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Theorem cdleme22e 40346
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t v = p q, fz(s) fz(t) v. (Contributed by NM, 6-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l = (le‘𝐾)
cdleme22.j = (join‘𝐾)
cdleme22.m = (meet‘𝐾)
cdleme22.a 𝐴 = (Atoms‘𝐾)
cdleme22.h 𝐻 = (LHyp‘𝐾)
cdleme22e.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme22e.f 𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
cdleme22e.n 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))
cdleme22e.o 𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))
Assertion
Ref Expression
cdleme22e (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑁 (𝑂 𝑉))

Proof of Theorem cdleme22e
StepHypRef Expression
1 cdleme22e.n . . 3 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))
2 simp1l 1198 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐾 ∈ HL)
32hllatd 39365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐾 ∈ Lat)
4 simp21l 1291 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑃𝐴)
5 simp22l 1293 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑄𝐴)
6 eqid 2737 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 cdleme22.j . . . . . 6 = (join‘𝐾)
8 cdleme22.a . . . . . 6 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 39368 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
102, 4, 5, 9syl3anc 1373 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp1r 1199 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑊𝐻)
12 simp33l 1301 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑧𝐴)
13 cdleme22.l . . . . . . 7 = (le‘𝐾)
14 cdleme22.m . . . . . . 7 = (meet‘𝐾)
15 cdleme22.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
16 cdleme22e.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
17 cdleme22e.f . . . . . . 7 𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
1813, 7, 14, 8, 15, 16, 17, 6cdleme1b 40228 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑧𝐴)) → 𝐹 ∈ (Base‘𝐾))
192, 11, 4, 5, 12, 18syl23anc 1379 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐹 ∈ (Base‘𝐾))
20 simp23l 1295 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑆𝐴)
216, 7, 8hlatjcl 39368 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑧𝐴) → (𝑆 𝑧) ∈ (Base‘𝐾))
222, 20, 12, 21syl3anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑆 𝑧) ∈ (Base‘𝐾))
236, 15lhpbase 40000 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2411, 23syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑊 ∈ (Base‘𝐾))
256, 14latmcl 18485 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑆 𝑧) 𝑊) ∈ (Base‘𝐾))
263, 22, 24, 25syl3anc 1373 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑆 𝑧) 𝑊) ∈ (Base‘𝐾))
276, 7latjcl 18484 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑆 𝑧) 𝑊) ∈ (Base‘𝐾)) → (𝐹 ((𝑆 𝑧) 𝑊)) ∈ (Base‘𝐾))
283, 19, 26, 27syl3anc 1373 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐹 ((𝑆 𝑧) 𝑊)) ∈ (Base‘𝐾))
296, 13, 14latmle1 18509 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ((𝑆 𝑧) 𝑊)) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊))) (𝑃 𝑄))
303, 10, 28, 29syl3anc 1373 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊))) (𝑃 𝑄))
311, 30eqbrtrid 5178 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑁 (𝑃 𝑄))
32 simp1 1137 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
33 simp21 1207 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
34 simp23r 1296 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑇𝐴)
35 simp31 1210 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑉𝐴𝑉 𝑊))
36 simp32l 1299 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑃𝑄)
37 simp32r 1300 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑉) = (𝑃 𝑄))
