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Theorem islpln5 37998
Description: The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 24-Jun-2012.)
Hypotheses
Ref Expression
islpln5.b 𝐵 = (Base‘𝐾)
islpln5.l = (le‘𝐾)
islpln5.j = (join‘𝐾)
islpln5.a 𝐴 = (Atoms‘𝐾)
islpln5.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
islpln5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
Distinct variable groups:   𝑞,𝑝,𝑟,𝐴   𝐵,𝑝,𝑞,𝑟   ,𝑝,𝑞,𝑟   𝐾,𝑝,𝑞,𝑟   ,𝑝,𝑞,𝑟   𝑋,𝑝,𝑞,𝑟
Allowed substitution hints:   𝑃(𝑟,𝑞,𝑝)

Proof of Theorem islpln5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 islpln5.b . . 3 𝐵 = (Base‘𝐾)
2 islpln5.l . . 3 = (le‘𝐾)
3 islpln5.j . . 3 = (join‘𝐾)
4 islpln5.a . . 3 𝐴 = (Atoms‘𝐾)
5 eqid 2736 . . 3 (LLines‘𝐾) = (LLines‘𝐾)
6 islpln5.p . . 3 𝑃 = (LPlanes‘𝐾)
71, 2, 3, 4, 5, 6islpln3 37996 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
8 df-rex 3074 . . 3 (∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
9 r19.41v 3185 . . . . . . . . . 10 (∃𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
10 an13 645 . . . . . . . . . 10 ((∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
119, 10bitri 274 . . . . . . . . 9 (∃𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
1211exbii 1850 . . . . . . . 8 (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦(𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
13 ovex 7390 . . . . . . . . 9 (𝑝 𝑞) ∈ V
14 an12 643 . . . . . . . . . . . 12 ((𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ (𝑦𝐵 ∧ (𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
15 eleq1 2825 . . . . . . . . . . . . 13 (𝑦 = (𝑝 𝑞) → (𝑦𝐵 ↔ (𝑝 𝑞) ∈ 𝐵))
16 breq2 5109 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑝 𝑞) → (𝑟 𝑦𝑟 (𝑝 𝑞)))
1716notbid 317 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑝 𝑞) → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 (𝑝 𝑞)))
18 oveq1 7364 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑝 𝑞) → (𝑦 𝑟) = ((𝑝 𝑞) 𝑟))
1918eqeq2d 2747 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑝 𝑞) → (𝑋 = (𝑦 𝑟) ↔ 𝑋 = ((𝑝 𝑞) 𝑟)))
2017, 19anbi12d 631 . . . . . . . . . . . . . . 15 (𝑦 = (𝑝 𝑞) → ((¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
2120anbi2d 629 . . . . . . . . . . . . . 14 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑝𝑞 ∧ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
22 3anass 1095 . . . . . . . . . . . . . 14 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ (𝑝𝑞 ∧ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
2321, 22bitr4di 288 . . . . . . . . . . . . 13 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
2415, 23anbi12d 631 . . . . . . . . . . . 12 (𝑦 = (𝑝 𝑞) → ((𝑦𝐵 ∧ (𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
2514, 24bitrid 282 . . . . . . . . . . 11 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
2625rexbidv 3175 . . . . . . . . . 10 (𝑦 = (𝑝 𝑞) → (∃𝑟𝐴 (𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ∃𝑟𝐴 ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
27 r19.42v 3187 . . . . . . . . . 10 (∃𝑟𝐴 (𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
28 r19.42v 3187 . . . . . . . . . 10 (∃𝑟𝐴 ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
2926, 27, 283bitr3g 312 . . . . . . . . 9 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
3013, 29ceqsexv 3494 . . . . . . . 8 (∃𝑦(𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
3112, 30bitri 274 . . . . . . 7 (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
32 simpll 765 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝐾 ∈ HL)
33 simprl 769 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝑝𝐴)
34 simprr 771 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝑞𝐴)
351, 3, 4hlatjcl 37829 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝 𝑞) ∈ 𝐵)
3632, 33, 34, 35syl3anc 1371 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (𝑝 𝑞) ∈ 𝐵)
3736biantrurd 533 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
3831, 37bitr4id 289 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
39382rexbidva 3211 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
40 rexcom4 3271 . . . . . . 7 (∃𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
4140rexbii 3097 . . . . . 6 (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
42 rexcom4 3271 . . . . . 6 (∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
4341, 42bitri 274 . . . . 5 (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
4439, 43bitr3di 285 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
45 rexcom 3273 . . . . . . . . 9 (∃𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
4645rexbii 3097 . . . . . . . 8 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
47 rexcom 3273 . . . . . . . 8 (∃𝑝𝐴𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
4846, 47bitri 274 . . . . . . 7 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
491, 3, 4, 5islln2 37974 . . . . . . . . . . 11 (𝐾 ∈ HL → (𝑦 ∈ (LLines‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞)))))
5049adantr 481 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑦 ∈ (LLines‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞)))))
5150anbi1d 630 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
52 r19.42v 3187 . . . . . . . . . 10 (∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
53 r19.42v 3187 . . . . . . . . . . 11 (∃𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
5453rexbii 3097 . . . . . . . . . 10 (∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
55 an32 644 . . . . . . . . . 10 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
5652, 54, 553bitr4ri 303 . . . . . . . . 9 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5751, 56bitrdi 286 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
5857rexbidv 3175 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
5948, 58bitr4id 289 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
60 r19.42v 3187 . . . . . 6 (∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
6159, 60bitrdi 286 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
6261exbidv 1924 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
6344, 62bitrd 278 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
648, 63bitr4id 289 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
657, 64bitrd 278 1 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
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  Atomscatm 37725  HLchlt 37812  LLinesclln 37954  LPlanesclpl 37955
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
This theorem is referenced by:  islpln2  37999  lplni2  38000
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