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Mirrors > Home > MPE Home > Th. List > Mathboxes > dath | Structured version Visualization version GIF version |
Description: Desargues' 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 don't assume 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 33964. For a visual demonstration, see the "Desargue'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 34190 for Desargues 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 |
---|---|
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)) |
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)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dath.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
2 | 1 | eleq2i 2680 | . . . . 5 ⊢ (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ (Base‘𝐾)) |
3 | 2 | anbi2i 726 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵) ↔ (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾))) |
4 | 3 | 3anbi1i 1246 | . . 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 1246 | . 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 2610 | . 2 ⊢ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
12 | eqid 2610 | . 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 34039 | 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 Atomscatm 33568 HLchlt 33655 LPlanesclpl 33796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 |
This theorem is referenced by: dath2 34041 |
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