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Theorem 3atlem1 37946
Description: Lemma for 3at 37953. (Contributed by NM, 22-Jun-2012.)
Hypotheses
Ref Expression
3at.l = (le‘𝐾)
3at.j = (join‘𝐾)
3at.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3atlem1 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Proof of Theorem 3atlem1
StepHypRef Expression
1 simp11 1203 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝐾 ∈ HL)
2 simp131 1308 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑆𝐴)
3 simp132 1309 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑇𝐴)
4 simp133 1310 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑈𝐴)
5 3at.j . . . . . 6 = (join‘𝐾)
6 3at.a . . . . . 6 𝐴 = (Atoms‘𝐾)
75, 6hlatjass 37832 . . . . 5 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑆 𝑇) 𝑈) = (𝑆 (𝑇 𝑈)))
81, 2, 3, 4, 7syl13anc 1372 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) = (𝑆 (𝑇 𝑈)))
9 simp121 1305 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝐴)
10 simp122 1306 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄𝐴)
11 simp123 1307 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅𝐴)
125, 6hlatjass 37832 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))
131, 9, 10, 11, 12syl13anc 1372 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))
14 simp3 1138 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈))
1513, 14eqbrtrrd 5129 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 (𝑄 𝑅)) ((𝑆 𝑇) 𝑈))
161hllatd 37826 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝐾 ∈ Lat)
17 eqid 2736 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
1817, 6atbase 37751 . . . . . . . . . 10 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
199, 18syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃 ∈ (Base‘𝐾))
2017, 5, 6hlatjcl 37829 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
211, 10, 11, 20syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 𝑅) ∈ (Base‘𝐾))
2217, 5, 6hlatjcl 37829 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
231, 2, 3, 22syl3anc 1371 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑆 𝑇) ∈ (Base‘𝐾))
2417, 6atbase 37751 . . . . . . . . . . 11 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
254, 24syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑈 ∈ (Base‘𝐾))
2617, 5latjcl 18328 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))
2716, 23, 25, 26syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))
28 3at.l . . . . . . . . . 10 = (le‘𝐾)
2917, 28, 5latjle12 18339 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))) → ((𝑃 ((𝑆 𝑇) 𝑈) ∧ (𝑄 𝑅) ((𝑆 𝑇) 𝑈)) ↔ (𝑃 (𝑄 𝑅)) ((𝑆 𝑇) 𝑈)))
3016, 19, 21, 27, 29syl13anc 1372 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 ((𝑆 𝑇) 𝑈) ∧ (𝑄 𝑅) ((𝑆 𝑇) 𝑈)) ↔ (𝑃 (𝑄 𝑅)) ((𝑆 𝑇) 𝑈)))
3115, 30mpbird 256 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 ((𝑆 𝑇) 𝑈) ∧ (𝑄 𝑅) ((𝑆 𝑇) 𝑈)))
3231simpld 495 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃 ((𝑆 𝑇) 𝑈))
3332, 8breqtrd 5131 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃 (𝑆 (𝑇 𝑈)))
3417, 5, 6hlatjcl 37829 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
351, 3, 4, 34syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑇 𝑈) ∈ (Base‘𝐾))
36 simp22 1207 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑃 (𝑇 𝑈))
3717, 28, 5, 6hlexchb2 37848 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴 ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) ∧ ¬ 𝑃 (𝑇 𝑈)) → (𝑃 (𝑆 (𝑇 𝑈)) ↔ (𝑃 (𝑇 𝑈)) = (𝑆 (𝑇 𝑈))))
381, 9, 2, 35, 36, 37syl131anc 1383 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 (𝑆 (𝑇 𝑈)) ↔ (𝑃 (𝑇 𝑈)) = (𝑆 (𝑇 𝑈))))
3933, 38mpbid 231 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 (𝑇 𝑈)) = (𝑆 (𝑇 𝑈)))
405, 6hlatj12 37833 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑇𝐴𝑈𝐴)) → (𝑃 (𝑇 𝑈)) = (𝑇 (𝑃 𝑈)))
411, 9, 3, 4, 40syl13anc 1372 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 (𝑇 𝑈)) = (𝑇 (𝑃 𝑈)))
428, 39, 413eqtr2d 2782 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) = (𝑇 (𝑃 𝑈)))
435, 6hlatj12 37833 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))
441, 9, 10, 11, 43syl13anc 1372 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))
4515, 44, 423brtr3d 5136 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 (𝑃 𝑅)) (𝑇 (𝑃 𝑈)))
