| Step | Hyp | Ref
| Expression |
| 1 | | simplr 769 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑋 ∈ 𝑉) |
| 2 | | eqid 2737 |
. . . . . 6
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 3 | | eqid 2737 |
. . . . . 6
⊢ ( ⋖
‘𝐾) = ( ⋖
‘𝐾) |
| 4 | | eqid 2737 |
. . . . . 6
⊢
(LPlanes‘𝐾) =
(LPlanes‘𝐾) |
| 5 | | lvolnle3at.v |
. . . . . 6
⊢ 𝑉 = (LVols‘𝐾) |
| 6 | 2, 3, 4, 5 | islvol 39575 |
. . . . 5
⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))) |
| 7 | 6 | ad2antrr 726 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))) |
| 8 | 1, 7 | mpbid 232 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)) |
| 9 | 8 | simprd 495 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋) |
| 10 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄)) |
| 11 | 10 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑃 = 𝑄 → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑄) ∨ 𝑅)) |
| 12 | 11 | breq2d 5155 |
. . . . . . 7
⊢ (𝑃 = 𝑄 → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅))) |
| 13 | 12 | notbid 318 |
. . . . . 6
⊢ (𝑃 = 𝑄 → (¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅))) |
| 14 | | simp1l 1198 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL) |
| 15 | | simp3l 1202 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LPlanes‘𝐾)) |
| 16 | | simp21 1207 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑃 ∈ 𝐴) |
| 17 | | simp22 1208 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑄 ∈ 𝐴) |
| 18 | | lvolnle3at.l |
. . . . . . . . . . . . 13
⊢ ≤ =
(le‘𝐾) |
| 19 | | lvolnle3at.j |
. . . . . . . . . . . . 13
⊢ ∨ =
(join‘𝐾) |
| 20 | | lvolnle3at.a |
. . . . . . . . . . . . 13
⊢ 𝐴 = (Atoms‘𝐾) |
| 21 | 18, 19, 20, 4 | lplnnle2at 39543 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) |
| 22 | 14, 15, 16, 17, 21 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 ≤ (𝑃 ∨ 𝑄)) |
| 23 | 2, 4 | lplnbase 39536 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (LPlanes‘𝐾) → 𝑦 ∈ (Base‘𝐾)) |
| 24 | 15, 23 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (Base‘𝐾)) |
| 25 | | simp1r 1199 |
. . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ 𝑉) |
| 26 | 2, 5 | lvolbase 39580 |
. . . . . . . . . . . . . . 15
⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾)) |
| 27 | 25, 26 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾)) |
| 28 | | simp3r 1203 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋) |
| 29 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(lt‘𝐾) =
(lt‘𝐾) |
| 30 | 2, 29, 3 | cvrlt 39271 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋) |
| 31 | 14, 24, 27, 28, 30 | syl31anc 1375 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋) |
| 32 | | hlpos 39367 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
| 33 | 14, 32 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset) |
| 34 | 2, 19, 20 | hlatjcl 39368 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 35 | 14, 16, 17, 34 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 36 | 2, 18, 29 | pltletr 18388 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑃 ∨ 𝑄)) → 𝑦(lt‘𝐾)(𝑃 ∨ 𝑄))) |
| 37 | 33, 24, 27, 35, 36 | syl13anc 1374 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑃 ∨ 𝑄)) → 𝑦(lt‘𝐾)(𝑃 ∨ 𝑄))) |
| 38 | 31, 37 | mpand 695 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑃 ∨ 𝑄) → 𝑦(lt‘𝐾)(𝑃 ∨ 𝑄))) |
| 39 | 18, 29 | pltle 18378 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑃 ∨ 𝑄) → 𝑦 ≤ (𝑃 ∨ 𝑄))) |
| 40 | 14, 15, 35, 39 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)(𝑃 ∨ 𝑄) → 𝑦 ≤ (𝑃 ∨ 𝑄))) |
| 41 | 38, 40 | syld 47 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑃 ∨ 𝑄) → 𝑦 ≤ (𝑃 ∨ 𝑄))) |
| 42 | 22, 41 | mtod 198 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑃 ∨ 𝑄)) |
| 43 | 42 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑋 ≤ (𝑃 ∨ 𝑄)) |
| 44 | | simprr 773 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ (𝑃 ∨ 𝑄)) |
| 45 | 14 | hllatd 39365 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Lat) |
| 46 | | simp23 1209 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ 𝐴) |
| 47 | 2, 20 | atbase 39290 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
| 48 | 46, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾)) |
| 49 | 2, 18, 19 | latleeqj2 18497 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄))) |
