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Theorem 3atlem5 36490
Description: Lemma for 3at 36493. (Contributed by NM, 23-Jun-2012.)
Hypotheses
Ref Expression
3at.l = (le‘𝐾)
3at.j = (join‘𝐾)
3at.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3atlem5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))

Proof of Theorem 3atlem5
StepHypRef Expression
1 oveq2 7156 . . . . . 6 (𝑈 = 𝑃 → ((𝑆 𝑇) 𝑈) = ((𝑆 𝑇) 𝑃))
21eqcoms 2834 . . . . 5 (𝑃 = 𝑈 → ((𝑆 𝑇) 𝑈) = ((𝑆 𝑇) 𝑃))
32breq2d 5075 . . . 4 (𝑃 = 𝑈 → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃)))
42eqeq2d 2837 . . . 4 (𝑃 = 𝑈 → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃)))
53, 4imbi12d 346 . . 3 (𝑃 = 𝑈 → ((((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)) ↔ (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃))))
6 simp1l 1191 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → (𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)))
7 simp1r1 1263 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑅 (𝑃 𝑄))
8 simp2 1131 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝑈)
9 simp1r3 1265 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑄 (𝑃 𝑈))
10 simp3 1132 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈))
11 3at.l . . . . . 6 = (le‘𝐾)
12 3at.j . . . . . 6 = (join‘𝐾)
13 3at.a . . . . . 6 𝐴 = (Atoms‘𝐾)
1411, 12, 133atlem3 36488 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑈 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
156, 7, 8, 9, 10, 14syl131anc 1377 . . . 4 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
16153expia 1115 . . 3 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
17 simp11 1197 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝐾 ∈ HL)
18 simp123 1301 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅𝐴)
19 simp122 1300 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑄𝐴)
20 simp121 1299 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃𝐴)
2118, 19, 203jca 1122 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → (𝑅𝐴𝑄𝐴𝑃𝐴))
22 simp131 1302 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑆𝐴)
23 simp132 1303 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑇𝐴)
2422, 23jca 512 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → (𝑆𝐴𝑇𝐴))
25 simp21 1200 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ¬ 𝑅 (𝑃 𝑄))
26 simp22 1201 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃𝑄)
2711, 12, 13hlatexch2 36399 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑅 𝑄) → 𝑅 (𝑃 𝑄)))
2817, 20, 18, 19, 26, 27syl131anc 1377 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → (𝑃 (𝑅 𝑄) → 𝑅 (𝑃 𝑄)))
2925, 28mtod 199 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ¬ 𝑃 (𝑅 𝑄))
3017hllatd 36367 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝐾 ∈ Lat)
31 eqid 2826 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3231, 13atbase 36292 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
3318, 32syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅 ∈ (Base‘𝐾))
3431, 13atbase 36292 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3520, 34syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃 ∈ (Base‘𝐾))
3631, 13atbase 36292 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3719, 36syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑄 ∈ (Base‘𝐾))
3831, 11, 12latnlej1r 17670 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝑄)
3930, 33, 35, 37, 25, 38syl131anc 1377 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅𝑄)
40 simp3 1132 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃))
4111, 12, 133atlem4 36489 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ (𝑆𝐴𝑇𝐴)) ∧ (¬ 𝑃 (𝑅 𝑄) ∧ 𝑅𝑄) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃))
4217, 21, 24, 29, 39, 40, 41syl321anc 1386 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃))
43423expia 1115 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃) → ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃)))
44 simpl1 1185 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝐾 ∈ HL)
4544hllatd 36367 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝐾 ∈ Lat)
46 simpl21 1245 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑃𝐴)
4746, 34syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑃 ∈ (Base‘𝐾))
48 simpl22 1246 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑄𝐴)
4948, 36syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑄 ∈ (Base‘𝐾))
50 simpl23 1247 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑅𝐴)
5150, 32syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑅 ∈ (Base‘𝐾))
5231, 12latj31 17699 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑄) 𝑃))
5345, 47, 49, 51, 52syl13anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑄) 𝑃))
5453breq1d 5073 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃) ↔ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)))
5553eqeq1d 2828 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃) ↔ ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃)))
5643, 54, 553imtr4d 295 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃)))
575, 16, 56pm2.61ne 3107 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
58573impia 1111 1 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 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  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  cfv 6352  (class class class)co 7148  Basecbs 16473  lecple 16562  joincjn 17544  Latclat 17645  Atomscatm 36266  HLchlt 36353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354
This theorem is referenced by:  3atlem6  36491  3atlem7  36492
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