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Theorem 4atlem12b 37367
Description: Lemma for 4at 37369. 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 1205 . . . . 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 1307 . . . . . 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 1308 . . . . . 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 515 . . . . 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 1207 . . . . 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 1130 . . . 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 1201 . . . 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 515 . . 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 1247 . . . . 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 1248 . . . . . 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 1249 . . . . . 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 1304 . . . . . . . 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)
1312hllatd 37120 . . . . . . 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)
14 eqid 2737 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
15 4at.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
1614, 15atbase 37045 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
172, 16syl 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))
1814, 15atbase 37045 . . . . . . . 8 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
193, 18syl 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))
20 simp123 1309 . . . . . . . . 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)
21 simp131 1310 . . . . . . . . 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)
22 4at.j . . . . . . . . . 10 = (join‘𝐾)
2314, 22, 15hlatjcl 37123 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
2412, 20, 21, 23syl3anc 1373 . . . . . . . 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))
25 simp132 1311 . . . . . . . . 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)
26 simp133 1312 . . . . . . . . 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)
2714, 22, 15hlatjcl 37123 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑉𝐴𝑊𝐴) → (𝑉 𝑊) ∈ (Base‘𝐾))
2812, 25, 26, 27syl3anc 1373 . . . . . . . 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))
2914, 22latjcl 17950 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ (𝑉 𝑊) ∈ (Base‘𝐾)) → ((𝑇 𝑈) (𝑉 𝑊)) ∈ (Base‘𝐾))
3013, 24, 28, 29syl3anc 1373 . . . . . . 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))
31 4at.l . . . . . . . 8 = (le‘𝐾)
3214, 31, 22latjle12 17961 . . . . . . 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))))
3313, 17, 19, 30, 32syl13anc 1374 . . . . . 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))))
3410, 11, 33mpbi2and 712 . . . . 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)))
35 simp113 1306 . . . . . . 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)
3614, 15atbase 37045 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3735, 36syl 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))
3814, 22, 15hlatjcl 37123 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
3912, 2, 3, 38syl3anc 1373 . . . . . 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))
4014, 31, 22latjle12 17961 . . . . . 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))))
4113, 37, 39, 30, 40syl13anc 1374 . . . . 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))))
429, 34, 41mpbi2and 712 . . . 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)))
43 simp3ll 1246 . . . . 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)))
44 simp112 1305 . . . . . 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)
45 simp2r 1202 . . . . . 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))
4631, 22, 154atlem12a 37366 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴𝑊𝐴) ∧ ¬ 𝑃 ((𝑈 𝑉) 𝑊)) → (𝑃 ((𝑇 𝑈) (𝑉 𝑊)) ↔ ((𝑃 𝑈) (𝑉 𝑊)) = ((𝑇 𝑈) (𝑉 𝑊))))
4712, 44, 20, 5, 45, 46syl311anc 1386 . . . . 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))))
4843, 47mpbid 235 . . . 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)))
4942, 48breqtrrd 5086 . . 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)))
5031, 22, 154atlem11 37365 . . 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))))
518, 49, 50sylc 65 . 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)))
5251, 48eqtrd 2777 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940   class class class wbr 5058  cfv 6385  (class class class)co 7218  Basecbs 16765  lecple 16814  joincjn 17823  Latclat 17942  Atomscatm 37019  HLchlt 37106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5184  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-riota 7175  df-ov 7221  df-oprab 7222  df-proset 17807  df-poset 17825  df-plt 17841  df-lub 17857  df-glb 17858  df-join 17859  df-meet 17860  df-p0 17936  df-lat 17943  df-clat 18010  df-oposet 36932  df-ol 36934  df-oml 36935  df-covers 37022  df-ats 37023  df-atl 37054  df-cvlat 37078  df-hlat 37107  df-llines 37254  df-lplanes 37255  df-lvols 37256
This theorem is referenced by:  4atlem12  37368
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