Proof of Theorem dalem39
| Step | Hyp | Ref
| Expression |
| 1 | | dalem.ph |
. . . . 5
⊢ (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 | dalemkehl 39625 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ HL) |
| 3 | 2 | 3ad2ant1 1134 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
| 4 | 1 | dalemyeo 39634 |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ 𝑂) |
| 5 | 4 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ 𝑂) |
| 6 | | dalem.ps |
. . . . . 6
⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| 7 | 6 | dalemccea 39685 |
. . . . 5
⊢ (𝜓 → 𝑐 ∈ 𝐴) |
| 8 | 7 | 3ad2ant3 1136 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴) |
| 9 | 6 | dalem-ccly 39687 |
. . . . 5
⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌) |
| 10 | 9 | 3ad2ant3 1136 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌) |
| 11 | | dalem.l |
. . . . 5
⊢ ≤ =
(le‘𝐾) |
| 12 | | dalem.j |
. . . . 5
⊢ ∨ =
(join‘𝐾) |
| 13 | | dalem.a |
. . . . 5
⊢ 𝐴 = (Atoms‘𝐾) |
| 14 | | dalem38.o |
. . . . 5
⊢ 𝑂 = (LPlanes‘𝐾) |
| 15 | | eqid 2737 |
. . . . 5
⊢
(LVols‘𝐾) =
(LVols‘𝐾) |
| 16 | 11, 12, 13, 14, 15 | lvoli3 39579 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾)) |
| 17 | 3, 5, 8, 10, 16 | syl31anc 1375 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾)) |
| 18 | | dalem38.m |
. . . 4
⊢ ∧ =
(meet‘𝐾) |
| 19 | | dalem38.y |
. . . 4
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 20 | | dalem38.z |
. . . 4
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| 21 | | dalem38.i |
. . . 4
⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) |
| 22 | 1, 11, 12, 13, 6, 18, 14, 19, 20, 21 | dalem34 39708 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴) |
| 23 | | dalem38.g |
. . . 4
⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
| 24 | 1, 11, 12, 13, 6, 18, 14, 19, 20, 23 | dalem23 39698 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
| 25 | 11, 12, 13, 15 | lvolnle3at 39584 |
. . 3
⊢ (((𝐾 ∈ HL ∧ (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾)) ∧ (𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ¬ (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐)) |
| 26 | 3, 17, 22, 24, 8, 25 | syl23anc 1379 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐)) |
| 27 | | dalem38.h |
. . . . . . 7
⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) |
| 28 | 1, 11, 12, 13, 6, 18, 14, 19, 20, 23, 27, 21 | dalem38 39712 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) |
| 29 | 1 | dalemkelat 39626 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 30 | 29 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) |
| 31 | 1, 11, 12, 13, 6, 18, 14, 19, 20, 27 | dalem29 39703 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴) |
| 32 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 33 | 32, 12, 13 | hlatjcl 39368 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 34 | 3, 24, 31, 33 | syl3anc 1373 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 35 | 32, 13 | atbase 39290 |
. . . . . . . . 9
⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾)) |
| 36 | 22, 35 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾)) |
| 37 | 32, 12 | latjcl 18484 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) |
| 38 | 30, 34, 36, 37 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) |
| 39 | 6, 13 | dalemcceb 39691 |
. . . . . . . 8
⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾)) |
| 40 | 39 | 3ad2ant3 1136 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾)) |
| 41 | 32, 11, 12 | latlej2 18494 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) |
| 42 | 30, 38, 40, 41 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) |
| 43 | 1, 14 | dalemyeb 39651 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
| 44 | 43 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾)) |
| 45 | 32, 12 | latjcl 18484 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾)) |
| 46 | 30, 38, 40, 45 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾)) |
| 47 | 32, 11, 12 | latjle12 18495 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑌 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))) → ((𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∧ 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) ↔ (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))) |
| 48 | 30, 44, 40, 46, 47 | syl13anc 1374 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∧ 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) ↔ (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))) |
| 49 | 28, 42, 48 | mpbi2and 712 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) |
| 50 | 12, 13 | hlatjrot 39374 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻)) |
| 51 | 3, 24, 31, 22, 50 | syl13anc 1374 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻)) |
| 52 | 51 | oveq1d 7446 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐)) |
| 53 | 49, 52 | breqtrd 5169 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐)) |
| 54 | 53 | adantr 480 |
. . 3
⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐)) |
| 55 | 32, 13 | atbase 39290 |
. . . . . . 7
⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾)) |
| 56 | 31, 55 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾)) |
| 57 | 32, 12, 13 | hlatjcl 39368 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾)) |
| 58 | 3, 22, 24, 57 | syl3anc 1373 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾)) |
| 59 | 32, 11, 12 | latleeqj2 18497 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 ∨ 𝐺) ∈ (Base‘𝐾)) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺))) |
| 60 | 30, 56, 58, 59 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺))) |
| 61 | 60 | biimpa 476 |
. . . 4
⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺)) |
| 62 | 61 | oveq1d 7446 |
. . 3
⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐) = ((𝐼 ∨ 𝐺) ∨ 𝑐)) |
| 63 | 54, 62 | breqtrd 5169 |
. 2
⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐)) |
| 64 | 26, 63 | mtand 816 |
1
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐻 ≤ (𝐼 ∨ 𝐺)) |