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Theorem 3atlem5 38822
Description: Lemma for 3at 38825. (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 7420 . . . . . 6 (𝑈 = 𝑃 → ((𝑆 𝑇) 𝑈) = ((𝑆 𝑇) 𝑃))
21eqcoms 2739 . . . . 5 (𝑃 = 𝑈 → ((𝑆 𝑇) 𝑈) = ((𝑆 𝑇) 𝑃))
32breq2d 5160 . . . 4 (𝑃 = 𝑈 → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑃)))
42eqeq2d 2742 . . . 4 (𝑃 = 𝑈 → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑃)))
53, 4imbi12d 344 . . 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 1196 . . . . 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 1268 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑅 (𝑃 𝑄))
8 simp2 1136 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → 𝑃𝑈)
9 simp1r3 1270 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) ∧ 𝑃𝑈 ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ¬ 𝑄 (𝑃 𝑈))
10 simp3 1137 . . . . 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 38820 . . . . 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 1382 . . . 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 1120 . . 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 1202 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝐾 ∈ HL)
18 simp123 1306 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅𝐴)
19 simp122 1305 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑄𝐴)
20 simp121 1304 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃𝐴)
2118, 19, 203jca 1127 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → (𝑅𝐴𝑄𝐴𝑃𝐴))
22 simp131 1307 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑆𝐴)
23 simp132 1308 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑇𝐴)
2422, 23jca 511 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → (𝑆𝐴𝑇𝐴))
25 simp21 1205 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ¬ 𝑅 (𝑃 𝑄))
26 simp22 1206 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑃𝑄)
2711, 12, 13hlatexch2 38731 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 (𝑅 𝑄) → 𝑅 (𝑃 𝑄)))
2817, 20, 18, 19, 26, 27syl131anc 1382 . . . . . . 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 197 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → ¬ 𝑃 (𝑅 𝑄))
3017hllatd 38698 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝐾 ∈ Lat)
31 eqid 2731 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3231, 13atbase 38623 . . . . . . . 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 38623 . . . . . . . 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 38623 . . . . . . . 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 18421 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝑄)
3930, 33, 35, 37, 25, 38syl131anc 1382 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈)) ∧ ((𝑅 𝑄) 𝑃) ((𝑆 𝑇) 𝑃)) → 𝑅𝑄)
40 simp3 1137 . . . . . 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 38821 . . . . . 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 1391 . . . . 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 1120 . . . 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 1190 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝐾 ∈ HL)
4544hllatd 38698 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → 𝐾 ∈ Lat)
46 simpl21 1250 . . . . . . 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 1251 . . . . . . 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 1252 . . . . . . 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 18450 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑄) 𝑃))
5345, 47, 49, 51, 52syl13anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑈))) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑄) 𝑃))
5453breq1d 5158 . . . 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 2733 . . . 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 294 . . 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 3026 . 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 1116 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 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  Latclat 18394  Atomscatm 38597  HLchlt 38684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-covers 38600  df-ats 38601  df-atl 38632  df-cvlat 38656  df-hlat 38685
This theorem is referenced by:  3atlem6  38823  3atlem7  38824
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