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Theorem dalawlem5 39574
Description: Lemma for dalaw 39585. Special case to eliminate the requirement ¬ (𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) in dalawlem1 39570. (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 2726 . 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 39062 . 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 39065 . . . 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 39065 . . . 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 18465 . . 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 38987 . . . . . 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 18464 . . . . 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 38987 . . . . 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 18465 . . . 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 18464 . . . . 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 18465 . . . 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 18464 . . 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 39065 . . . . 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 39065 . . . . 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 18465 . . . 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 39065 . . . . 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 39065 . . . . 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 18465 . . . 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 18464 . . 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 39571 . . 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 39066 . . . . . . . 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 7439 . . . . . 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 39070 . . . . . . 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 2766 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑇) = ((𝑄 𝑇) 𝑃))
5655oveq1d 7439 . . . 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 39572 . . . 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 5175 . . 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 39070 . . . . . 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 7439 . . . 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 39573 . . . 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 5175 . . 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 18475 . . . 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 710 . 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 18471 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 205  wa 394  w3a 1084   = wceq 1534  wcel 2099   class class class wbr 5153  cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  meetcmee 18337  Latclat 18456  Atomscatm 38961  HLchlt 39048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-psubsp 39202  df-pmap 39203  df-padd 39495
This theorem is referenced by:  dalawlem10  39579
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