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Theorem dalawlem12 39865
Description: Lemma for dalaw 39869. Second part of dalawlem13 39866. (Contributed by NM, 17-Sep-2012.)
Hypotheses
Ref Expression
dalawlem.l = (le‘𝐾)
dalawlem.j = (join‘𝐾)
dalawlem.m = (meet‘𝐾)
dalawlem.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dalawlem12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Proof of Theorem dalawlem12
StepHypRef Expression
1 eqid 2735 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 dalawlem.l . . . 4 = (le‘𝐾)
3 simp11 1202 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
43hllatd 39346 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ Lat)
5 simp21 1205 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
6 simp22 1206 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄𝐴)
7 dalawlem.j . . . . . . 7 = (join‘𝐾)
8 dalawlem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
91, 7, 8hlatjcl 39349 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
103, 5, 6, 9syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp31 1208 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆𝐴)
12 simp32 1209 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇𝐴)
131, 7, 8hlatjcl 39349 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
143, 11, 12, 13syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑇) ∈ (Base‘𝐾))
15 dalawlem.m . . . . . 6 = (meet‘𝐾)
161, 15latmcl 18498 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
174, 10, 14, 16syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
181, 8atbase 39271 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1911, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ∈ (Base‘𝐾))
201, 7latjcl 18497 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
214, 10, 19, 20syl3anc 1370 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
221, 8atbase 39271 . . . . . . 7 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
2312, 22syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 ∈ (Base‘𝐾))
241, 15latmcl 18498 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
254, 21, 23, 24syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
261, 7latjcl 18497 . . . . 5 ((𝐾 ∈ Lat ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
274, 25, 19, 26syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
281, 8atbase 39271 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
296, 28syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 ∈ (Base‘𝐾))
30 simp33 1210 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈𝐴)
311, 7, 8hlatjcl 39349 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
323, 12, 30, 31syl3anc 1370 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑈) ∈ (Base‘𝐾))
331, 15latmcl 18498 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
344, 29, 32, 33syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
351, 7, 8hlatjcl 39349 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
363, 30, 11, 35syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑆) ∈ (Base‘𝐾))
371, 7latjcl 18497 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
384, 34, 36, 37syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
391, 2, 7latlej1 18506 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
404, 10, 19, 39syl3anc 1370 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
411, 7, 8hlatjcl 39349 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑆𝐴) → (𝑇 𝑆) ∈ (Base‘𝐾))
423, 12, 11, 41syl3anc 1370 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑆) ∈ (Base‘𝐾))
431, 2, 15latmlem1 18527 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ (𝑇 𝑆) ∈ (Base‘𝐾))) → ((𝑃 𝑄) ((𝑃 𝑄) 𝑆) → ((𝑃 𝑄) (𝑇 𝑆)) (((𝑃 𝑄) 𝑆) (𝑇 𝑆))))
444, 10, 21, 42, 43syl13anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) ((𝑃 𝑄) 𝑆) → ((𝑃 𝑄) (𝑇 𝑆)) (((𝑃 𝑄) 𝑆) (𝑇 𝑆))))
4540, 44mpd 15 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑇 𝑆)) (((𝑃 𝑄) 𝑆) (𝑇 𝑆)))
467, 8hlatjcom 39350 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) = (𝑇 𝑆))
473, 11, 12, 46syl3anc 1370 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑇) = (𝑇 𝑆))
4847oveq2d 7447 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑄) (𝑇 𝑆)))
491, 2, 7latlej2 18507 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
504, 10, 19, 49syl3anc 1370 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
511, 2, 7, 15, 8atmod2i2 39845 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑇𝐴 ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) ∧ 𝑆 ((𝑃 𝑄) 𝑆)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) = (((𝑃 𝑄) 𝑆) (𝑇 𝑆)))
523, 12, 21, 19, 50, 51syl131anc 1382 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) = (((𝑃 𝑄) 𝑆) (𝑇 𝑆)))
5345, 48, 523brtr4d 5180 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆))
54 hlol 39343 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ OL)
553, 54syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ OL)
561, 7, 8hlatjcl 39349 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
573, 5, 11, 56syl3anc 1370 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
581, 7latjcl 18497 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
