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Theorem dath 30372
 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 30296. 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 30522 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 2499 . . . . 5
32anbi2i 676 . . . 4
433anbi1i 1144 . . 3
543anbi1i 1144 . 2
6 dath.l . 2
7 dath.j . 2
8 dath.a . 2
9 dath.m . 2
10 dath.o . 2
11 eqid 2435 . 2
12 eqid 2435 . 2
13 dath.d . 2
14 dath.e . 2
15 dath.f . 2
165, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15dalem63 30371 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725   class class class wbr 4204  cfv 5445  (class class class)co 6072  cbs 13457  cple 13524  cjn 14389  cmee 14390  catm 29900  chlt 29987  clpl 30128 This theorem is referenced by:  dath2  30373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136
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