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Theorem dalem5 33765
Description: Lemma for dath 33834. Atom 𝑈 (in plane 𝑍 = 𝑆𝑇𝑈) belongs to the 3-dimensional volume formed by 𝑌 and 𝐶. (Contributed by NM, 21-Jul-2012.)
Hypotheses
Ref Expression
dalema.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalemc.l = (le‘𝐾)
dalemc.j = (join‘𝐾)
dalemc.a 𝐴 = (Atoms‘𝐾)
dalem5.o 𝑂 = (LPlanes‘𝐾)
dalem5.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem5.w 𝑊 = (𝑌 𝐶)
Assertion
Ref Expression
dalem5 (𝜑𝑈 𝑊)

Proof of Theorem dalem5
StepHypRef Expression
1 eqid 2610 . 2 (Base‘𝐾) = (Base‘𝐾)
2 dalemc.l . 2 = (le‘𝐾)
3 dalema.ph . . 3 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
43dalemkelat 33722 . 2 (𝜑𝐾 ∈ Lat)
5 dalemc.a . . 3 𝐴 = (Atoms‘𝐾)
63, 5dalemueb 33742 . 2 (𝜑𝑈 ∈ (Base‘𝐾))
73dalemkehl 33721 . . 3 (𝜑𝐾 ∈ HL)
83dalemrea 33726 . . 3 (𝜑𝑅𝐴)
9 dalemc.j . . . 4 = (join‘𝐾)
10 dalem5.o . . . 4 𝑂 = (LPlanes‘𝐾)
11 dalem5.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
123, 2, 9, 5, 10, 11dalemcea 33758 . . 3 (𝜑𝐶𝐴)
131, 9, 5hlatjcl 33465 . . 3 ((𝐾 ∈ HL ∧ 𝑅𝐴𝐶𝐴) → (𝑅 𝐶) ∈ (Base‘𝐾))
147, 8, 12, 13syl3anc 1318 . 2 (𝜑 → (𝑅 𝐶) ∈ (Base‘𝐾))
15 dalem5.w . . 3 𝑊 = (𝑌 𝐶)
163, 10dalemyeb 33747 . . . 4 (𝜑𝑌 ∈ (Base‘𝐾))
173, 5dalemceb 33736 . . . 4 (𝜑𝐶 ∈ (Base‘𝐾))
181, 9latjcl 16823 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 𝐶) ∈ (Base‘𝐾))
194, 16, 17, 18syl3anc 1318 . . 3 (𝜑 → (𝑌 𝐶) ∈ (Base‘𝐾))
2015, 19syl5eqel 2692 . 2 (𝜑𝑊 ∈ (Base‘𝐾))
213dalemclrju 33734 . . 3 (𝜑𝐶 (𝑅 𝑈))
223dalemuea 33729 . . . 4 (𝜑𝑈𝐴)
233dalempea 33724 . . . . 5 (𝜑𝑃𝐴)
24 simp313 1203 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))) → ¬ 𝐶 (𝑅 𝑃))
253, 24sylbi 206 . . . . 5 (𝜑 → ¬ 𝐶 (𝑅 𝑃))
262, 9, 5atnlej1 33477 . . . . 5 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑅𝐴𝑃𝐴) ∧ ¬ 𝐶 (𝑅 𝑃)) → 𝐶𝑅)
277, 12, 8, 23, 25, 26syl131anc 1331 . . . 4 (𝜑𝐶𝑅)
282, 9, 5hlatexch1 33493 . . . 4 ((𝐾 ∈ HL ∧ (𝐶𝐴𝑈𝐴𝑅𝐴) ∧ 𝐶𝑅) → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
297, 12, 22, 8, 27, 28syl131anc 1331 . . 3 (𝜑 → (𝐶 (𝑅 𝑈) → 𝑈 (𝑅 𝐶)))
3021, 29mpd 15 . 2 (𝜑𝑈 (𝑅 𝐶))
313, 9, 5dalempjqeb 33743 . . . . . 6 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
323, 5dalemreb 33739 . . . . . 6 (𝜑𝑅 ∈ (Base‘𝐾))
331, 2, 9latlej2 16833 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ((𝑃 𝑄) 𝑅))
344, 31, 32, 33syl3anc 1318 . . . . 5 (𝜑𝑅 ((𝑃 𝑄) 𝑅))
3534, 11syl6breqr 4620 . . . 4 (𝜑𝑅 𝑌)
361, 2, 9latjlej1 16837 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾))) → (𝑅 𝑌 → (𝑅 𝐶) (𝑌 𝐶)))
374, 32, 16, 17, 36syl13anc 1320 . . . 4 (𝜑 → (𝑅 𝑌 → (𝑅 𝐶) (𝑌 𝐶)))
3835, 37mpd 15 . . 3 (𝜑 → (𝑅 𝐶) (𝑌 𝐶))
3938, 15syl6breqr 4620 . 2 (𝜑 → (𝑅 𝐶) 𝑊)
401, 2, 4, 6, 14, 20, 30, 39lattrd 16830 1 (𝜑𝑈 𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  Latclat 16817  Atomscatm 33362  HLchlt 33449  LPlanesclpl 33590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-lat 16818  df-clat 16880  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-llines 33596  df-lplanes 33597
This theorem is referenced by:  dalem6  33766  dalem8  33768
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