Home Metamath Proof ExplorerTheorem List (p. 302 of 325) < 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 (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2lplnja 30101 The join of two different lattice planes in a lattice volume equals the volume (version of 2lplnj 30102 in terms of atoms). (Contributed by NM, 12-Jul-2012.)

Theorem2lplnj 30102 The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)

Theorem2lplnm2N 30103 The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theorem2lplnmj 30104 The meet of two lattice planes is a lattice line iff their join is a lattice volume. (Contributed by NM, 13-Jul-2012.)

Theoremdalemkehl 30105 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemkelat 30106 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemkeop 30107 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempea 30108 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqea 30109 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemrea 30110 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemsea 30111 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemtea 30112 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemuea 30113 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemyeo 30114 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemzeo 30115 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemclpjs 30116 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemclqjt 30117 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemclrju 30118 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalem-clpjq 30119 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemceb 30120 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempeb 30121 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqeb 30122 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemreb 30123 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemseb 30124 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemteb 30125 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemueb 30126 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempjqeb 30127 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemsjteb 30128 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemtjueb 30129 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqrprot 30130 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemyeb 30131 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemcnes 30132 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempnes 30133 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqnet 30134 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempjsen 30135 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemply 30136 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemsly 30137 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalemswapyz 30138 Lemma for dath 30218. Swap the role of planes and to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)

Theoremdalemrot 30139 Lemma for dath 30218. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)

Theoremdalemrotyz 30140 Lemma for dath 30218. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 19-Aug-2012.)

Theoremdalem1 30141 Lemma for dath 30218. Show the lines and are different. (Contributed by NM, 9-Aug-2012.)

Theoremdalemcea 30142 Lemma for dath 30218. 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.)

Theoremdalem2 30143 Lemma for dath 30218. Show the lines and form a plane. (Contributed by NM, 11-Aug-2012.)

Theoremdalemdea 30144 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.)

Theoremdalemeea 30145 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.)

Theoremdalem3 30146 Lemma for dalemdnee 30148. (Contributed by NM, 10-Aug-2012.)

Theoremdalem4 30147 Lemma for dalemdnee 30148. (Contributed by NM, 10-Aug-2012.)

Theoremdalemdnee 30148 Lemma for dath 30218. Axis of perspectivity points and are different. (Contributed by NM, 10-Aug-2012.)

Theoremdalem5 30149 Lemma for dath 30218. Atom (in plane ) belongs to the 3-dimensional volume formed by and . (Contributed by NM, 21-Jul-2012.)

Theoremdalem6 30150 Lemma for dath 30218. Analog of dalem5 30149 for . (Contributed by NM, 21-Jul-2012.)

Theoremdalem7 30151 Lemma for dath 30218. Analog of dalem5 30149 for . (Contributed by NM, 21-Jul-2012.)

Theoremdalem8 30152 Lemma for dath 30218. Plane belongs to the 3-dimensional space. (Contributed by NM, 21-Jul-2012.)

Theoremdalem-cly 30153 Lemma for dalem9 30154. Center of perspectivity is not in plane (when and are different planes). (Contributed by NM, 13-Aug-2012.)

Theoremdalem9 30154 Lemma for dath 30218. Since , the join forms a 3-dimensional space. (Contributed by NM, 20-Jul-2012.)

Theoremdalem10 30155 Lemma for dath 30218. Atom belongs to the axis of perspectivity . (Contributed by NM, 19-Jul-2012.)

Theoremdalem11 30156 Lemma for dath 30218. Analog of dalem10 30155 for . (Contributed by NM, 23-Jul-2012.)

Theoremdalem12 30157 Lemma for dath 30218. Analog of dalem10 30155 for . (Contributed by NM, 11-Aug-2012.)

Theoremdalem13 30158 Lemma for dalem14 30159. (Contributed by NM, 21-Jul-2012.)

Theoremdalem14 30159 Lemma for dath 30218. Planes and form a 3-dimensional space (when they are different). (Contributed by NM, 22-Jul-2012.)

Theoremdalem15 30160 Lemma for dath 30218. The axis of perspectivity is a line. (Contributed by NM, 21-Jul-2012.)

Theoremdalem16 30161 Lemma for dath 30218. 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.)

Theoremdalem17 30162 Lemma for dath 30218. When planes and are equal, the center of perspectivity is in . (Contributed by NM, 1-Aug-2012.)

Theoremdalem18 30163* Lemma for dath 30218. 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.)

Theoremdalem19 30164* Lemma for dath 30218. Show that a second dummy atom exists outside of the and planes (when those planes are equal). (Contributed by NM, 15-Aug-2012.)

Theoremdalemccea 30165 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalemddea 30166 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalem-ccly 30167 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalem-ddly 30168 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalemccnedd 30169 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalemclccjdd 30170 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalemcceb 30171 Lemma for dath 30218. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalemswapyzps 30172 Lemma for dath 30218. 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.)

Theoremdalemrotps 30173 Lemma for dath 30218. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 15-Aug-2012.)

Theoremdalemcjden 30174 Lemma for dath 30218. Show that the dummy atoms form a line. (Contributed by NM, 15-Aug-2012.)

Theoremdalem20 30175* Lemma for dath 30218. Show that a second dummy atom exists outside of the and planes (when those planes are equal). (Contributed by NM, 14-Aug-2012.)

Theoremdalem21 30176 Lemma for dath 30218. Show that lines and intersect at an atom. (Contributed by NM, 2-Aug-2012.)

Theoremdalem22 30177 Lemma for dath 30218. Show that lines and determine a plane. (Contributed by NM, 2-Aug-2012.)

Theoremdalem23 30178 Lemma for dath 30218. Show that auxiliary atom is an atom. (Contributed by NM, 2-Aug-2012.)

Theoremdalem24 30179 Lemma for dath 30218. Show that auxiliary atom is outside of plane . (Contributed by NM, 2-Aug-2012.)

Theoremdalem25 30180 Lemma for dath 30218. Show that the dummy center of perspectivity is different from auxiliary atom . (Contributed by NM, 3-Aug-2012.)

Theoremdalem27 30181 Lemma for dath 30218. Show that the line intersects the dummy center of perspectivity . (Contributed by NM, 8-Aug-2012.)

Theoremdalem28 30182 Lemma for dath 30218. Lemma dalem27 30181 expressed differently. (Contributed by NM, 4-Aug-2012.)

Theoremdalem29 30183 Lemma for dath 30218. Analog of dalem23 30178 for . (Contributed by NM, 2-Aug-2012.)