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Theorem dalawlem12 33984
Description: Lemma for dalaw 33988. Second part of dalawlem13 33985. (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 2604 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 dalawlem.l . . . 4 = (le‘𝐾)
3 simp11 1083 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
4 hllat 33466 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
53, 4syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ Lat)
6 simp21 1086 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
7 simp22 1087 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄𝐴)
8 dalawlem.j . . . . . . 7 = (join‘𝐾)
9 dalawlem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
101, 8, 9hlatjcl 33469 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
113, 6, 7, 10syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
12 simp31 1089 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆𝐴)
13 simp32 1090 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇𝐴)
141, 8, 9hlatjcl 33469 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
153, 12, 13, 14syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑇) ∈ (Base‘𝐾))
16 dalawlem.m . . . . . 6 = (meet‘𝐾)
171, 16latmcl 16816 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
185, 11, 15, 17syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
191, 9atbase 33392 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
2012, 19syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ∈ (Base‘𝐾))
211, 8latjcl 16815 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
225, 11, 20, 21syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
231, 9atbase 33392 . . . . . . 7 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
2413, 23syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 ∈ (Base‘𝐾))
251, 16latmcl 16816 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
265, 22, 24, 25syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
271, 8latjcl 16815 . . . . 5 ((𝐾 ∈ Lat ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
285, 26, 20, 27syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
291, 9atbase 33392 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
307, 29syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 ∈ (Base‘𝐾))
31 simp33 1091 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈𝐴)
321, 8, 9hlatjcl 33469 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
333, 13, 31, 32syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑈) ∈ (Base‘𝐾))
341, 16latmcl 16816 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
355, 30, 33, 34syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
361, 8, 9hlatjcl 33469 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
373, 31, 12, 36syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑆) ∈ (Base‘𝐾))
381, 8latjcl 16815 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
395, 35, 37, 38syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
401, 2, 8latlej1 16824 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
415, 11, 20, 40syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
421, 8, 9hlatjcl 33469 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑆𝐴) → (𝑇 𝑆) ∈ (Base‘𝐾))
433, 13, 12, 42syl3anc 1317 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑆) ∈ (Base‘𝐾))
441, 2, 16latmlem1 16845 . . . . . . 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))))
455, 11, 22, 43, 44syl13anc 1319 . . . . . 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))))
4641, 45mpd 15 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑇 𝑆)) (((𝑃 𝑄) 𝑆) (𝑇 𝑆)))
478, 9hlatjcom 33470 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) = (𝑇 𝑆))
483, 12, 13, 47syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑇) = (𝑇 𝑆))
4948oveq2d 6538 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑄) (𝑇 𝑆)))
501, 2, 8latlej2 16825 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
515, 11, 20, 50syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
521, 2, 8, 16, 9atmod2i2 33964 . . . . . 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)))
533, 13, 22, 20, 51, 52syl131anc 1330 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) = (((𝑃 𝑄) 𝑆) (𝑇 𝑆)))
5446, 49, 533brtr4d 4604 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆))
55 hlol 33464 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ OL)
563, 55syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ OL)
571, 8, 9hlatjcl 33469 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
583, 6, 12, 57syl3anc 1317 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
591, 8latjcl 16815 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
605, 30, 58, 59syl3anc 1317 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
611, 8, 9hlatjcl 33469 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
623, 7, 13, 61syl3anc 1317 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑇) ∈ (Base‘𝐾))
631, 16latmassOLD 33332 . . . . . . . . . 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)))
6456, 60, 62, 24, 63syl13anc 1319 . . . . . . . . 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)))
658, 9hlatjass 33472 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
663, 6, 7, 12, 65syl13anc 1319 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
678, 9hlatj12 33473 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
683, 6, 7, 12, 67syl13anc 1319 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
6966, 68eqtr2d 2639 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) = ((𝑃 𝑄) 𝑆))
702, 8, 9hlatlej2 33478 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑇 (𝑄 𝑇))
713, 7, 13, 70syl3anc 1317 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑄 𝑇))
721, 2, 16latleeqm2 16844 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
735, 24, 62, 72syl3anc 1317 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
7471, 73mpbid 220 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) 𝑇) = 𝑇)
7569, 74oveq12d 6540 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) ((𝑄 𝑇) 𝑇)) = (((𝑃 𝑄) 𝑆) 𝑇))
7664, 75eqtr2d 2639 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) = (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇))
772, 8, 9hlatlej1 33477 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
783, 7, 13, 77syl3anc 1317 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑄 𝑇))
791, 2, 8, 16, 9atmod1i1 33959 . . . . . . . . . . 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)))
803, 7, 58, 62, 78, 79syl131anc 1330 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) = ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)))
812, 8, 9hlatlej2 33478 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → 𝑄 (𝑈 𝑄))
