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Theorem dalawlem5 37170
Description: Lemma for dalaw 37181. Special case to eliminate the requirement ¬ (𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) in dalawlem1 37166. (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
dalawlem5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Proof of Theorem dalawlem5
StepHypRef Expression
1 eqid 2801 . 2 (Base‘𝐾) = (Base‘𝐾)
2 dalawlem.l . 2 = (le‘𝐾)
3 simp11 1200 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
43hllatd 36659 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ Lat)
5 simp21 1203 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
6 simp22 1204 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄𝐴)
7 dalawlem.j . . . . 5 = (join‘𝐾)
8 dalawlem.a . . . . 5 𝐴 = (Atoms‘𝐾)
91, 7, 8hlatjcl 36662 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
103, 5, 6, 9syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp31 1206 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆𝐴)
12 simp32 1207 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇𝐴)
131, 7, 8hlatjcl 36662 . . . 4 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
143, 11, 12, 13syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑇) ∈ (Base‘𝐾))
15 dalawlem.m . . . 4 = (meet‘𝐾)
161, 15latmcl 17658 . . 3 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
174, 10, 14, 16syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (Base‘𝐾))
181, 8atbase 36584 . . . . . 6 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1912, 18syl 17 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇 ∈ (Base‘𝐾))
201, 7latjcl 17657 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
214, 10, 19, 20syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾))
221, 8atbase 36584 . . . . 5 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
2311, 22syl 17 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆 ∈ (Base‘𝐾))
241, 15latmcl 17658 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑇) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑇) 𝑆) ∈ (Base‘𝐾))
254, 21, 23, 24syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑇) 𝑆) ∈ (Base‘𝐾))
261, 7latjcl 17657 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
274, 10, 23, 26syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾))
281, 15latmcl 17658 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
294, 27, 19, 28syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾))
301, 7latjcl 17657 . . 3 ((𝐾 ∈ Lat ∧ (((𝑃 𝑄) 𝑇) 𝑆) ∈ (Base‘𝐾) ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) ∈ (Base‘𝐾))
314, 25, 29, 30syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) ∈ (Base‘𝐾))
32 simp23 1205 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑅𝐴)
331, 7, 8hlatjcl 36662 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
343, 6, 32, 33syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑅) ∈ (Base‘𝐾))
35 simp33 1208 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈𝐴)
361, 7, 8hlatjcl 36662 . . . . 5 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
373, 12, 35, 36syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑈) ∈ (Base‘𝐾))
381, 15latmcl 17658 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾))
394, 34, 37, 38syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾))
401, 7, 8hlatjcl 36662 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑃𝐴) → (𝑅 𝑃) ∈ (Base‘𝐾))
413, 32, 5, 40syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑃) ∈ (Base‘𝐾))
421, 7, 8hlatjcl 36662 . . . . 5 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
433, 35, 11, 42syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑆) ∈ (Base‘𝐾))
441, 15latmcl 17658 . . . 4 ((𝐾 ∈ Lat ∧ (𝑅 𝑃) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾))
454, 41, 43, 44syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾))
461, 7latjcl 17657 . . 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))
474, 39, 45, 46syl3anc 1368 . 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))
482, 7, 15, 8dalawlem2 37167 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
493, 5, 6, 11, 12, 48syl122anc 1376 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))
507, 8hlatjcom 36663 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
513, 5, 6, 50syl3anc 1368 . . . . . . 7 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
5251oveq1d 7154 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑇) = ((𝑄 𝑃) 𝑇))
537, 8hlatj32 36667 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑃𝐴𝑇𝐴)) → ((𝑄 𝑃) 𝑇) = ((𝑄 𝑇) 𝑃))
543, 6, 5, 12, 53syl13anc 1369 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑃) 𝑇) = ((𝑄 𝑇) 𝑃))
5552, 54eqtrd 2836 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑇) = ((𝑄 𝑇) 𝑃))
5655oveq1d 7154 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑇) 𝑆) = (((𝑄 𝑇) 𝑃) 𝑆))
572, 7, 15, 8dalawlem3 37168 . . . 4 (((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))))
5856, 57eqbrtrd 5055 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑇) 𝑆) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
597, 8hlatj32 36667 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
603, 5, 6, 11, 59syl13anc 1369 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑆) = ((𝑃 𝑆) 𝑄))
6160oveq1d 7154 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) = (((𝑃 𝑆) 𝑄) 𝑇))
622, 7, 15, 8dalawlem4 37169 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) 𝑄) 𝑇) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
6361, 62eqbrtrd 5055 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
641, 2, 7latjle12 17668 . . . 4 ((𝐾 ∈ Lat ∧ ((((𝑃 𝑄) 𝑇) 𝑆) ∈ (Base‘𝐾) ∧ (((𝑃 𝑄) 𝑆) 𝑇) ∈ (Base‘𝐾) ∧ (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))) ∈ (Base‘𝐾))) → (((((𝑃 𝑄) 𝑇) 𝑆) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))) ∧ (((𝑃 𝑄) 𝑆) 𝑇) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))) ↔ ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
654, 25, 29, 47, 64syl13anc 1369 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((((𝑃 𝑄) 𝑇) 𝑆) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))) ∧ (((𝑃 𝑄) 𝑆) 𝑇) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))) ↔ ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
6658, 63, 65mpbi2and 711 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
671, 2, 4, 17, 31, 47, 49, 66lattrd 17664 1 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  meetcmee 17551  Latclat 17651  Atomscatm 36558  HLchlt 36645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-lat 17652  df-clat 17714  df-oposet 36471  df-ol 36473  df-oml 36474  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-psubsp 36798  df-pmap 36799  df-padd 37091
This theorem is referenced by:  dalawlem10  37175
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