594, 29, 57, 58syl3anc 1370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
601, 7, 8hlatjcl 39349 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
613, 6, 12, 60syl3anc 1370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑇) ∈ (Base‘𝐾))
621, 15latmassOLD 39211 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ ((𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾))) → (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇) = ((𝑄 (𝑃 𝑆)) ((𝑄 𝑇) 𝑇)))
6355, 59, 61, 23, 62syl13anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇) = ((𝑄 (𝑃 𝑆)) ((𝑄 𝑇) 𝑇)))
647, 8hlatjass 39352 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
653, 5, 6, 11, 64syl13anc 1371 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
667, 8hlatj12 39353 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
673, 5, 6, 11, 66syl13anc 1371 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
6865, 67eqtr2d 2776 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) = ((𝑃 𝑄) 𝑆))
692, 7, 8hlatlej2 39358 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑇 (𝑄 𝑇))
703, 6, 12, 69syl3anc 1370 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑄 𝑇))
711, 2, 15latleeqm2 18526 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
724, 23, 61, 71syl3anc 1370 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
7370, 72mpbid 232 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) 𝑇) = 𝑇)
7468, 73oveq12d 7449 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) ((𝑄 𝑇) 𝑇)) = (((𝑃 𝑄) 𝑆) 𝑇))
7563, 74eqtr2d 2776 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) = (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇))
762, 7, 8hlatlej1 39357 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
773, 6, 12, 76syl3anc 1370 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑄 𝑇))
781, 2, 7, 15, 8atmod1i1 39840 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) ∧ 𝑄 (𝑄 𝑇)) → (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) = ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)))
793, 6, 57, 61, 77, 78syl131anc 1382 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) = ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)))
802, 7, 8hlatlej2 39358 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → 𝑄 (𝑈 𝑄))
813, 30, 6, 80syl3anc 1370 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑈 𝑄))
82 simp13 1204 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
83 simp12 1203 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = 𝑅)
8483oveq1d 7446 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑅 𝑈))
857, 8hlatjcom 39350 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑈𝐴) → (𝑄 𝑈) = (𝑈 𝑄))
863, 6, 30, 85syl3anc 1370 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑈 𝑄))
8784, 86eqtr3d 2777 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑈) = (𝑈 𝑄))
8882, 87breqtrd 5174 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑄))
891, 15latmcl 18498 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
904, 57, 61, 89syl3anc 1370 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
911, 7, 8hlatjcl 39349 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → (𝑈 𝑄) ∈ (Base‘𝐾))
923, 30, 6, 91syl3anc 1370 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑄) ∈ (Base‘𝐾))
931, 2, 7latjle12 18508 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾) ∧ (𝑈 𝑄) ∈ (Base‘𝐾))) → ((𝑄 (𝑈 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑄)) ↔ (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) (𝑈 𝑄)))
944, 29, 90, 92, 93syl13anc 1371 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑈 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑄)) ↔ (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) (𝑈 𝑄)))
9581, 88, 94mpbi2and 712 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) (𝑈 𝑄))
9679, 95eqbrtrrd 5172 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) (𝑈 𝑄))
972, 7, 8hlatlej1 39357 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑇 (𝑇 𝑈))
983, 12, 30, 97syl3anc 1370 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑇 𝑈))
991, 15latmcl 18498 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1004, 59, 61, 99syl3anc 1370 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1011, 2, 15latmlem12 18529 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾) ∧ (𝑈 𝑄) ∈ (Base‘𝐾)) ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾))) → ((((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) (𝑈 𝑄) ∧ 𝑇 (𝑇 𝑈)) → (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇) ((𝑈 𝑄) (𝑇 𝑈))))
1024, 100, 92, 23, 32, 101syl122anc 1378 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) (𝑈 𝑄) ∧ 𝑇 (𝑇 𝑈)) → (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇) ((𝑈 𝑄) (𝑇 𝑈))))
10396, 98, 102mp2and 699 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇) ((𝑈 𝑄) (𝑇 𝑈)))
10475, 103eqbrtrd 5170 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑈 𝑄) (𝑇 𝑈)))
1052, 7, 8hlatlej2 39358 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑈 (𝑇 𝑈))
1063, 12, 30, 105syl3anc 1370 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 (𝑇 𝑈))
1071, 2, 7, 15, 8atmod1i1 39840 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) ∧ 𝑈 (𝑇 𝑈)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑈 𝑄) (𝑇 𝑈)))
