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Theorem 3atlem2 33582
Description: Lemma for 3at 33588. (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
3atlem2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Proof of Theorem 3atlem2
StepHypRef Expression
1 simp3 1056 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈))
2 simp11 1084 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝐾 ∈ HL)
3 hllat 33462 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
42, 3syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝐾 ∈ Lat)
5 simp121 1186 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝐴)
6 simp122 1187 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄𝐴)
7 eqid 2610 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
8 3at.j . . . . . . . . 9 = (join‘𝐾)
9 3at.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
107, 8, 9hlatjcl 33465 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
112, 5, 6, 10syl3anc 1318 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑄) ∈ (Base‘𝐾))
12 simp123 1188 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅𝐴)
137, 9atbase 33388 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ∈ (Base‘𝐾))
15 simp131 1189 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑆𝐴)
16 simp132 1190 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑇𝐴)
177, 8, 9hlatjcl 33465 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
182, 15, 16, 17syl3anc 1318 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑆 𝑇) ∈ (Base‘𝐾))
19 simp133 1191 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑈𝐴)
207, 9atbase 33388 . . . . . . . . 9 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
2119, 20syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑈 ∈ (Base‘𝐾))
227, 8latjcl 16823 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))
234, 18, 21, 22syl3anc 1318 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))
24 3at.l . . . . . . . 8 = (le‘𝐾)
257, 24, 8latjle12 16834 . . . . . . 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)))
264, 11, 14, 23, 25syl13anc 1320 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (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)))
271, 26mpbird 246 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) ((𝑆 𝑇) 𝑈) ∧ 𝑅 ((𝑆 𝑇) 𝑈)))
2827simprd 478 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ((𝑆 𝑇) 𝑈))
298, 9hlatjass 33468 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑆 𝑇) 𝑈) = (𝑆 (𝑇 𝑈)))
302, 15, 16, 19, 29syl13anc 1320 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) = (𝑆 (𝑇 𝑈)))
31 simp22r 1174 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃 (𝑇 𝑈))
32 simp22l 1173 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝑈)
3324, 8, 9hlatexchb2 33492 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑇𝐴𝑈𝐴) ∧ 𝑃𝑈) → (𝑃 (𝑇 𝑈) ↔ (𝑃 𝑈) = (𝑇 𝑈)))
342, 5, 16, 19, 32, 33syl131anc 1331 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 (𝑇 𝑈) ↔ (𝑃 𝑈) = (𝑇 𝑈)))
3531, 34mpbid 221 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑈) = (𝑇 𝑈))
3635oveq2d 6543 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑆 (𝑃 𝑈)) = (𝑆 (𝑇 𝑈)))
3730, 36eqtr4d 2647 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) = (𝑆 (𝑃 𝑈)))
388, 9hlatjass 33468 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑈) = (𝑃 (𝑄 𝑈)))
392, 5, 6, 19, 38syl13anc 1320 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑈) = (𝑃 (𝑄 𝑈)))
408, 9hlatj12 33469 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑈𝐴)) → (𝑃 (𝑄 𝑈)) = (𝑄 (𝑃 𝑈)))
412, 5, 6, 19, 40syl13anc 1320 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 (𝑄 𝑈)) = (𝑄 (𝑃 𝑈)))
428, 9hlatj32 33470 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑅) 𝑄))
432, 5, 6, 12, 42syl13anc 1320 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑅) 𝑄))
441, 43, 303brtr3d 4609 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑅) 𝑄) (𝑆 (𝑇 𝑈)))
457, 8, 9hlatjcl 33465 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
462, 5, 12, 45syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑅) ∈ (Base‘𝐾))
477, 9atbase 33388 . . . . . . . . . . . 12 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
486, 47syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 ∈ (Base‘𝐾))
497, 9atbase 33388 . . . . . . . . . . . . 13 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
5015, 49syl 17 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑆 ∈ (Base‘𝐾))
517, 8, 9hlatjcl 33465 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
522, 16, 19, 51syl3anc 1318 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑇 𝑈) ∈ (Base‘𝐾))
537, 8latjcl 16823 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → (𝑆 (𝑇 𝑈)) ∈ (Base‘𝐾))
544, 50, 52, 53syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑆 (𝑇 𝑈)) ∈ (Base‘𝐾))
557, 24, 8latjle12 16834 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ ((𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑆 (𝑇 𝑈)) ∈ (Base‘𝐾))) → (((𝑃 𝑅) (𝑆 (𝑇 𝑈)) ∧ 𝑄 (𝑆 (𝑇 𝑈))) ↔ ((𝑃 𝑅) 𝑄) (𝑆 (𝑇 𝑈))))
564, 46, 48, 54, 55syl13anc 1320 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (((𝑃 𝑅) (𝑆 (𝑇 𝑈)) ∧ 𝑄 (𝑆 (𝑇 𝑈))) ↔ ((𝑃 𝑅) 𝑄) (𝑆 (𝑇 𝑈))))
5744, 56mpbird 246 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑅) (𝑆 (𝑇 𝑈)) ∧ 𝑄 (𝑆 (𝑇 𝑈))))
5857simprd 478 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 (𝑆 (𝑇 𝑈)))
5958, 36breqtrrd 4606 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 (𝑆 (𝑃 𝑈)))
607, 8, 9hlatjcl 33465 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
612, 5, 19, 60syl3anc 1318 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑈) ∈ (Base‘𝐾))
62 simp23 1089 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑄 (𝑃 𝑈))
637, 24, 8, 9hlexchb2 33483 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑆𝐴 ∧ (𝑃 𝑈) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 (𝑃 𝑈)) → (𝑄 (𝑆 (𝑃 𝑈)) ↔ (𝑄 (𝑃 𝑈)) = (𝑆 (𝑃 𝑈))))
642, 6, 15, 61, 62, 63syl131anc 1331 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 (𝑆 (𝑃 𝑈)) ↔ (𝑄 (𝑃 𝑈)) = (𝑆 (𝑃 𝑈))))
6559, 64mpbid 221 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑄 (𝑃 𝑈)) = (𝑆 (𝑃 𝑈)))
6639, 41, 653eqtrd 2648 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑈) = (𝑆 (𝑃 𝑈)))
6737, 66eqtr4d 2647 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑆 𝑇) 𝑈) = ((𝑃 𝑄) 𝑈))
6828, 67breqtrd 4604 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ((𝑃 𝑄) 𝑈))
69 simp21 1087 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑅 (𝑃 𝑄))
707, 24, 8, 9hlexchb1 33482 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑈𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 ((𝑃 𝑄) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑃 𝑄) 𝑈)))
712, 12, 19, 11, 69, 70syl131anc 1331 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑅 ((𝑃 𝑄) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑃 𝑄) 𝑈)))
7268, 71mpbid 221 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑄) 𝑈))
7372, 67eqtr4d 2647 1 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  Latclat 16817  Atomscatm 33362  HLchlt 33449
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  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 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-lat 16818  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450
This theorem is referenced by:  3atlem3  33583
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