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Theorem dath 28614
 Description: Desargues' Theorem of projective geometry (proved for a Hilbert lattice). Assume each triple of atoms (points) and forms a triangle (i.e. determines a plane). Assume that lines , , and meet at a "center of perspectivity" . (We also assume that is not on any of the 6 lines forming the two triangles.) Then the atoms , , 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 is an atom to make this theorem slightly more general for easier future use. However, we prove that must be an atom in dalemcea 28538. 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 28764 for Desargues Law, which eliminates all of the preconditions on the atoms except for central perspectivity. (Contributed by NM, 20-Aug-2012.)
Hypotheses
Ref Expression
dath.b
dath.l
dath.j
dath.a
dath.m
dath.o
dath.d
dath.e
dath.f
Assertion
Ref Expression
dath

Proof of Theorem dath
StepHypRef Expression
1 dath.b . . . . . 6
21eleq2i 2317 . . . . 5
32anbi2i 678 . . . 4
433anbi1i 1147 . . 3
543anbi1i 1147 . 2
6 dath.l . 2
7 dath.j . 2
8 dath.a . 2
9 dath.m . 2
10 dath.o . 2
11 eqid 2253 . 2
12 eqid 2253 . 2
13 dath.d . 2
14 dath.e . 2
15 dath.f . 2
165, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15dalem63 28613 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wa 360   w3a 939   wceq 1619   wcel 1621   class class class wbr 3920  cfv 4592  (class class class)co 5710  cbs 13022  cple 13089  cjn 13922  cmee 13923  catm 28142  chlt 28229  clpl 28370 This theorem is referenced by:  dath2  28615 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378
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