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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheorempaddclN 30701 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 30702 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddssw2 30703 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddss 30704 Subset law for projective subspace sum. (unss 3523 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempmodlem1 30705* Lemma for pmod1i 30707. (Contributed by NM, 9-Mar-2012.)

Theorempmodlem2 30706 Lemma for pmod1i 30707. (Contributed by NM, 9-Mar-2012.)

Theorempmod1i 30707 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)

Theorempmod2iN 30708 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

TheorempmodN 30709 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)

Theorempmodl42N 30710 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempmapjoin 30711 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 30712 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)

Theorempmapjat2 30713 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)

Theorempmapjlln1 30714 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 30715 A version of the modular law pmod1i 30707 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)

Theorematmod1i1 30716 Version of modular law pmod1i 30707 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 30717 Version of modular law pmod1i 30707 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 30718 Version of modular law pmod1i 30707 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 30719 Version of modular law pmod1i 30707 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 30720 Version of modular law pmod2iN 30708 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 30721 Version of modular law pmod2iN 30708 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 30722 Version of modular law pmod1i 30707 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 30723 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 30724 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 30725 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 30726 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 30727 Lemma for llnexchb2 30728. (Contributed by NM, 17-Nov-2012.)

Theoremllnexchb2 30728 Line exchange property (compare cvlatexchb2 30195 for atoms). (Contributed by NM, 17-Nov-2012.)

Theoremllnexch2N 30729 Line exchange property (compare cvlatexch2 30197 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)

Theoremdalawlem1 30730 Lemma for dalaw 30745. Special case of dath2 30596, where is replaced by . The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 30596. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem2 30731 Lemma for dalaw 30745. Utility lemma that breaks into a join of two pieces. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem3 30732 Lemma for dalaw 30745. First piece of dalawlem5 30734. (Contributed by NM, 4-Oct-2012.)

Theoremdalawlem4 30733 Lemma for dalaw 30745. Second piece of dalawlem5 30734. (Contributed by NM, 4-Oct-2012.)

Theoremdalawlem5 30734 Lemma for dalaw 30745. Special case to eliminate the requirement in dalawlem1 30730. (Contributed by NM, 4-Oct-2012.)

Theoremdalawlem6 30735 Lemma for dalaw 30745. First piece of dalawlem8 30737. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem7 30736 Lemma for dalaw 30745. Second piece of dalawlem8 30737. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem8 30737 Lemma for dalaw 30745. Special case to eliminate the requirement in dalawlem1 30730. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem9 30738 Lemma for dalaw 30745. Special case to eliminate the requirement in dalawlem1 30730. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem10 30739 Lemma for dalaw 30745. Combine dalawlem5 30734, dalawlem8 30737, and dalawlem9 . (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem11 30740 Lemma for dalaw 30745. First part of dalawlem13 30742. (Contributed by NM, 17-Sep-2012.)

Theoremdalawlem12 30741 Lemma for dalaw 30745. Second part of dalawlem13 30742. (Contributed by NM, 17-Sep-2012.)

Theoremdalawlem13 30742 Lemma for dalaw 30745. Special case to eliminate the requirement in dalawlem1 30730. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem14 30743 Lemma for dalaw 30745. Combine dalawlem10 30739 and dalawlem13 30742. (Contributed by NM, 6-Oct-2012.)

Theoremdalawlem15 30744 Lemma for dalaw 30745. Swap variable triples and in dalawlem14 30743, to obtain the elimination of the remaining conditions in dalawlem1 30730. (Contributed by NM, 6-Oct-2012.)

Theoremdalaw 30745 Desargues' law, derived from Desargues' theorem dath 30595 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 30746 Extend class notation with projective subspace closure.

Definitiondf-pclN 30747* 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 30794.) (Contributed by NM, 7-Sep-2013.)

TheorempclfvalN 30748* The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)

TheorempclvalN 30749* Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)

TheorempclclN 30750 Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

TheoremelpclN 30751* Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

TheoremelpcliN 30752 Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

TheorempclssN 30753 Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

TheorempclssidN 30754 A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

TheorempclidN 30755 The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

TheorempclbtwnN 30756 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 30757 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 30758 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 30759* 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 30809. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)

TheorempclcmpatN 30760* 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 30761 Extend class notation with polarity of projective subspace \$m\$.

Definitiondf-polarityN 30762* 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 30763* The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)

TheorempolvalN 30764* Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)

Theorempolval2N 30765 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 30766 The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempolssatN 30767 The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorempol0N 30768 The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)

Theorempol1N 30769 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 30770 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

TheorempolpmapN 30771 The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorem2polpmapN 30772 Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorem2polvalN 30773 Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

Theorem2polssN 30774 A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)

Theorem3polN 30775 Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

Theorempolcon3N 30776 Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theorem2polcon4bN 30777 Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

Theorempolcon2N 30778 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theorempolcon2bN 30779 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theorempclss2polN 30780 The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

Theorempcl0N 30781 The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

Theorempcl0bN 30782 The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

TheorempmaplubN 30783 The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)

TheoremsspmaplubN 30784 A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

Theorem2pmaplubN 30785 Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

TheorempaddunN 30786 The closure of the projective sum of two sets of atoms is the same as the closure of their union. (Closure is actually double polarity, which can be trivially inferred from this theorem using fveq2d 5734.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

Theorempoldmj1N 30787 De Morgan's law for polarity of projective sum. (oldmj1 30081 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)

Theorempmapj2N 30788 The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)

TheorempmapocjN 30789 The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)

TheorempolatN 30790 The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorem2polatN 30791 Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

TheorempnonsingN 30792 The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)

SyntaxcpscN 30793 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.

Definitiondf-psubclN 30794* Define set of all closed projective subspaces, which are those sets of atoms that equal their double polarity. Based on definition in [Holland95] p. 223. (Contributed by NM, 23-Jan-2012.)

TheorempsubclsetN 30795* The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheoremispsubclN 30796 The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempsubcliN 30797 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

Theorempsubcli2N 30798 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempsubclsubN 30799 A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempsubclssatN 30800 A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

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