![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 39127. For a visual demonstration, see the "Desargues's theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html 39127. The points I, J, and K there define the axis of perspectivity. See Theorems dalaw 39353 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 2821 | . . . . 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 1155 | . . 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 1155 | . 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 2728 | . 2 β’ ((π β¨ π) β¨ π ) = ((π β¨ π) β¨ π ) | |
12 | eqid 2728 | . 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 39202 | 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 1085 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 (class class class)co 7414 Basecbs 17173 lecple 17233 joincjn 18296 meetcmee 18297 Atomscatm 38729 HLchlt 38816 LPlanesclpl 38959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 |
This theorem is referenced by: dath2 39204 |
Copyright terms: Public domain | W3C validator |