823, 31, 7, 81syl3anc 1317 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑈 𝑄))
83 simp13 1085 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
84 simp12 1084 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = 𝑅)
8584oveq1d 6537 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑅 𝑈))
868, 9hlatjcom 33470 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑈𝐴) → (𝑄 𝑈) = (𝑈 𝑄))
873, 7, 31, 86syl3anc 1317 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑈 𝑄))
8885, 87eqtr3d 2640 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑈) = (𝑈 𝑄))
8983, 88breqtrd 4598 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑄))
901, 16latmcl 16816 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
915, 58, 62, 90syl3anc 1317 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
921, 8, 9hlatjcl 33469 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → (𝑈 𝑄) ∈ (Base‘𝐾))
933, 31, 7, 92syl3anc 1317 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑄) ∈ (Base‘𝐾))
941, 2, 8latjle12 16826 . . . . . . . . . . . 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)))
955, 30, 91, 93, 94syl13anc 1319 . . . . . . . . . . 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)))
9682, 89, 95mpbi2and 957 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) (𝑈 𝑄))
9780, 96eqbrtrrd 4596 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) (𝑈 𝑄))
982, 8, 9hlatlej1 33477 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑇 (𝑇 𝑈))
993, 13, 31, 98syl3anc 1317 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑇 𝑈))
1001, 16latmcl 16816 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1015, 60, 62, 100syl3anc 1317 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1021, 2, 16latmlem12 16847 . . . . . . . . . 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))))
1035, 101, 93, 24, 33, 102syl122anc 1326 . . . . . . . . 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))))
10497, 99, 103mp2and 710 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) 𝑇) ((𝑈 𝑄) (𝑇 𝑈)))
10576, 104eqbrtrd 4594 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑈 𝑄) (𝑇 𝑈)))
1062, 8, 9hlatlej2 33478 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑈 (𝑇 𝑈))
1073, 13, 31, 106syl3anc 1317 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 (𝑇 𝑈))
1081, 2, 8, 16, 9atmod1i1 33959 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) ∧ 𝑈 (𝑇 𝑈)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑈 𝑄) (𝑇 𝑈)))
1093, 31, 30, 33, 107, 108syl131anc 1330 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑈 𝑄) (𝑇 𝑈)))
1101, 9atbase 33392 . . . . . . . . . 10 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
11131, 110syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 ∈ (Base‘𝐾))
1121, 8latjcom 16823 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
1135, 111, 35, 112syl3anc 1317 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
114109, 113eqtr3d 2640 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑄) (𝑇 𝑈)) = ((𝑄 (𝑇 𝑈)) 𝑈))
115105, 114breqtrd 4598 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑄 (𝑇 𝑈)) 𝑈))
1161, 8latjcl 16815 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1175, 35, 111, 116syl3anc 1317 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1181, 2, 8latjlej1 16829 . . . . . . 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)))
1195, 26, 117, 20, 118syl13anc 1319 . . . . . 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)))
120115, 119mpd 15 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆))
1211, 8latjass 16859 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆) = ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
1225, 35, 111, 20, 121syl13anc 1319 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 (𝑇 𝑈)) 𝑈) 𝑆) = ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
123120, 122breqtrd 4598 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
1241, 2, 5, 18, 28, 39, 54, 123lattrd 16822 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)))
1251, 2, 16latmle1 16840 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1265, 11, 15, 125syl3anc 1317 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1271, 2, 16latlem12 16842 . . . 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))))
1285, 18, 39, 11, 127syl13anc 1319 . . 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))))
129124, 126, 128mpbi2and 957 . 2 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) (𝑃 𝑄)))
1301, 9atbase 33392 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1316, 130syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃 ∈ (Base‘𝐾))
1321, 2, 8, 16latmlej12 16855 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1335, 30, 33, 131, 132syl13anc 1319 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1341, 2, 8, 16, 9llnmod1i2 33962 . . . 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)))
1353, 35, 11, 31, 12, 133, 134syl321anc 1339 . . 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)))
1368, 9hlatjidm 33471 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
1373, 7, 136syl2anc 690 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = 𝑄)
13884oveq2d 6538 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = (𝑄 𝑅))
139137, 138eqtr3d 2640 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = (𝑄 𝑅))
140139oveq1d 6537 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) = ((𝑄 𝑅) (𝑇 𝑈)))
1411, 16latmcom 16839 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑈 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1425, 37, 11, 141syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1438, 9hlatjcom 33470 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
1443, 6, 7, 143syl3anc 1317 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
14584oveq1d 6537 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑃) = (𝑅 𝑃))
146144, 145eqtrd 2638 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑅 𝑃))
147146oveq1d 6537 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑈 𝑆)) = ((𝑅 𝑃) (𝑈 𝑆)))
148142, 147eqtrd 2638 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑅 𝑃) (𝑈 𝑆)))
149140, 148oveq12d 6540 . . 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))))
150135, 149eqtr3d 2640 . 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))))
151129, 150breqtrd 4598 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1975   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  lecple 15716  joincjn 16708  meetcmee 16709  Latclat 16809  OLcol 33277  Atomscatm 33366  HLchlt 33453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-iin 4447  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-psubsp 33605  df-pmap 33606  df-padd 33898
This theorem is referenced by:  dalawlem13  33985
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