Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lplnnle2at Structured version   Visualization version   GIF version

Theorem lplnnle2at 35349
Description: A lattice line (or atom) cannot majorize a lattice plane. (Contributed by NM, 8-Jul-2012.)
Hypotheses
Ref Expression
lplnnle2at.l = (le‘𝐾)
lplnnle2at.j = (join‘𝐾)
lplnnle2at.a 𝐴 = (Atoms‘𝐾)
lplnnle2at.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
lplnnle2at ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → ¬ 𝑋 (𝑄 𝑅))

Proof of Theorem lplnnle2at
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpr1 1233 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → 𝑋𝑃)
2 eqid 2771 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2771 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2771 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
5 lplnnle2at.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
62, 3, 4, 5islpln 35338 . . . . 5 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
76adantr 466 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
81, 7mpbid 222 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
98simprd 483 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10 oveq1 6800 . . . . . . . . 9 (𝑄 = 𝑅 → (𝑄 𝑅) = (𝑅 𝑅))
1110breq2d 4798 . . . . . . . 8 (𝑄 = 𝑅 → (𝑋 (𝑄 𝑅) ↔ 𝑋 (𝑅 𝑅)))
1211notbid 307 . . . . . . 7 (𝑄 = 𝑅 → (¬ 𝑋 (𝑄 𝑅) ↔ ¬ 𝑋 (𝑅 𝑅)))
13 simpl1 1227 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ HL)
14 simpl3l 1286 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦 ∈ (LLines‘𝐾))
15 simpl22 1322 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑄𝐴)
16 simpl23 1324 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑅𝐴)
17 simpr 471 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑄𝑅)
18 lplnnle2at.j . . . . . . . . . . 11 = (join‘𝐾)
19 lplnnle2at.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
2018, 19, 4llni2 35320 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
2113, 15, 16, 17, 20syl31anc 1479 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
22 eqid 2771 . . . . . . . . . 10 (lt‘𝐾) = (lt‘𝐾)
2322, 4llnnlt 35331 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ (𝑄 𝑅) ∈ (LLines‘𝐾)) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
2413, 14, 21, 23syl3anc 1476 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
252, 4llnbase 35317 . . . . . . . . . . 11 (𝑦 ∈ (LLines‘𝐾) → 𝑦 ∈ (Base‘𝐾))
2614, 25syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦 ∈ (Base‘𝐾))
27 simpl21 1320 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋𝑃)
282, 5lplnbase 35342 . . . . . . . . . . 11 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
2927, 28syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋 ∈ (Base‘𝐾))
30 simpl3r 1288 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦( ⋖ ‘𝐾)𝑋)
312, 22, 3cvrlt 35079 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3213, 26, 29, 30, 31syl31anc 1479 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦(lt‘𝐾)𝑋)
33 hlpos 35174 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3413, 33syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ Poset)
352, 18, 19hlatjcl 35175 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3613, 15, 16, 35syl3anc 1476 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (Base‘𝐾))
37 lplnnle2at.l . . . . . . . . . . 11 = (le‘𝐾)
382, 37, 22pltletr 17179 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
3934, 26, 29, 36, 38syl13anc 1478 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
4032, 39mpand 675 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑋 (𝑄 𝑅) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
4124, 40mtod 189 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ¬ 𝑋 (𝑄 𝑅))
42 simp1 1130 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
43 simp3l 1243 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LLines‘𝐾))
44 simp23 1250 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
4537, 19, 4llnnleat 35321 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → ¬ 𝑦 𝑅)
4642, 43, 44, 45syl3anc 1476 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑦 𝑅)
4743, 25syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (Base‘𝐾))
48 simp21 1248 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋𝑃)
4948, 28syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑋 ∈ (Base‘𝐾))
50 simp3r 1244 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
5142, 47, 49, 50, 31syl31anc 1479 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
52333ad2ant1 1127 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
532, 19atbase 35098 . . . . . . . . . . . . 13 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
5444, 53syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
552, 37, 22pltletr 17179 . . . . . . . . . . . 12 ((𝐾 ∈ Poset ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → ((𝑦(lt‘𝐾)𝑋𝑋 𝑅) → 𝑦(lt‘𝐾)𝑅))
5652, 47, 49, 54, 55syl13anc 1478 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 𝑅) → 𝑦(lt‘𝐾)𝑅))
5751, 56mpand 675 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 𝑅𝑦(lt‘𝐾)𝑅))
5837, 22pltle 17169 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → (𝑦(lt‘𝐾)𝑅𝑦 𝑅))
5942, 43, 44, 58syl3anc 1476 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑦(lt‘𝐾)𝑅𝑦 𝑅))
6057, 59syld 47 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 𝑅𝑦 𝑅))
6146, 60mtod 189 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 𝑅)
6218, 19hlatjidm 35177 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑅𝐴) → (𝑅 𝑅) = 𝑅)
6342, 44, 62syl2anc 573 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 𝑅) = 𝑅)
6463breq2d 4798 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑅 𝑅) ↔ 𝑋 𝑅))
6561, 64mtbird 314 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑅 𝑅))
6612, 41, 65pm2.61ne 3028 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
67663exp 1112 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → ((𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 (𝑄 𝑅))))
6867exp4a 418 . . . 4 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅)))))
6968imp 393 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅))))
7069rexlimdv 3178 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅)))
719, 70mpd 15 1 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → ¬ 𝑋 (𝑄 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  Posetcpo 17148  ltcplt 17149  joincjn 17152  ccvr 35071  Atomscatm 35072  HLchlt 35159  LLinesclln 35299  LPlanesclpl 35300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307
This theorem is referenced by:  lplnnleat  35350  lplnnlelln  35351  2atnelpln  35352  lvolnle3at  35390
  Copyright terms: Public domain W3C validator