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Theorem 4atlem3b 37258
Description: Lemma for 4at 37273. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)
Hypotheses
Ref Expression
4at.l = (le‘𝐾)
4at.j = (join‘𝐾)
4at.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
4atlem3b (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))

Proof of Theorem 4atlem3b
StepHypRef Expression
1 simp1 1137 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
2 simp21 1207 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑅𝐴)
3 simp22 1208 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑆𝐴)
42, 3jca 515 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑅𝐴𝑆𝐴))
5 simp13 1206 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄𝐴)
6 simp23 1209 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑉𝐴)
75, 6jca 515 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑄𝐴𝑉𝐴))
8 simp3 1139 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅)))
9 4at.l . . . . 5 = (le‘𝐾)
10 4at.j . . . . 5 = (join‘𝐾)
11 4at.a . . . . 5 𝐴 = (Atoms‘𝐾)
129, 10, 114atlem3a 37257 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑄𝐴𝑉𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑄 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))
131, 4, 7, 8, 12syl31anc 1374 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑄 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))
14 3orass 1091 . . 3 ((¬ 𝑄 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)) ↔ (¬ 𝑄 ((𝑃 𝑄) 𝑉) ∨ (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉))))
1513, 14sylib 221 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑄 ((𝑃 𝑄) 𝑉) ∨ (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉))))
16 simp11 1204 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ HL)
1716hllatd 37024 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝐾 ∈ Lat)
18 simp12 1205 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑃𝐴)
19 eqid 2739 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2019, 10, 11hlatjcl 37027 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑉𝐴) → (𝑃 𝑉) ∈ (Base‘𝐾))
2116, 18, 6, 20syl3anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (𝑃 𝑉) ∈ (Base‘𝐾))
2219, 11atbase 36949 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
235, 22syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄 ∈ (Base‘𝐾))
2419, 9, 10latlej2 17790 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑉) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑄 ((𝑃 𝑉) 𝑄))
2517, 21, 23, 24syl3anc 1372 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄 ((𝑃 𝑉) 𝑄))
2610, 11hlatj32 37032 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑉𝐴𝑄𝐴)) → ((𝑃 𝑉) 𝑄) = ((𝑃 𝑄) 𝑉))
2716, 18, 6, 5, 26syl13anc 1373 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((𝑃 𝑉) 𝑄) = ((𝑃 𝑄) 𝑉))
2825, 27breqtrd 5057 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → 𝑄 ((𝑃 𝑄) 𝑉))
29 biortn 937 . . 3 (𝑄 ((𝑃 𝑄) 𝑉) → ((¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)) ↔ (¬ 𝑄 ((𝑃 𝑄) 𝑉) ∨ (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))))
3028, 29syl 17 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → ((¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)) ↔ (¬ 𝑄 ((𝑃 𝑄) 𝑉) ∨ (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))))
3115, 30mpbird 260 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑉𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 ((𝑃 𝑄) 𝑅))) → (¬ 𝑅 ((𝑃 𝑄) 𝑉) ∨ ¬ 𝑆 ((𝑃 𝑄) 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  w3o 1087  w3a 1088   = wceq 1542  wcel 2114  wne 2935   class class class wbr 5031  cfv 6340  (class class class)co 7173  Basecbs 16589  lecple 16678  joincjn 17673  Latclat 17774  Atomscatm 36923  HLchlt 37010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7130  df-ov 7176  df-oprab 7177  df-proset 17657  df-poset 17675  df-plt 17687  df-lub 17703  df-glb 17704  df-join 17705  df-meet 17706  df-p0 17768  df-lat 17775  df-clat 17837  df-oposet 36836  df-ol 36838  df-oml 36839  df-covers 36926  df-ats 36927  df-atl 36958  df-cvlat 36982  df-hlat 37011  df-llines 37158  df-lplanes 37159  df-lvols 37160
This theorem is referenced by:  4atlem10  37266
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