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Theorem dalem27 39299
Description: Lemma for dath 39336. Show that the line 𝐺𝑃 intersects the dummy center of perspectivity 𝑐. (Contributed by NM, 8-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))))
dalem23.m = (meet‘𝐾)
dalem23.o 𝑂 = (LPlanes‘𝐾)
dalem23.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem23.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem23.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
Assertion
Ref Expression
dalem27 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))

Proof of Theorem dalem27
StepHypRef Expression
1 dalem23.g . . 3 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
2 dalem.ph . . . . . 6 (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)))))
32dalemkelat 39224 . . . . 5 (𝜑𝐾 ∈ Lat)
433ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
52dalemkehl 39223 . . . . . 6 (𝜑𝐾 ∈ HL)
653ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
7 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemccea 39283 . . . . . 6 (𝜓𝑐𝐴)
983ad2ant3 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
102dalempea 39226 . . . . . 6 (𝜑𝑃𝐴)
11103ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
12 eqid 2725 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
13 dalem.j . . . . . 6 = (join‘𝐾)
14 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
1512, 13, 14hlatjcl 38966 . . . . 5 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
166, 9, 11, 15syl3anc 1368 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
177dalemddea 39284 . . . . . 6 (𝜓𝑑𝐴)
18173ad2ant3 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
192dalemsea 39229 . . . . . 6 (𝜑𝑆𝐴)
20193ad2ant1 1130 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2112, 13, 14hlatjcl 38966 . . . . 5 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
226, 18, 20, 21syl3anc 1368 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
23 dalem.l . . . . 5 = (le‘𝐾)
24 dalem23.m . . . . 5 = (meet‘𝐾)
2512, 23, 24latmle1 18459 . . . 4 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑐 𝑃))
264, 16, 22, 25syl3anc 1368 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑐 𝑃))
271, 26eqbrtrid 5184 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑐 𝑃))
28 dalem23.o . . . 4 𝑂 = (LPlanes‘𝐾)
29 dalem23.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
30 dalem23.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
312, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem23 39296 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
322, 23, 13, 14, 28, 29dalemply 39254 . . . . 5 (𝜑𝑃 𝑌)
33323ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
342, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem24 39297 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
35 nbrne2 5169 . . . . 5 ((𝑃 𝑌 ∧ ¬ 𝐺 𝑌) → 𝑃𝐺)
3635necomd 2985 . . . 4 ((𝑃 𝑌 ∧ ¬ 𝐺 𝑌) → 𝐺𝑃)
3733, 34, 36syl2anc 582 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝑃)
3823, 13, 14hlatexch2 38996 . . 3 ((𝐾 ∈ HL ∧ (𝐺𝐴𝑐𝐴𝑃𝐴) ∧ 𝐺𝑃) → (𝐺 (𝑐 𝑃) → 𝑐 (𝐺 𝑃)))
396, 31, 9, 11, 37, 38syl131anc 1380 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 (𝑐 𝑃) → 𝑐 (𝐺 𝑃)))
4027, 39mpd 15 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 (𝐺 𝑃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  meetcmee 18307  Latclat 18426  Atomscatm 38862  HLchlt 38949  LPlanesclpl 39092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-lat 18427  df-clat 18494  df-oposet 38775  df-ol 38777  df-oml 38778  df-covers 38865  df-ats 38866  df-atl 38897  df-cvlat 38921  df-hlat 38950  df-llines 39098  df-lplanes 39099
This theorem is referenced by:  dalem28  39300  dalem32  39304  dalem51  39323  dalem52  39324
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