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Theorem 4atlem12b 33915
 Description: Lemma for 4at 33917. 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 1084 . . . . 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 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))))) → 𝑅𝐴)
3 simp122 1187 . . . . . 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 553 . . . . 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 1086 . . . . 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 1235 . . . 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 1080 . . . 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 553 . . 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 1126 . . . . 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 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))))) → 𝑅 ((𝑇 𝑈) (𝑉 𝑊)))
11 simp3rr 1128 . . . . . 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 1183 . . . . . . . 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 33668 . . . . . . . 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 2610 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
16 4at.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
1715, 16atbase 33594 . . . . . . . 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 33594 . . . . . . . 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 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))))) → 𝑇𝐴)
22 simp131 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))))) → 𝑈𝐴)
23 4at.j . . . . . . . . . 10 = (join‘𝐾)
2415, 23, 16hlatjcl 33671 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
2512, 21, 22, 24syl3anc 1318 . . . . . . . 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 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))))) → 𝑉𝐴)
27 simp133 1191 . . . . . . . . 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 33671 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑊𝐴) → (𝑉 𝑊) ∈ (Base‘𝐾))
2912, 26, 27, 28syl3anc 1318 . . . . . . . 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 16874 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ (𝑉 𝑊) ∈ (Base‘𝐾)) → ((𝑇 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))
3114, 25, 29, 30syl3anc 1318 . . . . . . 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 16885 . . . . . . 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 1320 . . . . . 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 958 . . . . 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 1185 . . . . . . 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 33594 . . . . . . 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 33671 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
4012, 2, 3, 39syl3anc 1318 . . . . . 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 16885 . . . . . 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 1320 . . . . 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 958 . . . 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 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))))) → 𝑃 ((𝑇 𝑈) (𝑉 𝑊)))
45 simp112 1184 . . . . . 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 1081 . . . . . 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 33914 . . . . . 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 1332 . . . . 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 221 . . . 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 4611 . . 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 33913 . . 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 63 . 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 2644 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868  Atomscatm 33568  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804 This theorem is referenced by:  4atlem12  33916
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