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Theorem islvol5 36823
Description: The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
islvol5.b 𝐵 = (Base‘𝐾)
islvol5.l = (le‘𝐾)
islvol5.j = (join‘𝐾)
islvol5.a 𝐴 = (Atoms‘𝐾)
islvol5.v 𝑉 = (LVols‘𝐾)
Assertion
Ref Expression
islvol5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝐴   𝐵,𝑝,𝑞,𝑟,𝑠   ,𝑝,𝑞,𝑟,𝑠   𝐾,𝑝,𝑞,𝑟,𝑠   ,𝑝,𝑞,𝑟,𝑠   𝑋,𝑝,𝑞,𝑟,𝑠
Allowed substitution hints:   𝑉(𝑠,𝑟,𝑞,𝑝)

Proof of Theorem islvol5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 islvol5.b . . 3 𝐵 = (Base‘𝐾)
2 islvol5.l . . 3 = (le‘𝐾)
3 islvol5.j . . 3 = (join‘𝐾)
4 islvol5.a . . 3 𝐴 = (Atoms‘𝐾)
5 eqid 2824 . . 3 (LPlanes‘𝐾) = (LPlanes‘𝐾)
6 islvol5.v . . 3 𝑉 = (LVols‘𝐾)
71, 2, 3, 4, 5, 6islvol3 36820 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8 hllat 36607 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
98ad3antrrr 729 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ Lat)
10 simplll 774 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
11 simplrl 776 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑝𝐴)
12 simplrr 777 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑞𝐴)
131, 3, 4hlatjcl 36611 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝 𝑞) ∈ 𝐵)
1410, 11, 12, 13syl3anc 1368 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → (𝑝 𝑞) ∈ 𝐵)
151, 4atbase 36533 . . . . . . . . . . 11 (𝑟𝐴𝑟𝐵)
1615adantl 485 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑟𝐵)
171, 3latjcl 17661 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑝 𝑞) ∈ 𝐵𝑟𝐵) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
189, 14, 16, 17syl3anc 1368 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
1918biantrurd 536 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → (∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)) ↔ (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))))
20 r19.41v 3338 . . . . . . . . . . 11 (∃𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ (∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
21 df-3an 1086 . . . . . . . . . . . . 13 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))
2221anbi2i 625 . . . . . . . . . . . 12 ((∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ (∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
23 an13 646 . . . . . . . . . . . 12 ((∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ (𝑦 = ((𝑝 𝑞) 𝑟) ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))))))
2422, 23bitri 278 . . . . . . . . . . 11 ((∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ (𝑦 = ((𝑝 𝑞) 𝑟) ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))))))
2520, 24bitri 278 . . . . . . . . . 10 (∃𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ (𝑦 = ((𝑝 𝑞) 𝑟) ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))))))
2625exbii 1849 . . . . . . . . 9 (∃𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑦(𝑦 = ((𝑝 𝑞) 𝑟) ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))))))
27 ovex 7182 . . . . . . . . . 10 ((𝑝 𝑞) 𝑟) ∈ V
28 an12 644 . . . . . . . . . . . . 13 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ (𝑦𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
29 eleq1 2903 . . . . . . . . . . . . . 14 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦𝐵 ↔ ((𝑝 𝑞) 𝑟) ∈ 𝐵))
30 breq2 5056 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑠 𝑦𝑠 ((𝑝 𝑞) 𝑟)))
3130notbid 321 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (¬ 𝑠 𝑦 ↔ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
32 oveq1 7156 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦 𝑠) = (((𝑝 𝑞) 𝑟) 𝑠))
3332eqeq2d 2835 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑋 = (𝑦 𝑠) ↔ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))
3431, 33anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑝 𝑞) 𝑟) → ((¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ (¬ 𝑠 ((𝑝 𝑞) 𝑟) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
3534anbi2d 631 . . . . . . . . . . . . . . 15 (𝑦 = ((𝑝 𝑞) 𝑟) → (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 ((𝑝 𝑞) 𝑟) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))))
36 anass 472 . . . . . . . . . . . . . . . 16 ((((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 ((𝑝 𝑞) 𝑟) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
37 df-3an 1086 . . . . . . . . . . . . . . . . . 18 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
3837bicomi 227 . . . . . . . . . . . . . . . . 17 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ↔ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
3938anbi1i 626 . . . . . . . . . . . . . . . 16 ((((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))
4036, 39bitr3i 280 . . . . . . . . . . . . . . 15 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 ((𝑝 𝑞) 𝑟) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))
4135, 40syl6bb 290 . . . . . . . . . . . . . 14 (𝑦 = ((𝑝 𝑞) 𝑟) → (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
4229, 41anbi12d 633 . . . . . . . . . . . . 13 (𝑦 = ((𝑝 𝑞) 𝑟) → ((𝑦𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))))
4328, 42syl5bb 286 . . . . . . . . . . . 12 (𝑦 = ((𝑝 𝑞) 𝑟) → (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))))
4443rexbidv 3289 . . . . . . . . . . 11 (𝑦 = ((𝑝 𝑞) 𝑟) → (∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ ∃𝑠𝐴 (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))))
45 r19.42v 3341 . . . . . . . . . . 11 (∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
46 r19.42v 3341 . . . . . . . . . . 11 (∃𝑠𝐴 (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))) ↔ (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
