Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2lplnj Structured version   Visualization version   GIF version

Theorem 2lplnj 39622
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 2737 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2lplnj.l . . . . . . . 8 = (le‘𝐾)
3 2lplnj.j . . . . . . . 8 = (join‘𝐾)
4 eqid 2737 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
5 2lplnj.p . . . . . . . 8 𝑃 = (LPlanes‘𝐾)
61, 2, 3, 4, 5islpln2 39538 . . . . . . 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 39538 . . . . . . 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 1172 . . 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 1133 . 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 1257 . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 1212 . . . . . . . . . . . 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 1204 . . . . . . . . . . . . . . 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 1308 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . 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 1243 . . . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . . 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 1129 . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 1255 . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 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))) → 𝑡 ∈ (Atoms‘𝐾))
37 simp2l 1200 . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . 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 1129 . . . . . . . . . . . 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 1210 . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . 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 1251 . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 5146 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋 𝑊 ↔ ((𝑞 𝑟) 𝑠) 𝑊))
46 neeq1 3003 . . . . . . . . . . . . . . . 16 (𝑋 = ((𝑞 𝑟) 𝑠) → (𝑋𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ 𝑌))
4745, 463anbi13d 1440 . . . . . . . . . . . . . . 15 (𝑋 = ((𝑞 𝑟) 𝑠) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ (((𝑞 𝑟) 𝑠) 𝑊𝑌 𝑊 ∧ ((𝑞 𝑟) 𝑠) ≠ 𝑌)))
48 breq1 5146 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (𝑌 𝑊 ↔ ((𝑡 𝑢) 𝑣) 𝑊))
49 neeq2 3004 . . . . . . . . . . . . . . . 16 (𝑌 = ((𝑡 𝑢) 𝑣) → (((𝑞 𝑟) 𝑠) ≠ 𝑌 ↔ ((𝑞 𝑟) 𝑠) ≠ ((𝑡 𝑢) 𝑣)))
5048, 493anbi23d 1441 . . . . . . . . . . . . . . 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 39621 . . . . . . . . . . . 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 1394 . . . . . . . . . . 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 2777 . . . . . . . . . 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 1120 . . . . . . . . 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 3212 . . . . . . . 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 3155 . . . . . . 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 1120 . . . . . 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 3212 . . . 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 3155 . . 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 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  Atomscatm 39264  HLchlt 39351  LPlanesclpl 39494  LVolsclvol 39495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502
This theorem is referenced by:  2lplnm2N  39623  dalem13  39678
  Copyright terms: Public domain W3C validator