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Theorem islpln5 33839
 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 2610 . . 3 (LLines‘𝐾) = (LLines‘𝐾)
6 islpln5.p . . 3 𝑃 = (LPlanes‘𝐾)
71, 2, 3, 4, 5, 6islpln3 33837 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
8 rexcom4 3198 . . . . . . 7 (∃𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
98rexbii 3023 . . . . . 6 (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
10 rexcom4 3198 . . . . . 6 (∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
119, 10bitri 263 . . . . 5 (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
12 simpll 786 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝐾 ∈ HL)
13 simprl 790 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝑝𝐴)
14 simprr 792 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝑞𝐴)
151, 3, 4hlatjcl 33671 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝 𝑞) ∈ 𝐵)
1612, 13, 14, 15syl3anc 1318 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (𝑝 𝑞) ∈ 𝐵)
1716biantrurd 528 . . . . . . 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)))))
18 r19.41v 3070 . . . . . . . . . 10 (∃𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
19 an13 836 . . . . . . . . . 10 ((∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
2018, 19bitri 263 . . . . . . . . 9 (∃𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
2120exbii 1764 . . . . . . . 8 (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦(𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
22 ovex 6577 . . . . . . . . 9 (𝑝 𝑞) ∈ V
23 an12 834 . . . . . . . . . . . 12 ((𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ (𝑦𝐵 ∧ (𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
24 eleq1 2676 . . . . . . . . . . . . 13 (𝑦 = (𝑝 𝑞) → (𝑦𝐵 ↔ (𝑝 𝑞) ∈ 𝐵))
25 breq2 4587 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑝 𝑞) → (𝑟 𝑦𝑟 (𝑝 𝑞)))
2625notbid 307 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑝 𝑞) → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 (𝑝 𝑞)))
27 oveq1 6556 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑝 𝑞) → (𝑦 𝑟) = ((𝑝 𝑞) 𝑟))
2827eqeq2d 2620 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑝 𝑞) → (𝑋 = (𝑦 𝑟) ↔ 𝑋 = ((𝑝 𝑞) 𝑟)))
2926, 28anbi12d 743 . . . . . . . . . . . . . . 15 (𝑦 = (𝑝 𝑞) → ((¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
3029anbi2d 736 . . . . . . . . . . . . . 14 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑝𝑞 ∧ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
31 3anass 1035 . . . . . . . . . . . . . 14 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ (𝑝𝑞 ∧ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
3230, 31syl6bbr 277 . . . . . . . . . . . . 13 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
3324, 32anbi12d 743 . . . . . . . . . . . 12 (𝑦 = (𝑝 𝑞) → ((𝑦𝐵 ∧ (𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
3423, 33syl5bb 271 . . . . . . . . . . 11 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
3534rexbidv 3034 . . . . . . . . . 10 (𝑦 = (𝑝 𝑞) → (∃𝑟𝐴 (𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ∃𝑟𝐴 ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
36 r19.42v 3073 . . . . . . . . . 10 (∃𝑟𝐴 (𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
37 r19.42v 3073 . . . . . . . . . 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))))
3835, 36, 373bitr3g 301 . . . . . . . . 9 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
3922, 38ceqsexv 3215 . . . . . . . 8 (∃𝑦(𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
4021, 39bitri 263 . . . . . . 7 (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
4117, 40syl6rbbr 278 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
42412rexbidva 3038 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
4311, 42syl5rbbr 274 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
441, 3, 4, 5islln2 33815 . . . . . . . . . . 11 (𝐾 ∈ HL → (𝑦 ∈ (LLines‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞)))))
4544adantr 480 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑦 ∈ (LLines‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞)))))
4645anbi1d 737 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
47 r19.42v 3073 . . . . . . . . . 10 (∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
48 r19.42v 3073 . . . . . . . . . . 11 (∃𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
4948rexbii 3023 . . . . . . . . . 10 (∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
50 an32 835 . . . . . . . . . 10 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
5147, 49, 503bitr4ri 292 . . . . . . . . 9 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5246, 51syl6bb 275 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
5352rexbidv 3034 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
54 rexcom 3080 . . . . . . . . 9 (∃𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5554rexbii 3023 . . . . . . . 8 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
56 rexcom 3080 . . . . . . . 8 (∃𝑝𝐴𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5755, 56bitri 263 . . . . . . 7 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5853, 57syl6rbbr 278 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
59 r19.42v 3073 . . . . . 6 (∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
6058, 59syl6bb 275 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
6160exbidv 1837 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
6243, 61bitrd 267 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
63 df-rex 2902 . . 3 (∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
6462, 63syl6rbbr 278 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
657, 64bitrd 267 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Atomscatm 33568  HLchlt 33655  LLinesclln 33795  LPlanesclpl 33796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803 This theorem is referenced by:  islpln2  33840  lplni2  33841
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