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Theorem 2lplnj 35400
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 2806 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2lplnj.l . . . . . . . 8 = (le‘𝐾)
3 2lplnj.j . . . . . . . 8 = (join‘𝐾)
4 eqid 2806 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
5 2lplnj.p . . . . . . . 8 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 35316 . . . . . . 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 473 . . . . . . 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 244 . . . . . 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 35316 . . . . . . 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 473 . . . . . . 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 244 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑃 → ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))))
128, 11anim12d 598 . . . . 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 395 . . . 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 1205 . . 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 1155 . 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 1338 . . . . . . . . . . . . 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 1156 . . . . . . . . . . . 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 1261 . . . . . . . . . . . 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 6892 . . . . . . . . . . 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 1253 . . . . . . . . . . . . . . 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 1399 . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 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 1156 . . . . . . . . . . . 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 1249 . . . . . . . . . . . . . . 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 1316 . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . 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 1151 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 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 1156 . . . . . . . . . . . 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 1334 . . . . . . . . . . . . . 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 1156 . . . . . . . . . . . . 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 1336 . . . . . . . . . . . . . 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 1156 . . . . . . . . . . . . 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 503 . . . . . . . . . . . 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 1248 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 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 1250 . . . . . . . . . . . . 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 1151 . . . . . . . . . . . 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 1259 . . . . . . . . . . . . 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 1260 . . . . . . . . . . . . 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 503 . . . . . . . . . . . 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 1326 . . . . . . . . . . . . . 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 1156 . . . . . . . . . . . . 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 4847 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋 𝑊 ↔ ((𝑞 𝑟) 𝑠) 𝑊))
46 neeq1 3040 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ 𝑌))
4745, 463anbi13d 1555 . . . . . . . . . . . . . . 15 (𝑋 = ((𝑞 𝑟) 𝑠) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊𝑌 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ 𝑌)))
48 breq1 4847 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (𝑌 𝑊 ↔ ((𝑡 𝑢) 𝑣) 𝑊))
49 neeq2 3041 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (((𝑞 𝑟) 𝑠) ≠ 𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣)))
5048, 493anbi23d 1556 . . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . 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 223 . . . . . . . . . . . 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 35399 . . . . . . . . . . . 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 1504 . . . . . . . . . . 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 2840 . . . . . . . . . 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 1141 . . . . . . . . 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 3225 . . . . . . . 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 3219 . . . . . . 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 1141 . . . . . 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 442 . . . . 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 3225 . . . 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 3219 . . 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 398 . 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wrex 3097   class class class wbr 4844  cfv 6101  (class class class)co 6874  Basecbs 16068  lecple 16160  joincjn 17149  Atomscatm 35043  HLchlt 35130  LPlanesclpl 35272  LVolsclvol 35273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-proset 17133  df-poset 17151  df-plt 17163  df-lub 17179  df-glb 17180  df-join 17181  df-meet 17182  df-p0 17244  df-lat 17251  df-clat 17313  df-oposet 34956  df-ol 34958  df-oml 34959  df-covers 35046  df-ats 35047  df-atl 35078  df-cvlat 35102  df-hlat 35131  df-llines 35278  df-lplanes 35279  df-lvols 35280
This theorem is referenced by:  2lplnm2N  35401  dalem13  35456
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