Proof of Theorem lhp2atne
| Step | Hyp | Ref
| Expression |
| 1 | | simp11 1204 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 2 | | simp12 1205 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
| 3 | | simp3 1139 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → 𝑈 ≠ 𝑉) |
| 4 | | simp2l 1200 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) |
| 5 | | simp2r 1201 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) |
| 6 | | lhp2atnle.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
| 7 | | lhp2atnle.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
| 8 | | lhp2atnle.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
| 9 | | lhp2atnle.h |
. . . 4
⊢ 𝐻 = (LHyp‘𝐾) |
| 10 | 6, 7, 8, 9 | lhp2atnle 40035 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑈 ≠ 𝑉) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ¬ 𝑉 ≤ (𝑃 ∨ 𝑈)) |
| 11 | 1, 2, 3, 4, 5, 10 | syl311anc 1386 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → ¬ 𝑉 ≤ (𝑃 ∨ 𝑈)) |
| 12 | | simp11l 1285 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → 𝐾 ∈ HL) |
| 13 | | simp13 1206 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → 𝑄 ∈ 𝐴) |
| 14 | | simp2rl 1243 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → 𝑉 ∈ 𝐴) |
| 15 | 6, 7, 8 | hlatlej2 39377 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ (𝑄 ∨ 𝑉)) |
| 16 | 12, 13, 14, 15 | syl3anc 1373 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → 𝑉 ≤ (𝑄 ∨ 𝑉)) |
| 17 | 16 | adantr 480 |
. . . . 5
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) ∧ (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑉)) → 𝑉 ≤ (𝑄 ∨ 𝑉)) |
| 18 | | simpr 484 |
. . . . 5
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) ∧ (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑉)) → (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑉)) |
| 19 | 17, 18 | breqtrrd 5171 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) ∧ (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈)) |
| 20 | 19 | ex 412 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → ((𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑉) → 𝑉 ≤ (𝑃 ∨ 𝑈))) |
| 21 | 20 | necon3bd 2954 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → (¬ 𝑉 ≤ (𝑃 ∨ 𝑈) → (𝑃 ∨ 𝑈) ≠ (𝑄 ∨ 𝑉))) |
| 22 | 11, 21 | mpd 15 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑈 ≠ 𝑉) → (𝑃 ∨ 𝑈) ≠ (𝑄 ∨ 𝑉)) |