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 ) | |
| 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 | dath | ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ 𝑂 ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ 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 ) | |
| 2 | dath.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
| 3 | dath.j | ⊢ ∨ = ( join ‘ 𝐾 ) | |
| 4 | dath.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | |
| 5 | dath.m | ⊢ ∧ = ( meet ‘ 𝐾 ) | |
| 6 | dath.o | ⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) | |
| 7 | dath.d | ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) | |
| 8 | dath.e | ⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) ) | |
| 9 | dath.f | ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑈 ∨ 𝑆 ) ) | |
| 10 | 1 | eleq2i | ⊢ ( 𝐶 ∈ 𝐵 ↔ 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
| 11 | 10 | anbi2i | ⊢ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵 ) ↔ ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ) |
| 12 | 11 | 3anbi1i | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ↔ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ) |
| 13 | 12 | 3anbi1i | ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ 𝑂 ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ 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 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ 𝑂 ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ 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 ) | |
| 15 | eqid | ⊢ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) | |
| 16 | 13 2 3 4 5 6 14 15 7 8 9 | dalem63 | ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ 𝑂 ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ 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 ) ) |