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Theorem 3atlem2 36728
Description: Lemma for 3at 36734. (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 1135 . . . . . 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 1200 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝐾 ∈ HL)
32hllatd 36608 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝐾 ∈ Lat)
4 simp121 1302 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝐴)
5 simp122 1303 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄𝐴)
6 eqid 2824 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
7 3at.j . . . . . . . . 9 = (join‘𝐾)
8 3at.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
96, 7, 8hlatjcl 36611 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
102, 4, 5, 9syl3anc 1368 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑄) ∈ (Base‘𝐾))
11 simp123 1304 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅𝐴)
126, 8atbase 36533 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1311, 12syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ∈ (Base‘𝐾))
14 simp131 1305 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑆𝐴)
15 simp132 1306 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑇𝐴)
166, 7, 8hlatjcl 36611 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
172, 14, 15, 16syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑆 𝑇) ∈ (Base‘𝐾))
18 simp133 1307 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑈𝐴)
196, 8atbase 36533 . . . . . . . . 9 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
2018, 19syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑈 ∈ (Base‘𝐾))
216, 7latjcl 17661 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))
223, 17, 20, 21syl3anc 1368 . . . . . . 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))
23 3at.l . . . . . . . 8 = (le‘𝐾)
246, 23, 7latjle12 17672 . . . . . . 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)))
253, 10, 13, 22, 24syl13anc 1369 . . . . . 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)))
261, 25mpbird 260 . . . . 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)))
2726simprd 499 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ((𝑆 𝑇) 𝑈))
287, 8hlatjass 36614 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑆 𝑇) 𝑈) = (𝑆 (𝑇 𝑈)))
292, 14, 15, 18, 28syl13anc 1369 . . . . . 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)))
30 simp22r 1290 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃 (𝑇 𝑈))
31 simp22l 1289 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝑈)
3223, 7, 8hlatexchb2 36638 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑇𝐴𝑈𝐴) ∧ 𝑃𝑈) → (𝑃 (𝑇 𝑈) ↔ (𝑃 𝑈) = (𝑇 𝑈)))
332, 4, 15, 18, 31, 32syl131anc 1380 . . . . . . . 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)))
3430, 33mpbid 235 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑈) = (𝑇 𝑈))
3534oveq2d 7165 . . . . . 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)))
3629, 35eqtr4d 2862 . . . . 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)))
377, 8hlatjass 36614 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑈𝐴)) → ((𝑃 𝑄) 𝑈) = (𝑃 (𝑄 𝑈)))
382, 4, 5, 18, 37syl13anc 1369 . . . . . 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)))
397, 8hlatj12 36615 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑈𝐴)) → (𝑃 (𝑄 𝑈)) = (𝑄 (𝑃 𝑈)))
402, 4, 5, 18, 39syl13anc 1369 . . . . . 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)))
417, 8hlatj32 36616 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑅) 𝑄))
422, 4, 5, 11, 41syl13anc 1369 . . . . . . . . . . 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))
431, 42, 293brtr3d 5083 . . . . . . . . . 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)))
446, 7, 8hlatjcl 36611 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
452, 4, 11, 44syl3anc 1368 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑅) ∈ (Base‘𝐾))
466, 8atbase 36533 . . . . . . . . . . . 12 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
475, 46syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 ∈ (Base‘𝐾))
486, 8atbase 36533 . . . . . . . . . . . . 13 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
4914, 48syl 17 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑆 ∈ (Base‘𝐾))
506, 7, 8hlatjcl 36611 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
512, 15, 18, 50syl3anc 1368 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑇 𝑈) ∈ (Base‘𝐾))
526, 7latjcl 17661 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → (𝑆 (𝑇 𝑈)) ∈ (Base‘𝐾))
533, 49, 51, 52syl3anc 1368 . . . . . . . . . . 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))
546, 23, 7latjle12 17672 . . . . . . . . . . 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))))
553, 45, 47, 53, 54syl13anc 1369 . . . . . . . . . 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))))
5643, 55mpbird 260 . . . . . . . . 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))))
5756simprd 499 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 (𝑆 (𝑇 𝑈)))
5857, 35breqtrrd 5080 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑄 (𝑆 (𝑃 𝑈)))
596, 7, 8hlatjcl 36611 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
602, 4, 18, 59syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝑃 𝑈) ∈ (Base‘𝐾))
61 simp23 1205 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑄 (𝑃 𝑈))
626, 23, 7, 8hlexchb2 36629 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑆𝐴 ∧ (𝑃 𝑈) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 (𝑃 𝑈)) → (𝑄 (𝑆 (𝑃 𝑈)) ↔ (𝑄 (𝑃 𝑈)) = (𝑆 (𝑃 𝑈))))
632, 5, 14, 60, 61, 62syl131anc 1380 . . . . . . 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))))
6458, 63mpbid 235 . . . . . 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)))
6538, 40, 643eqtrd 2863 . . . . 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)))
6636, 65eqtr4d 2862 . . . 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))
6727, 66breqtrd 5078 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑅 ((𝑃 𝑄) 𝑈))
68 simp21 1203 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ (𝑃𝑈𝑃 (𝑇 𝑈)) ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑅 (𝑃 𝑄))
696, 23, 7, 8hlexchb1 36628 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑈𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 ((𝑃 𝑄) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑃 𝑄) 𝑈)))
702, 11, 18, 10, 68, 69syl131anc 1380 . . 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)))
7167, 70mpbid 235 . 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))
7271, 66eqtr4d 2862 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  Latclat 17655  Atomscatm 36507  HLchlt 36594
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-covers 36510  df-ats 36511  df-atl 36542  df-cvlat 36566  df-hlat 36595
This theorem is referenced by:  3atlem3  36729
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