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Theorem islpln5 34340
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 2621 . . 3 (LLines‘𝐾) = (LLines‘𝐾)
6 islpln5.p . . 3 𝑃 = (LPlanes‘𝐾)
71, 2, 3, 4, 5, 6islpln3 34338 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
8 rexcom4 3215 . . . . . . 7 (∃𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
98rexbii 3036 . . . . . 6 (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
10 rexcom4 3215 . . . . . 6 (∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
119, 10bitri 264 . . . . 5 (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
12 simpll 789 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝐾 ∈ HL)
13 simprl 793 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝑝𝐴)
14 simprr 795 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → 𝑞𝐴)
151, 3, 4hlatjcl 34172 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑝𝐴𝑞𝐴) → (𝑝 𝑞) ∈ 𝐵)
1612, 13, 14, 15syl3anc 1323 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (𝑝 𝑞) ∈ 𝐵)
1716biantrurd 529 . . . . . . 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 3083 . . . . . . . . . 10 (∃𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
19 an13 839 . . . . . . . . . 10 ((∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
2018, 19bitri 264 . . . . . . . . 9 (∃𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
2120exbii 1771 . . . . . . . 8 (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦(𝑦 = (𝑝 𝑞) ∧ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))))))
22 ovex 6643 . . . . . . . . 9 (𝑝 𝑞) ∈ V
23 an12 837 . . . . . . . . . . . 12 ((𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ (𝑦𝐵 ∧ (𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
24 eleq1 2686 . . . . . . . . . . . . 13 (𝑦 = (𝑝 𝑞) → (𝑦𝐵 ↔ (𝑝 𝑞) ∈ 𝐵))
25 breq2 4627 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑝 𝑞) → (𝑟 𝑦𝑟 (𝑝 𝑞)))
2625notbid 308 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑝 𝑞) → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 (𝑝 𝑞)))
27 oveq1 6622 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑝 𝑞) → (𝑦 𝑟) = ((𝑝 𝑞) 𝑟))
2827eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑝 𝑞) → (𝑋 = (𝑦 𝑟) ↔ 𝑋 = ((𝑝 𝑞) 𝑟)))
2926, 28anbi12d 746 . . . . . . . . . . . . . . 15 (𝑦 = (𝑝 𝑞) → ((¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
3029anbi2d 739 . . . . . . . . . . . . . 14 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑝𝑞 ∧ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
31 3anass 1040 . . . . . . . . . . . . . 14 ((𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ (𝑝𝑞 ∧ (¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
3230, 31syl6bbr 278 . . . . . . . . . . . . 13 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
3324, 32anbi12d 746 . . . . . . . . . . . 12 (𝑦 = (𝑝 𝑞) → ((𝑦𝐵 ∧ (𝑝𝑞 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
3423, 33syl5bb 272 . . . . . . . . . . 11 (𝑦 = (𝑝 𝑞) → ((𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)))))
3534rexbidv 3047 . . . . . . . . . 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 3086 . . . . . . . . . 10 (∃𝑟𝐴 (𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
37 r19.42v 3086 . . . . . . . . . 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 302 . . . . . . . . 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 3232 . . . . . . . 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 264 . . . . . . 7 (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑝 𝑞) ∈ 𝐵 ∧ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
4117, 40syl6rbbr 279 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑝𝐴𝑞𝐴)) → (∃𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
42412rexbidva 3051 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
4311, 42syl5rbbr 275 . . . 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 34316 . . . . . . . . . . 11 (𝐾 ∈ HL → (𝑦 ∈ (LLines‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞)))))
4544adantr 481 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑦 ∈ (LLines‘𝐾) ↔ (𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞)))))
4645anbi1d 740 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
47 r19.42v 3086 . . . . . . . . . 10 (∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
48 r19.42v 3086 . . . . . . . . . . 11 (∃𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
4948rexbii 3036 . . . . . . . . . 10 (∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
50 an32 838 . . . . . . . . . 10 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
5147, 49, 503bitr4ri 293 . . . . . . . . 9 (((𝑦𝐵 ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5246, 51syl6bb 276 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
5352rexbidv 3047 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
54 rexcom 3093 . . . . . . . . 9 (∃𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5554rexbii 3036 . . . . . . . 8 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
56 rexcom 3093 . . . . . . . 8 (∃𝑝𝐴𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5755, 56bitri 264 . . . . . . 7 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5853, 57syl6rbbr 279 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
59 r19.42v 3086 . . . . . 6 (∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ (𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
6058, 59syl6bb 276 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ (𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
6160exbidv 1847 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
6243, 61bitrd 268 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟)) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)))))
63 df-rex 2914 . . 3 (∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ ∃𝑦(𝑦 ∈ (LLines‘𝐾) ∧ ∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
6462, 63syl6rbbr 279 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟)) ↔ ∃𝑝𝐴𝑞𝐴𝑟𝐴 (𝑝𝑞 ∧ ¬ 𝑟 (𝑝 𝑞) ∧ 𝑋 = ((𝑝 𝑞) 𝑟))))
657, 64bitrd 268 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 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2909   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  joincjn 16884  Atomscatm 34069  HLchlt 34156  LLinesclln 34296  LPlanesclpl 34297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-preset 16868  df-poset 16886  df-plt 16898  df-lub 16914  df-glb 16915  df-join 16916  df-meet 16917  df-p0 16979  df-lat 16986  df-clat 17048  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304
This theorem is referenced by:  islpln2  34341  lplni2  34342
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