Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalawlem12 Structured version   Visualization version   GIF version

Theorem dalawlem12 35838
Description: Lemma for dalaw 35842. Second part of dalawlem13 35839. (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 2765 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 dalawlem.l . . . 4 = (le‘𝐾)
3 simp11 1260 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
43hllatd 35320 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ Lat)
5 simp21 1263 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
6 simp22 1264 . . . . . 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 35323 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
103, 5, 6, 9syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp31 1266 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆𝐴)
12 simp32 1267 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇𝐴)
131, 7, 8hlatjcl 35323 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
143, 11, 12, 13syl3anc 1490 . . . . 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 17318 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
174, 10, 14, 16syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
181, 8atbase 35245 . . . . . . . 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 17317 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
214, 10, 19, 20syl3anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
221, 8atbase 35245 . . . . . . 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 17318 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
254, 21, 23, 24syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
261, 7latjcl 17317 . . . . 5 ((𝐾 ∈ Lat ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
274, 25, 19, 26syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
281, 8atbase 35245 . . . . . . 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 1268 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈𝐴)
311, 7, 8hlatjcl 35323 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
323, 12, 30, 31syl3anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑈) ∈ (Base‘𝐾))
331, 15latmcl 17318 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
344, 29, 32, 33syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
351, 7, 8hlatjcl 35323 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
363, 30, 11, 35syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑆) ∈ (Base‘𝐾))
371, 7latjcl 17317 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
384, 34, 36, 37syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
391, 2, 7latlej1 17326 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
404, 10, 19, 39syl3anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
411, 7, 8hlatjcl 35323 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑆𝐴) → (𝑇 𝑆) ∈ (Base‘𝐾))
423, 12, 11, 41syl3anc 1490 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑆) ∈ (Base‘𝐾))
431, 2, 15latmlem1 17347 . . . . . . 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 1491 . . . . . 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 35324 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) = (𝑇 𝑆))
473, 11, 12, 46syl3anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑇) = (𝑇 𝑆))
4847oveq2d 6858 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑄) (𝑇 𝑆)))
491, 2, 7latlej2 17327 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
504, 10, 19, 49syl3anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
511, 2, 7, 15, 8atmod2i2 35818 . . . . . 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 1502 . . . . 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 4841 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆))
54 hlol 35317 . . . . . . . . . . 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 35323 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
573, 5, 11, 56syl3anc 1490 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
581, 7latjcl 17317 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
594, 29, 57, 58syl3anc 1490 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
601, 7, 8hlatjcl 35323 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
613, 6, 12, 60syl3anc 1490 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑇) ∈ (Base‘𝐾))
621, 15latmassOLD 35185 . . . . . . . . . 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 1491 . . . . . . . . 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 35326 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
653, 5, 6, 11, 64syl13anc 1491 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
667, 8hlatj12 35327 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
673, 5, 6, 11, 66syl13anc 1491 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
6865, 67eqtr2d 2800 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) = ((𝑃 𝑄) 𝑆))
692, 7, 8hlatlej2 35332 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑇 (𝑄 𝑇))
703, 6, 12, 69syl3anc 1490 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑄 𝑇))
711, 2, 15latleeqm2 17346 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
724, 23, 61, 71syl3anc 1490 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
7370, 72mpbid 223 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) 𝑇) = 𝑇)
7468, 73oveq12d 6860 . . . . . . . . 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 2800 . . . . . . . 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 35331 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
773, 6, 12, 76syl3anc 1490 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑄 𝑇))
781, 2, 7, 15, 8atmod1i1 35813 . . . . . . . . . . 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 1502 . . . . . . . . . 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 35332 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → 𝑄 (𝑈 𝑄))
