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Theorem dath 36754
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 36678.

For a visual demonstration, see the "Desargues's theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html 36678. The points I, J, and K there define the axis of perspectivity.

See theorem dalaw 36904 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 2904 . . . . 5 (𝐶𝐵𝐶 ∈ (Base‘𝐾))
32anbi2i 622 . . . 4 ((𝐾 ∈ HL ∧ 𝐶𝐵) ↔ (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
433anbi1i 1149 . . 3 (((𝐾 ∈ HL ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ↔ ((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)))
543anbi1i 1149 . 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 2821 . 2 ((𝑃 𝑄) 𝑅) = ((𝑃 𝑄) 𝑅)
12 eqid 2821 . 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 36753 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 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5058  cfv 6349  (class class class)co 7145  Basecbs 16473  lecple 16562  joincjn 17544  meetcmee 17545  Atomscatm 36281  HLchlt 36368  LPlanesclpl 36510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-p1 17640  df-lat 17646  df-clat 17708  df-oposet 36194  df-ol 36196  df-oml 36197  df-covers 36284  df-ats 36285  df-atl 36316  df-cvlat 36340  df-hlat 36369  df-llines 36516  df-lplanes 36517  df-lvols 36518
This theorem is referenced by:  dath2  36755
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