1083, 30, 29, 32, 106, 107syl131anc 1382 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑈 𝑄) (𝑇 𝑈)))
1091, 8atbase 39271 . . . . . . . . . 10 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
11030, 109syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 ∈ (Base‘𝐾))
1111, 7latjcom 18505 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
1124, 110, 34, 111syl3anc 1370 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
113108, 112eqtr3d 2777 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑄) (𝑇 𝑈)) = ((𝑄 (𝑇 𝑈)) 𝑈))
114104, 113breqtrd 5174 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑄 (𝑇 𝑈)) 𝑈))
1151, 7latjcl 18497 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1164, 34, 110, 115syl3anc 1370 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1171, 2, 7latjlej1 18511 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾) ∧ ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((((𝑃 𝑄) 𝑆) 𝑇) ((𝑄 (𝑇 𝑈)) 𝑈) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆)))
1184, 25, 116, 19, 117syl13anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) ((𝑄 (𝑇 𝑈)) 𝑈) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆)))
119114, 118mpd 15 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆))
1201, 7latjass 18541 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆) = ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
1214, 34, 110, 19, 120syl13anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆) = ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
122119, 121breqtrd 5174 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
1231, 2, 4, 17, 27, 38, 53, 122lattrd 18504 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
1241, 2, 15latmle1 18522 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1254, 10, 14, 124syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1261, 2, 15latlem12 18524 . . . 4 ((𝐾 ∈ Lat ∧ (((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾) ∧ ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((((𝑃 𝑄) (𝑆 𝑇)) ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∧ ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄)) ↔ ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) (𝑃 𝑄))))
1274, 17, 38, 10, 126syl13anc 1371 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) (𝑆 𝑇)) ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∧ ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄)) ↔ ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) (𝑃 𝑄))))
128123, 125, 127mpbi2and 712 . 2 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) (𝑃 𝑄)))
1291, 8atbase 39271 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1305, 129syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃 ∈ (Base‘𝐾))
1311, 2, 7, 15latmlej12 18537 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1324, 29, 32, 130, 131syl13anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1331, 2, 7, 15, 8llnmod1i2 39843 . . . 4 (((𝐾 ∈ HL ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) ∧ (𝑈𝐴𝑆𝐴) ∧ (𝑄 (𝑇 𝑈)) (𝑃 𝑄)) → ((𝑄 (𝑇 𝑈)) ((𝑈 𝑆) (𝑃 𝑄))) = (((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) (𝑃 𝑄)))
1343, 34, 10, 30, 11, 132, 133syl321anc 1391 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) ((𝑈 𝑆) (𝑃 𝑄))) = (((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) (𝑃 𝑄)))
1357, 8hlatjidm 39351 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
1363, 6, 135syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = 𝑄)
13783oveq2d 7447 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = (𝑄 𝑅))
138136, 137eqtr3d 2777 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = (𝑄 𝑅))
139138oveq1d 7446 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) = ((𝑄 𝑅) (𝑇 𝑈)))
1401, 15latmcom 18521 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑈 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1414, 36, 10, 140syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1427, 8hlatjcom 39350 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
1433, 5, 6, 142syl3anc 1370 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
14483oveq1d 7446 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑃) = (𝑅 𝑃))
145143, 144eqtrd 2775 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑅 𝑃))
146145oveq1d 7446 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑈 𝑆)) = ((𝑅 𝑃) (𝑈 𝑆)))
147141, 146eqtrd 2775 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑅 𝑃) (𝑈 𝑆)))
148139, 147oveq12d 7449 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) ((𝑈 𝑆) (𝑃 𝑄))) = (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
149134, 148eqtr3d 2777 . 2 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) (𝑃 𝑄)) = (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
150128, 149breqtrd 5174 1 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  OLcol 39156  Atomscatm 39245  HLchlt 39332
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-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-psubsp 39486  df-pmap 39487  df-padd 39779
This theorem is referenced by:  dalawlem13  39866
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