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Theorem dalem48 36736
Description: Lemma for dath 36752. Analogue of dalem45 36733 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 36640 . . 3 (𝜑𝐾 ∈ Lat)
32adantr 481 . 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 36705 . . 3 (𝜓𝑐 ∈ (Base‘𝐾))
76adantl 482 . 2 ((𝜑𝜓) → 𝑐 ∈ (Base‘𝐾))
8 dalem.j . . . 4 = (join‘𝐾)
91, 8, 5dalempjqeb 36661 . . 3 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
109adantr 481 . 2 ((𝜑𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
111, 5dalemreb 36657 . . 3 (𝜑𝑅 ∈ (Base‘𝐾))
1211adantr 481 . 2 ((𝜑𝜓) → 𝑅 ∈ (Base‘𝐾))
134dalem-ccly 36701 . . . 4 (𝜓 → ¬ 𝑐 𝑌)
14 dalem44.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
1514breq2i 5065 . . . 4 (𝑐 𝑌𝑐 ((𝑃 𝑄) 𝑅))
1613, 15sylnib 329 . . 3 (𝜓 → ¬ 𝑐 ((𝑃 𝑄) 𝑅))
1716adantl 482 . 2 ((𝜑𝜓) → ¬ 𝑐 ((𝑃 𝑄) 𝑅))
18 eqid 2818 . . 3 (Base‘𝐾) = (Base‘𝐾)
19 dalem.l . . 3 = (le‘𝐾)
2018, 19, 8latnlej2l 17670 . 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 1375 1 ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  Latclat 17643  Atomscatm 36279  HLchlt 36366  LPlanesclpl 36508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-poset 17544  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-lat 17644  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367
This theorem is referenced by:  dalem49  36737  dalem51  36739  dalem52  36740
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