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Theorem dalawlem3 34177
Description: Lemma for dalaw 34190. First piece of dalawlem5 34179. (Contributed by NM, 4-Oct-2012.)
Hypotheses
Ref Expression
dalawlem.l = (le‘𝐾)
dalawlem.j = (join‘𝐾)
dalawlem.m = (meet‘𝐾)
dalawlem.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
dalawlem3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Proof of Theorem dalawlem3
StepHypRef Expression
1 eqid 2610 . 2 (Base‘𝐾) = (Base‘𝐾)
2 dalawlem.l . 2 = (le‘𝐾)
3 simp11 1084 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
4 hllat 33668 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
53, 4syl 17 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ Lat)
6 simp22 1088 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄𝐴)
7 simp32 1091 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇𝐴)
8 dalawlem.j . . . . . 6 = (join‘𝐾)
9 dalawlem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
101, 8, 9hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
113, 6, 7, 10syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑇) ∈ (Base‘𝐾))
12 simp21 1087 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
131, 9atbase 33594 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1412, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃 ∈ (Base‘𝐾))
151, 8latjcl 16874 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → ((𝑄 𝑇) 𝑃) ∈ (Base‘𝐾))
165, 11, 14, 15syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) 𝑃) ∈ (Base‘𝐾))
17 simp31 1090 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆𝐴)
181, 9atbase 33594 . . . 4 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1917, 18syl 17 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ∈ (Base‘𝐾))
20 dalawlem.m . . . 4 = (meet‘𝐾)
211, 20latmcl 16875 . . 3 ((𝐾 ∈ Lat ∧ ((𝑄 𝑇) 𝑃) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (((𝑄 𝑇) 𝑃) 𝑆) ∈ (Base‘𝐾))
225, 16, 19, 21syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) ∈ (Base‘𝐾))
23 simp23 1089 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑅𝐴)
241, 8, 9hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
253, 6, 23, 24syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑅) ∈ (Base‘𝐾))
26 simp33 1092 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈𝐴)
271, 9atbase 33594 . . . . 5 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
2826, 27syl 17 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 ∈ (Base‘𝐾))
291, 20latmcl 16875 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 𝑅) 𝑈) ∈ (Base‘𝐾))
305, 25, 28, 29syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) 𝑈) ∈ (Base‘𝐾))
311, 8, 9hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑃𝐴) → (𝑅 𝑃) ∈ (Base‘𝐾))
323, 23, 12, 31syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑃) ∈ (Base‘𝐾))
331, 8, 9hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
343, 26, 17, 33syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑆) ∈ (Base‘𝐾))
351, 20latmcl 16875 . . . 4 ((𝐾 ∈ Lat ∧ (𝑅 𝑃) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾))
365, 32, 34, 35syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾))
371, 8latjcl 16874 . . 3 ((𝐾 ∈ Lat ∧ ((𝑄 𝑅) 𝑈) ∈ (Base‘𝐾) ∧ ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾)) → (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))) ∈ (Base‘𝐾))
385, 30, 36, 37syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))) ∈ (Base‘𝐾))
391, 8, 9hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
403, 7, 26, 39syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑈) ∈ (Base‘𝐾))
411, 20latmcl 16875 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾))
425, 25, 40, 41syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾))
431, 8latjcl 16874 . . 3 ((𝐾 ∈ Lat ∧ ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾)) → (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))) ∈ (Base‘𝐾))
445, 42, 36, 43syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))) ∈ (Base‘𝐾))