4617, 6atbase 37751 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4710, 46syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 ∈ (Base‘𝐾))
4817, 5, 6hlatjcl 37829 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
491, 9, 11, 48syl3anc 1371 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑅) ∈ (Base‘𝐾))
5017, 6atbase 37751 . . . . . . . . 9 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
513, 50syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑇 ∈ (Base‘𝐾))
5217, 5, 6hlatjcl 37829 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
531, 9, 4, 52syl3anc 1371 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑈) ∈ (Base‘𝐾))
5417, 5latjcl 18328 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑃 𝑈) ∈ (Base‘𝐾)) → (𝑇 (𝑃 𝑈)) ∈ (Base‘𝐾))
5516, 51, 53, 54syl3anc 1371 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑇 (𝑃 𝑈)) ∈ (Base‘𝐾))
5617, 28, 5latjle12 18339 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 (𝑃 𝑈)) ∈ (Base‘𝐾))) → ((𝑄 (𝑇 (𝑃 𝑈)) ∧ (𝑃 𝑅) (𝑇 (𝑃 𝑈))) ↔ (𝑄 (𝑃 𝑅)) (𝑇 (𝑃 𝑈))))
5716, 47, 49, 55, 56syl13anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑄 (𝑇 (𝑃 𝑈)) ∧ (𝑃 𝑅) (𝑇 (𝑃 𝑈))) ↔ (𝑄 (𝑃 𝑅)) (𝑇 (𝑃 𝑈))))
5845, 57mpbird 256 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 (𝑇 (𝑃 𝑈)) ∧ (𝑃 𝑅) (𝑇 (𝑃 𝑈))))
5958simpld 495 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 (𝑇 (𝑃 𝑈)))
60 simp23 1208 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑄 (𝑃 𝑈))
6117, 28, 5, 6hlexchb2 37848 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑇𝐴 ∧ (𝑃 𝑈) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 (𝑃 𝑈)) → (𝑄 (𝑇 (𝑃 𝑈)) ↔ (𝑄 (𝑃 𝑈)) = (𝑇 (𝑃 𝑈))))
621, 10, 3, 53, 60, 61syl131anc 1383 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 (𝑇 (𝑃 𝑈)) ↔ (𝑄 (𝑃 𝑈)) = (𝑇 (𝑃 𝑈))))
6359, 62mpbid 231 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 (𝑃 𝑈)) = (𝑇 (𝑃 𝑈)))
6417, 5latj13 18375 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑄 (𝑃 𝑈)) = (𝑈 (𝑃 𝑄)))
6516, 47, 19, 25, 64syl13anc 1372 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 (𝑃 𝑈)) = (𝑈 (𝑃 𝑄)))
6642, 63, 653eqtr2d 2782 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) = (𝑈 (𝑃 𝑄)))
6717, 5, 6hlatjcl 37829 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
681, 9, 10, 67syl3anc 1371 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑄) ∈ (Base‘𝐾))
6917, 6atbase 37751 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
7011, 69syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ∈ (Base‘𝐾))
7117, 28, 5latjle12 18339 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))) → (((𝑃 𝑄) ((𝑆 𝑇) 𝑈) ∧ 𝑅 ((𝑆 𝑇) 𝑈)) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
7216, 68, 70, 27, 71syl13anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (((𝑃 𝑄) ((𝑆 𝑇) 𝑈) ∧ 𝑅 ((𝑆 𝑇) 𝑈)) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
7314, 72mpbird 256 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) ((𝑆 𝑇) 𝑈) ∧ 𝑅 ((𝑆 𝑇) 𝑈)))
7473simprd 496 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ((𝑆 𝑇) 𝑈))
7574, 66breqtrd 5131 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 (𝑈 (𝑃 𝑄)))
76 simp21 1206 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑅 (𝑃 𝑄))
7717, 28, 5, 6hlexchb2 37848 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑈𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 (𝑈 (𝑃 𝑄)) ↔ (𝑅 (𝑃 𝑄)) = (𝑈 (𝑃 𝑄))))
781, 11, 4, 68, 76, 77syl131anc 1383 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑅 (𝑈 (𝑃 𝑄)) ↔ (𝑅 (𝑃 𝑄)) = (𝑈 (𝑃 𝑄))))
7975, 78mpbid 231 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑅 (𝑃 𝑄)) = (𝑈 (𝑃 𝑄)))
8017, 5latjcom 18336 . . 3 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑅 (𝑃 𝑄)) = ((𝑃 𝑄) 𝑅))
8116, 70, 68, 80syl3anc 1371 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑅 (𝑃 𝑄)) = ((𝑃 𝑄) 𝑅))
8266, 79, 813eqtr2rd 2783 1 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑃 (𝑇 𝑈) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  Latclat 18320  Atomscatm 37725  HLchlt 37812
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-lat 18321  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813
This theorem is referenced by:  3atlem3  37948
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