4744, 45, 463bitr3g 316 . . . . . . . . . 10 (𝑦 = ((𝑝 𝑞) 𝑟) → (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))))
4827, 47ceqsexv 3527 . . . . . . . . 9 (∃𝑦(𝑦 = ((𝑝 𝑞) 𝑟) ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))))) ↔ (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
4926, 48bitri 278 . . . . . . . 8 (∃𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ (((𝑝 𝑞) 𝑟) ∈ 𝐵 ∧ ∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
5019, 49syl6rbbr 293 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → (∃𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
5150rexbidva 3288 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (∃𝑟𝐴𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
52512rexbidva 3291 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
53 rexcom4 3243 . . . . . . . . 9 (∃𝑟𝐴𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑦𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
5453rexbii 3241 . . . . . . . 8 (∃𝑞𝐴𝑟𝐴𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑞𝐴𝑦𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
55 rexcom4 3243 . . . . . . . 8 (∃𝑞𝐴𝑦𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑦𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
5654, 55bitri 278 . . . . . . 7 (∃𝑞𝐴𝑟𝐴𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑦𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
5756rexbii 3241 . . . . . 6 (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
58 rexcom4 3243 . . . . . 6 (∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
5957, 58bitri 278 . . . . 5 (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑦𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
6052, 59bitr3di 289 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
611, 2, 3, 4, 5islpln2 36780 . . . . . . . . . . 11 (𝐾 ∈ HL → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
6261adantr 484 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
6362anbi1d 632 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ ((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
64 r19.42v 3341 . . . . . . . . . 10 (∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
65 r19.42v 3341 . . . . . . . . . . . . 13 (∃𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
6665rexbii 3241 . . . . . . . . . . . 12 (∃𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
67 r19.42v 3341 . . . . . . . . . . . 12 (∃𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
6866, 67bitri 278 . . . . . . . . . . 11 (∃𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
6968rexbii 3241 . . . . . . . . . 10 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
70 an32 645 . . . . . . . . . 10 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
7164, 69, 703bitr4ri 307 . . . . . . . . 9 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
7263, 71syl6bb 290 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
7372rexbidv 3289 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑠𝐴 (𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ ∃𝑠𝐴𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
74 rexcom 3346 . . . . . . . . . . 11 (∃𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑠𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
7574rexbii 3241 . . . . . . . . . 10 (∃𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑞𝐴𝑠𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
76 rexcom 3346 . . . . . . . . . 10 (∃𝑞𝐴𝑠𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑠𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
7775, 76bitri 278 . . . . . . . . 9 (∃𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑠𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
7877rexbii 3241 . . . . . . . 8 (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑝𝐴𝑠𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
79 rexcom 3346 . . . . . . . 8 (∃𝑝𝐴𝑠𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑠𝐴𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
8078, 79bitri 278 . . . . . . 7 (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑠𝐴𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))))
8173, 80syl6rbbr 293 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑠𝐴 (𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
82 r19.42v 3341 . . . . . 6 (∃𝑠𝐴 (𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ (𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8381, 82syl6bb 290 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ (𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)))))
8483exbidv 1923 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟))) ↔ ∃𝑦(𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)))))
8560, 84bitrd 282 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)) ↔ ∃𝑦(𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)))))
86 df-rex 3139 . . 3 (∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ ∃𝑦(𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8785, 86syl6rbbr 293 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
887, 87bitrd 282 1 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wne 3014  wrex 3134   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  Latclat 17655  Atomscatm 36507  HLchlt 36594  LPlanesclpl 36736  LVolsclvol 36737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-clat 17718  df-oposet 36420  df-ol 36422  df-oml 36423  df-covers 36510  df-ats 36511  df-atl 36542  df-cvlat 36566  df-hlat 36595  df-llines 36742  df-lplanes 36743  df-lvols 36744
This theorem is referenced by:  islvol2  36824  lvoli2  36825
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