Proof of Theorem atmod2i1
Step | Hyp | Ref
| Expression |
1 | | hllat 37377 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
2 | 1 | 3ad2ant1 1132 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ Lat) |
3 | | simp22 1206 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵) |
4 | | simp23 1207 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → 𝑌 ∈ 𝐵) |
5 | | atmod.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐾) |
6 | | atmod.m |
. . . . 5
⊢ ∧ =
(meet‘𝐾) |
7 | 5, 6 | latmcom 18181 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
8 | 2, 3, 4, 7 | syl3anc 1370 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
9 | 8 | oveq2d 7291 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = (𝑃 ∨ (𝑌 ∧ 𝑋))) |
10 | | simp21 1205 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴) |
11 | | atmod.a |
. . . . 5
⊢ 𝐴 = (Atoms‘𝐾) |
12 | 5, 11 | atbase 37303 |
. . . 4
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
13 | 10, 12 | syl 17 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵) |
14 | 5, 6 | latmcl 18158 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
15 | 2, 3, 4, 14 | syl3anc 1370 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
16 | | atmod.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
17 | 5, 16 | latjcom 18165 |
. . 3
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = ((𝑋 ∧ 𝑌) ∨ 𝑃)) |
18 | 2, 13, 15, 17 | syl3anc 1370 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = ((𝑋 ∧ 𝑌) ∨ 𝑃)) |
19 | 5, 16 | latjcl 18157 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃 ∨ 𝑌) ∈ 𝐵) |
20 | 2, 13, 4, 19 | syl3anc 1370 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ 𝑌) ∈ 𝐵) |
21 | 5, 6 | latmcom 18181 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ 𝑌) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑌))) |
22 | 2, 20, 3, 21 | syl3anc 1370 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → ((𝑃 ∨ 𝑌) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑌))) |
23 | | simp1 1135 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ HL) |
24 | | simp3 1137 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → 𝑃 ≤ 𝑋) |
25 | | atmod.l |
. . . . 5
⊢ ≤ =
(le‘𝐾) |
26 | 5, 25, 16, 6, 11 | atmod1i1 37871 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑌 ∧ 𝑋)) = ((𝑃 ∨ 𝑌) ∧ 𝑋)) |
27 | 23, 10, 4, 3, 24, 26 | syl131anc 1382 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑌 ∧ 𝑋)) = ((𝑃 ∨ 𝑌) ∧ 𝑋)) |
28 | 5, 16 | latjcom 18165 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑌 ∨ 𝑃) = (𝑃 ∨ 𝑌)) |
29 | 2, 4, 13, 28 | syl3anc 1370 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑌 ∨ 𝑃) = (𝑃 ∨ 𝑌)) |
30 | 29 | oveq2d 7291 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑋 ∧ (𝑌 ∨ 𝑃)) = (𝑋 ∧ (𝑃 ∨ 𝑌))) |
31 | 22, 27, 30 | 3eqtr4d 2788 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑌 ∧ 𝑋)) = (𝑋 ∧ (𝑌 ∨ 𝑃))) |
32 | 9, 18, 31 | 3eqtr3d 2786 |
1
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∨ 𝑃) = (𝑋 ∧ (𝑌 ∨ 𝑃))) |