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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | dalemclrju 33201 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalem-clpjq 33202 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemceb 33203 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalempeb 33204 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemqeb 33205 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemreb 33206 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemseb 33207 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemteb 33208 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemueb 33209 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalempjqeb 33210 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemsjteb 33211 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemtjueb 33212 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemqrprot 33213 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemyeb 33214 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemcnes 33215 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalempnes 33216 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemqnet 33217 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalempjsen 33218 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemply 33219 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
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| Theorem | dalemsly 33220 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalemswapyz 33221 |
Lemma for dath 33301. Swap the role of planes and to allow
reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)
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| Theorem | dalemrot 33222 |
Lemma for dath 33301. Rotate triangles and
to allow reuse of analogous proofs. (Contributed by NM,
14-Aug-2012.)
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| Theorem | dalemrotyz 33223 |
Lemma for dath 33301. Rotate triangles and
to allow reuse of analogous proofs. (Contributed by NM,
19-Aug-2012.)
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| Theorem | dalem1 33224 |
Lemma for dath 33301. Show the lines and are different.
(Contributed by NM, 9-Aug-2012.)
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| Theorem | dalemcea 33225 |
Lemma for dath 33301. Frequently-used utility lemma. Here we
show that
must be an
atom. This is an assumption in most presentations of
Desargue's theorem; instead, we assume only the is a lattice
element, in order to make later substitutions for easier.
(Contributed by NM, 23-Sep-2012.)
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| Theorem | dalem2 33226 |
Lemma for dath 33301. Show the lines and form a plane.
(Contributed by NM, 11-Aug-2012.)
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| Theorem | dalemdea 33227 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.) |
![./\](_.wedge.gif "./\"){kind=small} 
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| Theorem | dalemeea 33228 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.) |
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| Theorem | dalem3 33229 | Lemma for dalemdnee 33231. (Contributed by NM, 10-Aug-2012.) |
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| Theorem | dalem4 33230 | Lemma for dalemdnee 33231. (Contributed by NM, 10-Aug-2012.) |
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| Theorem | dalemdnee 33231 |
Lemma for dath 33301. Axis of perspectivity points and are
different. (Contributed by NM, 10-Aug-2012.)
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| Theorem | dalem5 33232 |
Lemma for dath 33301. Atom (in plane ) belongs to
the 3-dimensional volume formed by and . (Contributed by
NM, 21-Jul-2012.)
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| Theorem | dalem6 33233 |
Lemma for dath 33301. Analogue of dalem5 33232 for . (Contributed by
NM, 21-Jul-2012.)
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| Theorem | dalem7 33234 |
Lemma for dath 33301. Analogue of dalem5 33232 for . (Contributed by
NM, 21-Jul-2012.)
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| Theorem | dalem8 33235 |
Lemma for dath 33301. Plane belongs to the 3-dimensional space.
(Contributed by NM, 21-Jul-2012.)
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| Theorem | dalem-cly 33236 |
Lemma for dalem9 33237. Center of perspectivity is not in plane
(when and are different planes). (Contributed by
NM, 13-Aug-2012.)
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| Theorem | dalem9 33237 |
Lemma for dath 33301. Since
, the join forms
a 3-dimensional space. (Contributed by NM, 20-Jul-2012.)
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| Theorem | dalem10 33238 |
Lemma for dath 33301. Atom belongs to the axis of perspectivity
 .
(Contributed by NM, 19-Jul-2012.)
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| Theorem | dalem11 33239 |
Lemma for dath 33301. Analogue of dalem10 33238 for . (Contributed by
NM, 23-Jul-2012.)
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| Theorem | dalem12 33240 |
Lemma for dath 33301. Analogue of dalem10 33238 for  . (Contributed by
NM, 11-Aug-2012.)
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| Theorem | dalem13 33241 | Lemma for dalem14 33242. (Contributed by NM, 21-Jul-2012.) |
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| Theorem | dalem14 33242 |
Lemma for dath 33301. Planes and form a 3-dimensional space
(when they are different). (Contributed by NM, 22-Jul-2012.)
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| Theorem | dalem15 33243 |
Lemma for dath 33301. The axis of perspectivity is a line.
(Contributed by NM, 21-Jul-2012.)
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| Theorem | dalem16 33244 |
Lemma for dath 33301. The atoms , , and form
a line of
perspectivity. This is Desargue's Theorem for the special case where
planes and
are different.
(Contributed by NM,
7-Aug-2012.)
