54.h |
. . . . . 6
⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) |
| 17 | 1, 6, 7, 8, 9, 10,
11, 12, 13, 16 | dalem29 39703 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴) |
| 18 | | eqid 2737 |
. . . . . 6
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 19 | 18, 7, 8 | hlatjcl 39368 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 20 | 5, 15, 17, 19 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 21 | 1, 7, 8 | dalempjqeb 39647 |
. . . . 5
⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 22 | 21 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 23 | 18, 6, 10 | latmle1 18509 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻)) |
| 24 | 3, 20, 22, 23 | syl3anc 1373 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻)) |
| 25 | | dalem54.i |
. . . . . . . 8
⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) |
| 26 | 1, 6, 7, 8, 9, 10,
11, 12, 13, 25 | dalem34 39708 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴) |
| 27 | 18, 8 | atbase 39290 |
. . . . . . 7
⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾)) |
| 28 | 26, 27 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾)) |
| 29 | 18, 6, 7 | latlej1 18493 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |
| 30 | 3, 20, 28, 29 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |
| 31 | 1, 8 | dalemreb 39643 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾)) |
| 32 | 18, 6, 7 | latlej1 18493 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 33 | 2, 21, 31, 32 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 34 | 33, 12 | breqtrrdi 5185 |
. . . . . 6
⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ 𝑌) |
| 35 | 34 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ 𝑌) |
| 36 | 1, 6, 7, 8, 9, 10,
11, 12, 13, 14, 16, 25 | dalem42 39716 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂) |
| 37 | 18, 11 | lplnbase 39536 |
. . . . . . 7
⊢ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) |
| 38 | 36, 37 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) |
| 39 | 1, 11 | dalemyeb 39651 |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
| 40 | 39 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾)) |
| 41 | 18, 6, 10 | latmlem12 18516 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ (𝑃 ∨ 𝑄) ≤ 𝑌) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌))) |
| 42 | 3, 20, 38, 22, 40, 41 | syl122anc 1381 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ (𝑃 ∨ 𝑄) ≤ 𝑌) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌))) |
| 43 | 30, 35, 42 | mp2and 699 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)) |
| 44 | | dalem54.b1 |
. . . 4
⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌) |
| 45 | 43, 44 | breqtrrdi 5185 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵) |
| 46 | 18, 10 | latmcl 18485 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾)) |
| 47 | 3, 20, 22, 46 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾)) |
| 48 | | eqid 2737 |
. . . . . 6
⊢
(LLines‘𝐾) =
(LLines‘𝐾) |
| 49 | 1, 6, 7, 8, 9, 10,
48, 11, 12, 13, 14, 16, 25, 44 | dalem53 39727 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (LLines‘𝐾)) |
| 50 | 18, 48 | llnbase 39511 |
. . . . 5
⊢ (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾)) |
| 51 | 49, 50 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (Base‘𝐾)) |
| 52 | 18, 6, 10 | latlem12 18511 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵))) |
| 53 | 3, 47, 20, 51, 52 | syl13anc 1374 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵))) |
| 54 | 24, 45, 53 | mpbi2and 712 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵)) |
| 55 | | hlatl 39361 |
. . . 4
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 56 | 5, 55 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ AtLat) |
| 57 | 1, 6, 7, 8, 9, 10,
11, 12, 13, 14, 16, 25 | dalem52 39726 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴) |
| 58 | 1, 6, 7, 8, 9, 10,
11, 12, 13, 14, 16, 25, 44 | dalem54 39728 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴) |
| 59 | 6, 8 | atcmp 39312 |
. . 3
⊢ ((𝐾 ∈ AtLat ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴) → (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))) |
| 60 | 56, 57, 58, 59 | syl3anc 1373 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))) |
| 61 | 54, 60 | mpbid 232 |
1
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵)) |