Description: Desargues's 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 do not assume that 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 .
For a visual demonstration, see the "Desargues's theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html . The points I, J, and K there define the axis of perspectivity.
See Theorems dalaw for Desargues's 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)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dath.b | |- B = ( Base ` K ) |
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| dath.l | |- .<_ = ( le ` K ) |
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| dath.j | |- .\/ = ( join ` K ) |
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| dath.a | |- A = ( Atoms ` K ) |
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| dath.m | |- ./\ = ( meet ` K ) |
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| dath.o | |- O = ( LPlanes ` K ) |
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| dath.d | |- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) ) |
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| dath.e | |- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) ) |
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| dath.f | |- F = ( ( R .\/ P ) ./\ ( U .\/ S ) ) |
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| Assertion | 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 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dath.b | |- B = ( Base ` K ) |
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| 2 | dath.l | |- .<_ = ( le ` K ) |
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| 3 | dath.j | |- .\/ = ( join ` K ) |
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| 4 | dath.a | |- A = ( Atoms ` K ) |
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| 5 | dath.m | |- ./\ = ( meet ` K ) |
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| 6 | dath.o | |- O = ( LPlanes ` K ) |
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| 7 | dath.d | |- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) ) |
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| 8 | dath.e | |- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) ) |
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| 9 | dath.f | |- F = ( ( R .\/ P ) ./\ ( U .\/ S ) ) |
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| 10 | 1 | eleq2i | |- ( C e. B <-> C e. ( Base ` K ) ) |
| 11 | 10 | anbi2i | |- ( ( K e. HL /\ C e. B ) <-> ( K e. HL /\ C e. ( Base ` K ) ) ) |
| 12 | 11 | 3anbi1i | |- ( ( ( 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 ) ) ) |
| 13 | 12 | 3anbi1i | |- ( ( ( ( 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 ) ) ) ) ) |
| 14 | eqid | |- ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ Q ) .\/ R ) |
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| 15 | eqid | |- ( ( S .\/ T ) .\/ U ) = ( ( S .\/ T ) .\/ U ) |
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| 16 | 13 2 3 4 5 6 14 15 7 8 9 | dalem63 | |- ( ( ( ( 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 ) ) |