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Theorem 4atlem3 36618
Description: Lemma for 4at 36635. 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 1242 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ HL)
2 simpl1 1185 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
3 simpl21 1245 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝐴)
4 simpl22 1246 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆𝐴)
5 simpr 485 . . . . 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 2826 . . . . . 6 (LVols‘𝐾) = (LVols‘𝐾)
106, 7, 8, 9lvoli2 36603 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾))
112, 3, 4, 5, 10syl121anc 1369 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾))
12 simpl23 1247 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑇𝐴)
13 simpl3l 1222 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑈𝐴)
14 simpl3r 1223 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑉𝐴)
156, 7, 8, 9lvolnle3at 36604 . . . 4 (((𝐾 ∈ HL ∧ (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾)) ∧ (𝑇𝐴𝑈𝐴𝑉𝐴)) → ¬ (((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉))
161, 11, 12, 13, 14, 15syl23anc 1371 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ¬ (((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉))
171hllatd 36386 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ Lat)
18 eqid 2826 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 36389 . . . . . 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 36389 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
221, 3, 4, 21syl3anc 1365 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑅 𝑆) ∈ (Base‘𝐾))
2318, 7, 8hlatjcl 36389 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
241, 12, 13, 23syl3anc 1365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑇 𝑈) ∈ (Base‘𝐾))
2518, 8atbase 36311 . . . . . . 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 17656 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))
2817, 24, 26, 27syl3anc 1365 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))
2918, 6, 7latjle12 17667 . . . . 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 1366 . . . 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 1243 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝐴)
3218, 8atbase 36311 . . . . . . 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 1244 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄𝐴)
3518, 8atbase 36311 . . . . . . 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 17667 . . . . . 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 1366 . . . . 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 36311 . . . . . . 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 36311 . . . . . . 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 17667 . . . . . 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 1366 . . . . 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 17700 . . . . . 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 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
4847breq1d 5073 . . . 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 312 . . 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 326 . 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 977 . . 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 977 . . . 4 (¬ (𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ↔ (¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)))
53 ianor 977 . . . 4 (¬ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉)) ↔ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉)))
5452, 53orbi12i 910 . . 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 276 . 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 219 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 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  cfv 6354  (class class class)co 7150  Basecbs 16478  lecple 16567  joincjn 17549  Latclat 17650  Atomscatm 36285  HLchlt 36372  LVolsclvol 36515
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 7455
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 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-llines 36520  df-lplanes 36521  df-lvols 36522
This theorem is referenced by:  4atlem3a  36619  4atlem12  36634
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