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Theorem 2lplnj 39603
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 2735 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2lplnj.l . . . . . . . 8 = (le‘𝐾)
3 2lplnj.j . . . . . . . 8 = (join‘𝐾)
4 eqid 2735 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
5 2lplnj.p . . . . . . . 8 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 39519 . . . . . . 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 484 . . . . . . 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, 7biimtrdi 253 . . . . . 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 39519 . . . . . . 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 484 . . . . . . 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, 10biimtrdi 253 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑃 → ∃𝑡 ∈ (Atoms‘𝐾)∃𝑢 ∈ (Atoms‘𝐾)∃𝑣 ∈ (Atoms‘𝐾)(𝑡𝑢 ∧ ¬ 𝑣 (𝑡 𝑢) ∧ 𝑌 = ((𝑡 𝑢) 𝑣))))
128, 11anim12d 609 . . . . 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 406 . . . 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 1170 . . 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 1131 . 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 1255 . . . . . . . . . . . . 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 1132 . . . . . . . . . . . 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 1210 . . . . . . . . . . . 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 7449 . . . . . . . . . . 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 1202 . . . . . . . . . . . . . . 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 1306 . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . 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 1198 . . . . . . . . . . . . . . 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 1241 . . . . . . . . . . . . . . 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 1242 . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . 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 1253 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 1254 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 511 . . . . . . . . . . . 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 1197 . . . . . . . . . . . . 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 1198 . . . . . . . . . . . . 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 1199 . . . . . . . . . . . . 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 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))) → (𝑡 ∈ (Atoms‘𝐾) ∧ 𝑢 ∈ (Atoms‘𝐾) ∧ 𝑣 ∈ (Atoms‘𝐾)))
40 simp31 1208 . . . . . . . . . . . . 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 1209 . . . . . . . . . . . . 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 511 . . . . . . . . . . . 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 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)) → (𝑋 𝑊𝑌 𝑊𝑋𝑌))
44433ad2ant1 1132 . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋 𝑊 ↔ ((𝑞 𝑟) 𝑠) 𝑊))
46 neeq1 3001 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ 𝑌))
4745, 463anbi13d 1437 . . . . . . . . . . . . . . 15 (𝑋 = ((𝑞 𝑟) 𝑠) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊𝑌 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ 𝑌)))
48 breq1 5151 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (𝑌 𝑊 ↔ ((𝑡 𝑢) 𝑣) 𝑊))
49 neeq2 3002 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (((𝑞 𝑟) 𝑠) ≠ 𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣)))
5048, 493anbi23d 1438 . . . . . . . . . . . . . . 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 509 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 39602 . . . . . . . . . . . 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 1391 . . . . . . . . . . 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 2775 . . . . . . . . . 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 1118 . . . . . . . . 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 3210 . . . . . . . 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 3153 . . . . . . 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 1118 . . . . . 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 3210 . . . 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 3153 . . 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 410 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Atomscatm 39245  HLchlt 39332  LPlanesclpl 39475  LVolsclvol 39476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483
This theorem is referenced by:  2lplnm2N  39604  dalem13  39659
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