Proof of Theorem 4atexlemswapqr
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 4thatlem.ph | . . . 4
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) | 
| 2 |  | simp11 1204 | . . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 3 | 1, 2 | sylbi 217 | . . 3
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 4 | 1 | 4atexlempw 40051 | . . 3
⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | 
| 5 |  | simp22 1208 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) | 
| 6 |  | 3simpa 1149 | . . . . 5
⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) | 
| 7 | 5, 6 | syl 17 | . . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) | 
| 8 | 1, 7 | sylbi 217 | . . 3
⊢ (𝜑 → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) | 
| 9 | 3, 4, 8 | 3jca 1129 | . 2
⊢ (𝜑 → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) | 
| 10 | 1 | 4atexlems 40054 | . . 3
⊢ (𝜑 → 𝑆 ∈ 𝐴) | 
| 11 | 1 | 4atexlemq 40053 | . . . 4
⊢ (𝜑 → 𝑄 ∈ 𝐴) | 
| 12 |  | simp13r 1290 | . . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑄 ≤ 𝑊) | 
| 13 | 1, 12 | sylbi 217 | . . . 4
⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊) | 
| 14 | 1 | 4atexlemkc 40060 | . . . . 5
⊢ (𝜑 → 𝐾 ∈ CvLat) | 
| 15 | 1 | 4atexlemp 40052 | . . . . 5
⊢ (𝜑 → 𝑃 ∈ 𝐴) | 
| 16 | 8 | simpld 494 | . . . . 5
⊢ (𝜑 → 𝑅 ∈ 𝐴) | 
| 17 | 1 | 4atexlempnq 40057 | . . . . 5
⊢ (𝜑 → 𝑃 ≠ 𝑄) | 
| 18 |  | simp223 1317 | . . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) | 
| 19 | 1, 18 | sylbi 217 | . . . . 5
⊢ (𝜑 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) | 
| 20 |  | 4thatlemslps.a | . . . . . 6
⊢ 𝐴 = (Atoms‘𝐾) | 
| 21 |  | 4thatlemslps.j | . . . . . 6
⊢  ∨ =
(join‘𝐾) | 
| 22 | 20, 21 | cvlsupr7 39349 | . . . . 5
⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) | 
| 23 | 14, 15, 11, 16, 17, 19, 22 | syl132anc 1390 | . . . 4
⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) | 
| 24 | 11, 13, 23 | 3jca 1129 | . . 3
⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))) | 
| 25 | 1 | 4atexlemt 40055 | . . . 4
⊢ (𝜑 → 𝑇 ∈ 𝐴) | 
| 26 |  | 4thatlemsw.u | . . . . . . 7
⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | 
| 27 | 20, 21 | cvlsupr8 39350 | . . . . . . . . 9
⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)) | 
| 28 | 14, 15, 11, 16, 17, 19, 27 | syl132anc 1390 | . . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)) | 
| 29 | 28 | oveq1d 7446 | . . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊)) | 
| 30 | 26, 29 | eqtrid 2789 | . . . . . 6
⊢ (𝜑 → 𝑈 = ((𝑃 ∨ 𝑅) ∧ 𝑊)) | 
| 31 | 30 | oveq1d 7446 | . . . . 5
⊢ (𝜑 → (𝑈 ∨ 𝑇) = (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇)) | 
| 32 | 1 | 4atexlemutvt 40056 | . . . . 5
⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇)) | 
| 33 | 31, 32 | eqtr3d 2779 | . . . 4
⊢ (𝜑 → (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇)) | 
| 34 | 25, 33 | jca 511 | . . 3
⊢ (𝜑 → (𝑇 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇))) | 
| 35 | 10, 24, 34 | 3jca 1129 | . 2
⊢ (𝜑 → (𝑆 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇)))) | 
| 36 | 20, 21 | cvlsupr5 39347 | . . . . 5
⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑃) | 
| 37 | 36 | necomd 2996 | . . . 4
⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑃 ≠ 𝑅) | 
| 38 | 14, 15, 11, 16, 17, 19, 37 | syl132anc 1390 | . . 3
⊢ (𝜑 → 𝑃 ≠ 𝑅) | 
| 39 | 1 | 4atexlemnslpq 40058 | . . . 4
⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) | 
| 40 | 28 | eqcomd 2743 | . . . . 5
⊢ (𝜑 → (𝑃 ∨ 𝑅) = (𝑃 ∨ 𝑄)) | 
| 41 | 40 | breq2d 5155 | . . . 4
⊢ (𝜑 → (𝑆 ≤ (𝑃 ∨ 𝑅) ↔ 𝑆 ≤ (𝑃 ∨ 𝑄))) | 
| 42 | 39, 41 | mtbird 325 | . . 3
⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑅)) | 
| 43 | 38, 42 | jca 511 | . 2
⊢ (𝜑 → (𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑅))) | 
| 44 | 9, 35, 43 | 3jca 1129 | 1
⊢ (𝜑 → (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑅)))) |