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Theorem dalem48 39703
Description: Lemma for dath 39719. Analogue of dalem45 39700 for 𝑃𝑄. (Contributed by NM, 16-Aug-2012.)
Hypotheses
Ref Expression
dalem.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)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem44.m = (meet‘𝐾)
dalem44.o 𝑂 = (LPlanes‘𝐾)
dalem44.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem44.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem44.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem44.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem44.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
Assertion
Ref Expression
dalem48 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))

Proof of Theorem dalem48
StepHypRef Expression
1 dalem.ph . . . 4 (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)))))
21dalemkelat 39607 . . 3 (𝜑𝐾 ∈ Lat)
32adantr 480 . 2 ((𝜑𝜓) → 𝐾 ∈ Lat)
4 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
5 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
64, 5dalemcceb 39672 . . 3 (𝜓𝑐 ∈ (Base‘𝐾))
76adantl 481 . 2 ((𝜑𝜓) → 𝑐 ∈ (Base‘𝐾))
8 dalem.j . . . 4 = (join‘𝐾)
91, 8, 5dalempjqeb 39628 . . 3 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
109adantr 480 . 2 ((𝜑𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
111, 5dalemreb 39624 . . 3 (𝜑𝑅 ∈ (Base‘𝐾))
1211adantr 480 . 2 ((𝜑𝜓) → 𝑅 ∈ (Base‘𝐾))
134dalem-ccly 39668 . . . 4 (𝜓 → ¬ 𝑐 𝑌)
14 dalem44.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
1514breq2i 5156 . . . 4 (𝑐 𝑌𝑐 ((𝑃 𝑄) 𝑅))
1613, 15sylnib 328 . . 3 (𝜓 → ¬ 𝑐 ((𝑃 𝑄) 𝑅))
1716adantl 481 . 2 ((𝜑𝜓) → ¬ 𝑐 ((𝑃 𝑄) 𝑅))
18 eqid 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
19 dalem.l . . 3 = (le‘𝐾)
2018, 19, 8latnlej2l 18518 . 2 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑐 ((𝑃 𝑄) 𝑅)) → ¬ 𝑐 (𝑃 𝑄))
213, 7, 10, 12, 17, 20syl131anc 1382 1 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  HLchlt 39332  LPlanesclpl 39475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333
This theorem is referenced by:  dalem49  39704  dalem51  39706  dalem52  39707
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