Proof of Theorem dalem1
| Step | Hyp | Ref
| Expression |
| 1 | | dalema.ph |
. . 3
⊢ (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 | dalemclpjs 39636 |
. 2
⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆)) |
| 3 | 1 | dalem-clpjq 39639 |
. . . . . 6
⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) |
| 4 | 3 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) |
| 5 | 1 | dalemkehl 39625 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ HL) |
| 6 | 1 | dalempea 39628 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
| 7 | 1 | dalemsea 39631 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ 𝐴) |
| 8 | | dalemc.l |
. . . . . . . . . . 11
⊢ ≤ =
(le‘𝐾) |
| 9 | | dalemc.j |
. . . . . . . . . . 11
⊢ ∨ =
(join‘𝐾) |
| 10 | | dalemc.a |
. . . . . . . . . . 11
⊢ 𝐴 = (Atoms‘𝐾) |
| 11 | 8, 9, 10 | hlatlej1 39376 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑆)) |
| 12 | 5, 6, 7, 11 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑆)) |
| 13 | 12 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑃 ≤ (𝑃 ∨ 𝑆)) |
| 14 | 1 | dalemqea 39629 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ 𝐴) |
| 15 | 1 | dalemtea 39632 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ∈ 𝐴) |
| 16 | 8, 9, 10 | hlatlej1 39376 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑇)) |
| 17 | 5, 14, 15, 16 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ (𝑄 ∨ 𝑇)) |
| 18 | 17 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑄 ∨ 𝑇)) |
| 19 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) |
| 20 | 18, 19 | breqtrrd 5171 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑃 ∨ 𝑆)) |
| 21 | 1 | dalemkelat 39626 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ Lat) |
| 22 | 1, 10 | dalempeb 39641 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
| 23 | 1, 10 | dalemqeb 39642 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
| 24 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 25 | 24, 9, 10 | hlatjcl 39368 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 26 | 5, 6, 7, 25 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 27 | 24, 8, 9 | latjle12 18495 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆))) |
| 28 | 21, 22, 23, 26, 27 | syl13anc 1374 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆))) |
| 29 | 28 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆))) |
| 30 | 13, 20, 29 | mpbi2and 712 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)) |
| 31 | 1 | dalemrea 39630 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ 𝐴) |
| 32 | 1 | dalemyeo 39634 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ 𝑂) |
| 33 | | dalem1.o |
. . . . . . . . . . 11
⊢ 𝑂 = (LPlanes‘𝐾) |
| 34 | | dalem1.y |
. . . . . . . . . . 11
⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| 35 | 9, 10, 33, 34 | lplnri1 39555 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑌 ∈ 𝑂) → 𝑃 ≠ 𝑄) |
| 36 | 5, 6, 14, 31, 32, 35 | syl131anc 1385 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ≠ 𝑄) |
| 37 | 8, 9, 10 | ps-1 39479 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))) |
| 38 | 5, 6, 14, 36, 6, 7, 37 | syl132anc 1390 |
. . . . . . . 8
⊢ (𝜑 → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))) |
| 39 | 38 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))) |
| 40 | 30, 39 | mpbid 232 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)) |
| 41 | 40 | breq2d 5155 |
. . . . 5
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝐶 ≤ (𝑃 ∨ 𝑄) ↔ 𝐶 ≤ (𝑃 ∨ 𝑆))) |
| 42 | 4, 41 | mtbid 324 |
. . . 4
⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑆)) |
| 43 | 42 | ex 412 |
. . 3
⊢ (𝜑 → ((𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑆))) |
| 44 | 43 | necon2ad 2955 |
. 2
⊢ (𝜑 → (𝐶 ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))) |
| 45 | 2, 44 | mpd 15 |
1
⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇)) |