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Theorem lplnnle2at 39524
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 1193 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → 𝑋𝑃)
2 eqid 2735 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3 eqid 2735 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
4 eqid 2735 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
5 lplnnle2at.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
62, 3, 4, 5islpln 39513 . . . . 5 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
76adantr 480 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)))
81, 7mpbid 232 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋))
98simprd 495 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → ∃𝑦 ∈ (LLines‘𝐾)𝑦( ⋖ ‘𝐾)𝑋)
10 oveq1 7438 . . . . . . . . 9 (𝑄 = 𝑅 → (𝑄 𝑅) = (𝑅 𝑅))
1110breq2d 5160 . . . . . . . 8 (𝑄 = 𝑅 → (𝑋 (𝑄 𝑅) ↔ 𝑋 (𝑅 𝑅)))
1211notbid 318 . . . . . . 7 (𝑄 = 𝑅 → (¬ 𝑋 (𝑄 𝑅) ↔ ¬ 𝑋 (𝑅 𝑅)))
13 simpl1 1190 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ HL)
14 simpl3l 1227 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦 ∈ (LLines‘𝐾))
15 simpl22 1251 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑄𝐴)
16 simpl23 1252 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑅𝐴)
17 simpr 484 . . . . . . . . . 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 39495 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
2113, 15, 16, 17, 20syl31anc 1372 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (LLines‘𝐾))
22 eqid 2735 . . . . . . . . . 10 (lt‘𝐾) = (lt‘𝐾)
2322, 4llnnlt 39506 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ (𝑄 𝑅) ∈ (LLines‘𝐾)) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
2413, 14, 21, 23syl3anc 1370 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ¬ 𝑦(lt‘𝐾)(𝑄 𝑅))
252, 4llnbase 39492 . . . . . . . . . . 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 1250 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋𝑃)
282, 5lplnbase 39517 . . . . . . . . . . 11 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
2927, 28syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑋 ∈ (Base‘𝐾))
30 simpl3r 1228 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦( ⋖ ‘𝐾)𝑋)
312, 22, 3cvrlt 39252 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → 𝑦(lt‘𝐾)𝑋)
3213, 26, 29, 30, 31syl31anc 1372 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝑦(lt‘𝐾)𝑋)
33 hlpos 39348 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3413, 33syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → 𝐾 ∈ Poset)
352, 18, 19hlatjcl 39349 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3613, 15, 16, 35syl3anc 1370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑄 𝑅) ∈ (Base‘𝐾))
37 lplnnle2at.l . . . . . . . . . . 11 = (le‘𝐾)
382, 37, 22pltletr 18401 . . . . . . . . . 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 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ((𝑦(lt‘𝐾)𝑋𝑋 (𝑄 𝑅)) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
4032, 39mpand 695 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → (𝑋 (𝑄 𝑅) → 𝑦(lt‘𝐾)(𝑄 𝑅)))
4124, 40mtod 198 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) ∧ 𝑄𝑅) → ¬ 𝑋 (𝑄 𝑅))
42 simp1 1135 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ HL)
43 simp3l 1200 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦 ∈ (LLines‘𝐾))
44 simp23 1207 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅𝐴)
4537, 19, 4llnnleat 39496 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → ¬ 𝑦 𝑅)
4642, 43, 44, 45syl3anc 1370 . . . . . . . . 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 1205 . . . . . . . . . . . . 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 1201 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦( ⋖ ‘𝐾)𝑋)
5142, 47, 49, 50, 31syl31anc 1372 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑦(lt‘𝐾)𝑋)
52333ad2ant1 1132 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝐾 ∈ Poset)
532, 19atbase 39271 . . . . . . . . . . . . 13 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
5444, 53syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → 𝑅 ∈ (Base‘𝐾))
552, 37, 22pltletr 18401 . . . . . . . . . . . 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 1371 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ((𝑦(lt‘𝐾)𝑋𝑋 𝑅) → 𝑦(lt‘𝐾)𝑅))
5751, 56mpand 695 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 𝑅𝑦(lt‘𝐾)𝑅))
5837, 22pltle 18391 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑦 ∈ (LLines‘𝐾) ∧ 𝑅𝐴) → (𝑦(lt‘𝐾)𝑅𝑦 𝑅))
5942, 43, 44, 58syl3anc 1370 . . . . . . . . . 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 198 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 𝑅)
6218, 19hlatjidm 39351 . . . . . . . . . 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 5160 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → (𝑋 (𝑅 𝑅) ↔ 𝑋 𝑅))
6561, 64mtbird 325 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑅 𝑅))
6612, 41, 65pm2.61ne 3025 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴) ∧ (𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋)) → ¬ 𝑋 (𝑄 𝑅))
67663exp 1118 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → ((𝑦 ∈ (LLines‘𝐾) ∧ 𝑦( ⋖ ‘𝐾)𝑋) → ¬ 𝑋 (𝑄 𝑅))))
6867exp4a 431 . . . 4 (𝐾 ∈ HL → ((𝑋𝑃𝑄𝐴𝑅𝐴) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅)))))
6968imp 406 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑄𝐴𝑅𝐴)) → (𝑦 ∈ (LLines‘𝐾) → (𝑦( ⋖ ‘𝐾)𝑋 → ¬ 𝑋 (𝑄 𝑅))))
7069rexlimdv 3151 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  Posetcpo 18365  ltcplt 18366  joincjn 18369  ccvr 39244  Atomscatm 39245  HLchlt 39332  LLinesclln 39474  LPlanesclpl 39475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482
This theorem is referenced by:  lplnnleat  39525  lplnnlelln  39526  2atnelpln  39527  lvolnle3at  39565
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