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Theorem 2lplnj 33924
 Description: The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)
Hypotheses
Ref Expression
2lplnj.l = (le‘𝐾)
2lplnj.j = (join‘𝐾)
2lplnj.p 𝑃 = (LPlanes‘𝐾)
2lplnj.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
2lplnj ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) = 𝑊)

Proof of Theorem 2lplnj
Dummy variables 𝑟 𝑞 𝑠 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2lplnj.l . . . . . . . 8 = (le‘𝐾)
3 2lplnj.j . . . . . . . 8 = (join‘𝐾)
4 eqid 2610 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
5 2lplnj.p . . . . . . . 8 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 33840 . . . . . . 7 (𝐾 ∈ HL → (𝑋𝑃 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)))))
7 simpr 476 . . . . . . 7 ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)))
86, 7syl6bi 242 . . . . . 6 (𝐾 ∈ HL → (𝑋𝑃 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))))
91, 2, 3, 4, 5islpln2 33840 . . . . . . 7 (𝐾 ∈ HL → (𝑌𝑃 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)))))
10 simpr 476 . . . . . . 7 ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)))
119, 10syl6bi 242 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑃 → ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))))
128, 11anim12d 584 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑃𝑌𝑃) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)))))
1312imp 444 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))))
14133adantr3 1215 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))))
15143adant3 1074 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))))
16 simpl33 1137 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → 𝑋 = ((𝑞 𝑟) 𝑠))
17163ad2ant1 1075 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → 𝑋 = ((𝑞 𝑟) 𝑠))
18 simp33 1092 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → 𝑌 = ((𝑡 𝑢) 𝑣))
1917, 18oveq12d 6567 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑋 𝑌) = (((𝑞 𝑟) 𝑠) ((𝑡 𝑢) 𝑣)))
20 simp11 1084 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → 𝐾 ∈ HL)
21 simp123 1188 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → 𝑊𝑉)
2220, 21jca 553 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → (𝐾 ∈ HL ∧ 𝑊𝑉))
2322adantr 480 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊𝑉))
24233ad2ant1 1075 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝐾 ∈ HL ∧ 𝑊𝑉))
25 simp2l 1080 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → 𝑞 ∈ (Atoms‘𝐾))
26 simp2rl 1123 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → 𝑟 ∈ (Atoms‘𝐾))
27 simp2rr 1124 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → 𝑠 ∈ (Atoms‘𝐾))
2825, 26, 273jca 1235 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)))
2928adantr 480 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)))
30293ad2ant1 1075 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)))
31 simpl31 1135 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → 𝑞𝑟)
32313ad2ant1 1075 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → 𝑞𝑟)
33 simpl32 1136 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → ¬ 𝑠 (𝑞 𝑟))
34333ad2ant1 1075 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → ¬ 𝑠 (𝑞 𝑟))
3532, 34jca 553 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟)))
36 simp1r 1079 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → 𝑡 ∈ (Atoms‘𝐾))
37 simp2l 1080 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → 𝑢 ∈ (Atoms‘𝐾))
38 simp2r 1081 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → 𝑣 ∈ (Atoms‘𝐾))
3936, 37, 383jca 1235 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)))
40 simp31 1090 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → 𝑡𝑢)
41 simp32 1091 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → ¬ 𝑣 (𝑡 𝑢))
4240, 41jca 553 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢)))
43 simpl13 1131 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (𝑋 𝑊𝑌 𝑊𝑋𝑌))
44433ad2ant1 1075 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑋 𝑊𝑌 𝑊𝑋𝑌))
45 breq1 4586 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋 𝑊 ↔ ((𝑞 𝑟) 𝑠) 𝑊))
46 neeq1 2844 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ 𝑌))
4745, 463anbi13d 1393 . . . . . . . . . . . . . . 15 (𝑋 = ((𝑞 𝑟) 𝑠) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊𝑌 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ 𝑌)))
48 breq1 4586 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (𝑌 𝑊 ↔ ((𝑡 𝑢) 𝑣) 𝑊))
49 neeq2 2845 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (((𝑞 𝑟) 𝑠) ≠ 𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣)))
5048, 493anbi23d 1394 . . . . . . . . . . . . . . 15 (𝑌 = ((𝑡 𝑢) 𝑣) → ((((𝑞 𝑟) 𝑠) 𝑊𝑌 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ 𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊 ∧ ((𝑡 𝑢) 𝑣) 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣))))
5147, 50sylan9bb 732 . . . . . . . . . . . . . 14 ((𝑋 = ((𝑞 𝑟) 𝑠) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊 ∧ ((𝑡 𝑢) 𝑣) 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣))))
5217, 18, 51syl2anc 691 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊 ∧ ((𝑡 𝑢) 𝑣) 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣))))
5344, 52mpbid 221 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (((𝑞 𝑟) 𝑠) 𝑊 ∧ ((𝑡 𝑢) 𝑣) 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣)))
54 2lplnj.v . . . . . . . . . . . . 13 𝑉 = (LVols‘𝐾)
552, 3, 4, 542lplnja 33923 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝑉) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟))) ∧ ((𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢))) ∧ (((𝑞 𝑟) 𝑠) 𝑊 ∧ ((𝑡 𝑢) 𝑣) 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣))) → (((𝑞 𝑟) 𝑠) ((𝑡 𝑢) 𝑣)) = 𝑊)
5624, 30, 35, 39, 42, 53, 55syl321anc 1340 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (((𝑞 𝑟) 𝑠) ((𝑡 𝑢) 𝑣)) = 𝑊)
5719, 56eqtrd 2644 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) ∧ (𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑋 𝑌) = 𝑊)
58573exp 1256 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → ((𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)) → ((𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → (𝑋 𝑌) = 𝑊)))
5958rexlimdvv 3019 . . . . . . . 8 ((((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) ∧ 𝑡 ∈ (Atoms‘𝐾)) → (∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → (𝑋 𝑌) = 𝑊))
6059rexlimdva 3013 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) ∧ (𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠))) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → (𝑋 𝑌) = 𝑊))
61603exp 1256 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → ((𝑞 ∈ (Atoms‘𝐾) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾))) → ((𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → (𝑋 𝑌) = 𝑊))))
6261expdimp 452 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑟 ∈ (Atoms‘𝐾) ∧ 𝑠 ∈ (Atoms‘𝐾)) → ((𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → (𝑋 𝑌) = 𝑊))))
6362rexlimdvv 3019 . . . 4 (((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ 𝑞 ∈ (Atoms‘𝐾)) → (∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → (𝑋 𝑌) = 𝑊)))
6463rexlimdva 3013 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) → (∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣)) → (𝑋 𝑌) = 𝑊)))
6564impd 446 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → ((∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)∃𝑠 ∈ (Atoms‘𝐾)(𝑞𝑟 ∧ ¬ 𝑠 (𝑞 𝑟) ∧ 𝑋 = ((𝑞 𝑟) 𝑠)) ∧ ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))) → (𝑋 𝑌) = 𝑊))
6615, 65mpd 15 1 ((𝐾 ∈ HL ∧ (𝑋𝑃𝑌𝑃𝑊𝑉) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) = 𝑊)
 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  joincjn 16767  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:  2lplnm2N  33925  dalem13  33980
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