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Theorem cdleme18d 39656
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 2813 . . . . . . . 8 (𝑅 = 𝑃 → (𝑅𝐴𝑃𝐴))
2 breq1 5141 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑊𝑃 𝑊))
32notbid 318 . . . . . . . 8 (𝑅 = 𝑃 → (¬ 𝑅 𝑊 ↔ ¬ 𝑃 𝑊))
41, 3anbi12d 630 . . . . . . 7 (𝑅 = 𝑃 → ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ↔ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)))
543anbi1d 1436 . . . . . 6 (𝑅 = 𝑃 → (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ↔ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊))))
653anbi2d 1437 . . . . 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 1200 . . . . . . 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 1203 . . . . . . 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 1285 . . . . . . 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 1204 . . . . . . 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 1321 . . . . . . 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 2724 . . . . . . . 8 ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))
2012, 13, 14, 15, 16, 17, 18, 19cdleme17d1 39650 . . . . . . 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 1380 . . . . . 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 1205 . . . . . . 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 1322 . . . . . . 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 2724 . . . . . . . 8 ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊)))
2612, 13, 14, 15, 16, 17, 24, 25cdleme17d1 39650 . . . . . . 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 1380 . . . . . 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 2767 . . . . 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 253 . . . 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 2742 . . . . 5 (𝐺 = 𝐸 ↔ ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
33 oveq1 7408 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑆) = (𝑃 𝑆))
3433oveq1d 7416 . . . . . . . 8 (𝑅 = 𝑃 → ((𝑅 𝑆) 𝑊) = ((𝑃 𝑆) 𝑊))
3534oveq2d 7417 . . . . . . 7 (𝑅 = 𝑃 → (𝐹 ((𝑅 𝑆) 𝑊)) = (𝐹 ((𝑃 𝑆) 𝑊)))
3635oveq2d 7417 . . . . . 6 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))))
37 oveq1 7408 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑇) = (𝑃 𝑇))
3837oveq1d 7416 . . . . . . . 8 (𝑅 = 𝑃 → ((𝑅 𝑇) 𝑊) = ((𝑃 𝑇) 𝑊))
3938oveq2d 7417 . . . . . . 7 (𝑅 = 𝑃 → (𝐷 ((𝑅 𝑇) 𝑊)) = (𝐷 ((𝑃 𝑇) 𝑊)))
4039oveq2d 7417 . . . . . 6 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))))
4136, 40eqeq12d 2740 . . . . 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, 41bitrid 283 . . . 4 (𝑅 = 𝑃 → (𝐺 = 𝐸 ↔ ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊)))))
4329, 42sylibrd 259 . . 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 2813 . . . . . . . 8 (𝑅 = 𝑄 → (𝑅𝐴𝑄𝐴))
46 breq1 5141 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑊𝑄 𝑊))
4746notbid 318 . . . . . . . 8 (𝑅 = 𝑄 → (¬ 𝑅 𝑊 ↔ ¬ 𝑄 𝑊))
4845, 47anbi12d 630 . . . . . . 7 (𝑅 = 𝑄 → ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ↔ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
49483anbi1d 1436 . . . . . 6 (𝑅 = 𝑄 → (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ↔ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊))))
50 breq1 5141 . . . . . . . 8 (𝑅 = 𝑄 → (𝑅 (𝑃 𝑄) ↔ 𝑄 (𝑃 𝑄)))
51503anbi1d 1436 . . . . . . 7 (𝑅 = 𝑄 → ((𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ↔ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))))
52513anbi2d 1437 . . . . . 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 1435 . . . . 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 1281 . . . . . . 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 1282 . . . . . . 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 1201 . . . . . . 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 1203 . . . . . . 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 1204 . . . . . . 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 1206 . . . . . . 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 1321 . . . . . . 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 1208 . . . . . . 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 2724 . . . . . . . 8 ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊)))
6312, 13, 14, 15, 16, 17, 18, 62cdleme18c 39654 . . . . . . 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 1396 . . . . . 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 1205 . . . . . . 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 1322 . . . . . . 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 2724 . . . . . . . 8 ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊)))
6812, 13, 14, 15, 16, 17, 24, 67cdleme18c 39654 . . . . . . 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 1396 . . . . . 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 2767 . . . . 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 253 . . . 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 7408 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑆) = (𝑄 𝑆))
7372oveq1d 7416 . . . . . . . 8 (𝑅 = 𝑄 → ((𝑅 𝑆) 𝑊) = ((𝑄 𝑆) 𝑊))
7473oveq2d 7417 . . . . . . 7 (𝑅 = 𝑄 → (𝐹 ((𝑅 𝑆) 𝑊)) = (𝐹 ((𝑄 𝑆) 𝑊)))
7574oveq2d 7417 . . . . . 6 (𝑅 = 𝑄 → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))))
76 oveq1 7408 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑇) = (𝑄 𝑇))
7776oveq1d 7416 . . . . . . . 8 (𝑅 = 𝑄 → ((𝑅 𝑇) 𝑊) = ((𝑄 𝑇) 𝑊))
7877oveq2d 7417 . . . . . . 7 (𝑅 = 𝑄 → (𝐷 ((𝑅 𝑇) 𝑊)) = (𝐷 ((𝑄 𝑇) 𝑊)))
7978oveq2d 7417 . . . . . 6 (𝑅 = 𝑄 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))))
8075, 79eqeq12d 2740 . . . . 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, 80bitrid 283 . . . 4 (𝑅 = 𝑄 → (𝐺 = 𝐸 ↔ ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊)))))
8271, 81sylibrd 259 . . 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 1281 . . 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 1320 . . 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 1208 . . 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 1283 . . 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 1285 . . 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 1206 . . 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 1287 . . 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 1288 . . 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 39651 . . 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 1398 . 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 857 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 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wrex 3062   class class class wbr 5138  cfv 6533  (class class class)co 7401  lecple 17203  joincjn 18266  meetcmee 18267  Atomscatm 38623  HLchlt 38710  LHypclh 39345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-oposet 38536  df-ol 38538  df-oml 38539  df-covers 38626  df-ats 38627  df-atl 38658  df-cvlat 38682  df-hlat 38711  df-llines 38859  df-lines 38862  df-psubsp 38864  df-pmap 38865  df-padd 39157  df-lhyp 39349
This theorem is referenced by:  cdleme21  39698
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