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Theorem lplnnle2at 34307
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 1065 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → 𝑋𝑃)
2 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2621 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2621 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
5 lplnnle2at.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
62, 3, 4, 5islpln 34296 . . . . 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 222 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
98simprd 479 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10 oveq1 6611 . . . . . . . . 9 (𝑄 = 𝑅 → (𝑄 𝑅) = (𝑅 𝑅))
1110breq2d 4625 . . . . . . . 8 (𝑄 = 𝑅 → (𝑋 (𝑄 𝑅) ↔ 𝑋 (𝑅 𝑅)))
1211notbid 308 . . . . . . 7 (𝑄 = 𝑅 → (¬ 𝑋 (𝑄 𝑅) ↔ ¬ 𝑋 (𝑅 𝑅)))
13 simpl1 1062 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ HL)
14 simpl3l 1114 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦 ∈ (LLines‘𝐾))
15 simpl22 1138 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑄𝐴)
16 simpl23 1139 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑅𝐴)
17 simpr 477 . . . . . . . . . 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 34278 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
2113, 15, 16, 17, 20syl31anc 1326 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
22 eqid 2621 . . . . . . . . . 10 (lt‘𝐾) = (lt‘𝐾)
2322, 4llnnlt 34289 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ (𝑄 𝑅) ∈ (LLines‘𝐾)) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
2413, 14, 21, 23syl3anc 1323 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
252, 4llnbase 34275 . . . . . . . . . . 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 1137 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋𝑃)
282, 5lplnbase 34300 . . . . . . . . . . 11 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
2927, 28syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋 ∈ (Base‘𝐾))
30 simpl3r 1115 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦( ⋖ ‘𝐾)𝑋)
312, 22, 3cvrlt 34037 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3213, 26, 29, 30, 31syl31anc 1326 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦(lt‘𝐾)𝑋)
33 hlpos 34132 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3413, 33syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ Poset)
352, 18, 19hlatjcl 34133 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3613, 15, 16, 35syl3anc 1323 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (Base‘𝐾))
37 lplnnle2at.l . . . . . . . . . . 11 = (le‘𝐾)
382, 37, 22pltletr 16892 . . . . . . . . . 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 1325 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
4032, 39mpand 710 . . . . . . . 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 1059 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
43 simp3l 1087 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LLines‘𝐾))
44 simp23 1094 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
4537, 19, 4llnnleat 34279 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → ¬ 𝑦 𝑅)
4642, 43, 44, 45syl3anc 1323 . . . . . . . . 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 1092 . . . . . . . . . . . . 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 1088 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
5142, 47, 49, 50, 31syl31anc 1326 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
52333ad2ant1 1080 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
532, 19atbase 34056 . . . . . . . . . . . . 13 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
5444, 53syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
552, 37, 22pltletr 16892 . . . . . . . . . . . 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 1325 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 𝑅) → 𝑦(lt‘𝐾)𝑅))
5751, 56mpand 710 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 𝑅𝑦(lt‘𝐾)𝑅))
5837, 22pltle 16882 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → (𝑦(lt‘𝐾)𝑅𝑦 𝑅))
5942, 43, 44, 58syl3anc 1323 . . . . . . . . . 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 34135 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑅𝐴) → (𝑅 𝑅) = 𝑅)
6342, 44, 62syl2anc 692 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑅 𝑅) = 𝑅)
6463breq2d 4625 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑅 𝑅) ↔ 𝑋 𝑅))
6561, 64mtbird 315 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑅 𝑅))
6612, 41, 65pm2.61ne 2875 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
67663exp 1261 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → ((𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 (𝑄 𝑅))))
6867exp4a 632 . . . 4 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅)))))
6968imp 445 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅))))
7069rexlimdv 3023 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  Posetcpo 16861  ltcplt 16862  joincjn 16865  ccvr 34029  Atomscatm 34030  HLchlt 34117  LLinesclln 34257  LPlanesclpl 34258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-llines 34264  df-lplanes 34265
This theorem is referenced by:  lplnnleat  34308  lplnnlelln  34309  2atnelpln  34310  lvolnle3at  34348
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