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Theorem dalem27 33802
Description: Lemma for dath 33839. 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 33727 . . . . 5 (𝜑𝐾 ∈ Lat)
433ad2ant1 1074 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
52dalemkehl 33726 . . . . . 6 (𝜑𝐾 ∈ HL)
653ad2ant1 1074 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
7 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemccea 33786 . . . . . 6 (𝜓𝑐𝐴)
983ad2ant3 1076 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
102dalempea 33729 . . . . . 6 (𝜑𝑃𝐴)
11103ad2ant1 1074 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
12 eqid 2605 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
13 dalem.j . . . . . 6 = (join‘𝐾)
14 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
1512, 13, 14hlatjcl 33470 . . . . 5 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
166, 9, 11, 15syl3anc 1317 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
177dalemddea 33787 . . . . . 6 (𝜓𝑑𝐴)
18173ad2ant3 1076 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
192dalemsea 33732 . . . . . 6 (𝜑𝑆𝐴)
20193ad2ant1 1074 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2112, 13, 14hlatjcl 33470 . . . . 5 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
226, 18, 20, 21syl3anc 1317 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
23 dalem.l . . . . 5 = (le‘𝐾)
24 dalem23.m . . . . 5 = (meet‘𝐾)
2512, 23, 24latmle1 16841 . . . 4 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑐 𝑃))
264, 16, 22, 25syl3anc 1317 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑐 𝑃))
271, 26syl5eqbr 4608 . 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 33799 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
322, 23, 13, 14, 28, 29dalemply 33757 . . . . 5 (𝜑𝑃 𝑌)
33323ad2ant1 1074 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝑃 𝑌)
342, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem24 33800 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝐺 𝑌)
35 nbrne2 4593 . . . . 5 ((𝑃 𝑌 ∧ ¬ 𝐺 𝑌) → 𝑃𝐺)
3635necomd 2832 . . . 4 ((𝑃 𝑌 ∧ ¬ 𝐺 𝑌) → 𝐺𝑃)
3733, 34, 36syl2anc 690 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝑃)
3823, 13, 14hlatexch2 33499 . . 3 ((𝐾 ∈ HL ∧ (𝐺𝐴𝑐𝐴𝑃𝐴) ∧ 𝐺𝑃) → (𝐺 (𝑐 𝑃) → 𝑐 (𝐺 𝑃)))
396, 31, 9, 11, 37, 38syl131anc 1330 . 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  Latclat 16810  Atomscatm 33367  HLchlt 33454  LPlanesclpl 33595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-llines 33601  df-lplanes 33602
This theorem is referenced by:  dalem28  33803  dalem32  33807  dalem51  33826  dalem52  33827
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