Proof of Theorem dalem44
| Step | Hyp | Ref
| Expression |
| 1 | | dalem.ph |
. . . 4
⊢ (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 | | dalem.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
| 3 | | dalem.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
| 4 | | dalem.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
| 5 | | dalem.ps |
. . . 4
⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| 6 | | dalem44.m |
. . . 4
⊢ ∧ =
(meet‘𝐾) |
| 7 | | dalem44.o |
. . . 4
⊢ 𝑂 = (LPlanes‘𝐾) |
| 8 | | dalem44.y |
. . . 4
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 9 | | dalem44.z |
. . . 4
⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| 10 | | dalem44.g |
. . . 4
⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
| 11 | | dalem44.h |
. . . 4
⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) |
| 12 | | dalem44.i |
. . . 4
⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12 | dalem43 39717 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) |
| 14 | 13 | necomd 2996 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≠ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |
| 15 | 1 | dalemkelat 39626 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 16 | 15 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) |
| 17 | 5, 4 | dalemcceb 39691 |
. . . . . . 7
⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾)) |
| 18 | 17 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾)) |
| 19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12 | dalem42 39716 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂) |
| 20 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 21 | 20, 7 | lplnbase 39536 |
. . . . . . 7
⊢ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) |
| 22 | 19, 21 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) |
| 23 | 20, 2, 3 | latleeqj1 18496 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 24 | 16, 18, 22, 23 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | dalem28 39702 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝐺 ∨ 𝑐)) |
| 26 | 1 | dalemkehl 39625 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐾 ∈ HL) |
| 27 | 26 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
| 28 | 5 | dalemccea 39685 |
. . . . . . . . . . . . . 14
⊢ (𝜓 → 𝑐 ∈ 𝐴) |
| 29 | 28 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴) |
| 30 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | dalem23 39698 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
| 31 | 3, 4 | hlatjcom 39369 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑐 ∨ 𝐺) = (𝐺 ∨ 𝑐)) |
| 32 | 27, 29, 30, 31 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐺) = (𝐺 ∨ 𝑐)) |
| 33 | 25, 32 | breqtrrd 5171 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝑐 ∨ 𝐺)) |
| 34 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11 | dalem33 39707 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐)) |
| 35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11 | dalem29 39703 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴) |
| 36 | 3, 4 | hlatjcom 39369 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑐 ∨ 𝐻) = (𝐻 ∨ 𝑐)) |
| 37 | 27, 29, 35, 36 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐻) = (𝐻 ∨ 𝑐)) |
| 38 | 34, 37 | breqtrrd 5171 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝑐 ∨ 𝐻)) |
| 39 | 1, 4 | dalempeb 39641 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
| 40 | 39 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾)) |
| 41 | 20, 3, 4 | hlatjcl 39368 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑐 ∨ 𝐺) ∈ (Base‘𝐾)) |
| 42 | 27, 29, 30, 41 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐺) ∈ (Base‘𝐾)) |
| 43 | 1, 4 | dalemqeb 39642 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
| 44 | 43 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ (Base‘𝐾)) |
| 45 | 20, 3, 4 | hlatjcl 39368 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑐 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 46 | 27, 29, 35, 45 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 47 | 20, 2, 3 | latjlej12 18500 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝐺) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝐻) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑐 ∨ 𝐺) ∧ 𝑄 ≤ (𝑐 ∨ 𝐻)) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻)))) |
| 48 | 16, 40, 42, 44, 46, 47 | syl122anc 1381 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ≤ (𝑐 ∨ 𝐺) ∧ 𝑄 ≤ (𝑐 ∨ 𝐻)) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻)))) |
| 49 | 33, 38, 48 | mp2and 699 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻))) |
| 50 | 20, 4 | atbase 39290 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾)) |
| 51 | 30, 50 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾)) |
| 52 | 20, 4 | atbase 39290 |
. . . . . . . . . . . 12
⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾)) |
| 53 | 35, 52 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾)) |
| 54 | 20, 3 | latjjdi 18536 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾))) → (𝑐 ∨ (𝐺 ∨ 𝐻)) = ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻))) |
| 55 | 16, 18, 51, 53, 54 | syl13anc 1374 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ (𝐺 ∨ 𝐻)) = ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻))) |
| 56 | 49, 55 | breqtrrd 5171 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻))) |
| 57 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 12 | dalem37 39711 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐)) |
| 58 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 12 | dalem34 39708 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴) |
| 59 | 3, 4 | hlatjcom 39369 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) → (𝑐 ∨ 𝐼) = (𝐼 ∨ 𝑐)) |
| 60 | 27, 29, 58, 59 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐼) = (𝐼 ∨ 𝑐)) |
| 61 | 57, 60 | breqtrrd 5171 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝑐 ∨ 𝐼)) |
| 62 | 1, 3, 4 | dalempjqeb 39647 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 63 | 62 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 64 | 20, 3, 4 | hlatjcl 39368 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 65 | 27, 30, 35, 64 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
| 66 | 20, 3 | latjcl 18484 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) → (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾)) |
| 67 | 16, 18, 65, 66 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾)) |
| 68 | 1, 4 | dalemreb 39643 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾)) |
| 69 | 68 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾)) |
| 70 | 20, 3, 4 | hlatjcl 39368 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) → (𝑐 ∨ 𝐼) ∈ (Base‘𝐾)) |
| 71 | 27, 29, 58, 70 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐼) ∈ (Base‘𝐾)) |
| 72 | 20, 2, 3 | latjlej12 18500 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝐼) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻)) ∧ 𝑅 ≤ (𝑐 ∨ 𝐼)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼)))) |
| 73 | 16, 63, 67, 69, 71, 72 | syl122anc 1381 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻)) ∧ 𝑅 ≤ (𝑐 ∨ 𝐼)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼)))) |
| 74 | 56, 61, 73 | mp2and 699 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼))) |
| 75 | 20, 4 | atbase 39290 |
. . . . . . . . . 10
⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾)) |
| 76 | 58, 75 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾)) |
| 77 | 20, 3 | latjjdi 18536 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾))) → (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼))) |
| 78 | 16, 18, 65, 76, 77 | syl13anc 1374 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼))) |
| 79 | 74, 78 | breqtrrd 5171 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 80 | 8, 79 | eqbrtrid 5178 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 81 | | breq2 5147 |
. . . . . 6
⊢ ((𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼) → (𝑌 ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) ↔ 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 82 | 80, 81 | syl5ibcom 245 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 83 | 24, 82 | sylbid 240 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 84 | 1 | dalemyeo 39634 |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ 𝑂) |
| 85 | 84 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ 𝑂) |
| 86 | 2, 7 | lplncmp 39564 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂) → (𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 87 | 27, 85, 19, 86 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 88 | 83, 87 | sylibd 239 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 89 | 88 | necon3ad 2953 |
. 2
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ≠ ((𝐺 ∨ 𝐻) ∨ 𝐼) → ¬ 𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))) |
| 90 | 14, 89 | mpd 15 |
1
⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |