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Theorem 4atlem11b 38467
Description: Lemma for 4at 38472. 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 1203 . . . 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 1204 . . . . 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 1309 . . . . 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 1310 . . . . 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 1128 . . . 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 1199 . . . 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 1128 . . 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 1210 . . . . 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 1211 . . . . 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 1302 . . . . . . 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)
1110hllatd 38222 . . . . . 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)
12 simp12l 1286 . . . . . . 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)
13 eqid 2732 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
14 4at.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1513, 14atbase 38147 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1612, 15syl 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))
17 simp12r 1287 . . . . . . 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)
1813, 14atbase 38147 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1917, 18syl 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))
20 simp112 1303 . . . . . . . 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)
21 simp131 1308 . . . . . . . 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)
22 4at.j . . . . . . . . 9 = (join‘𝐾)
2313, 22, 14hlatjcl 38225 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
2410, 20, 21, 23syl3anc 1371 . . . . . . 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))
2513, 22, 14hlatjcl 38225 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑊𝐴) → (𝑉 𝑊) ∈ (Base‘𝐾))
2610, 3, 4, 25syl3anc 1371 . . . . . . 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))
2713, 22latjcl 18388 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ (𝑉 𝑊) ∈ (Base‘𝐾)) → ((𝑃 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))
2811, 24, 26, 27syl3anc 1371 . . . . . 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))
29 4at.l . . . . . . 7 = (le‘𝐾)
3013, 29, 22latjle12 18399 . . . . . 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))))
3111, 16, 19, 28, 30syl13anc 1372 . . . . 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))))
328, 9, 31mpbi2and 710 . . . 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)))
33 simp31 1209 . . . . 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)))
34 simp13 1205 . . . . . 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))
35 simp2r 1200 . . . . . 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))
3629, 22, 144atlem11a 38466 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑄 ((𝑃 𝑉) 𝑊)) → (𝑄 ((𝑃 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑄) (𝑉 𝑊)) = ((𝑃 𝑈) (𝑉 𝑊))))
371, 34, 35, 36syl3anc 1371 . . . . 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))))
3833, 37mpbid 231 . . . 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)))
3932, 38breqtrrd 5175 . . 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)))
4029, 22, 144atlem10 38465 . . 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))))
417, 39, 40sylc 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)))
4241, 38eqtrd 2772 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940   class class class wbr 5147  cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  Latclat 18380  Atomscatm 38121  HLchlt 38208
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358  df-lvols 38359
This theorem is referenced by:  4atlem11  38468
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