Proof of Theorem cdlemblem
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simp132 1309 | . . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ¬ 𝑃 ≤ 𝑋) | 
| 2 |  | simp111 1302 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝐾 ∈ HL) | 
| 3 |  | simp2l 1199 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ∈ 𝐴) | 
| 4 |  | simp12l 1286 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑋 ∈ 𝐵) | 
| 5 | 2, 3, 4 | 3jca 1128 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝐾 ∈ HL ∧ 𝑢 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | 
| 6 |  | simp2rr 1243 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 < 𝑋) | 
| 7 |  | cdlemb.l | . . . . . . 7
⊢  ≤ =
(le‘𝐾) | 
| 8 |  | cdlemblem.s | . . . . . . 7
⊢  < =
(lt‘𝐾) | 
| 9 | 7, 8 | pltle 18379 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑢 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑢 < 𝑋 → 𝑢 ≤ 𝑋)) | 
| 10 | 5, 6, 9 | sylc 65 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ≤ 𝑋) | 
| 11 | 2 | hllatd 39366 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝐾 ∈ Lat) | 
| 12 |  | simp3l 1201 | . . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ∈ 𝐴) | 
| 13 |  | cdlemb.b | . . . . . . . . 9
⊢ 𝐵 = (Base‘𝐾) | 
| 14 |  | cdlemb.a | . . . . . . . . 9
⊢ 𝐴 = (Atoms‘𝐾) | 
| 15 | 13, 14 | atbase 39291 | . . . . . . . 8
⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵) | 
| 16 | 12, 15 | syl 17 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ∈ 𝐵) | 
| 17 | 13, 14 | atbase 39291 | . . . . . . . 8
⊢ (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵) | 
| 18 | 3, 17 | syl 17 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ∈ 𝐵) | 
| 19 |  | cdlemb.j | . . . . . . . 8
⊢  ∨ =
(join‘𝐾) | 
| 20 | 13, 7, 19 | latjle12 18496 | . . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑟 ≤ 𝑋 ∧ 𝑢 ≤ 𝑋) ↔ (𝑟 ∨ 𝑢) ≤ 𝑋)) | 
| 21 | 11, 16, 18, 4, 20 | syl13anc 1373 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑟 ≤ 𝑋 ∧ 𝑢 ≤ 𝑋) ↔ (𝑟 ∨ 𝑢) ≤ 𝑋)) | 
| 22 | 21 | biimpd 229 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑟 ≤ 𝑋 ∧ 𝑢 ≤ 𝑋) → (𝑟 ∨ 𝑢) ≤ 𝑋)) | 
| 23 | 10, 22 | mpan2d 694 | . . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ 𝑋 → (𝑟 ∨ 𝑢) ≤ 𝑋)) | 
| 24 |  | simp112 1303 | . . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ∈ 𝐴) | 
| 25 | 12, 24, 3 | 3jca 1128 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) | 
| 26 |  | simp3r2 1282 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ≠ 𝑢) | 
| 27 | 2, 25, 26 | 3jca 1128 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑟 ≠ 𝑢)) | 
| 28 |  | simp3r3 1283 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ≤ (𝑃 ∨ 𝑢)) | 
| 29 | 7, 19, 14 | hlatexch2 39399 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑟 ≠ 𝑢) → (𝑟 ≤ (𝑃 ∨ 𝑢) → 𝑃 ≤ (𝑟 ∨ 𝑢))) | 
| 30 | 27, 28, 29 | sylc 65 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ≤ (𝑟 ∨ 𝑢)) | 
| 31 | 13, 14 | atbase 39291 | . . . . . . 7
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) | 
| 32 | 24, 31 | syl 17 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ∈ 𝐵) | 
| 33 | 13, 19 | latjcl 18485 | . . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵) → (𝑟 ∨ 𝑢) ∈ 𝐵) | 
| 34 | 11, 16, 18, 33 | syl3anc 1372 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ∨ 𝑢) ∈ 𝐵) | 
| 35 | 13, 7 | lattr 18490 | . . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑟 ∨ 𝑢) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ (𝑟 ∨ 𝑢) ∧ (𝑟 ∨ 𝑢) ≤ 𝑋) → 𝑃 ≤ 𝑋)) | 
| 36 | 11, 32, 34, 4, 35 | syl13anc 1373 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑃 ≤ (𝑟 ∨ 𝑢) ∧ (𝑟 ∨ 𝑢) ≤ 𝑋) → 𝑃 ≤ 𝑋)) | 
| 37 | 30, 36 | mpand 695 | . . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑟 ∨ 𝑢) ≤ 𝑋 → 𝑃 ≤ 𝑋)) | 
| 38 | 23, 37 | syld 47 | . . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ 𝑋 → 𝑃 ≤ 𝑋)) | 
| 39 | 1, 38 | mtod 198 | . 2
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ¬ 𝑟 ≤ 𝑋) | 
| 40 |  | simp2rl 1242 | . . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ≠ 𝑉) | 
| 41 |  | simp113 1304 | . . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑄 ∈ 𝐴) | 
| 42 |  | simp3r1 1281 | . . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑟 ≠ 𝑃) | 
| 43 | 7, 19, 14 | hlatexchb1 39396 | . . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ≠ 𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄))) | 
| 44 | 2, 12, 41, 24, 42, 43 | syl131anc 1384 | . . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄))) | 
| 45 | 12, 3, 24 | 3jca 1128 | . . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) | 
| 46 | 2, 45, 42 | 3jca 1128 | . . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ≠ 𝑃)) | 
| 47 | 7, 19, 14 | hlatexch1 39398 | . . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑟 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ≠ 𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑢) → 𝑢 ≤ (𝑃 ∨ 𝑟))) | 
| 48 | 46, 28, 47 | sylc 65 | . . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑢 ≤ (𝑃 ∨ 𝑟)) | 
| 49 |  | breq2 5146 | . . . . . . . . 9
⊢ ((𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄) → (𝑢 ≤ (𝑃 ∨ 𝑟) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄))) | 
| 50 | 48, 49 | syl5ibcom 245 | . . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑄) → 𝑢 ≤ (𝑃 ∨ 𝑄))) | 
| 51 | 44, 50 | sylbid 240 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑢 ≤ (𝑃 ∨ 𝑄))) | 
| 52 | 51, 10 | jctird 526 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → (𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋))) | 
| 53 | 13, 14 | atbase 39291 | . . . . . . . . . 10
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) | 
| 54 | 41, 53 | syl 17 | . . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑄 ∈ 𝐵) | 
| 55 | 13, 19 | latjcl 18485 | . . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵) | 
| 56 | 11, 32, 54, 55 | syl3anc 1372 | . . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑃 ∨ 𝑄) ∈ 𝐵) | 
| 57 |  | cdlemblem.m | . . . . . . . . 9
⊢  ∧ =
(meet‘𝐾) | 
| 58 | 13, 7, 57 | latlem12 18512 | . . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑢 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋) ↔ 𝑢 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋))) | 
| 59 | 11, 18, 56, 4, 58 | syl13anc 1373 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋) ↔ 𝑢 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋))) | 
| 60 |  | cdlemblem.v | . . . . . . . 8
⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑋) | 
| 61 | 60 | breq2i 5150 | . . . . . . 7
⊢ (𝑢 ≤ 𝑉 ↔ 𝑢 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋)) | 
| 62 | 59, 61 | bitr4di 289 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑋) ↔ 𝑢 ≤ 𝑉)) | 
| 63 | 52, 62 | sylibd 239 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑢 ≤ 𝑉)) | 
| 64 |  | hlatl 39362 | . . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | 
| 65 | 2, 64 | syl 17 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝐾 ∈ AtLat) | 
| 66 |  | simp12r 1287 | . . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑃 ≠ 𝑄) | 
| 67 |  | simp131 1308 | . . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑋𝐶 1 ) | 
| 68 |  | cdlemb.u | . . . . . . . . 9
⊢  1 =
(1.‘𝐾) | 
| 69 |  | cdlemb.c | . . . . . . . . 9
⊢ 𝐶 = ( ⋖ ‘𝐾) | 
| 70 | 13, 7, 19, 57, 68, 69, 14 | 1cvrat 39479 | . . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴) | 
| 71 | 2, 24, 41, 4, 66, 67, 1, 70 | syl133anc 1394 | . . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴) | 
| 72 | 60, 71 | eqeltrid 2844 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → 𝑉 ∈ 𝐴) | 
| 73 | 7, 14 | atcmp 39313 | . . . . . 6
⊢ ((𝐾 ∈ AtLat ∧ 𝑢 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑢 ≤ 𝑉 ↔ 𝑢 = 𝑉)) | 
| 74 | 65, 3, 72, 73 | syl3anc 1372 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑢 ≤ 𝑉 ↔ 𝑢 = 𝑉)) | 
| 75 | 63, 74 | sylibd 239 | . . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑟 ≤ (𝑃 ∨ 𝑄) → 𝑢 = 𝑉)) | 
| 76 | 75 | necon3ad 2952 | . . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (𝑢 ≠ 𝑉 → ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))) | 
| 77 | 40, 76 | mpd 15 | . 2
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)) | 
| 78 | 39, 77 | jca 511 | 1
⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄))) |