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Theorem dath 29092
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 29016.

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 29242 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 2322 . . . . 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 2258 . 2  |-  ( ( P  .\/  Q ) 
.\/  R )  =  ( ( P  .\/  Q )  .\/  R )
12 eqid 2258 . 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 29091 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 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Atomscatm 28620   HLchlt 28707   LPlanesclpl 28848
This theorem is referenced by:  dath2  29093
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856
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