| 50 | 45, 48, 35, 49 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄))) |
| 51 | 50 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄))) |
| 52 | 44, 51 | mpbid 232 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ 𝑄)) |
| 53 | 52 | breq2d 5155 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑋 ≤ (𝑃 ∨ 𝑄))) |
| 54 | 43, 53 | mtbird 325 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 55 | 54 | anassrs 467 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 56 | | simpl1l 1225 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL) |
| 57 | | simpl3l 1229 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑦 ∈ (LPlanes‘𝐾)) |
| 58 | | simpl2 1193 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) |
| 59 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) |
| 60 | 18, 19, 20, 4 | lplni2 39539 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (LPlanes‘𝐾)) |
| 61 | 56, 58, 59, 60 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (LPlanes‘𝐾)) |
| 62 | 29, 4 | lplnnlt 39567 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (LPlanes‘𝐾)) → ¬ 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 63 | 56, 57, 61, 62 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 64 | 2, 19 | latjcl 18484 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾)) |
| 65 | 45, 35, 48, 64 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾)) |
| 66 | 2, 18, 29 | pltletr 18388 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅))) |
| 67 | 33, 24, 27, 65, 66 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅))) |
| 68 | 31, 67 | mpand 695 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅))) |
| 69 | 68 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) → 𝑦(lt‘𝐾)((𝑃 ∨ 𝑄) ∨ 𝑅))) |
| 70 | 63, 69 | mtod 198 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 71 | 70 | anassrs 467 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 72 | 55, 71 | pm2.61dan 813 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 73 | | eqid 2737 |
. . . . . . . . . 10
⊢
(le‘𝐾) =
(le‘𝐾) |
| 74 | 73, 19, 20, 4 | lplnnle2at 39543 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑦(le‘𝐾)(𝑄 ∨ 𝑅)) |
| 75 | 14, 15, 17, 46, 74 | syl13anc 1374 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦(le‘𝐾)(𝑄 ∨ 𝑅)) |
| 76 | 2, 19, 20 | hlatjcl 39368 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 77 | 14, 17, 46, 76 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 78 | 2, 18, 29 | pltletr 18388 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))) |
| 79 | 33, 24, 27, 77, 78 | syl13anc 1374 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))) |
| 80 | 31, 79 | mpand 695 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑄 ∨ 𝑅) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))) |
| 81 | 73, 29 | pltle 18378 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LPlanes‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) → (𝑦(lt‘𝐾)(𝑄 ∨ 𝑅) → 𝑦(le‘𝐾)(𝑄 ∨ 𝑅))) |
| 82 | 14, 15, 77, 81 | syl3anc 1373 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)(𝑄 ∨ 𝑅) → 𝑦(le‘𝐾)(𝑄 ∨ 𝑅))) |
| 83 | 80, 82 | syld 47 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑄 ∨ 𝑅) → 𝑦(le‘𝐾)(𝑄 ∨ 𝑅))) |
| 84 | 75, 83 | mtod 198 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)) |
| 85 | 19, 20 | hlatjidm 39370 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄) |
| 86 | 14, 17, 85 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑄 ∨ 𝑄) = 𝑄) |
| 87 | 86 | oveq1d 7446 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑄 ∨ 𝑄) ∨ 𝑅) = (𝑄 ∨ 𝑅)) |
| 88 | 87 | breq2d 5155 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅) ↔ 𝑋 ≤ (𝑄 ∨ 𝑅))) |
| 89 | 84, 88 | mtbird 325 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)) |
| 90 | 13, 72, 89 | pm2.61ne 3027 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
| 91 | 90 | 3expia 1122 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑦 ∈ (LPlanes‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) |
| 92 | 91 | expd 415 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑦 ∈ (LPlanes‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))) |
| 93 | 92 | rexlimdv 3153 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (∃𝑦 ∈ (LPlanes‘𝐾)𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) |
| 94 | 9, 93 | mpd 15 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |