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Theorem dath 35899
Description: Desargues's theorem of projective geometry (proved for a Hilbert lattice). Assume each triple of atoms (points) 𝑃𝑄𝑅 and 𝑆𝑇𝑈 forms a triangle (i.e. determines a plane). Assume that lines 𝑃𝑆, 𝑄𝑇, and 𝑅𝑈 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 35823.

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 theorem dalaw 36049 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.)

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 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((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))

Proof of Theorem dath
StepHypRef Expression
1 dath.b . . . . . 6 𝐵 = (Base‘𝐾)
21eleq2i 2851 . . . . 5 (𝐶𝐵𝐶 ∈ (Base‘𝐾))
32anbi2i 616 . . . 4 ((𝐾 ∈ HL ∧ 𝐶𝐵) ↔ (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
433anbi1i 1157 . . 3 (((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ↔ ((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)))
543anbi1i 1157 . 2 ((((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((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)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((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)))))
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 2778 . 2 ((𝑃 𝑄) 𝑅) = ((𝑃 𝑄) 𝑅)
12 eqid 2778 . 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 35898 1 ((((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((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))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107   class class class wbr 4888  cfv 6137  (class class class)co 6924  Basecbs 16266  lecple 16356  joincjn 17341  meetcmee 17342  Atomscatm 35426  HLchlt 35513  LPlanesclpl 35655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-proset 17325  df-poset 17343  df-plt 17355  df-lub 17371  df-glb 17372  df-join 17373  df-meet 17374  df-p0 17436  df-p1 17437  df-lat 17443  df-clat 17505  df-oposet 35339  df-ol 35341  df-oml 35342  df-covers 35429  df-ats 35430  df-atl 35461  df-cvlat 35485  df-hlat 35514  df-llines 35661  df-lplanes 35662  df-lvols 35663
This theorem is referenced by:  dath2  35900
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