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Theorem cdleme22e 37589
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 1194 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐾 ∈ HL)
32hllatd 36609 . . . 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 1287 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑃𝐴)
5 simp22l 1289 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑄𝐴)
6 eqid 2824 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 cdleme22.j . . . . . 6 = (join‘𝐾)
8 cdleme22.a . . . . . 6 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 36612 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
102, 4, 5, 9syl3anc 1368 . . . 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 1195 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑊𝐻)
12 simp33l 1297 . . . . . 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 37471 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑧𝐴)) → 𝐹 ∈ (Base‘𝐾))
192, 11, 4, 5, 12, 18syl23anc 1374 . . . . 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 1291 . . . . . . 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 36612 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑧𝐴) → (𝑆 𝑧) ∈ (Base‘𝐾))
222, 20, 12, 21syl3anc 1368 . . . . . 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 37243 . . . . . . 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 17662 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑆 𝑧) 𝑊) ∈ (Base‘𝐾))
263, 22, 24, 25syl3anc 1368 . . . . 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 17661 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑆 𝑧) 𝑊) ∈ (Base‘𝐾)) → (𝐹 ((𝑆 𝑧) 𝑊)) ∈ (Base‘𝐾))
283, 19, 26, 27syl3anc 1368 . . . 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 17686 . . . 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 1368 . . 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 5087 . 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 1133 . . . . . 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 1203 . . . . . 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 1292 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑇𝐴)
35 simp31 1206 . . . . . 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 1295 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑃𝑄)
37 simp32r 1296 . . . . . 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 37585 . . . . . 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 1390 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑉 = 𝑈)
4039oveq2d 7165 . . . 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 7159 . . . . 5 (𝑂 𝑈) = (((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊))) 𝑈)
43 simp21r 1288 . . . . . . 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 37456 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
452, 11, 4, 43, 5, 36, 44syl222anc 1383 . . . . . 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 36612 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑧𝐴) → (𝑇 𝑧) ∈ (Base‘𝐾))
472, 34, 12, 46syl3anc 1368 . . . . . . . 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 17662 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑇 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑇 𝑧) 𝑊) ∈ (Base‘𝐾))
493, 47, 24, 48syl3anc 1368 . . . . . . 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 17661 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑇 𝑧) 𝑊) ∈ (Base‘𝐾)) → (𝐹 ((𝑇 𝑧) 𝑊)) ∈ (Base‘𝐾))
513, 19, 49, 50syl3anc 1368 . . . . . 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 37465 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴)) → 𝑈 (𝑃 𝑄))
532, 11, 4, 5, 52syl22anc 837 . . . . . 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 37106 . . . . . 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 1380 . . . . 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, 55syl5req 2872 . . . 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 2862 . . 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 7165 . . . . . 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 2861 . . . . 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 36612 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
612, 34, 45, 60syl3anc 1368 . . . . . . 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 36534 . . . . . . . 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 17670 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑇 𝑈) ((𝑇 𝑈) 𝑧))
653, 61, 63, 64syl3anc 1368 . . . . . 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 36617 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑈𝐴𝑧𝐴)) → ((𝑇 𝑈) 𝑧) = ((𝑇 𝑧) 𝑈))
672, 34, 45, 12, 66syl13anc 1369 . . . . . . 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 36534 . . . . . . . . . 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 17707 . . . . . . . . 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 1369 . . . . . . . 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 17707 . . . . . . . . . 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 1369 . . . . . . . . 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 36612 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑧𝐴) → (𝑃 𝑧) ∈ (Base‘𝐾))
752, 4, 12, 74syl3anc 1368 . . . . . . . . . . . . . . . . . 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 36620 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑧𝐴) → 𝑃 (𝑃 𝑧))
772, 4, 12, 76syl3anc 1368 . . . . . . . . . . . . . . . . . 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 37109 . . . . . . . . . . . . . . . . . 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 1380 . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . . . 20 (1.‘𝐾) = (1.‘𝐾)
8113, 7, 80, 8, 15lhpjat2 37266 . . . . . . . . . . . . . . . . . . 19 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
822, 11, 33, 81syl21anc 836 . . . . . . . . . . . . . . . . . 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 7165 . . . . . . . . . . . . . . . . 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 36606 . . . . . . . . . . . . . . . . . . 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 36472 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ OL ∧ (𝑃 𝑧) ∈ (Base‘𝐾)) → ((𝑃 𝑧) (1.‘𝐾)) = (𝑃 𝑧))
8785, 75, 86syl2anc 587 . . . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . . . . 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 7164 . . . . . . . . . . . . . . 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 7160 . . . . . . . . . . . . . . . . . . 19 (𝑄 𝑈) = (𝑄 ((𝑃 𝑄) 𝑊))
9113, 7, 8hlatlej2 36621 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
