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Theorem dath 33347
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 33271.

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 33497 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 2532 . . . . 5  |-  ( C  e.  B  <->  C  e.  ( Base `  K )
)
32anbi2i 705 . . . 4  |-  ( ( K  e.  HL  /\  C  e.  B )  <->  ( K  e.  HL  /\  C  e.  ( Base `  K ) ) )
433anbi1i 1205 . . 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 1205 . 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 2462 . 2  |-  ( ( P  .\/  Q ) 
.\/  R )  =  ( ( P  .\/  Q )  .\/  R )
12 eqid 2462 . 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 33346 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 375    /\ w3a 991    = wceq 1455    e. wcel 1898   class class class wbr 4418   ` cfv 5605  (class class class)co 6320   Basecbs 15176   lecple 15252   joincjn 16244   meetcmee 16245   Atomscatm 32875   HLchlt 32962   LPlanesclpl 33103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-p1 16341  df-lat 16347  df-clat 16409  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963  df-llines 33109  df-lplanes 33110  df-lvols 33111
This theorem is referenced by:  dath2  33348
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