HomeTheorem List (p. 334 of 357)
Unicode version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 334 of 357) | < Previous Next > |
| Browser slow? Try the
Unicode version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-24339) |
Hilbert Space Explorer
(24340-25864) |
Users' Mathboxes
(25865-35640) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | dalemclpjs 33301 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
       ![/\](wedge.gif "/\"){kind=small}    
            
  | ||
| Theorem | dalemclqjt 33302 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemclrju 33303 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalem-clpjq 33304 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemceb 33305 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
   | ||
| Theorem | dalempeb 33306 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemqeb 33307 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemreb 33308 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemseb 33309 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemteb 33310 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemueb 33311 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalempjqeb 33312 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
 | ||
| Theorem | dalemsjteb 33313 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemtjueb 33314 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemqrprot 33315 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemyeb 33316 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
 | ||
| Theorem | dalemcnes 33317 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
 | ||
| Theorem | dalempnes 33318 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemqnet 33319 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalempjsen 33320 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
 | ||
| Theorem | dalemply 33321 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.) |
|
| ||
| Theorem | dalemsly 33322 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalemswapyz 33323 |
Lemma for dath 33403. Swap the role of planes and to allow
reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)
|
|
| ||
| Theorem | dalemrot 33324 |
Lemma for dath 33403. Rotate triangles and
to allow reuse of analogous proofs. (Contributed by NM,
14-Aug-2012.)
|
|
| ||
| Theorem | dalemrotyz 33325 |
Lemma for dath 33403. Rotate triangles and
to allow reuse of analogous proofs. (Contributed by NM,
19-Aug-2012.)
|
|
| ||
| Theorem | dalem1 33326 |
Lemma for dath 33403. Show the lines and are different.
(Contributed by NM, 9-Aug-2012.)
|
|
| ||
| Theorem | dalemcea 33327 |
Lemma for dath 33403. 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 33328 |
Lemma for dath 33403. Show the lines and form a plane.
(Contributed by NM, 11-Aug-2012.)
|
|
| ||
| Theorem | dalemdea 33329 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.) |
![./\](_.wedge.gif "./\"){kind=small} 
| ||
| Theorem | dalemeea 33330 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.) |
| ||
| Theorem | dalem3 33331 | Lemma for dalemdnee 33333. (Contributed by NM, 10-Aug-2012.) |
|
| ||
| Theorem | dalem4 33332 | Lemma for dalemdnee 33333. (Contributed by NM, 10-Aug-2012.) |
|
| ||
| Theorem | dalemdnee 33333 |
Lemma for dath 33403. Axis of perspectivity points and are
different. (Contributed by NM, 10-Aug-2012.)
|
|
| ||
| Theorem | dalem5 33334 |
Lemma for dath 33403. Atom (in plane ) belongs to
the 3-dimensional volume formed by and . (Contributed by
NM, 21-Jul-2012.)
|
 | ||
| Theorem | dalem6 33335 |
Lemma for dath 33403. Analog of dalem5 33334 for . (Contributed by
NM, 21-Jul-2012.)
|
|
| ||
| Theorem | dalem7 33336 |
Lemma for dath 33403. Analog of dalem5 33334 for . (Contributed by
NM, 21-Jul-2012.)
|
|
| ||
| Theorem | dalem8 33337 |
Lemma for dath 33403. Plane belongs to the 3-dimensional space.
(Contributed by NM, 21-Jul-2012.)
|
|
| ||
| Theorem | dalem-cly 33338 |
Lemma for dalem9 33339. Center of perspectivity is not in plane
(when and are different planes). (Contributed by
NM, 13-Aug-2012.)
|
|
| ||
| Theorem | dalem9 33339 |
Lemma for dath 33403. Since
, the join forms
a 3-dimensional space. (Contributed by NM, 20-Jul-2012.)
|
 
| ||
| Theorem | dalem10 33340 |
Lemma for dath 33403. Atom belongs to the axis of perspectivity
 .
