Proof of Theorem intnatN
| Step | Hyp | Ref
| Expression |
| 1 | | hlatl 39361 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 2 | 1 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ AtLat) |
| 3 | 2 | ad2antrr 726 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat) |
| 4 | | eqid 2737 |
. . . . . 6
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 5 | | intnat.a |
. . . . . 6
⊢ 𝐴 = (Atoms‘𝐾) |
| 6 | 4, 5 | atn0 39309 |
. . . . 5
⊢ ((𝐾 ∈ AtLat ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾)) |
| 7 | 3, 6 | sylancom 588 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾)) |
| 8 | 7 | ex 412 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∈ 𝐴 → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾))) |
| 9 | | simpll1 1213 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ HL) |
| 10 | 9 | hllatd 39365 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ Lat) |
| 11 | | simpll2 1214 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
| 12 | | simpll3 1215 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐵) |
| 13 | | intnat.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐾) |
| 14 | | intnat.m |
. . . . . . . 8
⊢ ∧ =
(meet‘𝐾) |
| 15 | 13, 14 | latmcom 18508 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
| 16 | 10, 11, 12, 15 | syl3anc 1373 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
| 17 | | simplr 769 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → ¬ 𝑌 ≤ 𝑋) |
| 18 | 9, 1 | syl 17 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ AtLat) |
| 19 | | simpr 484 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴) |
| 20 | | intnat.l |
. . . . . . . . 9
⊢ ≤ =
(le‘𝐾) |
| 21 | 13, 20, 14, 4, 5 | atnle 39318 |
. . . . . . . 8
⊢ ((𝐾 ∈ AtLat ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑌 ≤ 𝑋 ↔ (𝑌 ∧ 𝑋) = (0.‘𝐾))) |
| 22 | 18, 19, 11, 21 | syl3anc 1373 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (¬ 𝑌 ≤ 𝑋 ↔ (𝑌 ∧ 𝑋) = (0.‘𝐾))) |
| 23 | 17, 22 | mpbid 232 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∧ 𝑋) = (0.‘𝐾)) |
| 24 | 16, 23 | eqtrd 2777 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑋 ∧ 𝑌) = (0.‘𝐾)) |
| 25 | 24 | ex 412 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → (𝑌 ∈ 𝐴 → (𝑋 ∧ 𝑌) = (0.‘𝐾))) |
| 26 | 25 | necon3ad 2953 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ≠ (0.‘𝐾) → ¬ 𝑌 ∈ 𝐴)) |
| 27 | 8, 26 | syld 47 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∈ 𝐴 → ¬ 𝑌 ∈ 𝐴)) |
| 28 | 27 | impr 454 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑌 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ∈ 𝐴)) → ¬ 𝑌 ∈ 𝐴) |