Theorem dath 37032

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 36956.

For a visual demonstration, see the "Desargues's theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html 36956. The points I, J, and K there define the axis of perspectivity.

See theorem dalaw 37182 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 ∧ 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))

Proof of Theorem dath

Step	Hyp	Ref	Expression
1		dath.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2	1	eleq2i 2881	. . . . 5 ⊢ (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ (Base‘𝐾))
3	2	anbi2i 625	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵) ↔ (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
4	3	3anbi1i 1154	. . 3 ⊢ (((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)))
5	4	3anbi1i 1154	. 2 ⊢ ((((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)))))
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 2798	. 2 ⊢ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑄) ∨ 𝑅)
12		eqid 2798	. 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))
16	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	dalem63 37031	1 ⊢ ((((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))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36559  HLchlt 36646  LPlanesclpl 36788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647  df-llines 36794  df-lplanes 36795  df-lvols 36796

This theorem is referenced by:  dath2  37033