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Theorem 3atlem5 33574
Description: Lemma for 3at 33577. (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 6534 . . . . . 6 (𝑈 = 𝑃 → ((𝑆 𝑇) 𝑈) = ((𝑆 𝑇) 𝑃))
21eqcoms 2617 . . . . 5 (𝑃 = 𝑈 → ((𝑆 𝑇) 𝑈) = ((𝑆 𝑇) 𝑃))
32breq2d 4589 . . . 4 (𝑃 = 𝑈 → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃)))
42eqeq2d 2619 . . . 4 (𝑃 = 𝑈 → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃)))
53, 4imbi12d 332 . . 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 1077 . . . . 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 1149 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑅 (𝑃 𝑄))
8 simp2 1054 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝑈)
9 simp1r3 1151 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑄 (𝑃 𝑈))
10 simp3 1055 . . . . 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 33572 . . . . 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 1330 . . . 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 1258 . . 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 1083 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝐾 ∈ HL)
18 simp123 1187 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅𝐴)
19 simp122 1186 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑄𝐴)
20 simp121 1185 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃𝐴)
2118, 19, 203jca 1234 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → (𝑅𝐴𝑄𝐴𝑃𝐴))
22 simp131 1188 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑆𝐴)
23 simp132 1189 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑇𝐴)
2422, 23jca 552 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → (𝑆𝐴𝑇𝐴))
25 simp21 1086 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ¬ 𝑅 (𝑃 𝑄))
26 simp22 1087 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃𝑄)
2711, 12, 13hlatexch2 33483 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑅 𝑄) → 𝑅 (𝑃 𝑄)))
2817, 20, 18, 19, 26, 27syl131anc 1330 . . . . . . 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 187 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ¬ 𝑃 (𝑅 𝑄))
30 hllat 33451 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
3117, 30syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝐾 ∈ Lat)
32 eqid 2609 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3332, 13atbase 33377 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
3418, 33syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅 ∈ (Base‘𝐾))
3532, 13atbase 33377 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3620, 35syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃 ∈ (Base‘𝐾))
3732, 13atbase 33377 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3819, 37syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑄 ∈ (Base‘𝐾))
3932, 11, 12latnlej1r 16841 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝑄)
4031, 34, 36, 38, 25, 39syl131anc 1330 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅𝑄)
41 simp3 1055 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃))
4211, 12, 133atlem4 33573 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑅𝐴𝑄𝐴𝑃𝐴) ∧ (𝑆𝐴𝑇𝐴)) ∧ (¬ 𝑃 (𝑅 𝑄) ∧ 𝑅𝑄) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃))
4317, 21, 24, 29, 40, 41, 42syl321anc 1339 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃))
44433expia 1258 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃) → ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃)))
45 simpl1 1056 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝐾 ∈ HL)
4645, 30syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝐾 ∈ Lat)
47 simpl21 1131 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑃𝐴)
4847, 35syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑃 ∈ (Base‘𝐾))
49 simpl22 1132 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑄𝐴)
5049, 37syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑄 ∈ (Base‘𝐾))
51 simpl23 1133 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑅𝐴)
5251, 33syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝑅 ∈ (Base‘𝐾))
5332, 12latj31 16870 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑄) 𝑃))
5446, 48, 50, 52, 53syl13anc 1319 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑄) 𝑃))
5554breq1d 4587 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃) ↔ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)))
5654eqeq1d 2611 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃) ↔ ((𝑅 𝑄) 𝑃) = ((𝑆 𝑇) 𝑃)))
5744, 55, 563imtr4d 281 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃)))
585, 16, 57pm2.61ne 2866 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
59583impia 1252 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 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5789  (class class class)co 6526  Basecbs 15643  lecple 15723  joincjn 16715  Latclat 16816  Atomscatm 33351  HLchlt 33438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-preset 16699  df-poset 16717  df-plt 16729  df-lub 16745  df-glb 16746  df-join 16747  df-meet 16748  df-p0 16810  df-lat 16817  df-covers 33354  df-ats 33355  df-atl 33386  df-cvlat 33410  df-hlat 33439
This theorem is referenced by:  3atlem6  33575  3atlem7  33576
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