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Theorem dalawlem12 34994
Description: Lemma for dalaw 34998. Second part of dalawlem13 34995. (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 2621 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 dalawlem.l . . . 4 = (le‘𝐾)
3 simp11 1090 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
4 hllat 34476 . . . . 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 1093 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
7 simp22 1094 . . . . . 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 34479 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
113, 6, 7, 10syl3anc 1325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
12 simp31 1096 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆𝐴)
13 simp32 1097 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇𝐴)
141, 8, 9hlatjcl 34479 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
153, 12, 13, 14syl3anc 1325 . . . . 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 17046 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
185, 11, 15, 17syl3anc 1325 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
191, 9atbase 34402 . . . . . . . 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 17045 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
225, 11, 20, 21syl3anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
231, 9atbase 34402 . . . . . . 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 17046 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
265, 22, 24, 25syl3anc 1325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
271, 8latjcl 17045 . . . . 5 ((𝐾 ∈ Lat ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
285, 26, 20, 27syl3anc 1325 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆) ∈ (Base‘𝐾))
291, 9atbase 34402 . . . . . . 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 1098 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈𝐴)
321, 8, 9hlatjcl 34479 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
333, 13, 31, 32syl3anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑈) ∈ (Base‘𝐾))
341, 16latmcl 17046 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
355, 30, 33, 34syl3anc 1325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾))
361, 8, 9hlatjcl 34479 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
373, 31, 12, 36syl3anc 1325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑆) ∈ (Base‘𝐾))
381, 8latjcl 17045 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
395, 35, 37, 38syl3anc 1325 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) (𝑈 𝑆)) ∈ (Base‘𝐾))
401, 2, 8latlej1 17054 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
415, 11, 20, 40syl3anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑆))
421, 8, 9hlatjcl 34479 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑆𝐴) → (𝑇 𝑆) ∈ (Base‘𝐾))
433, 13, 12, 42syl3anc 1325 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑆) ∈ (Base‘𝐾))
441, 2, 16latmlem1 17075 . . . . . . 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 1327 . . . . . 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 34480 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) = (𝑇 𝑆))
483, 12, 13, 47syl3anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑇) = (𝑇 𝑆))
4948oveq2d 6663 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑄) (𝑇 𝑆)))
501, 2, 8latlej2 17055 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ((𝑃 𝑄) 𝑆))
515, 11, 20, 50syl3anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ((𝑃 𝑄) 𝑆))
521, 2, 8, 16, 9atmod2i2 34974 . . . . . 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 1338 . . . . 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 4683 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑆) 𝑇) 𝑆))
55 hlol 34474 . . . . . . . . . . 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 34479 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
583, 6, 12, 57syl3anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
591, 8latjcl 17045 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
605, 30, 58, 59syl3anc 1325 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
611, 8, 9hlatjcl 34479 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
623, 7, 13, 61syl3anc 1325 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑇) ∈ (Base‘𝐾))
631, 16latmassOLD 34342 . . . . . . . . . 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 1327 . . . . . . . . 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 34482 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
663, 6, 7, 12, 65syl13anc 1327 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) = (𝑃 (𝑄 𝑆)))
678, 9hlatj12 34483 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
683, 6, 7, 12, 67syl13anc 1327 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 (𝑄 𝑆)) = (𝑄 (𝑃 𝑆)))
6966, 68eqtr2d 2656 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑃 𝑆)) = ((𝑃 𝑄) 𝑆))
702, 8, 9hlatlej2 34488 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑇 (𝑄 𝑇))
713, 7, 13, 70syl3anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑄 𝑇))
721, 2, 16latleeqm2 17074 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
735, 24, 62, 72syl3anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 (𝑄 𝑇) ↔ ((𝑄 𝑇) 𝑇) = 𝑇))
7471, 73mpbid 222 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) 𝑇) = 𝑇)
7569, 74oveq12d 6665 . . . . . . . . 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 2656 . . . . . . . 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 34487 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
783, 7, 13, 77syl3anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑄 𝑇))
791, 2, 8, 16, 9atmod1i1 34969 . . . . . . . . . . 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 1338 . . . . . . . . . 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 34488 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → 𝑄 (𝑈 𝑄))
823, 31, 7, 81syl3anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑈 𝑄))
83 simp13 1092 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
