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Theorem 4atlem11b 34713
Description: Lemma for 4at 34718. Substitute 𝑈 for 𝑄 (cont.). (Contributed by NM, 10-Jul-2012.)
Hypotheses
Ref Expression
4at.l = (le‘𝐾)
4at.j = (join‘𝐾)
4at.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
4atlem11b ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑈) (𝑉 𝑊)))

Proof of Theorem 4atlem11b
StepHypRef Expression
1 simp11 1089 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
2 simp12 1090 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑅𝐴𝑆𝐴))
3 simp132 1195 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑉𝐴)
4 simp133 1196 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑊𝐴)
52, 3, 43jca 1240 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑅𝐴𝑆𝐴) ∧ 𝑉𝐴𝑊𝐴))
6 simp2l 1085 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)))
71, 5, 63jca 1240 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑅𝐴𝑆𝐴) ∧ 𝑉𝐴𝑊𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))))
8 simp32 1096 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑅 ((𝑃 𝑈) (𝑉 𝑊)))
9 simp33 1097 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))
10 simp111 1188 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝐾 ∈ HL)
11 hllat 34469 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1210, 11syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝐾 ∈ Lat)
13 simp12l 1172 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑅𝐴)
14 eqid 2620 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
15 4at.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1614, 15atbase 34395 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1713, 16syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑅 ∈ (Base‘𝐾))
18 simp12r 1173 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑆𝐴)
1914, 15atbase 34395 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
2018, 19syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑆 ∈ (Base‘𝐾))
21 simp112 1189 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑃𝐴)
22 simp131 1194 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑈𝐴)
23 4at.j . . . . . . . . 9 = (join‘𝐾)
2414, 23, 15hlatjcl 34472 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
2510, 21, 22, 24syl3anc 1324 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑃 𝑈) ∈ (Base‘𝐾))
2614, 23, 15hlatjcl 34472 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑊𝐴) → (𝑉 𝑊) ∈ (Base‘𝐾))
2710, 3, 4, 26syl3anc 1324 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑉 𝑊) ∈ (Base‘𝐾))
2814, 23latjcl 17032 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ (𝑉 𝑊) ∈ (Base‘𝐾)) → ((𝑃 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))
2912, 25, 27, 28syl3anc 1324 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑃 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))
30 4at.l . . . . . . 7 = (le‘𝐾)
3114, 30, 23latjle12 17043 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ ((𝑃 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))) → ((𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊))) ↔ (𝑅 𝑆) ((𝑃 𝑈) (𝑉 𝑊))))
3212, 17, 20, 29, 31syl13anc 1326 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊))) ↔ (𝑅 𝑆) ((𝑃 𝑈) (𝑉 𝑊))))
338, 9, 32mpbi2and 955 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑅 𝑆) ((𝑃 𝑈) (𝑉 𝑊)))
34 simp31 1095 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → 𝑄 ((𝑃 𝑈) (𝑉 𝑊)))
35 simp13 1091 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑈𝐴𝑉𝐴𝑊𝐴))
36 simp2r 1086 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ¬ 𝑄 ((𝑃 𝑉) 𝑊))
3730, 23, 154atlem11a 34712 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) → (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑉 𝑊)) = ((𝑃 𝑈) (𝑉 𝑊))))
381, 35, 36, 37syl3anc 1324 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑉 𝑊)) = ((𝑃 𝑈) (𝑉 𝑊))))
3934, 38mpbid 222 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑃 𝑄) (𝑉 𝑊)) = ((𝑃 𝑈) (𝑉 𝑊)))
4033, 39breqtrrd 4672 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → (𝑅 𝑆) ((𝑃 𝑄) (𝑉 𝑊)))
4130, 23, 154atlem10 34711 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ ((𝑅𝐴𝑆𝐴) ∧ 𝑉𝐴𝑊𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑅 𝑆) ((𝑃 𝑄) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) (𝑉 𝑊))))
427, 40, 41sylc 65 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑄) (𝑉 𝑊)))
4342, 39eqtrd 2654 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) ∧ (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑅 ((𝑃 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑃 𝑈) (𝑉 𝑊)))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑈) (𝑉 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  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-3or 1037  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-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-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-llines 34603  df-lplanes 34604  df-lvols 34605
This theorem is referenced by:  4atlem11  34714
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