Proof of Theorem hlatexch3N
Step | Hyp | Ref
| Expression |
1 | | simp1 1137 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝐾 ∈ HL) |
2 | | simp21 1207 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑃 ∈ 𝐴) |
3 | | simp22 1208 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑄 ∈ 𝐴) |
4 | | eqid 2738 |
. . . . . 6
⊢
(le‘𝐾) =
(le‘𝐾) |
5 | | hlatexch4.j |
. . . . . 6
⊢ ∨ =
(join‘𝐾) |
6 | | hlatexch4.a |
. . . . . 6
⊢ 𝐴 = (Atoms‘𝐾) |
7 | 4, 5, 6 | hlatlej2 37002 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄(le‘𝐾)(𝑃 ∨ 𝑄)) |
8 | 1, 2, 3, 7 | syl3anc 1372 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑄(le‘𝐾)(𝑃 ∨ 𝑄)) |
9 | | simp23 1209 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑅 ∈ 𝐴) |
10 | 4, 5, 6 | hlatlej2 37002 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → 𝑅(le‘𝐾)(𝑃 ∨ 𝑅)) |
11 | 1, 2, 9, 10 | syl3anc 1372 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑅(le‘𝐾)(𝑃 ∨ 𝑅)) |
12 | | simp3r 1203 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)) |
13 | 11, 12 | breqtrrd 5055 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑅(le‘𝐾)(𝑃 ∨ 𝑄)) |
14 | | hllat 36989 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
15 | 14 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝐾 ∈ Lat) |
16 | | eqid 2738 |
. . . . . . 7
⊢
(Base‘𝐾) =
(Base‘𝐾) |
17 | 16, 6 | atbase 36915 |
. . . . . 6
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
18 | 3, 17 | syl 17 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑄 ∈ (Base‘𝐾)) |
19 | 16, 6 | atbase 36915 |
. . . . . 6
⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
20 | 9, 19 | syl 17 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑅 ∈ (Base‘𝐾)) |
21 | 16, 5, 6 | hlatjcl 36993 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
22 | 1, 2, 3, 21 | syl3anc 1372 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
23 | 16, 4, 5 | latjle12 17781 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑄(le‘𝐾)(𝑃 ∨ 𝑄) ∧ 𝑅(le‘𝐾)(𝑃 ∨ 𝑄)) ↔ (𝑄 ∨ 𝑅)(le‘𝐾)(𝑃 ∨ 𝑄))) |
24 | 15, 18, 20, 22, 23 | syl13anc 1373 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → ((𝑄(le‘𝐾)(𝑃 ∨ 𝑄) ∧ 𝑅(le‘𝐾)(𝑃 ∨ 𝑄)) ↔ (𝑄 ∨ 𝑅)(le‘𝐾)(𝑃 ∨ 𝑄))) |
25 | 8, 13, 24 | mpbi2and 712 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → (𝑄 ∨ 𝑅)(le‘𝐾)(𝑃 ∨ 𝑄)) |
26 | | simp3l 1202 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → 𝑄 ≠ 𝑅) |
27 | 4, 5, 6 | ps-1 37103 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ≠ 𝑅) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ∨ 𝑅)(le‘𝐾)(𝑃 ∨ 𝑄) ↔ (𝑄 ∨ 𝑅) = (𝑃 ∨ 𝑄))) |
28 | 1, 3, 9, 26, 2, 3,
27 | syl132anc 1389 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → ((𝑄 ∨ 𝑅)(le‘𝐾)(𝑃 ∨ 𝑄) ↔ (𝑄 ∨ 𝑅) = (𝑃 ∨ 𝑄))) |
29 | 25, 28 | mpbid 235 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → (𝑄 ∨ 𝑅) = (𝑃 ∨ 𝑄)) |
30 | 29 | eqcomd 2744 |
1
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅)) |