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Theorem 2lplnj 36642
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 2826 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2lplnj.l . . . . . . . 8 = (le‘𝐾)
3 2lplnj.j . . . . . . . 8 = (join‘𝐾)
4 eqid 2826 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
5 2lplnj.p . . . . . . . 8 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 36558 . . . . . . 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 485 . . . . . . 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 254 . . . . . 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 36558 . . . . . . 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 485 . . . . . . 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 254 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑃 → ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))))
128, 11anim12d 608 . . . . 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 407 . . . 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 1165 . . 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 1126 . 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 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)) → 𝑋 = ((𝑞 𝑟) 𝑠))
17163ad2ant1 1127 . . . . . . . . . . . 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 1205 . . . . . . . . . . . 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 7168 . . . . . . . . . . 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 1197 . . . . . . . . . . . . . . 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 1301 . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 1127 . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . . . 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 1122 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 1127 . . . . . . . . . . . 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 1248 . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . 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 512 . . . . . . . . . . . 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 1192 . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . 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 1122 . . . . . . . . . . . 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 1203 . . . . . . . . . . . . 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 1204 . . . . . . . . . . . . 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 512 . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋 𝑊 ↔ ((𝑞 𝑟) 𝑠) 𝑊))
46 neeq1 3083 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ 𝑌))
4745, 463anbi13d 1431 . . . . . . . . . . . . . . 15 (𝑋 = ((𝑞 𝑟) 𝑠) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊𝑌 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ 𝑌)))
48 breq1 5066 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (𝑌 𝑊 ↔ ((𝑡 𝑢) 𝑣) 𝑊))
49 neeq2 3084 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (((𝑞 𝑟) 𝑠) ≠ 𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣)))
5048, 493anbi23d 1432 . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 233 . . . . . . . . . . . 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 36641 . . . . . . . . . . . 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 1386 . . . . . . . . . . 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 2861 . . . . . . . . . 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 1113 . . . . . . . . 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 3298 . . . . . . . 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 3289 . . . . . . 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 1113 . . . . . 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 453 . . . . 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 3298 . . . 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 3289 . . 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 411 . 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wrex 3144   class class class wbr 5063  cfv 6354  (class class class)co 7150  Basecbs 16478  lecple 16567  joincjn 17549  Atomscatm 36285  HLchlt 36372  LPlanesclpl 36514  LVolsclvol 36515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-llines 36520  df-lplanes 36521  df-lvols 36522
This theorem is referenced by:  2lplnm2N  36643  dalem13  36698
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