451, 9atbase 33594 . . . . . . . . . 10 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
466, 45syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 ∈ (Base‘𝐾))
471, 20latmcl 16875 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑄 𝑈) ∈ (Base‘𝐾))
485, 46, 28, 47syl3anc 1318 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) ∈ (Base‘𝐾))
491, 8, 9hlatjcl 33671 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
503, 12, 17, 49syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
511, 20latmcl 16875 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
525, 50, 46, 51syl3anc 1318 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))
531, 8latjcl 16874 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 𝑈) ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾)) → ((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) ∈ (Base‘𝐾))
545, 48, 52, 53syl3anc 1318 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) ∈ (Base‘𝐾))
551, 8latjcl 16874 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) ∈ (Base‘𝐾)) → (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))) ∈ (Base‘𝐾))
565, 14, 54, 55syl3anc 1318 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))) ∈ (Base‘𝐾))
571, 9atbase 33594 . . . . . . . . 9 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
5823, 57syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑅 ∈ (Base‘𝐾))
591, 8latjcl 16874 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ ((𝑄 𝑅) 𝑈) ∈ (Base‘𝐾)) → (𝑅 ((𝑄 𝑅) 𝑈)) ∈ (Base‘𝐾))
605, 58, 30, 59syl3anc 1318 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 ((𝑄 𝑅) 𝑈)) ∈ (Base‘𝐾))
611, 8latjcl 16874 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅 ((𝑄 𝑅) 𝑈)) ∈ (Base‘𝐾)) → (𝑃 (𝑅 ((𝑄 𝑅) 𝑈))) ∈ (Base‘𝐾))
625, 14, 60, 61syl3anc 1318 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 (𝑅 ((𝑄 𝑅) 𝑈))) ∈ (Base‘𝐾))
631, 8latjcl 16874 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑄 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → ((𝑄 𝑈) 𝑃) ∈ (Base‘𝐾))
645, 48, 14, 63syl3anc 1318 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) 𝑃) ∈ (Base‘𝐾))
651, 2, 8, 20latmlej22 16916 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ ((𝑄 𝑇) 𝑃) ∈ (Base‘𝐾) ∧ ((𝑄 𝑈) 𝑃) ∈ (Base‘𝐾))) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑈) 𝑃) 𝑆))
665, 19, 16, 64, 65syl13anc 1320 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑈) 𝑃) 𝑆))
671, 8latjass 16918 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ((𝑄 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (((𝑄 𝑈) 𝑃) 𝑆) = ((𝑄 𝑈) (𝑃 𝑆)))
685, 48, 14, 19, 67syl13anc 1320 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑈) 𝑃) 𝑆) = ((𝑄 𝑈) (𝑃 𝑆)))
6966, 68breqtrd 4609 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) ((𝑄 𝑈) (𝑃 𝑆)))
701, 20latmcl 16875 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) ∈ (Base‘𝐾))
715, 11, 50, 70syl3anc 1318 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) (𝑃 𝑆)) ∈ (Base‘𝐾))
721, 8latjcl 16874 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ((𝑄 𝑇) (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (((𝑄 𝑇) (𝑃 𝑆)) 𝑃) ∈ (Base‘𝐾))
735, 71, 14, 72syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) (𝑃 𝑆)) 𝑃) ∈ (Base‘𝐾))
741, 8, 9hlatjcl 33671 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
753, 12, 6, 74syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
762, 8, 9hlatlej2 33680 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑆 (𝑃 𝑆))
773, 12, 17, 76syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 (𝑃 𝑆))
781, 2, 20latmlem2 16905 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ ((𝑄 𝑇) 𝑃) ∈ (Base‘𝐾))) → (𝑆 (𝑃 𝑆) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑇) 𝑃) (𝑃 𝑆))))
795, 19, 50, 16, 78syl13anc 1320 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 (𝑃 𝑆) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑇) 𝑃) (𝑃 𝑆))))
8077, 79mpd 15 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑇) 𝑃) (𝑃 𝑆)))
812, 8, 9hlatlej1 33679 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑃 (𝑃 𝑆))
823, 12, 17, 81syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃 (𝑃 𝑆))
831, 2, 8, 20, 9atmod4i1 34170 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 𝑆)) → (((𝑄 𝑇) (𝑃 𝑆)) 𝑃) = (((𝑄 𝑇) 𝑃) (𝑃 𝑆)))
843, 12, 11, 50, 82, 83syl131anc 1331 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) (𝑃 𝑆)) 𝑃) = (((𝑄 𝑇) 𝑃) (𝑃 𝑆)))
8580, 84breqtrrd 4611 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑇) (𝑃 𝑆)) 𝑃))
861, 20latmcom 16898 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
875, 11, 50, 86syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
88 simp12 1085 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄))
8987, 88eqbrtrd 4605 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑄))
902, 8, 9hlatlej1 33679 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
913, 12, 6, 90syl3anc 1318 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃 (𝑃 𝑄))
921, 2, 8latjle12 16885 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (((𝑄 𝑇) (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑄) ∧ 𝑃 (𝑃 𝑄)) ↔ (((𝑄 𝑇) (𝑃 𝑆)) 𝑃) (𝑃 𝑄)))
935, 71, 14, 75, 92syl13anc 1320 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑄 𝑇) (𝑃 𝑆)) (𝑃 𝑄) ∧ 𝑃 (𝑃 𝑄)) ↔ (((𝑄 𝑇) (𝑃 𝑆)) 𝑃) (𝑃 𝑄)))
9489, 91, 93mpbi2and 958 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) (𝑃 𝑆)) 𝑃) (𝑃 𝑄))
951, 2, 5, 22, 73, 75, 85, 94lattrd 16881 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (𝑃 𝑄))
961, 8latjcl 16874 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 𝑈) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑈) (𝑃 𝑆)) ∈ (Base‘𝐾))
975, 48, 50, 96syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) (𝑃 𝑆)) ∈ (Base‘𝐾))
981, 2, 20latlem12 16901 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((((𝑄 𝑇) 𝑃) 𝑆) ∈ (Base‘𝐾) ∧ ((𝑄 𝑈) (𝑃 𝑆)) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → (((((𝑄 𝑇) 𝑃) 𝑆) ((𝑄 𝑈) (𝑃 𝑆)) ∧ (((𝑄 𝑇) 𝑃) 𝑆) (𝑃 𝑄)) ↔ (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑈) (𝑃 𝑆)) (𝑃 𝑄))))
995, 22, 97, 75, 98syl13anc 1320 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((((𝑄 𝑇) 𝑃) 𝑆) ((𝑄 𝑈) (𝑃 𝑆)) ∧ (((𝑄 𝑇) 𝑃) 𝑆) (𝑃 𝑄)) ↔ (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑈) (𝑃 𝑆)) (𝑃 𝑄))))
10069, 95, 99mpbi2and 958 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑈) (𝑃 𝑆)) (𝑃 𝑄)))
1011, 2, 8, 20, 9atmod3i1 34168 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 𝑆)) → (𝑃 ((𝑃 𝑆) 𝑄)) = ((𝑃 𝑆) (𝑃 𝑄)))
1023, 12, 50, 46, 82, 101syl131anc 1331 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 ((𝑃 𝑆) 𝑄)) = ((𝑃 𝑆) (𝑃 𝑄)))
103102oveq2d 6565 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) (𝑃 ((𝑃 𝑆) 𝑄))) = ((𝑄 𝑈) ((𝑃 𝑆) (𝑃 𝑄))))
1041, 8latj12 16919 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((𝑄 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾))) → ((𝑄 𝑈) (𝑃 ((𝑃 𝑆) 𝑄))) = (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))))
1055, 48, 14, 52, 104syl13anc 1320 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) (𝑃 ((𝑃 𝑆) 𝑄))) = (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))))
1061, 2, 8, 20latmlej12 16914 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → (𝑄 𝑈) (𝑃 𝑄))
1075, 46, 28, 14, 106syl13anc 1320 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑈) (𝑃 𝑄))
1081, 2, 8, 20, 9atmod1i1m 34162 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑈𝐴) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) ∧ (𝑄 𝑈) (𝑃 𝑄)) → ((𝑄 𝑈) ((𝑃 𝑆) (𝑃 𝑄))) = (((𝑄 𝑈) (𝑃 𝑆)) (𝑃 𝑄)))
1093, 26, 46, 50, 75, 107, 108syl231anc 1338 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) ((𝑃 𝑆) (𝑃 𝑄))) = (((𝑄 𝑈) (𝑃 𝑆)) (𝑃 𝑄)))
110103, 105, 1093eqtr3rd 2653 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑈) (𝑃 𝑆)) (𝑃 𝑄)) = (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))))
111100, 110breqtrd 4609 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))))
1122, 8, 9hlatlej1 33679 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → 𝑄 (𝑄 𝑅))
1133, 6, 23, 112syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑄 𝑅))
1142, 8, 9hlatlej2 33680 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → 𝑈 (𝑅 𝑈))
1153, 23, 26, 114syl3anc 1318 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 (𝑅 𝑈))
1161, 20latmcl 16875 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
1175, 50, 11, 116syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (Base‘𝐾))
1181, 8, 9hlatjcl 33671 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → (𝑅 𝑈) ∈ (Base‘𝐾))
1193, 23, 26, 118syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑈) ∈ (Base‘𝐾))
1202, 8, 9hlatlej1 33679 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → 𝑄 (𝑄 𝑇))
1213, 6, 7, 120syl3anc 1318 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄 (𝑄 𝑇))
1221, 2, 20latmlem2 16905 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾))) → (𝑄 (𝑄 𝑇) → ((𝑃 𝑆) 𝑄) ((𝑃 𝑆) (𝑄 𝑇))))
1235, 46, 11, 50, 122syl13anc 1320 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑄 𝑇) → ((𝑃 𝑆) 𝑄) ((𝑃 𝑆) (𝑄 𝑇))))
124121, 123mpd 15 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) 𝑄) ((𝑃 𝑆) (𝑄 𝑇)))
125 simp13 1086 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
1261, 2, 5, 52, 117, 119, 124, 125lattrd 16881 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) 𝑄) (𝑅 𝑈))
1271, 2, 8latjle12 16885 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 𝑈) ∈ (Base‘𝐾))) → ((𝑈 (𝑅 𝑈) ∧ ((𝑃 𝑆) 𝑄) (𝑅 𝑈)) ↔ (𝑈 ((𝑃 𝑆) 𝑄)) (𝑅 𝑈)))
1285, 28, 52, 119, 127syl13anc 1320 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑈 (𝑅 𝑈) ∧ ((𝑃 𝑆) 𝑄) (𝑅 𝑈)) ↔ (𝑈 ((𝑃 𝑆) 𝑄)) (𝑅 𝑈)))
129115, 126, 128mpbi2and 958 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 ((𝑃 𝑆) 𝑄)) (𝑅 𝑈))
1301, 8latjcl 16874 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾)) → (𝑈 ((𝑃 𝑆) 𝑄)) ∈ (Base‘𝐾))
1315, 28, 52, 130syl3anc 1318 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 ((𝑃 𝑆) 𝑄)) ∈ (Base‘𝐾))
1321, 2, 20latmlem12 16906 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) ∧ ((𝑈 ((𝑃 𝑆) 𝑄)) ∈ (Base‘𝐾) ∧ (𝑅 𝑈) ∈ (Base‘𝐾))) → ((𝑄 (𝑄 𝑅) ∧ (𝑈 ((𝑃 𝑆) 𝑄)) (𝑅 𝑈)) → (𝑄 (𝑈 ((𝑃 𝑆) 𝑄))) ((𝑄 𝑅) (𝑅 𝑈))))
1335, 46, 25, 131, 119, 132syl122anc 1327 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 (𝑄 𝑅) ∧ (𝑈 ((𝑃 𝑆) 𝑄)) (𝑅 𝑈)) → (𝑄 (𝑈 ((𝑃 𝑆) 𝑄))) ((𝑄 𝑅) (𝑅 𝑈))))
134113, 129, 133mp2and 711 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 (𝑈 ((𝑃 𝑆) 𝑄))) ((𝑄 𝑅) (𝑅 𝑈)))
1351, 2, 20latmle2 16900 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑄) 𝑄)
1365, 50, 46, 135syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) 𝑄) 𝑄)
1371, 2, 8, 20, 9atmod2i2 34166 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑄) ∈ (Base‘𝐾)) ∧ ((𝑃 𝑆) 𝑄) 𝑄) → ((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) = (𝑄 (𝑈 ((𝑃 𝑆) 𝑄))))
1383, 26, 46, 52, 136, 137syl131anc 1331 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) = (𝑄 (𝑈 ((𝑃 𝑆) 𝑄))))
1392, 8, 9hlatlej2 33680 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → 𝑅 (𝑄 𝑅))
1403, 6, 23, 139syl3anc 1318 . . . . . . . . 9 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑅 (𝑄 𝑅))
1411, 2, 8, 20, 9atmod3i2 34169 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑅 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) ∧ 𝑅 (𝑄 𝑅)) → (𝑅 ((𝑄 𝑅) 𝑈)) = ((𝑄 𝑅) (𝑅 𝑈)))
1423, 26, 58, 25, 140, 141syl131anc 1331 . . . . . . . 8 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 ((𝑄 𝑅) 𝑈)) = ((𝑄 𝑅) (𝑅 𝑈)))
143134, 138, 1423brtr4d 4615 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) (𝑅 ((𝑄 𝑅) 𝑈)))
1441, 2, 8latjlej2 16889 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) ∈ (Base‘𝐾) ∧ (𝑅 ((𝑄 𝑅) 𝑈)) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → (((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) (𝑅 ((𝑄 𝑅) 𝑈)) → (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))) (𝑃 (𝑅 ((𝑄 𝑅) 𝑈)))))
1455, 54, 60, 14, 144syl13anc 1320 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑈) ((𝑃 𝑆) 𝑄)) (𝑅 ((𝑄 𝑅) 𝑈)) → (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))) (𝑃 (𝑅 ((𝑄 𝑅) 𝑈)))))
146143, 145mpd 15 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 ((𝑄 𝑈) ((𝑃 𝑆) 𝑄))) (𝑃 (𝑅 ((𝑄 𝑅) 𝑈))))
1471, 2, 5, 22, 56, 62, 111, 146lattrd 16881 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (𝑃 (𝑅 ((𝑄 𝑅) 𝑈))))
1481, 8latj13 16921 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ ((𝑄 𝑅) 𝑈) ∈ (Base‘𝐾))) → (𝑃 (𝑅 ((𝑄 𝑅) 𝑈))) = (((𝑄 𝑅) 𝑈) (𝑅 𝑃)))
1495, 14, 58, 30, 148syl13anc 1320 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 (𝑅 ((𝑄 𝑅) 𝑈))) = (((𝑄 𝑅) 𝑈) (𝑅 𝑃)))
150147, 149breqtrd 4609 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑅) 𝑈) (𝑅 𝑃)))
1511, 2, 8, 20latmlej22 16916 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ ((𝑄 𝑇) 𝑃) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (((𝑄 𝑇) 𝑃) 𝑆) (𝑈 𝑆))
1525, 19, 16, 28, 151syl13anc 1320 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (𝑈 𝑆))
1531, 8latjcl 16874 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑄 𝑅) 𝑈) ∈ (Base‘𝐾) ∧ (𝑅 𝑃) ∈ (Base‘𝐾)) → (((𝑄 𝑅) 𝑈) (𝑅 𝑃)) ∈ (Base‘𝐾))
1545, 30, 32, 153syl3anc 1318 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) 𝑈) (𝑅 𝑃)) ∈ (Base‘𝐾))
1551, 2, 20latlem12 16901 . . . . 5 ((𝐾 ∈ Lat ∧ ((((𝑄 𝑇) 𝑃) 𝑆) ∈ (Base‘𝐾) ∧ (((𝑄 𝑅) 𝑈) (𝑅 𝑃)) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾))) → (((((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑅) 𝑈) (𝑅 𝑃)) ∧ (((𝑄 𝑇) 𝑃) 𝑆) (𝑈 𝑆)) ↔ (((𝑄 𝑇) 𝑃) 𝑆) ((((𝑄 𝑅) 𝑈) (𝑅 𝑃)) (𝑈 𝑆))))
1565, 22, 154, 34, 155syl13anc 1320 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑅) 𝑈) (𝑅 𝑃)) ∧ (((𝑄 𝑇) 𝑃) 𝑆) (𝑈 𝑆)) ↔ (((𝑄 𝑇) 𝑃) 𝑆) ((((𝑄 𝑅) 𝑈) (𝑅 𝑃)) (𝑈 𝑆))))
157150, 152, 156mpbi2and 958 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) ((((𝑄 𝑅) 𝑈) (𝑅 𝑃)) (𝑈 𝑆)))
1581, 2, 8, 20latmlej21 16915 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((𝑄 𝑅) 𝑈) (𝑈 𝑆))
1595, 28, 25, 19, 158syl13anc 1320 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) 𝑈) (𝑈 𝑆))
1601, 2, 8, 20, 9atmod1i1m 34162 . . . 4 (((𝐾 ∈ HL ∧ 𝑈𝐴) ∧ ((𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑅 𝑃) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) ∧ ((𝑄 𝑅) 𝑈) (𝑈 𝑆)) → (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))) = ((((𝑄 𝑅) 𝑈) (𝑅 𝑃)) (𝑈 𝑆)))
1613, 26, 25, 32, 34, 159, 160syl231anc 1338 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))) = ((((𝑄 𝑅) 𝑈) (𝑅 𝑃)) (𝑈 𝑆)))
162157, 161breqtrrd 4611 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))))
1632, 8, 9hlatlej2 33680 . . . . 5 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → 𝑈 (𝑇 𝑈))
1643, 7, 26, 163syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈 (𝑇 𝑈))
1651, 2, 20latmlem2 16905 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾))) → (𝑈 (𝑇 𝑈) → ((𝑄 𝑅) 𝑈) ((𝑄 𝑅) (𝑇 𝑈))))
1665, 28, 40, 25, 165syl13anc 1320 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 (𝑇 𝑈) → ((𝑄 𝑅) 𝑈) ((𝑄 𝑅) (𝑇 𝑈))))
167164, 166mpd 15 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) 𝑈) ((𝑄 𝑅) (𝑇 𝑈)))
1681, 2, 8latjlej1 16888 . . . 4 ((𝐾 ∈ Lat ∧ (((𝑄 𝑅) 𝑈) ∈ (Base‘𝐾) ∧ ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾) ∧ ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾))) → (((𝑄 𝑅) 𝑈) ((𝑄 𝑅) (𝑇 𝑈)) → (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
1695, 30, 42, 36, 168syl13anc 1320 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) 𝑈) ((𝑄 𝑅) (𝑇 𝑈)) → (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
170167, 169mpd 15 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑅) 𝑈) ((𝑅 𝑃) (𝑈 𝑆))) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
1711, 2, 5, 22, 38, 44, 162, 170lattrd 16881 1 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  HLchlt 33655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-padd 34100
This theorem is referenced by:  dalawlem4  34178  dalawlem5  34179
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