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Theorem 4atlem3 35385
Description: Lemma for 4at 35402. 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 1315 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ HL)
2 simpl1 1228 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
3 simpl21 1321 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝐴)
4 simpl22 1323 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆𝐴)
5 simpr 479 . . . . 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 2760 . . . . . 6 (LVols‘𝐾) = (LVols‘𝐾)
106, 7, 8, 9lvoli2 35370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾))
112, 3, 4, 5, 10syl121anc 1482 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾))
12 simpl23 1325 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑇𝐴)
13 simpl3l 1287 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑈𝐴)
14 simpl3r 1289 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑉𝐴)
156, 7, 8, 9lvolnle3at 35371 . . . 4 (((𝐾 ∈ HL ∧ (((𝑃 𝑄) 𝑅) 𝑆) ∈ (LVols‘𝐾)) ∧ (𝑇𝐴𝑈𝐴𝑉𝐴)) → ¬ (((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉))
161, 11, 12, 13, 14, 15syl23anc 1484 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ¬ (((𝑃 𝑄) 𝑅) 𝑆) ((𝑇 𝑈) 𝑉))
17 hllat 35153 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
181, 17syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ Lat)
19 eqid 2760 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2019, 7, 8hlatjcl 35156 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
212, 20syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃 𝑄) ∈ (Base‘𝐾))
2219, 7, 8hlatjcl 35156 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
231, 3, 4, 22syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑅 𝑆) ∈ (Base‘𝐾))
2419, 7, 8hlatjcl 35156 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑈𝐴) → (𝑇 𝑈) ∈ (Base‘𝐾))
251, 12, 13, 24syl3anc 1477 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑇 𝑈) ∈ (Base‘𝐾))
2619, 8atbase 35079 . . . . . . 7 (𝑉𝐴𝑉 ∈ (Base‘𝐾))
2714, 26syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑉 ∈ (Base‘𝐾))
2819, 7latjcl 17252 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑇 𝑈) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))
2918, 25, 27, 28syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑇 𝑈) 𝑉) ∈ (Base‘𝐾))
3019, 6, 7latjle12 17263 . . . . 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)))
3118, 21, 23, 29, 30syl13anc 1479 . . . 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)))
32 simpl12 1317 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝐴)
3319, 8atbase 35079 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3432, 33syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃 ∈ (Base‘𝐾))
35 simpl13 1319 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄𝐴)
3619, 8atbase 35079 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3735, 36syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄 ∈ (Base‘𝐾))
3819, 6, 7latjle12 17263 . . . . . 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)))
3918, 34, 37, 29, 38syl13anc 1479 . . . . 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)))
4019, 8atbase 35079 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
413, 40syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅 ∈ (Base‘𝐾))
4219, 8atbase 35079 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
434, 42syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆 ∈ (Base‘𝐾))
4419, 6, 7latjle12 17263 . . . . . 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)))
4518, 41, 43, 29, 44syl13anc 1479 . . . . 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)))
4639, 45anbi12d 749 . . . 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))))
4719, 7latjass 17296 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
4818, 21, 41, 43, 47syl13anc 1479 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑈𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (((𝑃 𝑄) 𝑅) 𝑆) = ((𝑃 𝑄) (𝑅 𝑆)))
4948breq1d 4814 . . . 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)))
5031, 46, 493bitr4d 300 . . 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)))
5116, 50mtbird 314 . 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))))
52 ianor 510 . . 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))))
53 ianor 510 . . . 4 (¬ (𝑃 ((𝑇 𝑈) 𝑉) ∧ 𝑄 ((𝑇 𝑈) 𝑉)) ↔ (¬ 𝑃 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑄 ((𝑇 𝑈) 𝑉)))
54 ianor 510 . . . 4 (¬ (𝑅 ((𝑇 𝑈) 𝑉) ∧ 𝑆 ((𝑇 𝑈) 𝑉)) ↔ (¬ 𝑅 ((𝑇 𝑈) 𝑉) ∨ ¬ 𝑆 ((𝑇 𝑈) 𝑉)))
5553, 54orbi12i 544 . . 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))))
5652, 55bitri 264 . 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))))
5751, 56sylib 208 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 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  Latclat 17246  Atomscatm 35053  HLchlt 35140  LVolsclvol 35282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-lplanes 35288  df-lvols 35289
This theorem is referenced by:  4atlem3a  35386  4atlem12  35401
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