3813, 7, 14, 8, 15, 16cdleme22a 40342 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑇𝐴) ∧ ((𝑉𝐴𝑉 𝑊) ∧ 𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄))) → 𝑉 = 𝑈)
3932, 33, 5, 34, 35, 36, 37, 38syl133anc 1395 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑉 = 𝑈)
4039oveq2d 7447 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑂 𝑉) = (𝑂 𝑈))
41 cdleme22e.o . . . . . 6 𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))
4241oveq1i 7441 . . . . 5 (𝑂 𝑈) = (((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊))) 𝑈)
43 simp21r 1292 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ¬ 𝑃 𝑊)
4413, 7, 14, 8, 15, 16cdleme0a 40213 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
452, 11, 4, 43, 5, 36, 44syl222anc 1388 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑈𝐴)
466, 7, 8hlatjcl 39368 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑧𝐴) → (𝑇 𝑧) ∈ (Base‘𝐾))
472, 34, 12, 46syl3anc 1373 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑧) ∈ (Base‘𝐾))
486, 14latmcl 18485 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑇 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑇 𝑧) 𝑊) ∈ (Base‘𝐾))
493, 47, 24, 48syl3anc 1373 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑇 𝑧) 𝑊) ∈ (Base‘𝐾))
506, 7latjcl 18484 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑇 𝑧) 𝑊) ∈ (Base‘𝐾)) → (𝐹 ((𝑇 𝑧) 𝑊)) ∈ (Base‘𝐾))
513, 19, 49, 50syl3anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐹 ((𝑇 𝑧) 𝑊)) ∈ (Base‘𝐾))
5213, 7, 14, 8, 15, 16cdlemeulpq 40222 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴)) → 𝑈 (𝑃 𝑄))
532, 11, 4, 5, 52syl22anc 839 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑈 (𝑃 𝑄))
546, 13, 7, 14, 8atmod2i1 39863 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑈𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ((𝑇 𝑧) 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑈 (𝑃 𝑄)) → (((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊))) 𝑈) = ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈)))
552, 45, 10, 51, 53, 54syl131anc 1385 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊))) 𝑈) = ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈)))
5642, 55eqtr2id 2790 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈)) = (𝑂 𝑈))
5740, 56eqtr4d 2780 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑂 𝑉) = ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈)))
5839oveq2d 7447 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑉) = (𝑇 𝑈))
5937, 58eqtr3d 2779 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 𝑄) = (𝑇 𝑈))
606, 7, 8hlatjcl 39368 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
612, 34, 45, 60syl3anc 1373 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑈) ∈ (Base‘𝐾))
626, 8atbase 39290 . . . . . . . 8 (𝑧𝐴𝑧 ∈ (Base‘𝐾))
6312, 62syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑧 ∈ (Base‘𝐾))
646, 13, 7latlej1 18493 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑇 𝑈) ((𝑇 𝑈) 𝑧))
653, 61, 63, 64syl3anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑈) ((𝑇 𝑈) 𝑧))
667, 8hlatj32 39373 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑈𝐴𝑧𝐴)) → ((𝑇 𝑈) 𝑧) = ((𝑇 𝑧) 𝑈))
672, 34, 45, 12, 66syl13anc 1374 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑇 𝑈) 𝑧) = ((𝑇 𝑧) 𝑈))
686, 8atbase 39290 . . . . . . . . . 10 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
6945, 68syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑈 ∈ (Base‘𝐾))
706, 7latj32 18530 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑇 𝑧) 𝑊) ∈ (Base‘𝐾))) → ((𝑧 𝑈) ((𝑇 𝑧) 𝑊)) = ((𝑧 ((𝑇 𝑧) 𝑊)) 𝑈))
713, 63, 69, 49, 70syl13anc 1374 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑧 𝑈) ((𝑇 𝑧) 𝑊)) = ((𝑧 ((𝑇 𝑧) 𝑊)) 𝑈))
726, 7latj32 18530 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝐹 ∈ (Base‘𝐾) ∧ ((𝑇 𝑧) 𝑊) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) = ((𝐹 𝑈) ((𝑇 𝑧) 𝑊)))
