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Theorem lplncvrlvol2 33919
Description: A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)
Hypotheses
Ref Expression
lplncvrlvol2.l = (le‘𝐾)
lplncvrlvol2.c 𝐶 = ( ⋖ ‘𝐾)
lplncvrlvol2.p 𝑃 = (LPlanes‘𝐾)
lplncvrlvol2.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
lplncvrlvol2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑋𝐶𝑌)

Proof of Theorem lplncvrlvol2
Dummy variables 𝑞 𝑝 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑋 𝑌)
2 simpl1 1057 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝐾 ∈ HL)
3 simpl3 1059 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑌𝑉)
4 lplncvrlvol2.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
5 lplncvrlvol2.v . . . . . 6 𝑉 = (LVols‘𝐾)
64, 5lvolnelpln 33894 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑉) → ¬ 𝑌𝑃)
72, 3, 6syl2anc 691 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → ¬ 𝑌𝑃)
8 simpl2 1058 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑋𝑃)
9 eleq1 2676 . . . . . 6 (𝑋 = 𝑌 → (𝑋𝑃𝑌𝑃))
108, 9syl5ibcom 234 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → (𝑋 = 𝑌𝑌𝑃))
1110necon3bd 2796 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → (¬ 𝑌𝑃𝑋𝑌))
127, 11mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑋𝑌)
13 lplncvrlvol2.l . . . . 5 = (le‘𝐾)
14 eqid 2610 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
1513, 14pltval 16783 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
1615adantr 480 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
171, 12, 16mpbir2and 959 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑋(lt‘𝐾)𝑌)
18 simpl1 1057 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝐾 ∈ HL)
19 simpl2 1058 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝑃)
20 eqid 2610 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2120, 4lplnbase 33838 . . . . 5 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
2219, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ∈ (Base‘𝐾))
23 simpl3 1059 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌𝑉)
2420, 5lvolbase 33882 . . . . 5 (𝑌𝑉𝑌 ∈ (Base‘𝐾))
2523, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑌 ∈ (Base‘𝐾))
26 simpr 476 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋(lt‘𝐾)𝑌)
27 eqid 2610 . . . . 5 (join‘𝐾) = (join‘𝐾)
28 lplncvrlvol2.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
29 eqid 2610 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3020, 13, 14, 27, 28, 29hlrelat3 33716 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))
3118, 22, 25, 26, 30syl31anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → ∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))
3220, 13, 27, 29, 5islvol2 33884 . . . . . . . 8 (𝐾 ∈ HL → (𝑌𝑉 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡𝑢 ∧ ¬ 𝑣 (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))))
3332adantr 480 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑃) → (𝑌𝑉 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡𝑢 ∧ ¬ 𝑣 (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))))
34 simpr 476 . . . . . . . . . . 11 (((𝑡𝑢 ∧ ¬ 𝑣 (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
3520, 13, 27, 29, 4islpln2 33840 . . . . . . . . . . . . 13 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)))))
36 simp3rl 1127 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑋𝐶(𝑋(join‘𝐾)𝑠))
37 simp3rr 1128 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (𝑋(join‘𝐾)𝑠) 𝑌)
38 simp133 1191 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
3938oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (𝑋(join‘𝐾)𝑠) = (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠))
40 simp23 1089 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
4137, 39, 403brtr3d 4614 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
42 simp11 1084 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)))
43 simp12 1085 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑟 ∈ (Atoms‘𝐾))
44 simp3l 1082 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑠 ∈ (Atoms‘𝐾))
45 simp21l 1171 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑡 ∈ (Atoms‘𝐾))
4643, 44, 453jca 1235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)))
47 simp21r 1172 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑢 ∈ (Atoms‘𝐾))
48 simp22l 1173 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑣 ∈ (Atoms‘𝐾))
49 simp22r 1174 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑤 ∈ (Atoms‘𝐾))
5047, 48, 493jca 1235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)))
51 simp131 1189 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑝𝑞)
52 simp132 1190 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → ¬ 𝑟 (𝑝(join‘𝐾)𝑞))
5336, 38, 393brtr3d 4614 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)𝐶(((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠))
54 simp111 1183 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝐾 ∈ HL)
55 hllat 33668 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐾 ∈ HL → 𝐾 ∈ Lat)
5654, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝐾 ∈ Lat)
5720, 27, 29hlatjcl 33671 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
5842, 57syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
5920, 29atbase 33594 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
