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Theorem islvol5 35654
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 2825 . . 3 (LPlanes‘𝐾) = (LPlanes‘𝐾)
6 islvol5.v . . 3 𝑉 = (LVols‘𝐾)
71, 2, 3, 4, 5, 6islvol3 35651 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8 rexcom4 3442 . . . . . . . . 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))))
98rexbii 3251 . . . . . . . 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))))
10 rexcom4 3442 . . . . . . . 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))))
119, 10bitri 267 . . . . . . 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))))
1211rexbii 3251 . . . . . 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))))
13 rexcom4 3442 . . . . . 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))))
1412, 13bitri 267 . . . . 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))))
15 hllat 35438 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1615ad3antrrr 723 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ Lat)
17 simplll 793 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
18 simplrl 797 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑝𝐴)
19 simplrr 798 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑞𝐴)
201, 3, 4hlatjcl 35442 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝 𝑞) ∈ 𝐵)
2117, 18, 19, 20syl3anc 1496 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → (𝑝 𝑞) ∈ 𝐵)
221, 4atbase 35364 . . . . . . . . . . 11 (𝑟𝐴𝑟𝐵)
2322adantl 475 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑟𝐵)
241, 3latjcl 17404 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑝 𝑞) ∈ 𝐵𝑟𝐵) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
2516, 21, 23, 24syl3anc 1496 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
2625biantrurd 530 . . . . . . . 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)))))
27 r19.41v 3299 . . . . . . . . . . 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))))
28 df-3an 1115 . . . . . . . . . . . . 13 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))
2928anbi2i 618 . . . . . . . . . . . 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))))
30 an13 639 . . . . . . . . . . . 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))))))
3129, 30bitri 267 . . . . . . . . . . 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))))))
3227, 31bitri 267 . . . . . . . . . 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))))))
3332exbii 1949 . . . . . . . . 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))))))
34 ovex 6937 . . . . . . . . . 10 ((𝑝 𝑞) 𝑟) ∈ V
35 an12 637 . . . . . . . . . . . . 13 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ (𝑦𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
36 eleq1 2894 . . . . . . . . . . . . . 14 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦𝐵 ↔ ((𝑝 𝑞) 𝑟) ∈ 𝐵))
37 breq2 4877 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑠 𝑦𝑠 ((𝑝 𝑞) 𝑟)))
3837notbid 310 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (¬ 𝑠 𝑦 ↔ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
39 oveq1 6912 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦 𝑠) = (((𝑝 𝑞) 𝑟) 𝑠))
4039eqeq2d 2835 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑋 = (𝑦 𝑠) ↔ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))
4138, 40anbi12d 626 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑝 𝑞) 𝑟) → ((¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ (¬ 𝑠 ((𝑝 𝑞) 𝑟) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
4241anbi2d 624 . . . . . . . . . . . . . . 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)))))
43 anass 462 . . . . . . . . . . . . . . . 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))))
44 df-3an 1115 . . . . . . . . . . . . . . . . . 18 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
4544bicomi 216 . . . . . . . . . . . . . . . . 17 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ↔ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
4645anbi1i 619 . . . . . . . . . . . . . . . 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)))
4743, 46bitr3i 269 . . . . . . . . . . . . . . 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)))
4842, 47syl6bb 279 . . . . . . . . . . . . . 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))))
4936, 48anbi12d 626 . . . . . . . . . . . . 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)))))
5035, 49syl5bb 275 . . . . . . . . . . . 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)))))
5150rexbidv 3262 . . . . . . . . . . 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)))))
52 r19.42v 3302 . . . . . . . . . . 11 (∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
53 r19.42v 3302 . . . . . . . . . . 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))))
5451, 52, 533bitr3g 305 . . . . . . . . . 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)))))
5534, 54ceqsexv 3459 . . . . . . . . 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))))
5633, 55bitri 267 . . . . . . . 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))))
5726, 56syl6rbbr 282 . . . . . . 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))))
5857rexbidva 3259 . . . . . 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))))
59582rexbidva 3266 . . . . 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))))
6014, 59syl5rbbr 278 . . . 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 35611 . . . . . . . . . . 11 (𝐾 ∈ HL → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
6261adantr 474 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
6362anbi1d 625 . . . . . . . . 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 3302 . . . . . . . . . 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 3302 . . . . . . . . . . . . 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 3251 . . . . . . . . . . . 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 3302 . . . . . . . . . . . 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 267 . . . . . . . . . . 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 3251 . . . . . . . . . 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 638 . . . . . . . . . 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 296 . . . . . . . . 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 279 . . . . . . . 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 3262 . . . . . . 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 3309 . . . . . . . . . . 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 3251 . . . . . . . . . 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 3309 . . . . . . . . . 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 267 . . . . . . . . 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 3251 . . . . . . . 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 3309 . . . . . . . 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 267 . . . . . . 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 282 . . . . . 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 3302 . . . . . 6 (∃𝑠𝐴 (𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ (𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8381, 82syl6bb 279 . . . . 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 2022 . . . 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 271 . . 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 3123 . . 3 (∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ ∃𝑦(𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8785, 86syl6rbbr 282 . 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 271 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 198  wa 386  w3a 1113   = wceq 1658  wex 1880  wcel 2166  wne 2999  wrex 3118   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  Latclat 17398  Atomscatm 35338  HLchlt 35425  LPlanesclpl 35567  LVolsclvol 35568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35251  df-ol 35253  df-oml 35254  df-covers 35341  df-ats 35342  df-atl 35373  df-cvlat 35397  df-hlat 35426  df-llines 35573  df-lplanes 35574  df-lvols 35575
This theorem is referenced by:  islvol2  35655  lvoli2  35656
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