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Theorem islpln5 34298
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 34296 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑋𝑃 ↔ ∃𝑦 ∈ (LLines‘𝐾)∃𝑟𝐴𝑟 𝑦𝑋 = (𝑦 𝑟))))
8 rexcom4 3211 . . . . . . 7 (∃𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
98rexbii 3034 . . . . . 6 (∃𝑝𝐴𝑞𝐴𝑦𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑦𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
10 rexcom4 3211 . . . . . 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 34130 . . . . . . . . 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 3081 . . . . . . . . . 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 6632 . . . . . . . . 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 4617 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑝 𝑞) → (𝑟 𝑦𝑟 (𝑝 𝑞)))
2625notbid 308 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑝 𝑞) → (¬ 𝑟 𝑦 ↔ ¬ 𝑟 (𝑝 𝑞)))
27 oveq1 6611 . . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . 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 3084 . . . . . . . . . 10 (∃𝑟𝐴 (𝑝𝑞 ∧ (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))) ↔ (𝑝𝑞 ∧ ∃𝑟𝐴 (𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟)))))
37 r19.42v 3084 . . . . . . . . . 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 3228 . . . . . . . 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 3049 . . . . 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 34274 . . . . . . . . . . 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 3084 . . . . . . . . . 10 (∃𝑝𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
48 r19.42v 3084 . . . . . . . . . . 11 (∃𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ ∃𝑞𝐴 (𝑝𝑞𝑦 = (𝑝 𝑞))))
4948rexbii 3034 . . . . . . . . . 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 3045 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑟𝐴 (𝑦 ∈ (LLines‘𝐾) ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ↔ ∃𝑟𝐴𝑝𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞)))))
54 rexcom 3091 . . . . . . . . 9 (∃𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
5554rexbii 3034 . . . . . . . 8 (∃𝑝𝐴𝑞𝐴𝑟𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))) ↔ ∃𝑝𝐴𝑟𝐴𝑞𝐴 ((𝑦𝐵 ∧ (¬ 𝑟 𝑦𝑋 = (𝑦 𝑟))) ∧ (𝑝𝑞𝑦 = (𝑝 𝑞))))
56 rexcom 3091 . . . . . . . 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 3084 . . . . . 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 2913 . . 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 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  Atomscatm 34027  HLchlt 34114  LLinesclln 34254  LPlanesclpl 34255
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-llines 34261  df-lplanes 34262
This theorem is referenced by:  islpln2  34299  lplni2  34300
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