6043, 59syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑟 ∈ (Base‘𝐾))
6120, 27latjcl 16874 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐾 ∈ Lat ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ∈ (Base‘𝐾))
6256, 58, 60, 61syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ∈ (Base‘𝐾))
6320, 13, 27, 28, 29cvr1 33714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ HL ∧ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) → (¬ 𝑠 ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)𝐶(((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠)))
6454, 62, 44, 63syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (¬ 𝑠 ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟) ↔ ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)𝐶(((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠)))
6553, 64mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → ¬ 𝑠 ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))
6613, 27, 294at2 33918 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ ¬ 𝑠 ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → ((((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) ↔ (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))
6742, 46, 50, 51, 52, 65, 66syl33anc 1333 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → ((((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) ↔ (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)))
6841, 67mpbid 221 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)(join‘𝐾)𝑠) = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))
6968, 39, 403eqtr4d 2654 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → (𝑋(join‘𝐾)𝑠) = 𝑌)
7036, 69breqtrd 4609 . . . . . . . . . . . . . . . . . . . 20 ((((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌))) → 𝑋𝐶𝑌)
71703exp 1256 . . . . . . . . . . . . . . . . . . 19 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → (((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → ((𝑠 ∈ (Atoms‘𝐾) ∧ (𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌)) → 𝑋𝐶𝑌)))
7271exp4a 631 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → (((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) ∧ (𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))))
73723expd 1276 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))))))
7473rexlimdv3a 3015 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)(𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌)))))))
75743expib 1260 . . . . . . . . . . . . . . 15 (𝐾 ∈ HL → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)(𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))))))))
7675rexlimdvv 3019 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟)) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌)))))))
7776adantld 482 . . . . . . . . . . . . 13 (𝐾 ∈ HL → ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑝𝑞 ∧ ¬ 𝑟 (𝑝(join‘𝐾)𝑞) ∧ 𝑋 = ((𝑝(join‘𝐾)𝑞)(join‘𝐾)𝑟))) → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌)))))))
7835, 77sylbid 229 . . . . . . . . . . . 12 (𝐾 ∈ HL → (𝑋𝑃 → ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾)) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌)))))))
7978imp31 447 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌)))))
8034, 79syl7 72 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) → ((𝑣 ∈ (Atoms‘𝐾) ∧ 𝑤 ∈ (Atoms‘𝐾)) → (((𝑡𝑢 ∧ ¬ 𝑣 (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌)))))
8180rexlimdvv 3019 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑃) ∧ (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾))) → (∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡𝑢 ∧ ¬ 𝑣 (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))))
8281rexlimdvva 3020 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑃) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡𝑢 ∧ ¬ 𝑣 (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤)) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))))
8382adantld 482 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑃) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)∃𝑤 ∈ (Atoms‘𝐾)((𝑡𝑢 ∧ ¬ 𝑣 (𝑡(join‘𝐾)𝑢) ∧ ¬ 𝑤 ((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)) ∧ 𝑌 = (((𝑡(join‘𝐾)𝑢)(join‘𝐾)𝑣)(join‘𝐾)𝑤))) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))))
8433, 83sylbid 229 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑃) → (𝑌𝑉 → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))))
85843impia 1253 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) → (𝑠 ∈ (Atoms‘𝐾) → ((𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌)))
8685rexlimdv 3012 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) → (∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌) → 𝑋𝐶𝑌))
8786imp 444 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ ∃𝑠 ∈ (Atoms‘𝐾)(𝑋𝐶(𝑋(join‘𝐾)𝑠) ∧ (𝑋(join‘𝐾)𝑠) 𝑌)) → 𝑋𝐶𝑌)
8831, 87syldan 486 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋𝐶𝑌)
8917, 88syldan 486 1 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑌𝑉) ∧ 𝑋 𝑌) → 𝑋𝐶𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  ltcplt 16764  joincjn 16767  Latclat 16868  ccvr 33567  Atomscatm 33568  HLchlt 33655  LPlanesclpl 33796  LVolsclvol 33797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804
This theorem is referenced by:  lplncvrlvol  33920  lvolcmp  33921  2lplnm2N  33925  2lplnmj  33926
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