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Theorem dath 28829
Description: Desargues' Theorem of projective geometry (proved for a Hilbert lattice). Assume each triple of atoms (points)  P Q R and  S T U forms a triangle (i.e. determines a plane). Assume that lines  P S,  Q T, and  R U 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 28753.

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 28979 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  |-  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  e.  HL  /\  C  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  (
( ( P  .\/  Q )  .\/  R )  e.  O  /\  (
( S  .\/  T
)  .\/  U )  e.  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 2317 . . . . 5  |-  ( C  e.  B  <->  C  e.  ( Base `  K )
)
32anbi2i 678 . . . 4  |-  ( ( K  e.  HL  /\  C  e.  B )  <->  ( K  e.  HL  /\  C  e.  ( Base `  K ) ) )
433anbi1i 1147 . . 3  |-  ( ( ( K  e.  HL  /\  C  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  <->  ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) ) )
543anbi1i 1147 . 2  |-  ( ( ( ( K  e.  HL  /\  C  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  (
( ( P  .\/  Q )  .\/  R )  e.  O  /\  (
( S  .\/  T
)  .\/  U )  e.  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  e.  HL  /\  C  e.  ( Base `  K ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( S  .\/  T )  .\/  U )  e.  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 2253 . 2  |-  ( ( P  .\/  Q ) 
.\/  R )  =  ( ( P  .\/  Q )  .\/  R )
12 eqid 2253 . 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 28828 1  |-  ( ( ( ( K  e.  HL  /\  C  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  (
( ( P  .\/  Q )  .\/  R )  e.  O  /\  (
( S  .\/  T
)  .\/  U )  e.  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 set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   HLchlt 28444   LPlanesclpl 28585
This theorem is referenced by:  dath2  28830
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 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593
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