733, 19, 49, 69, 72syl13anc 1374 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) = ((𝐹 𝑈) ((𝑇 𝑧) 𝑊)))
746, 7, 8hlatjcl 39368 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑧𝐴) → (𝑃 𝑧) ∈ (Base‘𝐾))
752, 4, 12, 74syl3anc 1373 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 𝑧) ∈ (Base‘𝐾))
7613, 7, 8hlatlej1 39376 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑧𝐴) → 𝑃 (𝑃 𝑧))
772, 4, 12, 76syl3anc 1373 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑃 (𝑃 𝑧))
786, 13, 7, 14, 8atmod3i1 39866 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 𝑧)) → (𝑃 ((𝑃 𝑧) 𝑊)) = ((𝑃 𝑧) (𝑃 𝑊)))
792, 4, 75, 24, 77, 78syl131anc 1385 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 ((𝑃 𝑧) 𝑊)) = ((𝑃 𝑧) (𝑃 𝑊)))
80 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (1.‘𝐾) = (1.‘𝐾)
8113, 7, 80, 8, 15lhpjat2 40023 . . . . . . . . . . . . . . . . . . 19 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
822, 11, 33, 81syl21anc 838 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 𝑊) = (1.‘𝐾))
8382oveq2d 7447 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑧) (𝑃 𝑊)) = ((𝑃 𝑧) (1.‘𝐾)))
84 hlol 39362 . . . . . . . . . . . . . . . . . . 19 (𝐾 ∈ HL → 𝐾 ∈ OL)
852, 84syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐾 ∈ OL)
866, 14, 80olm11 39228 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ OL ∧ (𝑃 𝑧) ∈ (Base‘𝐾)) → ((𝑃 𝑧) (1.‘𝐾)) = (𝑃 𝑧))
8785, 75, 86syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑧) (1.‘𝐾)) = (𝑃 𝑧))
8879, 83, 873eqtrd 2781 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 ((𝑃 𝑧) 𝑊)) = (𝑃 𝑧))
8988oveq1d 7446 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 ((𝑃 𝑧) 𝑊)) 𝑄) = ((𝑃 𝑧) 𝑄))
9016oveq2i 7442 . . . . . . . . . . . . . . . . . . 19 (𝑄 𝑈) = (𝑄 ((𝑃 𝑄) 𝑊))
9113, 7, 8hlatlej2 39377 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
922, 4, 5, 91syl3anc 1373 . . . . . . . . . . . . . . . . . . . 20 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑄 (𝑃 𝑄))
936, 13, 7, 14, 8atmod3i1 39866 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑄 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑄 𝑊)))
942, 5, 10, 24, 92, 93syl131anc 1385 . . . . . . . . . . . . . . . . . . 19 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑄 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑄 𝑊)))
9590, 94eqtrid 2789 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑄 𝑈) = ((𝑃 𝑄) (𝑄 𝑊)))
96 simp22 1208 . . . . . . . . . . . . . . . . . . . 20 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
9713, 7, 80, 8, 15lhpjat2 40023 . . . . . . . . . . . . . . . . . . . 20 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
982, 11, 96, 97syl21anc 838 . . . . . . . . . . . . . . . . . . 19 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑄 𝑊) = (1.‘𝐾))
9998oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) (𝑄 𝑊)) = ((𝑃 𝑄) (1.‘𝐾)))
1006, 14, 80olm11 39228 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
10185, 10, 100syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
10295, 99, 1013eqtrd 2781 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))
103102oveq1d 7446 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑄 𝑈) ((𝑃 𝑧) 𝑊)) = ((𝑃 𝑄) ((𝑃 𝑧) 𝑊)))
1046, 8atbase 39290 . . . . . . . . . . . . . . . . . 18 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1054, 104syl 17 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑃 ∈ (Base‘𝐾))
1066, 14latmcl 18485 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Lat ∧ (𝑃 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑧) 𝑊) ∈ (Base‘𝐾))
1073, 75, 24, 106syl3anc 1373 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑧) 𝑊) ∈ (Base‘𝐾))
1086, 8atbase 39290 . . . . . . . . . . . . . . . . . 18 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1095, 108syl 17 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑄 ∈ (Base‘𝐾))
