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Theorem 4atlem3 39196
Description: Lemma for 4at 39213. Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012.)
Hypotheses
Ref Expression
4at.l = (le‘𝐾)
4at.j = (join‘𝐾)
4at.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
4atlem3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)) ∨ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉))))

Proof of Theorem 4atlem3
StepHypRef Expression
1 simpl11 1245 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ HL)
2 simpl1 1188 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
3 simpl21 1248 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝐴)
4 simpl22 1249 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆𝐴)
5 simpr 483 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)))
6 4at.l . . . . . 6 = (le‘𝐾)
7 4at.j . . . . . 6 = (join‘𝐾)
8 4at.a . . . . . 6 𝐴 = (Atoms‘𝐾)
9 eqid 2725 . . . . . 6 (LVols‘𝐾) = (LVols‘𝐾)
106, 7, 8, 9lvoli2 39181 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾))
112, 3, 4, 5, 10syl121anc 1372 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾))
12 simpl23 1250 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑇𝐴)
13 simpl3l 1225 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑈𝐴)
14 simpl3r 1226 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑉𝐴)
156, 7, 8, 9lvolnle3at 39182 . . . 4 (((𝐾 ∈ HL ∧ (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾)) ∧ (𝑇𝐴𝑈𝐴𝑉𝐴)) → ¬ (((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉))
161, 11, 12, 13, 14, 15syl23anc 1374 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ¬ (((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉))
171hllatd 38963 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ Lat)
18 eqid 2725 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 38966 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
202, 19syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃 𝑄) ∈ (Base‘𝐾))
2118, 7, 8hlatjcl 38966 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
221, 3, 4, 21syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑅 𝑆) ∈ (Base‘𝐾))
2318, 7, 8hlatjcl 38966 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
241, 12, 13, 23syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑇 𝑈) ∈ (Base‘𝐾))
2518, 8atbase 38888 . . . . . . 7 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
2614, 25syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑉 ∈ (Base‘𝐾))
2718, 7latjcl 18434 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))
2817, 24, 26, 27syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))
2918, 6, 7latjle12 18445 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾) ∧ ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))) → (((𝑃 𝑄) ((𝑇 𝑈) 𝑉) ∧ (𝑅 𝑆) ((𝑇 𝑈) 𝑉)) ↔ ((𝑃 𝑄) (𝑅 𝑆)) ((𝑇 𝑈) 𝑉)))
3017, 20, 22, 28, 29syl13anc 1369 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) ((𝑇 𝑈) 𝑉) ∧ (𝑅 𝑆) ((𝑇 𝑈) 𝑉)) ↔ ((𝑃 𝑄) (𝑅 𝑆)) ((𝑇 𝑈) 𝑉)))
31 simpl12 1246 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝐴)
3218, 8atbase 38888 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3331, 32syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃 ∈ (Base‘𝐾))
34 simpl13 1247 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄𝐴)
3518, 8atbase 38888 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3634, 35syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄 ∈ (Base‘𝐾))
3718, 6, 7latjle12 18445 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))) → ((𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ↔ (𝑃 𝑄) ((𝑇 𝑈) 𝑉)))
3817, 33, 36, 28, 37syl13anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ↔ (𝑃 𝑄) ((𝑇 𝑈) 𝑉)))
3918, 8atbase 38888 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
403, 39syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅 ∈ (Base‘𝐾))
4118, 8atbase 38888 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
424, 41syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆 ∈ (Base‘𝐾))
4318, 6, 7latjle12 18445 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))) → ((𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉)) ↔ (𝑅 𝑆) ((𝑇 𝑈) 𝑉)))
4417, 40, 42, 28, 43syl13anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉)) ↔ (𝑅 𝑆) ((𝑇 𝑈) 𝑉)))
4538, 44anbi12d 630 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ∧ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉))) ↔ ((𝑃 𝑄) ((𝑇 𝑈) 𝑉) ∧ (𝑅 𝑆) ((𝑇 𝑈) 𝑉))))
4618, 7latjass 18478 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
4717, 20, 40, 42, 46syl13anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
4847breq1d 5159 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉) ↔ ((𝑃 𝑄) (𝑅 𝑆)) ((𝑇 𝑈) 𝑉)))
4930, 45, 483bitr4d 310 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ∧ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉))) ↔ (((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉)))
5016, 49mtbird 324 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ¬ ((𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ∧ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉))))
51 ianor 979 . . 3 (¬ ((𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ∧ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉))) ↔ (¬ (𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ∨ ¬ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉))))
52 ianor 979 . . . 4 (¬ (𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ↔ (¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)))
53 ianor 979 . . . 4 (¬ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉)) ↔ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉)))
5452, 53orbi12i 912 . . 3 ((¬ (𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ∨ ¬ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉))) ↔ ((¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)) ∨ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉))))
5551, 54bitri 274 . 2 (¬ ((𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ∧ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉))) ↔ ((¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)) ∨ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉))))
5650, 55sylib 217 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)) ∨ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2929   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  Latclat 18426  Atomscatm 38862  HLchlt 38949  LVolsclvol 39093
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-lat 18427  df-clat 18494  df-oposet 38775  df-ol 38777  df-oml 38778  df-covers 38865  df-ats 38866  df-atl 38897  df-cvlat 38921  df-hlat 38950  df-llines 39098  df-lplanes 39099  df-lvols 39100
This theorem is referenced by:  4atlem3a  39197  4atlem12  39212
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