| Step | Hyp | Ref
| Expression |
| 1 | | simpr1 1195 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑋 ∈ 𝑃) |
| 2 | | eqid 2737 |
. . . . . 6
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 3 | | eqid 2737 |
. . . . . 6
⊢ ( ⋖
‘𝐾) = ( ⋖
‘𝐾) |
| 4 | | eqid 2737 |
. . . . . 6
⊢
(LLines‘𝐾) =
(LLines‘𝐾) |
| 5 | | lplnnle2at.p |
. . . . . 6
⊢ 𝑃 = (LPlanes‘𝐾) |
| 6 | 2, 3, 4, 5 | islpln 39532 |
. . . . 5
⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))) |
| 7 | 6 | adantr 480 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))) |
| 8 | 1, 7 | mpbid 232 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)) |
| 9 | 8 | simprd 495 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋) |
| 10 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑄 = 𝑅 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑅)) |
| 11 | 10 | breq2d 5155 |
. . . . . . . 8
⊢ (𝑄 = 𝑅 → (𝑋 ≤ (𝑄 ∨ 𝑅) ↔ 𝑋 ≤ (𝑅 ∨ 𝑅))) |
| 12 | 11 | notbid 318 |
. . . . . . 7
⊢ (𝑄 = 𝑅 → (¬ 𝑋 ≤ (𝑄 ∨ 𝑅) ↔ ¬ 𝑋 ≤ (𝑅 ∨ 𝑅))) |
| 13 | | simpl1 1192 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝐾 ∈ HL) |
| 14 | | simpl3l 1229 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦 ∈ (LLines‘𝐾)) |
| 15 | | simpl22 1253 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ∈ 𝐴) |
| 16 | | simpl23 1254 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑅 ∈ 𝐴) |
| 17 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑄 ≠ 𝑅) |
| 18 | | lplnnle2at.j |
. . . . . . . . . . 11
⊢ ∨ =
(join‘𝐾) |
| 19 | | lplnnle2at.a |
. . . . . . . . . . 11
⊢ 𝐴 = (Atoms‘𝐾) |
| 20 | 18, 19, 4 | llni2 39514 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾)) |
| 21 | 13, 15, 16, 17, 20 | syl31anc 1375 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾)) |
| 22 | | eqid 2737 |
. . . . . . . . . 10
⊢
(lt‘𝐾) =
(lt‘𝐾) |
| 23 | 22, 4 | llnnlt 39525 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (LLines‘𝐾)) → ¬ 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)) |
| 24 | 13, 14, 21, 23 | syl3anc 1373 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → ¬ 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅)) |
| 25 | 2, 4 | llnbase 39511 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (LLines‘𝐾) → 𝑦 ∈ (Base‘𝐾)) |
| 26 | 14, 25 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦 ∈ (Base‘𝐾)) |
| 27 | | simpl21 1252 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑋 ∈ 𝑃) |
| 28 | 2, 5 | lplnbase 39536 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾)) |
| 29 | 27, 28 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑋 ∈ (Base‘𝐾)) |
| 30 | | simpl3r 1230 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦( ⋖ ‘𝐾)𝑋) |
| 31 | 2, 22, 3 | cvrlt 39271 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋) |
| 32 | 13, 26, 29, 30, 31 | syl31anc 1375 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝑦(lt‘𝐾)𝑋) |
| 33 | | hlpos 39367 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) |
| 34 | 13, 33 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → 𝐾 ∈ Poset) |
| 35 | 2, 18, 19 | hlatjcl 39368 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 36 | 13, 15, 16, 35 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) |
| 37 | | lplnnle2at.l |
. . . . . . . . . . 11
⊢ ≤ =
(le‘𝐾) |
| 38 | 2, 37, 22 | pltletr 18388 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))) |
| 39 | 34, 26, 29, 36, 38 | syl13anc 1374 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ (𝑄 ∨ 𝑅)) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))) |
| 40 | 32, 39 | mpand 695 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → (𝑋 ≤ (𝑄 ∨ 𝑅) → 𝑦(lt‘𝐾)(𝑄 ∨ 𝑅))) |
| 41 | 24, 40 | mtod 198 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄 ≠ 𝑅) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)) |
| 42 | | simp1 1137 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL) |
| 43 | | simp3l 1202 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LLines‘𝐾)) |
| 44 | | simp23 1209 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ 𝐴) |
| 45 | 37, 19, 4 | llnnleat 39515 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅 ∈ 𝐴) → ¬ 𝑦 ≤ 𝑅) |
| 46 | 42, 43, 44, 45 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 ≤ 𝑅) |
| 47 | 43, 25 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (Base‘𝐾)) |
| 48 | | simp21 1207 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ 𝑃) |
| 49 | 48, 28 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾)) |
| 50 | | simp3r 1203 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋) |
| 51 | 42, 47, 49, 50, 31 | syl31anc 1375 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋) |
| 52 | 33 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset) |
| 53 | 2, 19 | atbase 39290 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
| 54 | 44, 53 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾)) |
| 55 | 2, 37, 22 | pltletr 18388 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑅) → 𝑦(lt‘𝐾)𝑅)) |
| 56 | 52, 47, 49, 54, 55 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑅) → 𝑦(lt‘𝐾)𝑅)) |
| 57 | 51, 56 | mpand 695 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ 𝑅 → 𝑦(lt‘𝐾)𝑅)) |
| 58 | 37, 22 | pltle 18378 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅 ∈ 𝐴) → (𝑦(lt‘𝐾)𝑅 → 𝑦 ≤ 𝑅)) |
| 59 | 42, 43, 44, 58 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)𝑅 → 𝑦 ≤ 𝑅)) |
| 60 | 57, 59 | syld 47 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ 𝑅 → 𝑦 ≤ 𝑅)) |
| 61 | 46, 60 | mtod 198 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ 𝑅) |
| 62 | 18, 19 | hlatjidm 39370 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) → (𝑅 ∨ 𝑅) = 𝑅) |
| 63 | 42, 44, 62 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 ∨ 𝑅) = 𝑅) |
| 64 | 63 | breq2d 5155 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 ≤ (𝑅 ∨ 𝑅) ↔ 𝑋 ≤ 𝑅)) |
| 65 | 61, 64 | mtbird 325 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑅 ∨ 𝑅)) |
| 66 | 12, 41, 65 | pm2.61ne 3027 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)) |
| 67 | 66 | 3exp 1120 |
. . . . 5
⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ((𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)))) |
| 68 | 67 | exp4a 431 |
. . . 4
⊢ (𝐾 ∈ HL → ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))))) |
| 69 | 68 | imp 406 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)))) |
| 70 | 69 | rexlimdv 3153 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅))) |
| 71 | 9, 70 | mpd 15 |
1
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ 𝑋 ≤ (𝑄 ∨ 𝑅)) |