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Theorem dalawlem4 34979
 Description: Lemma for dalaw 34991. Second piece of dalawlem5 34980. (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
dalawlem4 (((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))))

Proof of Theorem dalawlem4
StepHypRef Expression
1 simp11 1089 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ HL)
2 simp12 1090 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄))
3 hllat 34469 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
41, 3syl 17 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝐾 ∈ Lat)
5 simp22 1093 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑄𝐴)
6 simp32 1096 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑇𝐴)
7 eqid 2620 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 dalawlem.j . . . . . . 7 = (join‘𝐾)
9 dalawlem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
107, 8, 9hlatjcl 34472 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
111, 5, 6, 10syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑇) ∈ (Base‘𝐾))
12 simp21 1092 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑃𝐴)
13 simp31 1095 . . . . . 6 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑆𝐴)
147, 8, 9hlatjcl 34472 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
151, 12, 13, 14syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
16 dalawlem.m . . . . . 6 = (meet‘𝐾)
177, 16latmcom 17056 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
184, 11, 15, 17syl3anc 1324 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) (𝑃 𝑆)) = ((𝑃 𝑆) (𝑄 𝑇)))
198, 9hlatjcom 34473 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑃𝐴) → (𝑄 𝑃) = (𝑃 𝑄))
201, 5, 12, 19syl3anc 1324 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑃) = (𝑃 𝑄))
212, 18, 203brtr4d 4676 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑃))
22 simp13 1091 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))
2318, 22eqbrtrd 4666 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑇) (𝑃 𝑆)) (𝑅 𝑈))
24 simp23 1094 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑅𝐴)
25 simp33 1097 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → 𝑈𝐴)
26 dalawlem.l . . . 4 = (le‘𝐾)
2726, 8, 16, 9dalawlem3 34978 . . 3 (((𝐾 ∈ HL ∧ ((𝑄 𝑇) (𝑃 𝑆)) (𝑄 𝑃) ∧ ((𝑄 𝑇) (𝑃 𝑆)) (𝑅 𝑈)) ∧ (𝑄𝐴𝑃𝐴𝑅𝐴) ∧ (𝑇𝐴𝑆𝐴𝑈𝐴)) → (((𝑃 𝑆) 𝑄) 𝑇) (((𝑃 𝑅) (𝑆 𝑈)) ((𝑅 𝑄) (𝑈 𝑇))))
281, 21, 23, 5, 12, 24, 6, 13, 25, 27syl333anc 1356 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) 𝑄) 𝑇) (((𝑃 𝑅) (𝑆 𝑈)) ((𝑅 𝑄) (𝑈 𝑇))))
298, 9hlatjcom 34473 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) = (𝑅 𝑃))
301, 12, 24, 29syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑃 𝑅) = (𝑅 𝑃))
318, 9hlatjcom 34473 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑈𝐴) → (𝑆 𝑈) = (𝑈 𝑆))
321, 13, 25, 31syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑆 𝑈) = (𝑈 𝑆))
3330, 32oveq12d 6653 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑅) (𝑆 𝑈)) = ((𝑅 𝑃) (𝑈 𝑆)))
348, 9hlatjcom 34473 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑄𝐴) → (𝑅 𝑄) = (𝑄 𝑅))
351, 24, 5, 34syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑄) = (𝑄 𝑅))
368, 9hlatjcom 34473 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑇𝐴) → (𝑈 𝑇) = (𝑇 𝑈))
371, 25, 6, 36syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑇) = (𝑇 𝑈))
3835, 37oveq12d 6653 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑅 𝑄) (𝑈 𝑇)) = ((𝑄 𝑅) (𝑇 𝑈)))
3933, 38oveq12d 6653 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑅) (𝑆 𝑈)) ((𝑅 𝑄) (𝑈 𝑇))) = (((𝑅 𝑃) (𝑈 𝑆)) ((𝑄 𝑅) (𝑇 𝑈))))
407, 8, 9hlatjcl 34472 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑃𝐴) → (𝑅 𝑃) ∈ (Base‘𝐾))
411, 24, 12, 40syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑅 𝑃) ∈ (Base‘𝐾))
427, 8, 9hlatjcl 34472 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑆𝐴) → (𝑈 𝑆) ∈ (Base‘𝐾))
431, 25, 13, 42syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑈 𝑆) ∈ (Base‘𝐾))
447, 16latmcl 17033 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 𝑃) ∈ (Base‘𝐾) ∧ (𝑈 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾))
454, 41, 43, 44syl3anc 1324 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾))
467, 8, 9hlatjcl 34472 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
471, 5, 24, 46syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑄 𝑅) ∈ (Base‘𝐾))
487, 8, 9hlatjcl 34472 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
491, 6, 25, 48syl3anc 1324 . . . . 5 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (𝑇 𝑈) ∈ (Base‘𝐾))
507, 16latmcl 17033 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾))
514, 47, 49, 50syl3anc 1324 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾))
527, 8latjcom 17040 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑅 𝑃) (𝑈 𝑆)) ∈ (Base‘𝐾) ∧ ((𝑄 𝑅) (𝑇 𝑈)) ∈ (Base‘𝐾)) → (((𝑅 𝑃) (𝑈 𝑆)) ((𝑄 𝑅) (𝑇 𝑈))) = (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
534, 45, 51, 52syl3anc 1324 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑅 𝑃) (𝑈 𝑆)) ((𝑄 𝑅) (𝑇 𝑈))) = (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
5439, 53eqtrd 2654 . 2 (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑅) (𝑆 𝑈)) ((𝑅 𝑄) (𝑈 𝑇))) = (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
5528, 54breqtrd 4670 1 (((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))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   class class class wbr 4644  ‘cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  meetcmee 16926  Latclat 17026  Atomscatm 34369  HLchlt 34456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-lat 17027  df-clat 17089  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-psubsp 34608  df-pmap 34609  df-padd 34901 This theorem is referenced by:  dalawlem5  34980
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