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Theorem cdleme18d 40278
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 2827 . . . . . . . 8 (𝑅 = 𝑃 → (𝑅𝐴𝑃𝐴))
2 breq1 5151 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑊𝑃 𝑊))
32notbid 318 . . . . . . . 8 (𝑅 = 𝑃 → (¬ 𝑅 𝑊 ↔ ¬ 𝑃 𝑊))
41, 3anbi12d 632 . . . . . . 7 (𝑅 = 𝑃 → ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ↔ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)))
543anbi1d 1439 . . . . . 6 (𝑅 = 𝑃 → (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ↔ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊))))
653anbi2d 1440 . . . . 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 1202 . . . . . . 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 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)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
9 simp13l 1287 . . . . . . 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 1206 . . . . . . 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 1323 . . . . . . 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 2735 . . . . . . . 8 ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))
2012, 13, 14, 15, 16, 17, 18, 19cdleme17d1 40272 . . . . . . 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 1382 . . . . . 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 1207 . . . . . . 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 1324 . . . . . . 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 2735 . . . . . . . 8 ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊)))
2612, 13, 14, 15, 16, 17, 24, 25cdleme17d1 40272 . . . . . . 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 1382 . . . . . 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 2778 . . . . 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, 28biimtrdi 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 2753 . . . . 5 (𝐺 = 𝐸 ↔ ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))))
33 oveq1 7438 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑆) = (𝑃 𝑆))
3433oveq1d 7446 . . . . . . . 8 (𝑅 = 𝑃 → ((𝑅 𝑆) 𝑊) = ((𝑃 𝑆) 𝑊))
3534oveq2d 7447 . . . . . . 7 (𝑅 = 𝑃 → (𝐹 ((𝑅 𝑆) 𝑊)) = (𝐹 ((𝑃 𝑆) 𝑊)))
3635oveq2d 7447 . . . . . 6 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊))))
37 oveq1 7438 . . . . . . . . 9 (𝑅 = 𝑃 → (𝑅 𝑇) = (𝑃 𝑇))
3837oveq1d 7446 . . . . . . . 8 (𝑅 = 𝑃 → ((𝑅 𝑇) 𝑊) = ((𝑃 𝑇) 𝑊))
3938oveq2d 7447 . . . . . . 7 (𝑅 = 𝑃 → (𝐷 ((𝑅 𝑇) 𝑊)) = (𝐷 ((𝑃 𝑇) 𝑊)))
4039oveq2d 7447 . . . . . 6 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑃 𝑇) 𝑊))))
4136, 40eqeq12d 2751 . . . . 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 2827 . . . . . . . 8 (𝑅 = 𝑄 → (𝑅𝐴𝑄𝐴))
46 breq1 5151 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑊𝑄 𝑊))
4746notbid 318 . . . . . . . 8 (𝑅 = 𝑄 → (¬ 𝑅 𝑊 ↔ ¬ 𝑄 𝑊))
4845, 47anbi12d 632 . . . . . . 7 (𝑅 = 𝑄 → ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ↔ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
49483anbi1d 1439 . . . . . 6 (𝑅 = 𝑄 → (((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ↔ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊))))
50 breq1 5151 . . . . . . . 8 (𝑅 = 𝑄 → (𝑅 (𝑃 𝑄) ↔ 𝑄 (𝑃 𝑄)))
51503anbi1d 1439 . . . . . . 7 (𝑅 = 𝑄 → ((𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ↔ (𝑄 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))))
52513anbi2d 1440 . . . . . 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 1438 . . . . 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 1283 . . . . . . 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 1284 . . . . . . 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 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)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
57 simp21 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)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
58 simp22 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)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
59 simp31 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)))) → 𝑃𝑄)
60 simp322 1323 . . . . . . 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 1210 . . . . . . 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 2735 . . . . . . . 8 ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊)))
6312, 13, 14, 15, 16, 17, 18, 62cdleme18c 40276 . . . . . . 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 1398 . . . . . 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 1207 . . . . . . 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 1324 . . . . . . 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 2735 . . . . . . . 8 ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊)))
6812, 13, 14, 15, 16, 17, 24, 67cdleme18c 40276 . . . . . . 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 1398 . . . . . 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 2778 . . . . 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, 70biimtrdi 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 7438 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑆) = (𝑄 𝑆))
7372oveq1d 7446 . . . . . . . 8 (𝑅 = 𝑄 → ((𝑅 𝑆) 𝑊) = ((𝑄 𝑆) 𝑊))
7473oveq2d 7447 . . . . . . 7 (𝑅 = 𝑄 → (𝐹 ((𝑅 𝑆) 𝑊)) = (𝐹 ((𝑄 𝑆) 𝑊)))
7574oveq2d 7447 . . . . . 6 (𝑅 = 𝑄 → ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑄 𝑆) 𝑊))))
76 oveq1 7438 . . . . . . . . 9 (𝑅 = 𝑄 → (𝑅 𝑇) = (𝑄 𝑇))
7776oveq1d 7446 . . . . . . . 8 (𝑅 = 𝑄 → ((𝑅 𝑇) 𝑊) = ((𝑄 𝑇) 𝑊))
7877oveq2d 7447 . . . . . . 7 (𝑅 = 𝑄 → (𝐷 ((𝑅 𝑇) 𝑊)) = (𝐷 ((𝑄 𝑇) 𝑊)))
7978oveq2d 7447 . . . . . 6 (𝑅 = 𝑄 → ((𝑃 𝑄) (𝐷 ((𝑅 𝑇) 𝑊))) = ((𝑃 𝑄) (𝐷 ((𝑄 𝑇) 𝑊))))
8075, 79eqeq12d 2751 . . . . 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 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)))) → 𝐾 ∈ HL)
85 simp321 1322 . . 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 1210 . . 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 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)))) → 𝑃𝐴)
88 simp13l 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)))) → 𝑄𝐴)
89 simp31 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)))) → 𝑃𝑄)
90 simp21l 1289 . . 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 1290 . . 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 40273 . . 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 1400 . 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 860 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 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  HLchlt 39332  LHypclh 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971
This theorem is referenced by:  cdleme21  40320
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