84 simp12 1091 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = 𝑅)
8584oveq1d 6662 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑅 𝑈))
868, 9hlatjcom 34480 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑈𝐴) → (𝑄 𝑈) = (𝑈 𝑄))
873, 7, 31, 86syl3anc 1325 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) = (𝑈 𝑄))
8885, 87eqtr3d 2657 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑈) = (𝑈 𝑄))
8983, 88breqtrd 4677 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑄))
901, 16latmcl 17046 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
915, 58, 62, 90syl3anc 1325 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
921, 8, 9hlatjcl 34479 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑄𝐴) → (𝑈 𝑄) ∈ (Base‘𝐾))
933, 31, 7, 92syl3anc 1325 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑄) ∈ (Base‘𝐾))
941, 2, 8latjle12 17056 . . . . . . . . . . . 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 1327 . . . . . . . . . . 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 956 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 ((𝑃 𝑆) (𝑄 𝑇))) (𝑈 𝑄))
9780, 96eqbrtrrd 4675 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) (𝑈 𝑄))
982, 8, 9hlatlej1 34487 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑇 (𝑇 𝑈))
993, 13, 31, 98syl3anc 1325 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 (𝑇 𝑈))
1001, 16latmcl 17046 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1015, 60, 62, 100syl3anc 1325 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑃 𝑆)) (𝑄 𝑇)) ∈ (Base‘𝐾))
1021, 2, 16latmlem12 17077 . . . . . . . . . 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 1334 . . . . . . . . 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 715 . . . . . . . 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 4673 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑈 𝑄) (𝑇 𝑈)))
1062, 8, 9hlatlej2 34488 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑈 (𝑇 𝑈))
1073, 13, 31, 106syl3anc 1325 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 (𝑇 𝑈))
1081, 2, 8, 16, 9atmod1i1 34969 . . . . . . . . 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 1338 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑈 𝑄) (𝑇 𝑈)))
1101, 9atbase 34402 . . . . . . . . . 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 17053 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
1135, 111, 35, 112syl3anc 1325 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑄 (𝑇 𝑈))) = ((𝑄 (𝑇 𝑈)) 𝑈))
114109, 113eqtr3d 2657 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑄) (𝑇 𝑈)) = ((𝑄 (𝑇 𝑈)) 𝑈))
115105, 114breqtrd 4677 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ((𝑄 (𝑇 𝑈)) 𝑈))
1161, 8latjcl 17045 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1175, 35, 111, 116syl3anc 1325 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑇 𝑈)) 𝑈) ∈ (Base‘𝐾))
1181, 2, 8latjlej1 17059 . . . . . . 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 1327 . . . . . 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 17089 . . . . . 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 1327 . . . . 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 4677 . . . 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 17052 . . 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 17070 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1265, 11, 15, 125syl3anc 1325 . . 3 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (𝑃 𝑄))
1271, 2, 16latlem12 17072 . . . 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 1327 . . 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 956 . 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 34402 . . . . . 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 17085 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1335, 30, 33, 131, 132syl13anc 1327 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) (𝑃 𝑄))
1341, 2, 8, 16, 9llnmod1i2 34972 . . . 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 1347 . . 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 34481 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑄 𝑄) = 𝑄)
1373, 7, 136syl2anc 693 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = 𝑄)
13884oveq2d 6663 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑄) = (𝑄 𝑅))
139137, 138eqtr3d 2657 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 = (𝑄 𝑅))
140139oveq1d 6662 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑇 𝑈)) = ((𝑄 𝑅) (𝑇 𝑈)))
1411, 16latmcom 17069 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑈 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1425, 37, 11, 141syl3anc 1325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑃 𝑄) (𝑈 𝑆)))
1438, 9hlatjcom 34480 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
1443, 6, 7, 143syl3anc 1325 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
14584oveq1d 6662 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑃) = (𝑅 𝑃))
146144, 145eqtrd 2655 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑅 𝑃))
147146oveq1d 6662 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑈 𝑆)) = ((𝑅 𝑃) (𝑈 𝑆)))
148142, 147eqtrd 2655 . . . 4 (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 𝑆) (𝑃 𝑄)) = ((𝑅 𝑃) (𝑈 𝑆)))
149140, 148oveq12d 6665 . . 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 2657 . 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 4677 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 196  wa 384  w3a 1037   = wceq 1482  wcel 1989   class class class wbr 4651  cfv 5886  (class class class)co 6647  Basecbs 15851  lecple 15942  joincjn 16938  meetcmee 16939  Latclat 17039  OLcol 34287  Atomscatm 34376  HLchlt 34463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-preset 16922  df-poset 16940  df-plt 16952  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-p0 17033  df-lat 17040  df-clat 17102  df-oposet 34289  df-ol 34291  df-oml 34292  df-covers 34379  df-ats 34380  df-atl 34411  df-cvlat 34435  df-hlat 34464  df-psubsp 34615  df-pmap 34616  df-padd 34908
This theorem is referenced by:  dalawlem13  34995
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