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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlplnnlt 33201 Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.)

Theorem2llnjaN 33202 The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 33203 in terms of atoms). (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnjN 33203 The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Theorem2llnm2N 33204 The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnm3N 33205 Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnm4 33206 Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)

Theorem2llnmeqat 33207 An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)

Theoremlvolset 33208* The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)

Theoremislvol 33209* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremislvol4 33210* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremlvoli 33211 Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)

Theoremislvol3 33212* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremlvoli3 33213 Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)

Theoremlvolbase 33214 A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)

Theoremislvol5 33215* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)

Theoremislvol2 33216* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)

Theoremlvoli2 33217 The join of 4 different atoms is a lattice volume. (Contributed by NM, 8-Jul-2012.)

Theoremlvolnle3at 33218 A lattice plane (or lattice line or atom) cannot majorize a lattice volume. (Contributed by NM, 8-Jul-2012.)

Theoremlvolnleat 33219 An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theoremlvolnlelln 33220 A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theoremlvolnlelpln 33221 A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theorem3atnelvolN 33222 The join of 3 atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

Theorem2atnelvolN 33223 The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

TheoremlvolneatN 33224 No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)

Theoremlvolnelln 33225 No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.)

Theoremlvolnelpln 33226 No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.)

Theoremlvoln0N 33227 A lattice volume is non-zero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

Theoremislvol2aN 33228 The predicate "is a lattice volume". (Contributed by NM, 16-Jul-2012.) (New usage is discouraged.)

Theorem4atlem0a 33229 Lemma for 4at 33249. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem0ae 33230 Lemma for 4at 33249. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem0be 33231 Lemma for 4at 33249. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem3 33232 Lemma for 4at 33249. Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012.)

Theorem4atlem3a 33233 Lemma for 4at 33249. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem3b 33234 Lemma for 4at 33249. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4a 33235 Lemma for 4at 33249. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4b 33236 Lemma for 4at 33249. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4c 33237 Lemma for 4at 33249. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4d 33238 Lemma for 4at 33249. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem9 33239 Lemma for 4at 33249. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem10a 33240 Lemma for 4at 33249. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem10b 33241 Lemma for 4at 33249. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)

Theorem4atlem10 33242 Lemma for 4at 33249. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem11a 33243 Lemma for 4at 33249. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem11b 33244 Lemma for 4at 33249. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)

Theorem4atlem11 33245 Lemma for 4at 33249. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem12a 33246 Lemma for 4at 33249. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem12b 33247 Lemma for 4at 33249. Substitute for (cont.). (Contributed by NM, 11-Jul-2012.)

Theorem4atlem12 33248 Lemma for 4at 33249. Combine all four possible cases. (Contributed by NM, 11-Jul-2012.)

Theorem4at 33249 Four atoms determine a lattice volume uniquely. Three-dimensional analogue of ps-1 33113 and 3at 33126. (Contributed by NM, 11-Jul-2012.)

Theorem4at2 33250 Four atoms determine a lattice volume uniquely. (Contributed by NM, 11-Jul-2012.)

Theoremlplncvrlvol2 33251 A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)

Theoremlplncvrlvol 33252 An element covering a lattice plane is a lattice volume and vice-versa. (Contributed by NM, 15-Jul-2012.)

Theoremlvolcmp 33253 If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)

TheoremlvolnltN 33254 Two lattice volumes cannot satisfy the less than relation. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theorem2lplnja 33255 The join of two different lattice planes in a lattice volume equals the volume (version of 2lplnj 33256 in terms of atoms). (Contributed by NM, 12-Jul-2012.)

Theorem2lplnj 33256 The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)

Theorem2lplnm2N 33257 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 33258 The meet of two lattice planes is a lattice line iff their join is a lattice volume. (Contributed by NM, 13-Jul-2012.)

Theoremdalemkehl 33259 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemkelat 33260 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemkeop 33261 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempea 33262 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqea 33263 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemrea 33264 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemsea 33265 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemtea 33266 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemuea 33267 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemyeo 33268 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemzeo 33269 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemclpjs 33270 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemclqjt 33271 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemclrju 33272 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalem-clpjq 33273 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemceb 33274 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempeb 33275 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqeb 33276 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemreb 33277 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemseb 33278 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemteb 33279 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemueb 33280 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempjqeb 33281 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemsjteb 33282 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemtjueb 33283 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqrprot 33284 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemyeb 33285 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemcnes 33286 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempnes 33287 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqnet 33288 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempjsen 33289 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemply 33290 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemsly 33291 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)

Theoremdalemswapyz 33292 Lemma for dath 33372. Swap the role of planes and to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)

Theoremdalemrot 33293 Lemma for dath 33372. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)

Theoremdalemrotyz 33294 Lemma for dath 33372. Rotate triangles and to allow reuse of analogous proofs. (Contributed by NM, 19-Aug-2012.)

Theoremdalem1 33295 Lemma for dath 33372. Show the lines and are different. (Contributed by NM, 9-Aug-2012.)

Theoremdalemcea 33296 Lemma for dath 33372. 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 33297 Lemma for dath 33372. Show the lines and form a plane. (Contributed by NM, 11-Aug-2012.)

Theoremdalemdea 33298 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.)

Theoremdalemeea 33299 Lemma for dath 33372. Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012.)

Theoremdalem3 33300 Lemma for dalemdnee 33302. (Contributed by NM, 10-Aug-2012.)

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