813, 30, 6, 80syl3anc 1490 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑈 𝑄))
82 simp13 1262 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
83 simp12 1261 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = 𝑅)
8483oveq1d 6857 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑅 𝑈))
857, 8hlatjcom 35324 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑈𝐴) → (𝑄 𝑈) = (𝑈 𝑄))
863, 6, 30, 85syl3anc 1490 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑈 𝑄))
8784, 86eqtr3d 2801 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑈) = (𝑈 𝑄))
8882, 87breqtrd 4835 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑄))
891, 15latmcl 17318 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
904, 57, 61, 89syl3anc 1490 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
911, 7, 8hlatjcl 35323 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → (𝑈 𝑄) ∈ (Base‘𝐾))
923, 30, 6, 91syl3anc 1490 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑄) ∈ (Base‘𝐾))
931, 2, 7latjle12 17328 . . . . . . . . . . . 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 1491 . . . . . . . . . . 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 703 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) (𝑈 𝑄))
9679, 95eqbrtrrd 4833 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) (𝑈 𝑄))
972, 7, 8hlatlej1 35331 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑇 (𝑇 𝑈))
983, 12, 30, 97syl3anc 1490 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑇 𝑈))
991, 15latmcl 17318 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1004, 59, 61, 99syl3anc 1490 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1011, 2, 15latmlem12 17349 . . . . . . . . . 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 1498 . . . . . . . . 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 690 . . . . . . . 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 4831 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑈 𝑄) (𝑇 𝑈)))
1052, 7, 8hlatlej2 35332 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑈 (𝑇 𝑈))
1063, 12, 30, 105syl3anc 1490 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 (𝑇 𝑈))
1071, 2, 7, 15, 8atmod1i1 35813 . . . . . . . . 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 1502 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑈 𝑄) (𝑇 𝑈)))
1091, 8atbase 35245 . . . . . . . . . 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 17325 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
1124, 110, 34, 111syl3anc 1490 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
113108, 112eqtr3d 2801 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑄) (𝑇 𝑈)) = ((𝑄 (𝑇 𝑈)) 𝑈))
114104, 113breqtrd 4835 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑄 (𝑇 𝑈)) 𝑈))
1151, 7latjcl 17317 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1164, 34, 110, 115syl3anc 1490 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1171, 2, 7latjlej1 17331 . . . . . . 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 1491 . . . . . 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 17361 . . . . . 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 1491 . . . . 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 4835 . . . 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 17324 . . 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 17342 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1254, 10, 14, 124syl3anc 1490 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1261, 2, 15latlem12 17344 . . . 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 1491 . . 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 703 . 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 35245 . . . . . 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 17357 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1324, 29, 32, 130, 131syl13anc 1491 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1331, 2, 7, 15, 8llnmod1i2 35816 . . . 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 1511 . . 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 35325 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
1363, 6, 135syl2anc 579 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = 𝑄)
13783oveq2d 6858 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = (𝑄 𝑅))
138136, 137eqtr3d 2801 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = (𝑄 𝑅))
139138oveq1d 6857 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) = ((𝑄 𝑅) (𝑇 𝑈)))
1401, 15latmcom 17341 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑈 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1414, 36, 10, 140syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1427, 8hlatjcom 35324 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
1433, 5, 6, 142syl3anc 1490 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
14483oveq1d 6857 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑃) = (𝑅 𝑃))
145143, 144eqtrd 2799 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑅 𝑃))
146145oveq1d 6857 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑈 𝑆)) = ((𝑅 𝑃) (𝑈 𝑆)))
147141, 146eqtrd 2799 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑅 𝑃) (𝑈 𝑆)))
148139, 147oveq12d 6860 . . 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 2801 . 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 4835 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  joincjn 17210  meetcmee 17211  Latclat 17311  OLcol 35130  Atomscatm 35219  HLchlt 35306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-lat 17312  df-clat 17374  df-oposet 35132  df-ol 35134  df-oml 35135  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278  df-hlat 35307  df-psubsp 35459  df-pmap 35460  df-padd 35752
This theorem is referenced by:  dalawlem13  35839
  Copyright terms: Public domain W3C validator