Proof of Theorem 2atjm
| Step | Hyp | Ref
| Expression |
| 1 | | hllat 39364 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
| 2 | 1 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ Lat) |
| 3 | | simp21 1207 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐴) |
| 4 | | 2atjm.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐾) |
| 5 | | 2atjm.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
| 6 | 4, 5 | atbase 39290 |
. . . . . 6
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
| 7 | 3, 6 | syl 17 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐵) |
| 8 | | simp22 1208 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴) |
| 9 | 4, 5 | atbase 39290 |
. . . . . 6
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
| 10 | 8, 9 | syl 17 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐵) |
| 11 | | 2atjm.l |
. . . . . 6
⊢ ≤ =
(le‘𝐾) |
| 12 | | 2atjm.j |
. . . . . 6
⊢ ∨ =
(join‘𝐾) |
| 13 | 4, 11, 12 | latlej1 18493 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
| 14 | 2, 7, 10, 13 | syl3anc 1373 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ (𝑃 ∨ 𝑄)) |
| 15 | | simp3l 1202 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ 𝑋) |
| 16 | | simp1 1137 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL) |
| 17 | 4, 12, 5 | hlatjcl 39368 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
| 18 | 16, 3, 8, 17 | syl3anc 1373 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
| 19 | | simp23 1209 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵) |
| 20 | | 2atjm.m |
. . . . . 6
⊢ ∧ =
(meet‘𝐾) |
| 21 | 4, 11, 20 | latlem12 18511 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≤ 𝑋) ↔ 𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋))) |
| 22 | 2, 7, 18, 19, 21 | syl13anc 1374 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≤ 𝑋) ↔ 𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋))) |
| 23 | 14, 15, 22 | mpbi2and 712 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋)) |
| 24 | | hlatl 39361 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 25 | 24 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ AtLat) |
| 26 | 4, 20 | latmcom 18508 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄))) |
| 27 | 2, 18, 19, 26 | syl3anc 1373 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄))) |
| 28 | 19, 3, 8 | 3jca 1129 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) |
| 29 | | nbrne2 5163 |
. . . . . . 7
⊢ ((𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋) → 𝑃 ≠ 𝑄) |
| 30 | 29 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≠ 𝑄) |
| 31 | | simp3r 1203 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ¬ 𝑄 ≤ 𝑋) |
| 32 | 4, 12 | latjcl 18484 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵) |
| 33 | 2, 19, 10, 32 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑋 ∨ 𝑄) ∈ 𝐵) |
| 34 | 4, 11, 12 | latlej1 18493 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑄)) |
| 35 | 2, 19, 10, 34 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑋 ≤ (𝑋 ∨ 𝑄)) |
| 36 | 4, 11, 2, 7, 19, 33, 15, 35 | lattrd 18491 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ (𝑋 ∨ 𝑄)) |
| 37 | 4, 11, 12, 20, 5 | cvrat3 39444 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)) |
| 38 | 37 | imp 406 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴) |
| 39 | 16, 28, 30, 31, 36, 38 | syl23anc 1379 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴) |
| 40 | 27, 39 | eqeltrd 2841 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴) |
| 41 | 11, 5 | atcmp 39312 |
. . . 4
⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴) → (𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋))) |
| 42 | 25, 3, 40, 41 | syl3anc 1373 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋))) |
| 43 | 23, 42 | mpbid 232 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋)) |
| 44 | 43 | eqcomd 2743 |
1
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃) |