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Theorem cdleme18d 36076
Description: Part of proof of Lemma E in [Crawley] p. 114, 4th sentence of 4th paragraph. 𝐹, 𝐺, 𝐷, 𝐸 represent f(s), fs(r), f(t), ft(r) respectively. We show fs(r)=ft(r) for all possible r (which must equal p or q in the case of exactly 3 atoms in p q/0 i.e. when ¬ ∃𝑟𝐴...). (Contributed by NM, 12-Nov-2012.)
Hypotheses
Ref Expression
cdleme18d.l = (le‘𝐾)
cdleme18d.j = (join‘𝐾)
cdleme18d.m = (meet‘𝐾)
cdleme18d.a 𝐴 = (Atoms‘𝐾)
cdleme18d.h 𝐻 = (LHyp‘𝐾)
cdleme18d.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme18d.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme18d.g 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
cdleme18d.d 𝐷 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
cdleme18d.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊)))
Assertion
Ref Expression
cdleme18d ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐺 = 𝐸)
Distinct variable groups:   𝐴,𝑟   𝐷,𝑟   𝐹,𝑟   ,𝑟   ,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑅,𝑟   𝑆,𝑟   𝑇,𝑟   𝑊,𝑟
Allowed substitution hints:   𝑈(𝑟)   𝐸(𝑟)   𝐺(𝑟)   𝐻(𝑟)   𝐾(𝑟)

Proof of Theorem cdleme18d
StepHypRef Expression
1 eleq1 2873 . . . . . . . 8 (𝑅 = 𝑃 → (𝑅𝐴𝑃𝐴))
2 breq1 4847 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑊𝑃 𝑊))
32notbid 309 . . . . . . . 8 (𝑅 = 𝑃 → (¬ 𝑅 𝑊 ↔ ¬ 𝑃 𝑊))
41, 3anbi12d 618 . . . . . . 7 (𝑅 = 𝑃 → ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ↔ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)))
543anbi1d 1557 . . . . . 6 (𝑅 = 𝑃 → (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ↔ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊))))
653anbi2d 1558 . . . . 5 (𝑅 = 𝑃 → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))))))
7 simp11 1253 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
8 simp21 1256 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
9 simp13l 1380 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝐴)
10 simp22 1257 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
11 simp322 1416 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑆 (𝑃 𝑄))
12 cdleme18d.l . . . . . . . 8 = (le‘𝐾)
13 cdleme18d.j . . . . . . . 8 = (join‘𝐾)
14 cdleme18d.m . . . . . . . 8 = (meet‘𝐾)
15 cdleme18d.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
16 cdleme18d.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
17 cdleme18d.u . . . . . . . 8 𝑈 = ((𝑃 𝑄) 𝑊)
18 cdleme18d.f . . . . . . . 8 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
19 eqid 2806 . . . . . . . 8 ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))
2012, 13, 14, 15, 16, 17, 18, 19cdleme17d1 36070 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = 𝑄)
217, 8, 9, 10, 11, 20syl131anc 1495 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = 𝑄)
22 simp23 1258 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
23 simp323 1417 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑇 (𝑃 𝑄))
24 cdleme18d.d . . . . . . . 8 𝐷 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
25 eqid 2806 . . . . . . . 8 ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊)))
2612, 13, 14, 15, 16, 17, 24, 25cdleme17d1 36070 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ¬ 𝑇 (𝑃 𝑄)) → ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))) = 𝑄)
277, 8, 9, 22, 23, 26syl131anc 1495 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))) = 𝑄)
2821, 27eqtr4d 2843 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))))
296, 28syl6bi 244 . . . 4 (𝑅 = 𝑃 → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊)))))
30 cdleme18d.g . . . . . 6 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
31 cdleme18d.e . . . . . 6 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊)))
3230, 31eqeq12i 2820 . . . . 5 (𝐺 = 𝐸 ↔ ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
33 oveq1 6881 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑆) = (𝑃 𝑆))
3433oveq1d 6889 . . . . . . . 8 (𝑅 = 𝑃 → ((𝑅 𝑆) 𝑊) = ((𝑃 𝑆) 𝑊))
3534oveq2d 6890 . . . . . . 7 (𝑅 = 𝑃 → (𝐹 ((𝑅 𝑆) 𝑊)) = (𝐹 ((𝑃 𝑆) 𝑊)))
3635oveq2d 6890 . . . . . 6 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))))
37 oveq1 6881 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑇) = (𝑃 𝑇))
3837oveq1d 6889 . . . . . . . 8 (𝑅 = 𝑃 → ((𝑅 𝑇) 𝑊) = ((𝑃 𝑇) 𝑊))
3938oveq2d 6890 . . . . . . 7 (𝑅 = 𝑃 → (𝐷 ((𝑅 𝑇) 𝑊)) = (𝐷 ((𝑃 𝑇) 𝑊)))
4039oveq2d 6890 . . . . . 6 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))))
4136, 40eqeq12d 2821 . . . . 5 (𝑅 = 𝑃 → (((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) ↔ ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊)))))
4232, 41syl5bb 274 . . . 4 (𝑅 = 𝑃 → (𝐺 = 𝐸 ↔ ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊)))))
4329, 42sylibrd 250 . . 3 (𝑅 = 𝑃 → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐺 = 𝐸))
4443com12 32 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑅 = 𝑃𝐺 = 𝐸))
45 eleq1 2873 . . . . . . . 8 (𝑅 = 𝑄 → (𝑅𝐴𝑄𝐴))
46 breq1 4847 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑊𝑄 𝑊))
4746notbid 309 . . . . . . . 8 (𝑅 = 𝑄 → (¬ 𝑅 𝑊 ↔ ¬ 𝑄 𝑊))
4845, 47anbi12d 618 . . . . . . 7 (𝑅 = 𝑄 → ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ↔ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
49483anbi1d 1557 . . . . . 6 (𝑅 = 𝑄 → (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ↔ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊))))
50 breq1 4847 . . . . . . . 8 (𝑅 = 𝑄 → (𝑅 (𝑃 𝑄) ↔ 𝑄 (𝑃 𝑄)))
51503anbi1d 1557 . . . . . . 7 (𝑅 = 𝑄 → ((𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ↔ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))))
52513anbi2d 1558 . . . . . 6 (𝑅 = 𝑄 → ((𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ↔ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))))
5349, 523anbi23d 1556 . . . . 5 (𝑅 = 𝑄 → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))))))
54 simp11l 1376 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
55 simp11r 1377 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑊𝐻)
56 simp12 1254 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
57 simp21 1256 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
58 simp22 1257 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
59 simp31 1259 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝑄)
60 simp322 1416 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑆 (𝑃 𝑄))
61 simp33 1261 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
62 eqid 2806 . . . . . . . 8 ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊)))
6312, 13, 14, 15, 16, 17, 18, 62cdleme18c 36074 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = 𝑃)
6454, 55, 56, 57, 58, 59, 60, 61, 63syl233anc 1511 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = 𝑃)
65 simp23 1258 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
66 simp323 1417 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑇 (𝑃 𝑄))
67 eqid 2806 . . . . . . . 8 ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊)))
6812, 13, 14, 15, 16, 17, 24, 67cdleme18c 36074 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))) = 𝑃)
6954, 55, 56, 57, 65, 59, 66, 61, 68syl233anc 1511 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))) = 𝑃)
7064, 69eqtr4d 2843 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))))
7153, 70syl6bi 244 . . . 4 (𝑅 = 𝑄 → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊)))))
72 oveq1 6881 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑆) = (𝑄 𝑆))
7372oveq1d 6889 . . . . . . . 8 (𝑅 = 𝑄 → ((𝑅 𝑆) 𝑊) = ((𝑄 𝑆) 𝑊))
7473oveq2d 6890 . . . . . . 7 (𝑅 = 𝑄 → (𝐹 ((𝑅 𝑆) 𝑊)) = (𝐹 ((𝑄 𝑆) 𝑊)))
7574oveq2d 6890 . . . . . 6 (𝑅 = 𝑄 → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))))
76 oveq1 6881 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑇) = (𝑄 𝑇))
7776oveq1d 6889 . . . . . . . 8 (𝑅 = 𝑄 → ((𝑅 𝑇) 𝑊) = ((𝑄 𝑇) 𝑊))
7877oveq2d 6890 . . . . . . 7 (𝑅 = 𝑄 → (𝐷 ((𝑅 𝑇) 𝑊)) = (𝐷 ((𝑄 𝑇) 𝑊)))
7978oveq2d 6890 . . . . . 6 (𝑅 = 𝑄 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))))
8075, 79eqeq12d 2821 . . . . 5 (𝑅 = 𝑄 → (((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) ↔ ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊)))))
8132, 80syl5bb 274 . . . 4 (𝑅 = 𝑄 → (𝐺 = 𝐸 ↔ ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊)))))
8271, 81sylibrd 250 . . 3 (𝑅 = 𝑄 → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐺 = 𝐸))
8382com12 32 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑅 = 𝑄𝐺 = 𝐸))
84 simp11l 1376 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
85 simp321 1415 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑅 (𝑃 𝑄))
86 simp33 1261 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
87 simp12l 1378 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝐴)
88 simp13l 1380 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝐴)
89 simp31 1259 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝑄)
90 simp21l 1382 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑅𝐴)
91 simp21r 1383 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑅 𝑊)
9212, 13, 15cdleme0nex 36071 . . 3 (((𝐾 ∈ HL ∧ 𝑅 (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 = 𝑃𝑅 = 𝑄))
9384, 85, 86, 87, 88, 89, 90, 91, 92syl332anc 1513 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑅 = 𝑃𝑅 = 𝑄))
9444, 83, 93mpjaod 878 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐺 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wrex 3097   class class class wbr 4844  cfv 6101  (class class class)co 6874  lecple 16160  joincjn 17149  meetcmee 17150  Atomscatm 35043  HLchlt 35130  LHypclh 35764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-2nd 7399  df-proset 17133  df-poset 17151  df-plt 17163  df-lub 17179  df-glb 17180  df-join 17181  df-meet 17182  df-p0 17244  df-p1 17245  df-lat 17251  df-clat 17313  df-oposet 34956  df-ol 34958  df-oml 34959  df-covers 35046  df-ats 35047  df-atl 35078  df-cvlat 35102  df-hlat 35131  df-llines 35278  df-lines 35281  df-psubsp 35283  df-pmap 35284  df-padd 35576  df-lhyp 35768
This theorem is referenced by:  cdleme21  36118
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