Proof of Theorem atlelt
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simp3r 1203 | . . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 < 𝑋) | 
| 2 |  | breq1 5146 | . . 3
⊢ (𝑃 = 𝑄 → (𝑃 < 𝑋 ↔ 𝑄 < 𝑋)) | 
| 3 | 1, 2 | syl5ibrcom 247 | . 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 = 𝑄 → 𝑃 < 𝑋)) | 
| 4 |  | simp1 1137 | . . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝐾 ∈ HL) | 
| 5 |  | simp21 1207 | . . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 ∈ 𝐴) | 
| 6 |  | simp22 1208 | . . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 ∈ 𝐴) | 
| 7 |  | atlelt.s | . . . . 5
⊢  < =
(lt‘𝐾) | 
| 8 |  | eqid 2737 | . . . . 5
⊢
(join‘𝐾) =
(join‘𝐾) | 
| 9 |  | atlelt.a | . . . . 5
⊢ 𝐴 = (Atoms‘𝐾) | 
| 10 | 7, 8, 9 | atlt 39439 | . . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃 ≠ 𝑄)) | 
| 11 | 4, 5, 6, 10 | syl3anc 1373 | . . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃 ≠ 𝑄)) | 
| 12 |  | simp3l 1202 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 ≤ 𝑋) | 
| 13 |  | simp23 1209 | . . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑋 ∈ 𝐵) | 
| 14 | 4, 6, 13 | 3jca 1129 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | 
| 15 |  | atlelt.l | . . . . . . 7
⊢  ≤ =
(le‘𝐾) | 
| 16 | 15, 7 | pltle 18378 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑄 < 𝑋 → 𝑄 ≤ 𝑋)) | 
| 17 | 14, 1, 16 | sylc 65 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 ≤ 𝑋) | 
| 18 |  | hllat 39364 | . . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | 
| 19 | 18 | 3ad2ant1 1134 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝐾 ∈ Lat) | 
| 20 |  | atlelt.b | . . . . . . . 8
⊢ 𝐵 = (Base‘𝐾) | 
| 21 | 20, 9 | atbase 39290 | . . . . . . 7
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) | 
| 22 | 5, 21 | syl 17 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 ∈ 𝐵) | 
| 23 | 20, 9 | atbase 39290 | . . . . . . 7
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) | 
| 24 | 6, 23 | syl 17 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 ∈ 𝐵) | 
| 25 | 20, 15, 8 | latjle12 18495 | . . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑃(join‘𝐾)𝑄) ≤ 𝑋)) | 
| 26 | 19, 22, 24, 13, 25 | syl13anc 1374 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑃(join‘𝐾)𝑄) ≤ 𝑋)) | 
| 27 | 12, 17, 26 | mpbi2and 712 | . . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ≤ 𝑋) | 
| 28 |  | hlpos 39367 | . . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) | 
| 29 | 28 | 3ad2ant1 1134 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝐾 ∈ Poset) | 
| 30 | 20, 8 | latjcl 18484 | . . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵) | 
| 31 | 19, 22, 24, 30 | syl3anc 1373 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵) | 
| 32 | 20, 15, 7 | pltletr 18388 | . . . . 5
⊢ ((𝐾 ∈ Poset ∧ (𝑃 ∈ 𝐵 ∧ (𝑃(join‘𝐾)𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) ≤ 𝑋) → 𝑃 < 𝑋)) | 
| 33 | 29, 22, 31, 13, 32 | syl13anc 1374 | . . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) ≤ 𝑋) → 𝑃 < 𝑋)) | 
| 34 | 27, 33 | mpan2d 694 | . . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) → 𝑃 < 𝑋)) | 
| 35 | 11, 34 | sylbird 260 | . 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 ≠ 𝑄 → 𝑃 < 𝑋)) | 
| 36 | 3, 35 | pm2.61dne 3028 | 1
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 < 𝑋) |