922, 4, 5, 91syl3anc 1368 . . . . . . . . . . . . . . . . . . . 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 37109 . . . . . . . . . . . . . . . . . . . 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 1380 . . . . . . . . . . . . . . . . . . 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, 94syl5eq 2871 . . . . . . . . . . . . . . . . . 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 1204 . . . . . . . . . . . . . . . . . . . 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 37266 . . . . . . . . . . . . . . . . . . . 20 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
982, 11, 96, 97syl21anc 836 . . . . . . . . . . . . . . . . . . 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 7165 . . . . . . . . . . . . . . . . . 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 36472 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
10185, 10, 100syl2anc 587 . . . . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . . . . . 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 7164 . . . . . . . . . . . . . . . 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 36534 . . . . . . . . . . . . . . . . . 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 17662 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Lat ∧ (𝑃 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑧) 𝑊) ∈ (Base‘𝐾))
1073, 75, 24, 106syl3anc 1368 . . . . . . . . . . . . . . . . 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 36534 . . . . . . . . . . . . . . . . . 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 17707 . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . . . 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 36617 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑧𝐴)) → ((𝑃 𝑄) 𝑧) = ((𝑃 𝑧) 𝑄))
1142, 4, 5, 12, 113syl13anc 1369 . . . . . . . . . . . . . . 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 2870 . . . . . . . . . . . . . 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 17707 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . 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 7165 . . . . . . . . . . . 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 17661 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑧) ∈ (Base‘𝐾))
1213, 10, 63, 120syl3anc 1368 . . . . . . . . . . . . 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 17671 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧 ((𝑃 𝑄) 𝑧))
1233, 10, 63, 122syl3anc 1368 . . . . . . . . . . . . 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 37102 . . . . . . . . . . . . 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 1380 . . . . . . . . . . . 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 7159 . . . . . . . . . . . . 13 (𝐹 𝑈) = (((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊))) 𝑈)
1276, 7, 8hlatjcl 36612 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑧𝐴𝑈𝐴) → (𝑧 𝑈) ∈ (Base‘𝐾))
1282, 12, 45, 127syl3anc 1368 . . . . . . . . . . . . . 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 17661 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑧) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑧) 𝑊)) ∈ (Base‘𝐾))
1303, 109, 107, 129syl3anc 1368 . . . . . . . . . . . . . 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 36621 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑧𝐴𝑈𝐴) → 𝑈 (𝑧 𝑈))
1322, 12, 45, 131syl3anc 1368 . . . . . . . . . . . . . 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 37106 . . . . . . . . . . . . . 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 1380 . . . . . . . . . . . . 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, 134syl5eq 2871 . . . . . . . . . . . 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 2870 . . . . . . . . . . 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 17670 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑧))
1383, 10, 63, 137syl3anc 1368 . . . . . . . . . . . . . 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 17668 . . . . . . . . . . . . 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 17689 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑧) ∈ (Base‘𝐾)) → (𝑈 ((𝑃 𝑄) 𝑧) ↔ (𝑈 ((𝑃 𝑄) 𝑧)) = 𝑈))
1413, 69, 121, 140syl3anc 1368 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 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 7165 . . . . . . . . . . 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 2859 . . . . . . . . . 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 7164 . . . . . . . . 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 2859 . . . . . . . 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 36621 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑧𝐴) → 𝑧 (𝑇 𝑧))
1482, 34, 12, 147syl3anc 1368 . . . . . . . . . . . 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 37109 . . . . . . . . . . . 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 1380 . . . . . . . . . . 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 1208 . . . . . . . . . . . . 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 37266 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊)) → (𝑧 𝑊) = (1.‘𝐾))
1532, 11, 151, 152syl21anc 836 . . . . . . . . . . . 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 7165 . . . . . . . . . . 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 2859 . . . . . . . . . 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 36472 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ (𝑇 𝑧) ∈ (Base‘𝐾)) → ((𝑇 𝑧) (1.‘𝐾)) = (𝑇 𝑧))
15785, 47, 156syl2anc 587 . . . . . . . . . 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 2860 . . . . . . . . 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 7164 . . . . . . . 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 2870 . . . . . . 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 2859 . . . . . 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 5078 . . . . 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 5074 . . . 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 17661 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹 ((𝑇 𝑧) 𝑊)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝐹 ((𝑇 𝑧) 𝑊)) 𝑈) ∈ (Base‘𝐾))
1653, 51, 69, 164syl3anc 1368 . . . . 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 17689 . . . . 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 1368 . . . 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 235 . . 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 2860 . 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 5078 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  1.cp1 17648  Latclat 17655  OLcol 36419  Atomscatm 36508  HLchlt 36595  LHypclh 37229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-oposet 36421  df-ol 36423  df-oml 36424  df-covers 36511  df-ats 36512  df-atl 36543  df-cvlat 36567  df-hlat 36596  df-psubsp 36748  df-pmap 36749  df-padd 37041  df-lhyp 37233
This theorem is referenced by:  cdleme26e  37604
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