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Mirrors > Home > MPE Home > Th. List > Mathboxes > dath | Structured version Visualization version GIF version |
Description: Desargues's theorem of
projective geometry (proved for a Hilbert
lattice). Assume each triple of atoms (points) πππ
and πππ
forms a triangle (i.e. determines a plane). Assume that lines ππ,
ππ, and π
π meet at a "center of
perspectivity" πΆ. (We
also assume that πΆ is not on any of the 6 lines forming
the two
triangles.) Then the atoms π· = (π β¨ π) β§ (π β¨ π),
πΈ =
(π β¨ π
) β§ (π β¨ π),
πΉ =
(π
β¨ π) β§ (π β¨ π) 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 πΆ is an atom to make this theorem slightly more general for easier future use. However, we prove that πΆ must be an atom in dalemcea 39034. For a visual demonstration, see the "Desargues's theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html 39034. The points I, J, and K there define the axis of perspectivity. See Theorems dalaw 39260 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 |
---|---|
dath.b | β’ π΅ = (BaseβπΎ) |
dath.l | β’ β€ = (leβπΎ) |
dath.j | β’ β¨ = (joinβπΎ) |
dath.a | β’ π΄ = (AtomsβπΎ) |
dath.m | β’ β§ = (meetβπΎ) |
dath.o | β’ π = (LPlanesβπΎ) |
dath.d | β’ π· = ((π β¨ π) β§ (π β¨ π)) |
dath.e | β’ πΈ = ((π β¨ π ) β§ (π β¨ π)) |
dath.f | β’ πΉ = ((π β¨ π) β§ (π β¨ π)) |
Ref | Expression |
---|---|
dath | β’ ((((πΎ β HL β§ πΆ β π΅) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (((π β¨ π) β¨ π ) β π β§ ((π β¨ π) β¨ π) β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π)))) β πΉ β€ (π· β¨ πΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dath.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
2 | 1 | eleq2i 2817 | . . . . 5 β’ (πΆ β π΅ β πΆ β (BaseβπΎ)) |
3 | 2 | anbi2i 622 | . . . 4 β’ ((πΎ β HL β§ πΆ β π΅) β (πΎ β HL β§ πΆ β (BaseβπΎ))) |
4 | 3 | 3anbi1i 1154 | . . 3 β’ (((πΎ β HL β§ πΆ β π΅) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β ((πΎ β HL β§ πΆ β (BaseβπΎ)) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄))) |
5 | 4 | 3anbi1i 1154 | . 2 β’ ((((πΎ β HL β§ πΆ β π΅) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (((π β¨ π) β¨ π ) β π β§ ((π β¨ π) β¨ π) β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π)))) β (((πΎ β HL β§ πΆ β (BaseβπΎ)) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (((π β¨ π) β¨ π ) β π β§ ((π β¨ π) β¨ π) β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π))))) |
6 | dath.l | . 2 β’ β€ = (leβπΎ) | |
7 | dath.j | . 2 β’ β¨ = (joinβπΎ) | |
8 | dath.a | . 2 β’ π΄ = (AtomsβπΎ) | |
9 | dath.m | . 2 β’ β§ = (meetβπΎ) | |
10 | dath.o | . 2 β’ π = (LPlanesβπΎ) | |
11 | eqid 2724 | . 2 β’ ((π β¨ π) β¨ π ) = ((π β¨ π) β¨ π ) | |
12 | eqid 2724 | . 2 β’ ((π β¨ π) β¨ π) = ((π β¨ π) β¨ π) | |
13 | dath.d | . 2 β’ π· = ((π β¨ π) β§ (π β¨ π)) | |
14 | dath.e | . 2 β’ πΈ = ((π β¨ π ) β§ (π β¨ π)) | |
15 | dath.f | . 2 β’ πΉ = ((π β¨ π) β§ (π β¨ π)) | |
16 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | dalem63 39109 | 1 β’ ((((πΎ β HL β§ πΆ β π΅) β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ (((π β¨ π) β¨ π ) β π β§ ((π β¨ π) β¨ π) β π) β§ ((Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π ) β§ Β¬ πΆ β€ (π β¨ π)) β§ (Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π) β§ Β¬ πΆ β€ (π β¨ π)) β§ (πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π) β§ πΆ β€ (π β¨ π)))) β πΉ β€ (π· β¨ πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 (class class class)co 7402 Basecbs 17149 lecple 17209 joincjn 18272 meetcmee 18273 Atomscatm 38636 HLchlt 38723 LPlanesclpl 38866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-oposet 38549 df-ol 38551 df-oml 38552 df-covers 38639 df-ats 38640 df-atl 38671 df-cvlat 38695 df-hlat 38724 df-llines 38872 df-lplanes 38873 df-lvols 38874 |
This theorem is referenced by: dath2 39111 |
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