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Theorem islvol5 38042
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 2736 . . 3 (LPlanes‘𝐾) = (LPlanes‘𝐾)
6 islvol5.v . . 3 𝑉 = (LVols‘𝐾)
71, 2, 3, 4, 5, 6islvol3 38039 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8 df-rex 3074 . . 3 (∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ ∃𝑦(𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
9 r19.41v 3185 . . . . . . . . . . 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))))
10 df-3an 1089 . . . . . . . . . . . . 13 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))
1110anbi2i 623 . . . . . . . . . . . 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))))
12 an13 645 . . . . . . . . . . . 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))))))
1311, 12bitri 274 . . . . . . . . . . 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))))))
149, 13bitri 274 . . . . . . . . . 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))))))
1514exbii 1850 . . . . . . . . 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))))))
16 ovex 7390 . . . . . . . . . 10 ((𝑝 𝑞) 𝑟) ∈ V
17 an12 643 . . . . . . . . . . . . 13 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ (𝑦𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
18 eleq1 2825 . . . . . . . . . . . . . 14 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦𝐵 ↔ ((𝑝 𝑞) 𝑟) ∈ 𝐵))
19 breq2 5109 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑠 𝑦𝑠 ((𝑝 𝑞) 𝑟)))
2019notbid 317 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (¬ 𝑠 𝑦 ↔ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
21 oveq1 7364 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦 𝑠) = (((𝑝 𝑞) 𝑟) 𝑠))
2221eqeq2d 2747 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑋 = (𝑦 𝑠) ↔ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))
2320, 22anbi12d 631 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑝 𝑞) 𝑟) → ((¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ (¬ 𝑠 ((𝑝 𝑞) 𝑟) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
2423anbi2d 629 . . . . . . . . . . . . . . 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)))))
25 anass 469 . . . . . . . . . . . . . . . 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))))
26 df-3an 1089 . . . . . . . . . . . . . . . . . 18 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
2726bicomi 223 . . . . . . . . . . . . . . . . 17 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ↔ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
2827anbi1i 624 . . . . . . . . . . . . . . . 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)))
2925, 28bitr3i 276 . . . . . . . . . . . . . . 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)))
3024, 29bitrdi 286 . . . . . . . . . . . . . 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))))
3118, 30anbi12d 631 . . . . . . . . . . . . 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)))))
3217, 31bitrid 282 . . . . . . . . . . . 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)))))
3332rexbidv 3175 . . . . . . . . . . 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)))))
34 r19.42v 3187 . . . . . . . . . . 11 (∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
35 r19.42v 3187 . . . . . . . . . . 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))))
3633, 34, 353bitr3g 312 . . . . . . . . . 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)))))
3716, 36ceqsexv 3494 . . . . . . . . 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))))
3815, 37bitri 274 . . . . . . . 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))))
39 hllat 37825 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4039ad3antrrr 728 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ Lat)
41 simplll 773 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
42 simplrl 775 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑝𝐴)
43 simplrr 776 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑞𝐴)
441, 3, 4hlatjcl 37829 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝 𝑞) ∈ 𝐵)
4541, 42, 43, 44syl3anc 1371 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → (𝑝 𝑞) ∈ 𝐵)
461, 4atbase 37751 . . . . . . . . . . 11 (𝑟𝐴𝑟𝐵)
4746adantl 482 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑟𝐵)
481, 3latjcl 18328 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑝 𝑞) ∈ 𝐵𝑟𝐵) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
4940, 45, 47, 48syl3anc 1371 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
5049biantrurd 533 . . . . . . . 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)))))
5138, 50bitr4id 289 . . . . . . 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))))
5251rexbidva 3173 . . . . . 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))))
53522rexbidva 3211 . . . . 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))))
54 rexcom4 3271 . . . . . . . . 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))))
5554rexbii 3097 . . . . . . . 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))))
56 rexcom4 3271 . . . . . . . 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))))
5755, 56bitri 274 . . . . . . 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))))
5857rexbii 3097 . . . . . 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))))
59 rexcom4 3271 . . . . . 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))))
6058, 59bitri 274 . . . . 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))))
6153, 60bitr3di 285 . . . 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)))))
62 rexcom 3273 . . . . . . . . . . 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))))
6362rexbii 3097 . . . . . . . . . 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))))
64 rexcom 3273 . . . . . . . . . 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))))
6563, 64bitri 274 . . . . . . . . 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))))
6665rexbii 3097 . . . . . . . 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))))
67 rexcom 3273 . . . . . . . 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))))
6866, 67bitri 274 . . . . . . 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))))
691, 2, 3, 4, 5islpln2 37999 . . . . . . . . . . 11 (𝐾 ∈ HL → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
7069adantr 481 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
7170anbi1d 630 . . . . . . . . 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)))))
72 r19.42v 3187 . . . . . . . . . 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))))
73 r19.42v 3187 . . . . . . . . . . . . 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))))
7473rexbii 3097 . . . . . . . . . . . 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))))
75 r19.42v 3187 . . . . . . . . . . . 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))))
7674, 75bitri 274 . . . . . . . . . . 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))))
7776rexbii 3097 . . . . . . . . . 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))))
78 an32 644 . . . . . . . . . 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))))
7972, 77, 783bitr4ri 303 . . . . . . . . 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))))
8071, 79bitrdi 286 . . . . . . . 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)))))
8180rexbidv 3175 . . . . . . 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)))))
8268, 81bitr4id 289 . . . . . 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)))))
83 r19.42v 3187 . . . . . 6 (∃𝑠𝐴 (𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ (𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8482, 83bitrdi 286 . . . . 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)))))
8584exbidv 1924 . . . 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)))))
8661, 85bitrd 278 . . 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)))))
878, 86bitr4id 289 . 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 278 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  Latclat 18320  Atomscatm 37725  HLchlt 37812  LPlanesclpl 37955  LVolsclvol 37956
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963
This theorem is referenced by:  islvol2  38043  lvoli2  38044
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