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Theorem lplnnle2at 38004
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 1194 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → 𝑋𝑃)
2 eqid 2736 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2736 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2736 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
5 lplnnle2at.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
62, 3, 4, 5islpln 37993 . . . . 5 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
76adantr 481 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
81, 7mpbid 231 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
98simprd 496 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10 oveq1 7364 . . . . . . . . 9 (𝑄 = 𝑅 → (𝑄 𝑅) = (𝑅 𝑅))
1110breq2d 5117 . . . . . . . 8 (𝑄 = 𝑅 → (𝑋 (𝑄 𝑅) ↔ 𝑋 (𝑅 𝑅)))
1211notbid 317 . . . . . . 7 (𝑄 = 𝑅 → (¬ 𝑋 (𝑄 𝑅) ↔ ¬ 𝑋 (𝑅 𝑅)))
13 simpl1 1191 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ HL)
14 simpl3l 1228 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦 ∈ (LLines‘𝐾))
15 simpl22 1252 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑄𝐴)
16 simpl23 1253 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑅𝐴)
17 simpr 485 . . . . . . . . . 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 37975 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
2113, 15, 16, 17, 20syl31anc 1373 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
22 eqid 2736 . . . . . . . . . 10 (lt‘𝐾) = (lt‘𝐾)
2322, 4llnnlt 37986 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ (𝑄 𝑅) ∈ (LLines‘𝐾)) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
2413, 14, 21, 23syl3anc 1371 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
252, 4llnbase 37972 . . . . . . . . . . 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 1251 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋𝑃)
282, 5lplnbase 37997 . . . . . . . . . . 11 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
2927, 28syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋 ∈ (Base‘𝐾))
30 simpl3r 1229 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦( ⋖ ‘𝐾)𝑋)
312, 22, 3cvrlt 37732 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3213, 26, 29, 30, 31syl31anc 1373 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦(lt‘𝐾)𝑋)
33 hlpos 37828 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3413, 33syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ Poset)
352, 18, 19hlatjcl 37829 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3613, 15, 16, 35syl3anc 1371 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (Base‘𝐾))
37 lplnnle2at.l . . . . . . . . . . 11 = (le‘𝐾)
382, 37, 22pltletr 18232 . . . . . . . . . 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 1372 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
4032, 39mpand 693 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑋 (𝑄 𝑅) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
4124, 40mtod 197 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ¬ 𝑋 (𝑄 𝑅))
42 simp1 1136 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
43 simp3l 1201 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LLines‘𝐾))
44 simp23 1208 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
4537, 19, 4llnnleat 37976 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → ¬ 𝑦 𝑅)
4642, 43, 44, 45syl3anc 1371 . . . . . . . . 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 1206 . . . . . . . . . . . . 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 1202 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
5142, 47, 49, 50, 31syl31anc 1373 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
52333ad2ant1 1133 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
532, 19atbase 37751 . . . . . . . . . . . . 13 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
5444, 53syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
552, 37, 22pltletr 18232 . . . . . . . . . . . 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 1372 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 𝑅) → 𝑦(lt‘𝐾)𝑅))
5751, 56mpand 693 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 𝑅𝑦(lt‘𝐾)𝑅))
5837, 22pltle 18222 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → (𝑦(lt‘𝐾)𝑅𝑦 𝑅))
5942, 43, 44, 58syl3anc 1371 . . . . . . . . . 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 197 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 𝑅)
6218, 19hlatjidm 37831 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑅𝐴) → (𝑅 𝑅) = 𝑅)
6342, 44, 62syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 𝑅) = 𝑅)
6463breq2d 5117 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑅 𝑅) ↔ 𝑋 𝑅))
6561, 64mtbird 324 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑅 𝑅))
6612, 41, 65pm2.61ne 3030 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
67663exp 1119 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → ((𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 (𝑄 𝑅))))
6867exp4a 432 . . . 4 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅)))))
6968imp 407 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅))))
7069rexlimdv 3150 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  Posetcpo 18196  ltcplt 18197  joincjn 18200  ccvr 37724  Atomscatm 37725  HLchlt 37812  LLinesclln 37954  LPlanesclpl 37955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962
This theorem is referenced by:  lplnnleat  38005  lplnnlelln  38006  2atnelpln  38007  lvolnle3at  38045
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