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Theorem dath 29925
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 29849.

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 30075 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 2347 . . . . 5  |-  ( C  e.  B  <->  C  e.  ( Base `  K )
)
32anbi2i 675 . . . 4  |-  ( ( K  e.  HL  /\  C  e.  B )  <->  ( K  e.  HL  /\  C  e.  ( Base `  K ) ) )
433anbi1i 1142 . . 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 1142 . 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 2283 . 2  |-  ( ( P  .\/  Q ) 
.\/  R )  =  ( ( P  .\/  Q )  .\/  R )
12 eqid 2283 . 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 29924 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 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  dath2  29926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689
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