1106, 7latj32 18530 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑧) 𝑊) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → ((𝑃 ((𝑃 𝑧) 𝑊)) 𝑄) = ((𝑃 𝑄) ((𝑃 𝑧) 𝑊)))
1113, 105, 107, 109, 110syl13anc 1374 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 ((𝑃 𝑧) 𝑊)) 𝑄) = ((𝑃 𝑄) ((𝑃 𝑧) 𝑊)))
112103, 111eqtr4d 2780 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑄 𝑈) ((𝑃 𝑧) 𝑊)) = ((𝑃 ((𝑃 𝑧) 𝑊)) 𝑄))
1137, 8hlatj32 39373 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑧𝐴)) → ((𝑃 𝑄) 𝑧) = ((𝑃 𝑧) 𝑄))
1142, 4, 5, 12, 113syl13anc 1374 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) 𝑧) = ((𝑃 𝑧) 𝑄))
11589, 112, 1143eqtr4rd 2788 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) 𝑧) = ((𝑄 𝑈) ((𝑃 𝑧) 𝑊)))
1166, 7latj32 18530 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 𝑧) 𝑊) ∈ (Base‘𝐾))) → ((𝑄 𝑈) ((𝑃 𝑧) 𝑊)) = ((𝑄 ((𝑃 𝑧) 𝑊)) 𝑈))
1173, 109, 69, 107, 116syl13anc 1374 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑄 𝑈) ((𝑃 𝑧) 𝑊)) = ((𝑄 ((𝑃 𝑧) 𝑊)) 𝑈))
118115, 117eqtrd 2777 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) 𝑧) = ((𝑄 ((𝑃 𝑧) 𝑊)) 𝑈))
119118oveq2d 7447 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑧 𝑈) ((𝑃 𝑄) 𝑧)) = ((𝑧 𝑈) ((𝑄 ((𝑃 𝑧) 𝑊)) 𝑈)))
1206, 7latjcl 18484 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑧) ∈ (Base‘𝐾))
1213, 10, 63, 120syl3anc 1373 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) 𝑧) ∈ (Base‘𝐾))
1226, 13, 7latlej2 18494 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧 ((𝑃 𝑄) 𝑧))
1233, 10, 63, 122syl3anc 1373 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑧 ((𝑃 𝑄) 𝑧))
1246, 13, 7, 14, 8atmod1i1 39859 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑧𝐴𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑧) ∈ (Base‘𝐾)) ∧ 𝑧 ((𝑃 𝑄) 𝑧)) → (𝑧 (𝑈 ((𝑃 𝑄) 𝑧))) = ((𝑧 𝑈) ((𝑃 𝑄) 𝑧)))
1252, 12, 69, 121, 123, 124syl131anc 1385 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧 (𝑈 ((𝑃 𝑄) 𝑧))) = ((𝑧 𝑈) ((𝑃 𝑄) 𝑧)))
12617oveq1i 7441 . . . . . . . . . . . . 13 (𝐹 𝑈) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) 𝑈)
1276, 7, 8hlatjcl 39368 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑧𝐴𝑈𝐴) → (𝑧 𝑈) ∈ (Base‘𝐾))
1282, 12, 45, 127syl3anc 1373 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧 𝑈) ∈ (Base‘𝐾))
1296, 7latjcl 18484 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑧) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑧) 𝑊)) ∈ (Base‘𝐾))
1303, 109, 107, 129syl3anc 1373 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑄 ((𝑃 𝑧) 𝑊)) ∈ (Base‘𝐾))
13113, 7, 8hlatlej2 39377 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑧𝐴𝑈𝐴) → 𝑈 (𝑧 𝑈))
1322, 12, 45, 131syl3anc 1373 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑈 (𝑧 𝑈))
1336, 13, 7, 14, 8atmod2i1 39863 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑈𝐴 ∧ (𝑧 𝑈) ∈ (Base‘𝐾) ∧ (𝑄 ((𝑃 𝑧) 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑈 (𝑧 𝑈)) → (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) 𝑈) = ((𝑧 𝑈) ((𝑄 ((𝑃 𝑧) 𝑊)) 𝑈)))
1342, 45, 128, 130, 132, 133syl131anc 1385 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) 𝑈) = ((𝑧 𝑈) ((𝑄 ((𝑃 𝑧) 𝑊)) 𝑈)))
135126, 134eqtrid 2789 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐹 𝑈) = ((𝑧 𝑈) ((𝑄 ((𝑃 𝑧) 𝑊)) 𝑈)))
136119, 125, 1353eqtr4rd 2788 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐹 𝑈) = (𝑧 (𝑈 ((𝑃 𝑄) 𝑧))))
1376, 13, 7latlej1 18493 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑧))
1383, 10, 63, 137syl3anc 1373 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 𝑄) ((𝑃 𝑄) 𝑧))
1396, 13, 3, 69, 10, 121, 53, 138lattrd 18491 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑈 ((𝑃 𝑄) 𝑧))
1406, 13, 14latleeqm1 18512 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑧) ∈ (Base‘𝐾)) → (𝑈 ((𝑃 𝑄) 𝑧) ↔ (𝑈 ((𝑃 𝑄) 𝑧)) = 𝑈))
