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Type | Label | Description |
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Statement | ||
Theorem | dalemclrju 33201 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalem-clpjq 33202 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemceb 33203 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalempeb 33204 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemqeb 33205 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemreb 33206 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemseb 33207 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemteb 33208 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemueb 33209 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalempjqeb 33210 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemsjteb 33211 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemtjueb 33212 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemqrprot 33213 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemyeb 33214 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemcnes 33215 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalempnes 33216 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemqnet 33217 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalempjsen 33218 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemply 33219 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
Theorem | dalemsly 33220 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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.) |
Theorem | dalemrot 33222 | Lemma for dath 33301. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.) |
Theorem | dalemrotyz 33223 | Lemma for dath 33301. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 19-Aug-2012.) |
Theorem | dalem1 33224 | Lemma for dath 33301. Show the lines and are different. (Contributed by NM, 9-Aug-2012.) |
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.) |
Theorem | dalem2 33226 | Lemma for dath 33301. Show the lines and form a plane. (Contributed by NM, 11-Aug-2012.) |
Theorem | dalemdea 33227 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.) |
Theorem | dalemeea 33228 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.) |
Theorem | dalem3 33229 | Lemma for dalemdnee 33231. (Contributed by NM, 10-Aug-2012.) |
Theorem | dalem4 33230 | Lemma for dalemdnee 33231. (Contributed by NM, 10-Aug-2012.) |
Theorem | dalemdnee 33231 | Lemma for dath 33301. Axis of perspectivity points and are different. (Contributed by NM, 10-Aug-2012.) |
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.) |
Theorem | dalem6 33233 | Lemma for dath 33301. Analogue of dalem5 33232 for . (Contributed by NM, 21-Jul-2012.) |
Theorem | dalem7 33234 | Lemma for dath 33301. Analogue of dalem5 33232 for . (Contributed by NM, 21-Jul-2012.) |
Theorem | dalem8 33235 | Lemma for dath 33301. Plane belongs to the 3-dimensional space. (Contributed by NM, 21-Jul-2012.) |
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.) |
Theorem | dalem9 33237 | Lemma for dath 33301. Since , the join forms a 3-dimensional space. (Contributed by NM, 20-Jul-2012.) |
Theorem | dalem10 33238 | Lemma for dath 33301. Atom belongs to the axis of perspectivity . (Contributed by NM, 19-Jul-2012.) |
Theorem | dalem11 33239 | Lemma for dath 33301. Analogue of dalem10 33238 for . (Contributed by NM, 23-Jul-2012.) |
Theorem | dalem12 33240 | Lemma for dath 33301. Analogue of dalem10 33238 for . (Contributed by NM, 11-Aug-2012.) |
Theorem | dalem13 33241 | Lemma for dalem14 33242. (Contributed by NM, 21-Jul-2012.) |
Theorem | dalem14 33242 | Lemma for dath 33301. Planes and form a 3-dimensional space (when they are different). (Contributed by NM, 22-Jul-2012.) |
Theorem | dalem15 33243 | Lemma for dath 33301. The axis of perspectivity is a line. (Contributed by NM, 21-Jul-2012.) |
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.) |
Theorem | dalem17 33245 | Lemma for dath 33301. When planes and are equal, the center of perspectivity is in . (Contributed by NM, 1-Aug-2012.) |
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.) |
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.) |
Theorem | dalemccea 33248 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Theorem | dalemddea 33249 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Theorem | dalem-ccly 33250 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Theorem | dalem-ddly 33251 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Theorem | dalemccnedd 33252 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Theorem | dalemclccjdd 33253 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
Theorem | dalemcceb 33254 | Lemma for dath 33301. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
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.) |
Theorem | dalemrotps 33256 | Lemma for dath 33301. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 15-Aug-2012.) |
Theorem | dalemcjden 33257 | Lemma for dath 33301. Show that the dummy atoms form a line. (Contributed by NM, 15-Aug-2012.) |
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.) |
Theorem | dalem21 33259 | Lemma for dath 33301. Show that lines and intersect at an atom. (Contributed by NM, 2-Aug-2012.) |
Theorem | dalem22 33260 | Lemma for dath 33301. Show that lines and determine a plane. (Contributed by NM, 2-Aug-2012.) |
Theorem | dalem23 33261 | Lemma for dath 33301. Show that auxiliary atom is an atom. (Contributed by NM, 2-Aug-2012.) |
Theorem | dalem24 33262 | Lemma for dath 33301. Show that auxiliary atom is outside of plane . (Contributed by NM, 2-Aug-2012.) |
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.) |
Theorem | dalem27 33264 | Lemma for dath 33301. Show that the line intersects the dummy center of perspectivity . (Contributed by NM, 8-Aug-2012.) |
Theorem | dalem28 33265 | Lemma for dath 33301. Lemma dalem27 33264 expressed differently. (Contributed by NM, 4-Aug-2012.) |
Theorem | dalem29 33266 | Lemma for dath 33301. Analogue of dalem23 33261 for . (Contributed by NM, 2-Aug-2012.) |
Theorem | dalem30 33267 | Lemma for dath 33301. Analogue of dalem24 33262 for . (Contributed by NM, 3-Aug-2012.) |
Theorem | dalem31N 33268 | Lemma for dath 33301. Analogue of dalem25 33263 for . (Contributed by NM, 4-Aug-2012.) (New usage is discouraged.) |
Theorem | dalem32 33269 | Lemma for dath 33301. Analogue of dalem27 33264 for . (Contributed by NM, 8-Aug-2012.) |
Theorem | dalem33 33270 | Lemma for dath 33301. Analogue of dalem28 33265 for . (Contributed by NM, 4-Aug-2012.) |
Theorem | dalem34 33271 | Lemma for dath 33301. Analogue of dalem23 33261 for . (Contributed by NM, 2-Aug-2012.) |
Theorem | dalem35 33272 | Lemma for dath 33301. Analogue of dalem24 33262 for . (Contributed by NM, 3-Aug-2012.) |
Theorem | dalem36 33273 | Lemma for dath 33301. Analogue of dalem27 33264 for . (Contributed by NM, 8-Aug-2012.) |
Theorem | dalem37 33274 | Lemma for dath 33301. Analogue of dalem28 33265 for . (Contributed by NM, 4-Aug-2012.) |
Theorem |