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| Theorem | dalem17 33245 |
Lemma for dath 33301. When planes and are equal, the center
of perspectivity is in .
(Contributed by NM,
1-Aug-2012.)
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| Theorem | dalem18 33246* |
Lemma for dath 33301. Show that a dummy atom  exists outside of the
and planes (when those planes
are equal). This requires
that the projective space be 3-dimensional. (Desargue's theorem
doesn't always hold in 2 dimensions.) (Contributed by NM,
29-Jul-2012.)
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| Theorem | dalem19 33247* |
Lemma for dath 33301. Show that a second dummy atom  exists outside
of the and
planes (when those
planes are equal).
(Contributed by NM, 15-Aug-2012.)
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| Theorem | dalemccea 33248 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalemddea 33249 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalem-ccly 33250 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalem-ddly 33251 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalemccnedd 33252 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalemclccjdd 33253 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalemcceb 33254 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalemswapyzps 33255 |
Lemma for dath 33301. Swap the and planes, along with dummy
concurrency (center of perspectivity) atoms and , to allow
reuse of analogous proofs. (Contributed by NM, 17-Aug-2012.)
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| Theorem | dalemrotps 33256 |
Lemma for dath 33301. Rotate triangles and
to allow reuse of analogous proofs. (Contributed by NM,
15-Aug-2012.)
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| Theorem | dalemcjden 33257 | Lemma for dath 33301. Show that the dummy atoms form a line. (Contributed by NM, 15-Aug-2012.) |
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| Theorem | dalem20 33258* |
Lemma for dath 33301. Show that a second dummy atom exists outside
of the and
planes (when those
planes are equal).
(Contributed by NM, 14-Aug-2012.)
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| Theorem | dalem21 33259 |
Lemma for dath 33301. Show that lines and intersect at
an atom. (Contributed by NM, 2-Aug-2012.)
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| Theorem | dalem22 33260 |
Lemma for dath 33301. Show that lines and determine a
plane. (Contributed by NM, 2-Aug-2012.)
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| Theorem | dalem23 33261 |
Lemma for dath 33301. Show that auxiliary atom  is an atom.
(Contributed by NM, 2-Aug-2012.)
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| Theorem | dalem24 33262 |
Lemma for dath 33301. Show that auxiliary atom is outside of plane
.
(Contributed by NM, 2-Aug-2012.)
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| Theorem | dalem25 33263 |
Lemma for dath 33301. Show that the dummy center of
perspectivity
is different from auxiliary atom . (Contributed by NM,
3-Aug-2012.)
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| Theorem | dalem27 33264 |
Lemma for dath 33301. Show that the line intersects the dummy
center of perspectivity . (Contributed by NM, 8-Aug-2012.)
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| Theorem | dalem28 33265 | Lemma for dath 33301. Lemma dalem27 33264 expressed differently. (Contributed by NM, 4-Aug-2012.) |
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| Theorem | dalem29 33266 |
Lemma for dath 33301. Analogue of dalem23 33261 for  . (Contributed by
NM, 2-Aug-2012.)
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| Theorem | dalem30 33267 |
Lemma for dath 33301. Analogue of dalem24 33262 for . (Contributed by
NM, 3-Aug-2012.)
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| Theorem | dalem31N 33268 |
Lemma for dath 33301. Analogue of dalem25 33263 for . (Contributed by
NM, 4-Aug-2012.) (New usage is discouraged.)
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| Theorem | dalem32 33269 |
Lemma for dath 33301. Analogue of dalem27 33264 for . (Contributed by
NM, 8-Aug-2012.)
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| Theorem | dalem33 33270 |
Lemma for dath 33301. Analogue of dalem28 33265 for . (Contributed by
NM, 4-Aug-2012.)
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| Theorem | dalem34 33271 |
Lemma for dath 33301. Analogue of dalem23 33261 for  . (Contributed by
NM, 2-Aug-2012.)
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| Theorem | dalem35 33272 |
Lemma for dath 33301. Analogue of dalem24 33262 for . (Contributed by
NM, 3-Aug-2012.)
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| Theorem | dalem36 33273 |
Lemma for dath 33301. Analogue of dalem27 33264 for . (Contributed by
NM, 8-Aug-2012.)
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| Theorem | dalem37 33274 |
Lemma for dath 33301. Analogue of dalem28 33265 for . (Contributed by
NM, 4-Aug-2012.)
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| Theorem | ||