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Theorem dalem27 35773
 Description: Lemma for dath 35810. 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 35698 . . . . 5 (𝜑𝐾 ∈ Lat)
433ad2ant1 1167 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
52dalemkehl 35697 . . . . . 6 (𝜑𝐾 ∈ HL)
653ad2ant1 1167 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
7 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemccea 35757 . . . . . 6 (𝜓𝑐𝐴)
983ad2ant3 1169 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
102dalempea 35700 . . . . . 6 (𝜑𝑃𝐴)
11103ad2ant1 1167 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
12 eqid 2825 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
13 dalem.j . . . . . 6 = (join‘𝐾)
14 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
1512, 13, 14hlatjcl 35441 . . . . 5 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
166, 9, 11, 15syl3anc 1494 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
177dalemddea 35758 . . . . . 6 (𝜓𝑑𝐴)
18173ad2ant3 1169 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
192dalemsea 35703 . . . . . 6 (𝜑𝑆𝐴)
20193ad2ant1 1167 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2112, 13, 14hlatjcl 35441 . . . . 5 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
226, 18, 20, 21syl3anc 1494 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
23 dalem.l . . . . 5 = (le‘𝐾)
24 dalem23.m . . . . 5 = (meet‘𝐾)
2512, 23, 24latmle1 17436 . . . 4 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑐 𝑃))
264, 16, 22, 25syl3anc 1494 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑐 𝑃))
271, 26syl5eqbr 4910 . 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 35770 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
322, 23, 13, 14, 28, 29dalemply 35728 . . . . 5 (𝜑𝑃 𝑌)
33323ad2ant1 1167 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
342, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem24 35771 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
35 nbrne2 4895 . . . . 5 ((𝑃 𝑌 ∧ ¬ 𝐺 𝑌) → 𝑃𝐺)
3635necomd 3054 . . . 4 ((𝑃 𝑌 ∧ ¬ 𝐺 𝑌) → 𝐺𝑃)
3733, 34, 36syl2anc 579 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝑃)
3823, 13, 14hlatexch2 35470 . . 3 ((𝐾 ∈ HL ∧ (𝐺𝐴𝑐𝐴𝑃𝐴) ∧ 𝐺𝑃) → (𝐺 (𝑐 𝑃) → 𝑐 (𝐺 𝑃)))
396, 31, 9, 11, 37, 38syl131anc 1506 . 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 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999   class class class wbr 4875  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  meetcmee 17305  Latclat 17405  Atomscatm 35337  HLchlt 35424  LPlanesclpl 35566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-lat 17406  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-llines 35572  df-lplanes 35573 This theorem is referenced by:  dalem28  35774  dalem32  35778  dalem51  35797  dalem52  35798
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