Proof of Theorem hlatexch4
| Step | Hyp | Ref
| Expression |
| 1 | | simp11 1204 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝐾 ∈ HL) |
| 2 | | simp2l 1200 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑅 ∈ 𝐴) |
| 3 | | simp2r 1201 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑆 ∈ 𝐴) |
| 4 | | eqid 2737 |
. . . . . . . 8
⊢
(le‘𝐾) =
(le‘𝐾) |
| 5 | | hlatexch4.j |
. . . . . . . 8
⊢ ∨ =
(join‘𝐾) |
| 6 | | hlatexch4.a |
. . . . . . . 8
⊢ 𝐴 = (Atoms‘𝐾) |
| 7 | 4, 5, 6 | hlatlej2 39377 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑆(le‘𝐾)(𝑅 ∨ 𝑆)) |
| 8 | 1, 2, 3, 7 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑆(le‘𝐾)(𝑅 ∨ 𝑆)) |
| 9 | | simp33 1212 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)) |
| 10 | 8, 9 | breqtrrd 5171 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑆(le‘𝐾)(𝑃 ∨ 𝑄)) |
| 11 | | simp12 1205 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑃 ∈ 𝐴) |
| 12 | | simp13 1206 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑄 ∈ 𝐴) |
| 13 | | simp32 1211 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑄 ≠ 𝑆) |
| 14 | 13 | necomd 2996 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑆 ≠ 𝑄) |
| 15 | 4, 5, 6 | hlatexch2 39398 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑆 ≠ 𝑄) → (𝑆(le‘𝐾)(𝑃 ∨ 𝑄) → 𝑃(le‘𝐾)(𝑆 ∨ 𝑄))) |
| 16 | 1, 3, 11, 12, 14, 15 | syl131anc 1385 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑆(le‘𝐾)(𝑃 ∨ 𝑄) → 𝑃(le‘𝐾)(𝑆 ∨ 𝑄))) |
| 17 | 10, 16 | mpd 15 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑃(le‘𝐾)(𝑆 ∨ 𝑄)) |
| 18 | 5, 6 | hlatjcom 39369 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑆 ∨ 𝑄) = (𝑄 ∨ 𝑆)) |
| 19 | 1, 3, 12, 18 | syl3anc 1373 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑆 ∨ 𝑄) = (𝑄 ∨ 𝑆)) |
| 20 | 17, 19 | breqtrd 5169 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑃(le‘𝐾)(𝑄 ∨ 𝑆)) |
| 21 | 4, 5, 6 | hlatlej2 39377 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄(le‘𝐾)(𝑃 ∨ 𝑄)) |
| 22 | 1, 11, 12, 21 | syl3anc 1373 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑄(le‘𝐾)(𝑃 ∨ 𝑄)) |
| 23 | 22, 9 | breqtrd 5169 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑄(le‘𝐾)(𝑅 ∨ 𝑆)) |
| 24 | 4, 5, 6 | hlatexch2 39398 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑄 ≠ 𝑆) → (𝑄(le‘𝐾)(𝑅 ∨ 𝑆) → 𝑅(le‘𝐾)(𝑄 ∨ 𝑆))) |
| 25 | 1, 12, 2, 3, 13, 24 | syl131anc 1385 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑄(le‘𝐾)(𝑅 ∨ 𝑆) → 𝑅(le‘𝐾)(𝑄 ∨ 𝑆))) |
| 26 | 23, 25 | mpd 15 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑅(le‘𝐾)(𝑄 ∨ 𝑆)) |
| 27 | 1 | hllatd 39365 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝐾 ∈ Lat) |
| 28 | | eqid 2737 |
. . . . . 6
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 29 | 28, 6 | atbase 39290 |
. . . . 5
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
| 30 | 11, 29 | syl 17 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑃 ∈ (Base‘𝐾)) |
| 31 | 28, 6 | atbase 39290 |
. . . . 5
⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
| 32 | 2, 31 | syl 17 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑅 ∈ (Base‘𝐾)) |
| 33 | 28, 5, 6 | hlatjcl 39368 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑄 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 34 | 1, 12, 3, 33 | syl3anc 1373 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑄 ∨ 𝑆) ∈ (Base‘𝐾)) |
| 35 | 28, 4, 5 | latjle12 18495 |
. . . 4
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑃(le‘𝐾)(𝑄 ∨ 𝑆) ∧ 𝑅(le‘𝐾)(𝑄 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑆))) |
| 36 | 27, 30, 32, 34, 35 | syl13anc 1374 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → ((𝑃(le‘𝐾)(𝑄 ∨ 𝑆) ∧ 𝑅(le‘𝐾)(𝑄 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑆))) |
| 37 | 20, 26, 36 | mpbi2and 712 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑃 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑆)) |
| 38 | | simp31 1210 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → 𝑃 ≠ 𝑅) |
| 39 | 4, 5, 6 | ps-1 39479 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ≠ 𝑅) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑆) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆))) |
| 40 | 1, 11, 2, 38, 12, 3, 39 | syl132anc 1390 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → ((𝑃 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑆) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆))) |
| 41 | 37, 40 | mpbid 232 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑄 ≠ 𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆)) |