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Theorem llncvrlpln2 38020
Description: A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)
Hypotheses
Ref Expression
llncvrlpln2.l = (le‘𝐾)
llncvrlpln2.c 𝐶 = ( ⋖ ‘𝐾)
llncvrlpln2.n 𝑁 = (LLines‘𝐾)
llncvrlpln2.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
llncvrlpln2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋𝐶𝑌)

Proof of Theorem llncvrlpln2
Dummy variables 𝑞 𝑝 𝑟 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋 𝑌)
2 simpl1 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝐾 ∈ HL)
3 simpl3 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑌𝑃)
4 llncvrlpln2.n . . . . . 6 𝑁 = (LLines‘𝐾)
5 llncvrlpln2.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
64, 5lplnnelln 38009 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑃) → ¬ 𝑌𝑁)
72, 3, 6syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → ¬ 𝑌𝑁)
8 simpl2 1192 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋𝑁)
9 eleq1 2825 . . . . . 6 (𝑋 = 𝑌 → (𝑋𝑁𝑌𝑁))
108, 9syl5ibcom 244 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → (𝑋 = 𝑌𝑌𝑁))
1110necon3bd 2957 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → (¬ 𝑌𝑁𝑋𝑌))
127, 11mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋𝑌)
13 llncvrlpln2.l . . . . 5 = (le‘𝐾)
14 eqid 2736 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
1513, 14pltval 18221 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
1615adantr 481 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
171, 12, 16mpbir2and 711 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋(lt‘𝐾)𝑌)
18 simpl1 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝐾 ∈ HL)
19 simpl2 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝑁)
20 eqid 2736 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2120, 4llnbase 37972 . . . . 5 (𝑋𝑁𝑋 ∈ (Base‘𝐾))
2219, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ (Base‘𝐾))
23 simpl3 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌𝑃)
2420, 5lplnbase 37997 . . . . 5 (𝑌𝑃𝑌 ∈ (Base‘𝐾))
2523, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ (Base‘𝐾))
26 simpr 485 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋(lt‘𝐾)𝑌)
27 eqid 2736 . . . . 5 (join‘𝐾) = (join‘𝐾)
28 llncvrlpln2.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
29 eqid 2736 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3020, 13, 14, 27, 28, 29hlrelat3 37875 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌))
3118, 22, 25, 26, 30syl31anc 1373 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌))
3220, 13, 27, 29, 5islpln2 37999 . . . . . . . 8 (𝐾 ∈ HL → (𝑌𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠𝑡 ∧ ¬ 𝑢 (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))))
3332adantr 481 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑁) → (𝑌𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠𝑡 ∧ ¬ 𝑢 (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))))
34 simp3 1138 . . . . . . . . . . 11 ((𝑠𝑡 ∧ ¬ 𝑢 (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
3520, 27, 29, 4islln2 37974 . . . . . . . . . . . . 13 (𝐾 ∈ HL → (𝑋𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))))
36 simp3l 1201 . . . . . . . . . . . . . . . . . . . 20 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑋𝐶(𝑋(join‘𝐾)𝑟))
37 simp3r 1202 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (𝑋(join‘𝐾)𝑟) 𝑌)
38 simp12r 1287 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑋 = (𝑝(join‘𝐾)𝑞))
3938oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (𝑋(join‘𝐾)𝑟) = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
40 simp22 1207 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
4137, 39, 403brtr3d 5136 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
42 simp111 1302 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝐾 ∈ HL)
43 simp112 1303 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
44 simp113 1304 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑞 ∈ (Atoms‘𝐾))
45 simp23 1208 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑟 ∈ (Atoms‘𝐾))
4643, 44, 453jca 1128 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)))
47 simp13l 1288 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑠 ∈ (Atoms‘𝐾))
48 simp13r 1289 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑡 ∈ (Atoms‘𝐾))
49 simp21 1206 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑢 ∈ (Atoms‘𝐾))
5047, 48, 493jca 1128 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)))
5136, 38, 393brtr3d 5136 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (𝑝(join‘𝐾)𝑞)𝐶((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
5220, 27, 29hlatjcl 37829 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
5342, 43, 44, 52syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
5420, 13, 27, 28, 29cvr1 37873 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (¬ 𝑟 (𝑝(join‘𝐾)𝑞) ↔ (𝑝(join‘𝐾)𝑞)𝐶((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)))
5542, 53, 45, 54syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (¬ 𝑟 (𝑝(join‘𝐾)𝑞) ↔ (𝑝(join‘𝐾)𝑞)𝐶((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)))
5651, 55mpbird 256 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → ¬ 𝑟 (𝑝(join‘𝐾)𝑞))
57 simp12l 1286 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑝𝑞)
5813, 27, 293at 37953 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑝𝑞)) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))
5942, 46, 50, 56, 57, 58syl32anc 1378 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)))
6041, 59mpbid 231 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))
6160, 39, 403eqtr4d 2786 . . . . . . . . . . . . . . . . . . . 20 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → (𝑋(join‘𝐾)𝑟) = 𝑌)
6236, 61breqtrd 5131 . . . . . . . . . . . . . . . . . . 19 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑋𝐶𝑌)
63623exp 1119 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → ((𝑢 ∈ (Atoms‘𝐾) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))
64633expd 1353 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))))
65643exp 1119 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))))))
66653expib 1122 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))))))))
6766rexlimdvv 3204 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))))))
6867adantld 491 . . . . . . . . . . . . 13 (𝐾 ∈ HL → ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))))))
6935, 68sylbid 239 . . . . . . . . . . . 12 (𝐾 ∈ HL → (𝑋𝑁 → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))))))
7069imp31 418 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑁) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (𝑢 ∈ (Atoms‘𝐾) → (𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))))
7134, 70syl7 74 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑁) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (𝑢 ∈ (Atoms‘𝐾) → ((𝑠𝑡 ∧ ¬ 𝑢 (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))))
7271rexlimdv 3150 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾))) → (∃𝑢 ∈ (Atoms‘𝐾)(𝑠𝑡 ∧ ¬ 𝑢 (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))))
7372rexlimdvva 3205 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑁) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠𝑡 ∧ ¬ 𝑢 (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢)) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))))
7473adantld 491 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)(𝑠𝑡 ∧ ¬ 𝑢 (𝑠(join‘𝐾)𝑡) ∧ 𝑌 = ((𝑠(join‘𝐾)𝑡)(join‘𝐾)𝑢))) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))))
7533, 74sylbid 239 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑁) → (𝑌𝑃 → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))))
76753impia 1117 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))
7776rexlimdv 3150 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) → (∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))
7877imp 407 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ ∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑋𝐶𝑌)
7931, 78syldan 591 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝐶𝑌)
8017, 79syldan 591 1 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋𝐶𝑌)
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  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:  llncvrlpln  38021  2llnmj  38023  lplncmp  38025  lplnexatN  38026  2llnm2N  38031  2lplnmj  38085
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