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Theorem dath 33380
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 33304.

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 33530 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  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 2507 . . . . 5  |-  ( C  e.  B  <->  C  e.  ( Base `  K )
)
32anbi2i 694 . . . 4  |-  ( ( K  e.  HL  /\  C  e.  B )  <->  ( K  e.  HL  /\  C  e.  ( Base `  K ) ) )
433anbi1i 1178 . . 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 1178 . 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 2443 . 2  |-  ( ( P  .\/  Q ) 
.\/  R )  =  ( ( P  .\/  Q )  .\/  R )
12 eqid 2443 . 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 33379 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   Atomscatm 32908   HLchlt 32995   LPlanesclpl 33136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lplanes 33143  df-lvols 33144
This theorem is referenced by:  dath2  33381
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