(Contributed by NM, 19-Jul-2012.)
|
|
| ||
| Theorem | dalem11 33341 |
Lemma for dath 33403. Analog of dalem10 33340 for . (Contributed by
NM, 23-Jul-2012.)
|
|
| ||
| Theorem | dalem12 33342 |
Lemma for dath 33403. Analog of dalem10 33340 for  . (Contributed by
NM, 11-Aug-2012.)
|
|
| ||
| Theorem | dalem13 33343 | Lemma for dalem14 33344. (Contributed by NM, 21-Jul-2012.) |
|
| ||
| Theorem | dalem14 33344 |
Lemma for dath 33403. Planes and form a 3-dimensional space
(when they are different). (Contributed by NM, 22-Jul-2012.)
|
|
| ||
| Theorem | dalem15 33345 |
Lemma for dath 33403. The axis of perspectivity is a line.
(Contributed by NM, 21-Jul-2012.)
|
| ||
| Theorem | dalem16 33346 |
Lemma for dath 33403. 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 33347 |
Lemma for dath 33403. When planes and are equal, the center
of perspectivity is in .
(Contributed by NM,
1-Aug-2012.)
|
|
| ||
| Theorem | dalem18 33348* |
Lemma for dath 33403. 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 33349* |
Lemma for dath 33403. 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 33350 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
| ||
| Theorem | dalemddea 33351 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalem-ccly 33352 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalem-ddly 33353 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalemccnedd 33354 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalemclccjdd 33355 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalemcceb 33356 | Lemma for dath 33403. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalemswapyzps 33357 |
Lemma for dath 33403. 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 33358 |
Lemma for dath 33403. Rotate triangles and
to allow reuse of analogous proofs. (Contributed by NM,
15-Aug-2012.)
|
|
| ||
| Theorem | dalemcjden 33359 | Lemma for dath 33403. Show that the dummy atoms form a line. (Contributed by NM, 15-Aug-2012.) |
|
| ||
| Theorem | dalem20 33360* |
Lemma for dath 33403. 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 33361 |
Lemma for dath 33403. Show that lines and intersect at
an atom. (Contributed by NM, 2-Aug-2012.)
|
|
| ||
| Theorem | dalem22 33362 |
Lemma for dath 33403. Show that lines and determine a
plane. (Contributed by NM, 2-Aug-2012.)
|
|
| ||
| Theorem | dalem23 33363 |
Lemma for dath 33403. Show that auxiliary atom  is an atom.
(Contributed by NM, 2-Aug-2012.)
|
|
| ||
| Theorem | dalem24 33364 |
Lemma for dath 33403. Show that auxiliary atom is outside of plane
.
(Contributed by NM, 2-Aug-2012.)
|
|
| ||
| Theorem | dalem25 33365 |
Lemma for dath 33403. Show that the dummy center of
perspectivity
is different from auxiliary atom . (Contributed by NM,
3-Aug-2012.)
|
|
| ||
| Theorem | dalem27 33366 |
Lemma for dath 33403. Show that the line intersects the dummy
center of perspectivity . (Contributed by NM, 8-Aug-2012.)
|
|
| ||
| Theorem | dalem28 33367 | Lemma for dath 33403. Lemma dalem27 33366 expressed differently. (Contributed by NM, 4-Aug-2012.) |
|
| ||
| Theorem | dalem29 33368 |
Lemma for dath 33403. Analog of dalem23 33363 for  . (Contributed by
NM, 2-Aug-2012.)
|
|
| ||
| Theorem | dalem30 33369 |
Lemma for dath 33403. Analog of dalem24 33364 for . (Contributed by
NM, 3-Aug-2012.)
|
|
| ||
| Theorem | dalem31N 33370 |
Lemma for dath 33403. Analog of dalem25 33365 for . (Contributed by
NM, 4-Aug-2012.) (New usage is discouraged.)
|
|
| ||
| Theorem | dalem32 33371 |
Lemma for dath 33403. Analog of dalem27 33366 for . (Contributed by
NM, 8-Aug-2012.)
|
|
| ||
| Theorem | dalem33 33372 |
Lemma for dath 33403. Analog of dalem28 33367 for . (Contributed by
NM, 4-Aug-2012.)
|
|
| ||
| Theorem | dalem34 33373 |
Lemma for dath 33403. Analog of dalem23 33363 for  . (Contributed by
NM, 2-Aug-2012.)
|
|
| ||
| Theorem | dalem35 33374 |
Lemma for dath 33403. Analog of dalem24 33364 for . (Contributed by
NM, 3-Aug-2012.)
|
|
| ||
| Theorem | dalem36 33375 |
Lemma for dath 33403. Analog of dalem27 33366 for . (Contributed by
NM, 8-Aug-2012.)
|
|
| ||