Proof of Theorem 2llnmeqat
Step | Hyp | Ref
| Expression |
1 | | simp3r 1200 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑃 ≤ (𝑋 ∧ 𝑌)) |
2 | | hlatl 37353 |
. . . 4
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
3 | 2 | 3ad2ant1 1131 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝐾 ∈ AtLat) |
4 | | simp23 1206 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑃 ∈ 𝐴) |
5 | | simp1 1134 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝐾 ∈ HL) |
6 | | simp21 1204 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑋 ∈ 𝑁) |
7 | | simp22 1205 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑌 ∈ 𝑁) |
8 | | simp3l 1199 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑋 ≠ 𝑌) |
9 | | hllat 37356 |
. . . . . . . 8
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
10 | 9 | 3ad2ant1 1131 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝐾 ∈ Lat) |
11 | | eqid 2739 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
12 | | 2llnmeqat.a |
. . . . . . . . 9
⊢ 𝐴 = (Atoms‘𝐾) |
13 | 11, 12 | atbase 37282 |
. . . . . . . 8
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
14 | 4, 13 | syl 17 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑃 ∈ (Base‘𝐾)) |
15 | | 2llnmeqat.n |
. . . . . . . . 9
⊢ 𝑁 = (LLines‘𝐾) |
16 | 11, 15 | llnbase 37502 |
. . . . . . . 8
⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾)) |
17 | 6, 16 | syl 17 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑋 ∈ (Base‘𝐾)) |
18 | 11, 15 | llnbase 37502 |
. . . . . . . 8
⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾)) |
19 | 7, 18 | syl 17 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑌 ∈ (Base‘𝐾)) |
20 | | 2llnmeqat.l |
. . . . . . . 8
⊢ ≤ =
(le‘𝐾) |
21 | | 2llnmeqat.m |
. . . . . . . 8
⊢ ∧ =
(meet‘𝐾) |
22 | 11, 20, 21 | latlem12 18165 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌) ↔ 𝑃 ≤ (𝑋 ∧ 𝑌))) |
23 | 10, 14, 17, 19, 22 | syl13anc 1370 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → ((𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌) ↔ 𝑃 ≤ (𝑋 ∧ 𝑌))) |
24 | 1, 23 | mpbird 256 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) |
25 | | eqid 2739 |
. . . . . 6
⊢
(0.‘𝐾) =
(0.‘𝐾) |
26 | 20, 21, 25, 12, 15 | 2llnm4 37563 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾)) |
27 | 5, 4, 6, 7, 24, 26 | syl131anc 1381 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾)) |
28 | 21, 25, 12, 15 | 2llnmat 37517 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑋 ≠ 𝑌 ∧ (𝑋 ∧ 𝑌) ≠ (0.‘𝐾))) → (𝑋 ∧ 𝑌) ∈ 𝐴) |
29 | 5, 6, 7, 8, 27, 28 | syl32anc 1376 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → (𝑋 ∧ 𝑌) ∈ 𝐴) |
30 | 20, 12 | atcmp 37304 |
. . 3
⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑃 ≤ (𝑋 ∧ 𝑌) ↔ 𝑃 = (𝑋 ∧ 𝑌))) |
31 | 3, 4, 29, 30 | syl3anc 1369 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → (𝑃 ≤ (𝑋 ∧ 𝑌) ↔ 𝑃 = (𝑋 ∧ 𝑌))) |
32 | 1, 31 | mpbid 231 |
1
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 𝑌 ∧ 𝑃 ≤ (𝑋 ∧ 𝑌))) → 𝑃 = (𝑋 ∧ 𝑌)) |