1413, 69, 121, 140syl3anc 1373 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑈 ((𝑃 𝑄) 𝑧) ↔ (𝑈 ((𝑃 𝑄) 𝑧)) = 𝑈))
142139, 141mpbid 232 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑈 ((𝑃 𝑄) 𝑧)) = 𝑈)
143142oveq2d 7447 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧 (𝑈 ((𝑃 𝑄) 𝑧))) = (𝑧 𝑈))
144136, 143eqtrd 2777 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐹 𝑈) = (𝑧 𝑈))
145144oveq1d 7446 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝐹 𝑈) ((𝑇 𝑧) 𝑊)) = ((𝑧 𝑈) ((𝑇 𝑧) 𝑊)))
14673, 145eqtrd 2777 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) = ((𝑧 𝑈) ((𝑇 𝑧) 𝑊)))
14713, 7, 8hlatlej2 39377 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑧𝐴) → 𝑧 (𝑇 𝑧))
1482, 34, 12, 147syl3anc 1373 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑧 (𝑇 𝑧))
1496, 13, 7, 14, 8atmod3i1 39866 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑧𝐴 ∧ (𝑇 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑧 (𝑇 𝑧)) → (𝑧 ((𝑇 𝑧) 𝑊)) = ((𝑇 𝑧) (𝑧 𝑊)))
1502, 12, 47, 24, 148, 149syl131anc 1385 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧 ((𝑇 𝑧) 𝑊)) = ((𝑇 𝑧) (𝑧 𝑊)))
151 simp33 1212 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧𝐴 ∧ ¬ 𝑧 𝑊))
15213, 7, 80, 8, 15lhpjat2 40023 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊)) → (𝑧 𝑊) = (1.‘𝐾))
1532, 11, 151, 152syl21anc 838 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧 𝑊) = (1.‘𝐾))
154153oveq2d 7447 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑇 𝑧) (𝑧 𝑊)) = ((𝑇 𝑧) (1.‘𝐾)))
155150, 154eqtrd 2777 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧 ((𝑇 𝑧) 𝑊)) = ((𝑇 𝑧) (1.‘𝐾)))
1566, 14, 80olm11 39228 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ (𝑇 𝑧) ∈ (Base‘𝐾)) → ((𝑇 𝑧) (1.‘𝐾)) = (𝑇 𝑧))
15785, 47, 156syl2anc 584 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑇 𝑧) (1.‘𝐾)) = (𝑇 𝑧))
158155, 157eqtr2d 2778 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑧) = (𝑧 ((𝑇 𝑧) 𝑊)))
159158oveq1d 7446 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑇 𝑧) 𝑈) = ((𝑧 ((𝑇 𝑧) 𝑊)) 𝑈))
16071, 146, 1593eqtr4rd 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑇 𝑧) 𝑈) = ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈))
16167, 160eqtrd 2777 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑇 𝑈) 𝑧) = ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈))
16265, 161breqtrd 5169 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑈) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈))
16359, 162eqbrtrd 5165 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈))
1646, 7latjcl 18484 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹 ((𝑇 𝑧) 𝑊)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) ∈ (Base‘𝐾))
1653, 51, 69, 164syl3anc 1373 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) ∈ (Base‘𝐾))
1666, 13, 14latleeqm1 18512 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) ∈ (Base‘𝐾)) → ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) ↔ ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈)) = (𝑃 𝑄)))
1673, 10, 165, 166syl3anc 1373 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) ↔ ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈)) = (𝑃 𝑄)))
168163, 167mpbid 232 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ((𝑃 𝑄) ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈)) = (𝑃 𝑄))
16957, 168eqtr2d 2778 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃 𝑄) = (𝑂 𝑉))
17031, 169breqtrd 5169 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑁 (𝑂 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  1.cp1 18469  Latclat 18476  OLcol 39175  Atomscatm 39264  HLchlt 39351  LHypclh 39986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990
This theorem is referenced by:  cdleme26e  40361
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