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Theorem llncvrlpln2 36798
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 488 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋 𝑌)
2 simpl1 1188 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝐾 ∈ HL)
3 simpl3 1190 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑌𝑃)
4 llncvrlpln2.n . . . . . 6 𝑁 = (LLines‘𝐾)
5 llncvrlpln2.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
64, 5lplnnelln 36787 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑃) → ¬ 𝑌𝑁)
72, 3, 6syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → ¬ 𝑌𝑁)
8 simpl2 1189 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋𝑁)
9 eleq1 2903 . . . . . 6 (𝑋 = 𝑌 → (𝑋𝑁𝑌𝑁))
108, 9syl5ibcom 248 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → (𝑋 = 𝑌𝑌𝑁))
1110necon3bd 3028 . . . 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 2824 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
1513, 14pltval 17570 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
1615adantr 484 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
171, 12, 16mpbir2and 712 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋(lt‘𝐾)𝑌)
18 simpl1 1188 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝐾 ∈ HL)
19 simpl2 1189 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝑁)
20 eqid 2824 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2120, 4llnbase 36750 . . . . 5 (𝑋𝑁𝑋 ∈ (Base‘𝐾))
2219, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ (Base‘𝐾))
23 simpl3 1190 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌𝑃)
2420, 5lplnbase 36775 . . . . 5 (𝑌𝑃𝑌 ∈ (Base‘𝐾))
2523, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ (Base‘𝐾))
26 simpr 488 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋(lt‘𝐾)𝑌)
27 eqid 2824 . . . . 5 (join‘𝐾) = (join‘𝐾)
28 llncvrlpln2.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
29 eqid 2824 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3020, 13, 14, 27, 28, 29hlrelat3 36653 . . . 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 1370 . . 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 36777 . . . . . . . 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 484 . . . . . . 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 1135 . . . . . . . . . . 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 36752 . . . . . . . . . . . . 13 (𝐾 ∈ HL → (𝑋𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞)))))
36 simp3l 1198 . . . . . . . . . . . . . . . . . . . 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 1199 . . . . . . . . . . . . . . . . . . . . . . 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 1284 . . . . . . . . . . . . . . . . . . . . . . . 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 7164 . . . . . . . . . . . . . . . . . . . . . . 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 1204 . . . . . . . . . . . . . . . . . . . . . . 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 5083 . . . . . . . . . . . . . . . . . . . . . 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 1299 . . . . . . . . . . . . . . . . . . . . . . 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 1300 . . . . . . . . . . . . . . . . . . . . . . . 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 1301 . . . . . . . . . . . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . . . . . . . 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 1285 . . . . . . . . . . . . . . . . . . . . . . . 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 1286 . . . . . . . . . . . . . . . . . . . . . . . 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 1203 . . . . . . . . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . . . . . . . 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 5083 . . . . . . . . . . . . . . . . . . . . . . . 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 36608 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
5342, 43, 44, 52syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . 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 36651 . . . . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . . . . . . . . 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 1283 . . . . . . . . . . . . . . . . . . . . . . 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 36731 . . . . . . . . . . . . . . . . . . . . . . 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 1375 . . . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . . . 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 2869 . . . . . . . . . . . . . . . . . . . 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 5078 . . . . . . . . . . . . . . . . . . 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 1116 . . . . . . . . . . . . . . . . . 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 1350 . . . . . . . . . . . . . . . . 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 1116 . . . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . 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 3285 . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . 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 243 . . . . . . . . . . . 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 421 . . . . . . . . . . 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 3275 . . . . . . . . 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 3286 . . . . . . . 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 494 . . . . . . 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 243 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑁) → (𝑌𝑃 → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))))
76753impia 1114 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) → (𝑟 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌)))
7776rexlimdv 3275 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) → (∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌) → 𝑋𝐶𝑌))
7877imp 410 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ ∃𝑟 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑟) ∧ (𝑋(join‘𝐾)𝑟) 𝑌)) → 𝑋𝐶𝑌)
7931, 78syldan 594 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝐶𝑌)
8017, 79syldan 594 1 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑃) ∧ 𝑋 𝑌) → 𝑋𝐶𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wrex 3134   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  ltcplt 17551  joincjn 17554  ccvr 36503  Atomscatm 36504  HLchlt 36591  LLinesclln 36732  LPlanesclpl 36733
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-clat 17718  df-oposet 36417  df-ol 36419  df-oml 36420  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563  df-hlat 36592  df-llines 36739  df-lplanes 36740
This theorem is referenced by:  llncvrlpln  36799  2llnmj  36801  lplncmp  36803  lplnexatN  36804  2llnm2N  36809  2lplnmj  36863
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