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Theorem dath 34040
 Description: Desargues' Theorem of projective geometry (proved for a Hilbert lattice). Assume each triple of atoms (points) 𝑃𝑄𝑅 and 𝑆𝑇𝑈 forms a triangle (i.e. determines a plane). Assume that lines 𝑃𝑆, 𝑄𝑇, and 𝑅𝑈 meet at a "center of perspectivity" 𝐶. (We also assume that 𝐶 is not on any of the 6 lines forming the two triangles.) Then the atoms 𝐷 = (𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇), 𝐸 = (𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈), 𝐹 = (𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆) are colinear, forming an "axis of perspectivity". Our proof roughly follows Theorem 2.7.1, p. 78 in Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press (1988). Unlike them, we don't assume 𝐶 is an atom to make this theorem slightly more general for easier future use. However, we prove that 𝐶 must be an atom in dalemcea 33964. For a visual demonstration, see the "Desargue's Theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html. The points I, J, and K there define the axis of perspectivity. See theorem dalaw 34190 for Desargues Law, which eliminates all of the preconditions on the atoms except for central perspectivity. This is Metamath 100 proof #87. (Contributed by NM, 20-Aug-2012.)
Hypotheses
Ref Expression
dath.b 𝐵 = (Base‘𝐾)
dath.l = (le‘𝐾)
dath.j = (join‘𝐾)
dath.a 𝐴 = (Atoms‘𝐾)
dath.m = (meet‘𝐾)
dath.o 𝑂 = (LPlanes‘𝐾)
dath.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dath.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
dath.f 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
Assertion
Ref Expression
dath ((((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑆 𝑇) 𝑈) ∈ 𝑂) ∧ ((¬ 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)))) → 𝐹 (𝐷 𝐸))

Proof of Theorem dath
StepHypRef Expression
1 dath.b . . . . . 6 𝐵 = (Base‘𝐾)
21eleq2i 2680 . . . . 5 (𝐶𝐵𝐶 ∈ (Base‘𝐾))
32anbi2i 726 . . . 4 ((𝐾 ∈ HL ∧ 𝐶𝐵) ↔ (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
433anbi1i 1246 . . 3 (((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ↔ ((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)))
543anbi1i 1246 . 2 ((((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑆 𝑇) 𝑈) ∈ 𝑂) ∧ ((¬ 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)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑆 𝑇) 𝑈) ∈ 𝑂) ∧ ((¬ 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)))))
6 dath.l . 2 = (le‘𝐾)
7 dath.j . 2 = (join‘𝐾)
8 dath.a . 2 𝐴 = (Atoms‘𝐾)
9 dath.m . 2 = (meet‘𝐾)
10 dath.o . 2 𝑂 = (LPlanes‘𝐾)
11 eqid 2610 . 2 ((𝑃 𝑄) 𝑅) = ((𝑃 𝑄) 𝑅)
12 eqid 2610 . 2 ((𝑆 𝑇) 𝑈) = ((𝑆 𝑇) 𝑈)
13 dath.d . 2 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
14 dath.e . 2 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
15 dath.f . 2 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
165, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15dalem63 34039 1 ((((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑆 𝑇) 𝑈) ∈ 𝑂) ∧ ((¬ 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)))) → 𝐹 (𝐷 𝐸))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  HLchlt 33655  LPlanesclpl 33796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804 This theorem is referenced by:  dath2  34041
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