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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempaddss12 30301 Subset law for projective subspace sum. (unss12 3479 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddasslem3 30304* Lemma for paddass 30320. Restate projective space axiom ps-2 29960. (Contributed by NM, 8-Jan-2012.)

Theorempaddasslem11 30312 Lemma for paddass 30320. The case when . (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem12 30313 Lemma for paddass 30320. The case when . (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem13 30314 Lemma for paddass 30320. The case when . (Unlike the proof in Maeda and Maeda, we don't need .) (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem17 30318 Lemma for paddass 30320. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)

Theorempaddass 30320 Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be non-empty. (Contributed by NM, 29-Dec-2011.)

Theorempadd12N 30321 Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempadd4N 30322 Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempaddidm 30323 Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)

TheorempaddclN 30324 The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempaddssw1 30325 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddssw2 30326 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddss 30327 Subset law for projective subspace sum. (unss 3481 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempmodlem1 30328* Lemma for pmod1i 30330. (Contributed by NM, 9-Mar-2012.)

Theorempmodlem2 30329 Lemma for pmod1i 30330. (Contributed by NM, 9-Mar-2012.)

Theorempmod1i 30330 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)

Theorempmod2iN 30331 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

TheorempmodN 30332 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)

Theorempmodl42N 30333 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempmapjoin 30334 The projective map of the join of two lattice elements. Part of Equation 15.5.3 of [MaedaMaeda] p. 63. (Contributed by NM, 27-Jan-2012.)

Theorempmapjat1 30335 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)

Theorempmapjat2 30336 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)

Theorempmapjlln1 30337 The projective map of the join of a lattice element and a lattice line (expressed as the join of two atoms). (Contributed by NM, 16-Sep-2012.)

Theoremhlmod1i 30338 A version of the modular law pmod1i 30330 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)

Theorematmod1i1 30339 Version of modular law pmod1i 30330 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod1i1m 30340 Version of modular law pmod1i 30330 that holds in a Hilbert lattice, when an element meets an atom. (Contributed by NM, 2-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod1i2 30341 Version of modular law pmod1i 30330 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremllnmod1i2 30342 Version of modular law pmod1i 30330 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join ). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod2i1 30343 Version of modular law pmod2iN 30331 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod2i2 30344 Version of modular law pmod2iN 30331 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremllnmod2i2 30345 Version of modular law pmod1i 30330 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join ). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod3i1 30346 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod3i2 30347 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod4i1 30348 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod4i2 30349 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-Mar-2013.)

Theoremllnexchb2lem 30350 Lemma for llnexchb2 30351. (Contributed by NM, 17-Nov-2012.)

Theoremllnexchb2 30351 Line exchange property (compare cvlatexchb2 29818 for atoms). (Contributed by NM, 17-Nov-2012.)

Theoremllnexch2N 30352 Line exchange property (compare cvlatexch2 29820 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)

Theoremdalawlem1 30353 Lemma for dalaw 30368. Special case of dath2 30219, where is replaced by . The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 30219. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem2 30354 Lemma for dalaw 30368. Utility lemma that breaks into a join of two pieces. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem3 30355 Lemma for dalaw 30368. First piece of dalawlem5 30357. (Contributed by NM, 4-Oct-2012.)

Theoremdalawlem4 30356 Lemma for dalaw 30368. Second piece of dalawlem5 30357. (Contributed by NM, 4-Oct-2012.)

Theoremdalawlem5 30357 Lemma for dalaw 30368. Special case to eliminate the requirement in dalawlem1 30353. (Contributed by NM, 4-Oct-2012.)

Theoremdalawlem6 30358 Lemma for dalaw 30368. First piece of dalawlem8 30360. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem7 30359 Lemma for dalaw 30368. Second piece of dalawlem8 30360. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem8 30360 Lemma for dalaw 30368. Special case to eliminate the requirement in dalawlem1 30353. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem9 30361 Lemma for dalaw 30368. Special case to eliminate the requirement in dalawlem1 30353. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem10 30362 Lemma for dalaw 30368. Combine dalawlem5 30357, dalawlem8 30360, and dalawlem9 . (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem11 30363 Lemma for dalaw 30368. First part of dalawlem13 30365. (Contributed by NM, 17-Sep-2012.)

Theoremdalawlem12 30364 Lemma for dalaw 30368. Second part of dalawlem13 30365. (Contributed by NM, 17-Sep-2012.)

Theoremdalawlem13 30365 Lemma for dalaw 30368. Special case to eliminate the requirement in dalawlem1 30353. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem14 30366 Lemma for dalaw 30368. Combine dalawlem10 30362 and dalawlem13 30365. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem15 30367 Lemma for dalaw 30368. Swap variable triples and in dalawlem14 30366, to obtain the elimination of the remaining conditions in dalawlem1 30353. (Contributed by NM, 6-Oct-2012.)

Theoremdalaw 30368 Desargues' law, derived from Desargues' theorem dath 30218 and with no conditions on the atoms. If triples and are centrally perspective, i.e. , then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)

SyntaxcpclN 30369 Extend class notation with projective subspace closure.

Definitiondf-pclN 30370* Projective subspace closure, which is the smallest projective subspace containing an arbitrary set of atoms. The subspace closure of the union of a set of projective subspaces is their supremum in . Related to an analogous definition of closure used in Lemma 3.1.4 of [PtakPulmannova] p. 68. (Note that this closure is not necessarily one of the closed projective subspaces of df-psubclN 30417.) (Contributed by NM, 7-Sep-2013.)

TheorempclfvalN 30371* The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)

TheorempclvalN 30372* Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)

TheorempclclN 30373 Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

TheoremelpclN 30374* Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

TheoremelpcliN 30375 Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

TheorempclssN 30376 Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

TheorempclssidN 30377 A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

TheorempclidN 30378 The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

TheorempclbtwnN 30379 A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

TheorempclunN 30380 The projective subspace closure of the union of two sets of atoms equals the closure of their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

Theorempclun2N 30381 The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

TheorempclfinN 30382* The projective subspace closure of a set equals the union of the closures of its finite subsets. Analogous to Lemma 3.3.6 of [PtakPulmannova] p. 72. Compare the closed subspace version pclfinclN 30432. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)

TheorempclcmpatN 30383* The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of [PtakPulmannova] p. 74. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)

SyntaxcpolN 30384 Extend class notation with polarity of projective subspace \$m\$.

Definitiondf-polarityN 30385* Define polarity of projective subspace, which is a kind of complement of the subspace. Item 2 in [Holland95] p. 222 bottom. For more generality, we define it for all subsets of atoms, not just projective subspaces. The intersection with ensures it is defined when . (Contributed by NM, 23-Oct-2011.)

TheorempolfvalN 30386* The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)

TheorempolvalN 30387* Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)

Theorempolval2N 30388 Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)

TheorempolsubN 30389 The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempolssatN 30390 The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorempol0N 30391 The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)

Theorempol1N 30392 The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorem2pol0N 30393 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

TheorempolpmapN 30394 The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorem2polpmapN 30395 Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorem2polvalN 30396 Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

Theorem2polssN 30397 A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)

Theorem3polN 30398 Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

Theorempolcon3N 30399 Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theorem2polcon4bN 30400 Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

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