Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islvol5 Structured version   Visualization version   GIF version

Theorem islvol5 39562
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 2735 . . 3 (LPlanes‘𝐾) = (LPlanes‘𝐾)
6 islvol5.v . . 3 𝑉 = (LVols‘𝐾)
71, 2, 3, 4, 5, 6islvol3 39559 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑉 ↔ ∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8 df-rex 3069 . . 3 (∃𝑦 ∈ (LPlanes‘𝐾)∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ ∃𝑦(𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
9 r19.41v 3187 . . . . . . . . . . 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 1088 . . . . . . . . . . . . 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 647 . . . . . . . . . . . 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 275 . . . . . . . . . . 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 275 . . . . . . . . . 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 1845 . . . . . . . . 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 7464 . . . . . . . . . 10 ((𝑝 𝑞) 𝑟) ∈ V
17 an12 645 . . . . . . . . . . . . 13 (((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ (𝑦𝐵 ∧ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
18 eleq1 2827 . . . . . . . . . . . . . 14 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦𝐵 ↔ ((𝑝 𝑞) 𝑟) ∈ 𝐵))
19 breq2 5152 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑠 𝑦𝑠 ((𝑝 𝑞) 𝑟)))
2019notbid 318 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (¬ 𝑠 𝑦 ↔ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
21 oveq1 7438 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑦 𝑠) = (((𝑝 𝑞) 𝑟) 𝑠))
2221eqeq2d 2746 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑝 𝑞) 𝑟) → (𝑋 = (𝑦 𝑠) ↔ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠)))
2320, 22anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑝 𝑞) 𝑟) → ((¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)) ↔ (¬ 𝑠 ((𝑝 𝑞) 𝑟) ∧ 𝑋 = (((𝑝 𝑞) 𝑟) 𝑠))))
2423anbi2d 630 . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . 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 1088 . . . . . . . . . . . . . . . . . 18 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ¬ 𝑠 ((𝑝 𝑞) 𝑟)))
2726bicomi 224 . . . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . . . 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 287 . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . 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 283 . . . . . . . . . . . 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 3177 . . . . . . . . . . 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 3189 . . . . . . . . . . 11 (∃𝑠𝐴 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))) ↔ ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞)) ∧ ∃𝑠𝐴 (𝑦𝐵 ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠)))))
35 r19.42v 3189 . . . . . . . . . . 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 313 . . . . . . . . . 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 3530 . . . . . . . . 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 275 . . . . . . . 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 39345 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4039ad3antrrr 730 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ Lat)
41 simplll 775 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
42 simplrl 777 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑝𝐴)
43 simplrr 778 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑞𝐴)
441, 3, 4hlatjcl 39349 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝 𝑞) ∈ 𝐵)
4541, 42, 43, 44syl3anc 1370 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → (𝑝 𝑞) ∈ 𝐵)
461, 4atbase 39271 . . . . . . . . . . 11 (𝑟𝐴𝑟𝐵)
4746adantl 481 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → 𝑟𝐵)
481, 3latjcl 18497 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑝 𝑞) ∈ 𝐵𝑟𝐵) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
4940, 45, 47, 48syl3anc 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) ∧ 𝑟𝐴) → ((𝑝 𝑞) 𝑟) ∈ 𝐵)
5049biantrurd 532 . . . . . . . 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 290 . . . . . . 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 3175 . . . . . 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 3218 . . . . 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 3286 . . . . . . . . 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 3092 . . . . . . . 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 3286 . . . . . . . 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 275 . . . . . . 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 3092 . . . . . 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 3286 . . . . . 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 275 . . . . 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 286 . . . 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 3288 . . . . . . . . . . 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 3092 . . . . . . . . . 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 3288 . . . . . . . . . 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 275 . . . . . . . . 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 3092 . . . . . . . 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 3288 . . . . . . . 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 275 . . . . . . 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 39519 . . . . . . . . . . 11 (𝐾 ∈ HL → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
7069adantr 480 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑦 ∈ (LPlanes‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑦 = ((𝑝 𝑞) 𝑟)))))
7170anbi1d 631 . . . . . . . . 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 3189 . . . . . . . . . 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 3189 . . . . . . . . . . . . 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 3092 . . . . . . . . . . . 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 3189 . . . . . . . . . . . 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 275 . . . . . . . . . . 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 3092 . . . . . . . . . 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 646 . . . . . . . . . 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 304 . . . . . . . . 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 287 . . . . . . . 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 3177 . . . . . . 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 290 . . . . . 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 3189 . . . . . 6 (∃𝑠𝐴 (𝑦 ∈ (LPlanes‘𝐾) ∧ (¬ 𝑠 𝑦𝑋 = (𝑦 𝑠))) ↔ (𝑦 ∈ (LPlanes‘𝐾) ∧ ∃𝑠𝐴𝑠 𝑦𝑋 = (𝑦 𝑠))))
8482, 83bitrdi 287 . . . . 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 1919 . . . 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 279 . . 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 290 . 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 279 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 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wrex 3068   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  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-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:  islvol2  39563  lvoli2  39564
  Copyright terms: Public domain W3C validator