Proof of Theorem atcvrj2b
| Step | Hyp | Ref
| Expression |
| 1 | | simpl3l 1229 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄 ≠ 𝑅) |
| 2 | 1 | necomd 2996 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ≠ 𝑄) |
| 3 | | simpl1 1192 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL) |
| 4 | | simpl23 1254 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅 ∈ 𝐴) |
| 5 | | simpl22 1253 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄 ∈ 𝐴) |
| 6 | | atcvrj1x.j |
. . . . . . . 8
⊢ ∨ =
(join‘𝐾) |
| 7 | | atcvrj1x.c |
. . . . . . . 8
⊢ 𝐶 = ( ⋖ ‘𝐾) |
| 8 | | atcvrj1x.a |
. . . . . . . 8
⊢ 𝐴 = (Atoms‘𝐾) |
| 9 | 6, 7, 8 | atcvr2 39420 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑅 ≠ 𝑄 ↔ 𝑅𝐶(𝑄 ∨ 𝑅))) |
| 10 | 3, 4, 5, 9 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → (𝑅 ≠ 𝑄 ↔ 𝑅𝐶(𝑄 ∨ 𝑅))) |
| 11 | 2, 10 | mpbid 232 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 ∨ 𝑅)) |
| 12 | | breq1 5146 |
. . . . . 6
⊢ (𝑃 = 𝑅 → (𝑃𝐶(𝑄 ∨ 𝑅) ↔ 𝑅𝐶(𝑄 ∨ 𝑅))) |
| 13 | 12 | adantl 481 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → (𝑃𝐶(𝑄 ∨ 𝑅) ↔ 𝑅𝐶(𝑄 ∨ 𝑅))) |
| 14 | 11, 13 | mpbird 257 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 ∨ 𝑅)) |
| 15 | | simpl1 1192 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ HL) |
| 16 | | simpl2 1193 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) |
| 17 | | simpr 484 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≠ 𝑅) |
| 18 | | simpl3r 1230 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≤ (𝑄 ∨ 𝑅)) |
| 19 | | atcvrj1x.l |
. . . . . 6
⊢ ≤ =
(le‘𝐾) |
| 20 | 19, 6, 7, 8 | atcvrj1 39433 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅)) |
| 21 | 15, 16, 17, 18, 20 | syl112anc 1376 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) ∧ 𝑃 ≠ 𝑅) → 𝑃𝐶(𝑄 ∨ 𝑅)) |
| 22 | 14, 21 | pm2.61dane 3029 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃𝐶(𝑄 ∨ 𝑅)) |
| 23 | 22 | 3expia 1122 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅))) |
| 24 | | hlatl 39361 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
| 25 | 24 | ad2antrr 726 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ AtLat) |
| 26 | | simplr1 1216 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴) |
| 27 | | eqid 2737 |
. . . . . . 7
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 28 | 27, 8 | atn0 39309 |
. . . . . 6
⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
| 29 | 25, 26, 28 | syl2anc 584 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≠ (0.‘𝐾)) |
| 30 | | simpll 767 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ HL) |
| 31 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 32 | 31, 8 | atbase 39290 |
. . . . . . . 8
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
| 33 | 26, 32 | syl 17 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ (Base‘𝐾)) |
| 34 | | simplr2 1217 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴) |
| 35 | | simplr3 1218 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴) |
| 36 | | simpr 484 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅)) |
| 37 | 31, 6, 27, 7, 8 | atcvrj0 39430 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅)) |
| 38 | 30, 33, 34, 35, 36, 37 | syl131anc 1385 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅)) |
| 39 | 38 | necon3bid 2985 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄 ≠ 𝑅)) |
| 40 | 29, 39 | mpbid 232 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑅) |
| 41 | | hllat 39364 |
. . . . . . . 8
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
| 42 | 41 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ Lat) |
| 43 | 31, 8 | atbase 39290 |
. . . . . . . 8
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
| 44 | 34, 43 | syl 17 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ∈ (Base‘𝐾)) |
| 45 | 31, 8 | atbase 39290 |
. . . . . . . 8
⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
| 46 | 35, 45 | syl 17 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑅 ∈ (Base‘𝐾)) |
| 47 | 31, 6 | latjcl 18484 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 48 | 42, 44, 46, 47 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 49 | 30, 33, 48 | 3jca 1129 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))) |
| 50 | 31, 19, 7 | cvrle 39279 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅)) |
| 51 | 49, 50 | sylancom 588 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≤ (𝑄 ∨ 𝑅)) |
| 52 | 40, 51 | jca 511 |
. . 3
⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) |
| 53 | 52 | ex 412 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃𝐶(𝑄 ∨ 𝑅) → (𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)))) |
| 54 | 23, 53 | impbid 212 |
1
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑄 ≠ 𝑅 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐶(𝑄 ∨ 𝑅))) |