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Theorem 4atlem12b 33711
Description: Lemma for 4at 33713. Substitute 𝑇 for 𝑃 (cont.). (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
4at.l = (le‘𝐾)
4at.j = (join‘𝐾)
4at.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
4atlem12b ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑇 𝑈) (𝑉 𝑊)))

Proof of Theorem 4atlem12b
StepHypRef Expression
1 simp11 1083 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
2 simp121 1185 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑅𝐴)
3 simp122 1186 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑆𝐴)
42, 3jca 552 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑅𝐴𝑆𝐴))
5 simp13 1085 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑈𝐴𝑉𝐴𝑊𝐴))
61, 4, 53jca 1234 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)))
7 simp2l 1079 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)))
86, 7jca 552 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))))
9 simp3lr 1125 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑄 ((𝑇 𝑈) (𝑉 𝑊)))
10 simp3rl 1126 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑅 ((𝑇 𝑈) (𝑉 𝑊)))
11 simp3rr 1127 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑆 ((𝑇 𝑈) (𝑉 𝑊)))
12 simp111 1182 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝐾 ∈ HL)
13 hllat 33464 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1412, 13syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝐾 ∈ Lat)
15 eqid 2609 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
16 4at.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
1715, 16atbase 33390 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
182, 17syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑅 ∈ (Base‘𝐾))
1915, 16atbase 33390 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
203, 19syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑆 ∈ (Base‘𝐾))
21 simp123 1187 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑇𝐴)
22 simp131 1188 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑈𝐴)
23 4at.j . . . . . . . . . 10 = (join‘𝐾)
2415, 23, 16hlatjcl 33467 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
2512, 21, 22, 24syl3anc 1317 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑇 𝑈) ∈ (Base‘𝐾))
26 simp132 1189 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑉𝐴)
27 simp133 1190 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑊𝐴)
2815, 23, 16hlatjcl 33467 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑊𝐴) → (𝑉 𝑊) ∈ (Base‘𝐾))
2912, 26, 27, 28syl3anc 1317 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑉 𝑊) ∈ (Base‘𝐾))
3015, 23latjcl 16820 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ (𝑉 𝑊) ∈ (Base‘𝐾)) → ((𝑇 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))
3114, 25, 29, 30syl3anc 1317 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑇 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))
32 4at.l . . . . . . . 8 = (le‘𝐾)
3315, 32, 23latjle12 16831 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ ((𝑇 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))) → ((𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))) ↔ (𝑅 𝑆) ((𝑇 𝑈) (𝑉 𝑊))))
3414, 18, 20, 31, 33syl13anc 1319 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))) ↔ (𝑅 𝑆) ((𝑇 𝑈) (𝑉 𝑊))))
3510, 11, 34mpbi2and 957 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑅 𝑆) ((𝑇 𝑈) (𝑉 𝑊)))
36 simp113 1184 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑄𝐴)
3715, 16atbase 33390 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3836, 37syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑄 ∈ (Base‘𝐾))
3915, 23, 16hlatjcl 33467 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
4012, 2, 3, 39syl3anc 1317 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑅 𝑆) ∈ (Base‘𝐾))
4115, 32, 23latjle12 16831 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾) ∧ ((𝑇 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))) → ((𝑄 ((𝑇 𝑈) (𝑉 𝑊)) ∧ (𝑅 𝑆) ((𝑇 𝑈) (𝑉 𝑊))) ↔ (𝑄 (𝑅 𝑆)) ((𝑇 𝑈) (𝑉 𝑊))))
4214, 38, 40, 31, 41syl13anc 1319 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑄 ((𝑇 𝑈) (𝑉 𝑊)) ∧ (𝑅 𝑆) ((𝑇 𝑈) (𝑉 𝑊))) ↔ (𝑄 (𝑅 𝑆)) ((𝑇 𝑈) (𝑉 𝑊))))
439, 35, 42mpbi2and 957 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑄 (𝑅 𝑆)) ((𝑇 𝑈) (𝑉 𝑊)))
44 simp3ll 1124 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑃 ((𝑇 𝑈) (𝑉 𝑊)))
45 simp112 1183 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → 𝑃𝐴)
46 simp2r 1080 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ¬ 𝑃 ((𝑈 𝑉) 𝑊))
4732, 23, 164atlem12a 33710 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) → (𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑈) (𝑉 𝑊)) = ((𝑇 𝑈) (𝑉 𝑊))))
4812, 45, 21, 5, 46, 47syl311anc 1331 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑈) (𝑉 𝑊)) = ((𝑇 𝑈) (𝑉 𝑊))))
4944, 48mpbid 220 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑃 𝑈) (𝑉 𝑊)) = ((𝑇 𝑈) (𝑉 𝑊)))
5043, 49breqtrrd 4605 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → (𝑄 (𝑅 𝑆)) ((𝑃 𝑈) (𝑉 𝑊)))
5132, 23, 164atlem11 33709 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑄 (𝑅 𝑆)) ((𝑃 𝑈) (𝑉 𝑊)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑈) (𝑉 𝑊))))
528, 50, 51sylc 62 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑈) (𝑉 𝑊)))
5352, 49eqtrd 2643 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) ∧ ((𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑄 ((𝑇 𝑈) (𝑉 𝑊))) ∧ (𝑅 ((𝑇 𝑈) (𝑉 𝑊)) ∧ 𝑆 ((𝑇 𝑈) (𝑉 𝑊))))) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑇 𝑈) (𝑉 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  Latclat 16814  Atomscatm 33364  HLchlt 33451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33277  df-ol 33279  df-oml 33280  df-covers 33367  df-ats 33368  df-atl 33399  df-cvlat 33423  df-hlat 33452  df-llines 33598  df-lplanes 33599  df-lvols 33600
This theorem is referenced by:  4atlem12  33712
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