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Theorem List for Metamath Proof Explorer - 29901-30000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdalem40 29901 Lemma for dath 29925. Analog of dalem39 29900 for . (Contributed by NM, 4-Aug-2012.)

Theoremdalem41 29902 Lemma for dath 29925. (Contributed by NM, 4-Aug-2012.)

Theoremdalem42 29903 Lemma for dath 29925. Auxiliary atoms form a plane. (Contributed by NM, 4-Aug-2012.)

Theoremdalem43 29904 Lemma for dath 29925. Planes and are different. (Contributed by NM, 8-Aug-2012.)

Theoremdalem44 29905 Lemma for dath 29925. Dummy center of perspectivity lies outside of plane . (Contributed by NM, 16-Aug-2012.)

Theoremdalem45 29906 Lemma for dath 29925. Dummy center of perspectivity is not on the line . (Contributed by NM, 16-Aug-2012.)

Theoremdalem46 29907 Lemma for dath 29925. Analog of dalem45 29906 for . (Contributed by NM, 16-Aug-2012.)

Theoremdalem47 29908 Lemma for dath 29925. Analog of dalem45 29906 for . (Contributed by NM, 16-Aug-2012.)

Theoremdalem48 29909 Lemma for dath 29925. Analog of dalem45 29906 for . (Contributed by NM, 16-Aug-2012.)

Theoremdalem49 29910 Lemma for dath 29925. Analog of dalem45 29906 for . (Contributed by NM, 16-Aug-2012.)

Theoremdalem50 29911 Lemma for dath 29925. Analog of dalem45 29906 for . (Contributed by NM, 16-Aug-2012.)

Theoremdalem51 29912 Lemma for dath 29925. Construct the condition with , , and in place of , , and respectively. This lets us reuse the special case of Desargues' Theorem where , to eventually prove the case where . (Contributed by NM, 16-Aug-2012.)

Theoremdalem52 29913 Lemma for dath 29925. Lines and intersect at an atom. (Contributed by NM, 8-Aug-2012.)

Theoremdalem53 29914 Lemma for dath 29925. The auxliary axis of perspectivity is a line (analogous to the actual axis of perspectivity in dalem15 29867. (Contributed by NM, 8-Aug-2012.)

Theoremdalem54 29915 Lemma for dath 29925. Line intersects the auxiliary axis of perspectivity . (Contributed by NM, 8-Aug-2012.)

Theoremdalem55 29916 Lemma for dath 29925. Lines and intersect at the auxiliary line (later shown to be an axis of perspectivity; see dalem60 29921). (Contributed by NM, 8-Aug-2012.)

Theoremdalem56 29917 Lemma for dath 29925. Analog of dalem55 29916 for line . (Contributed by NM, 8-Aug-2012.)

Theoremdalem57 29918 Lemma for dath 29925. Axis of perspectivity point is on the auxiliary line . (Contributed by NM, 9-Aug-2012.)

Theoremdalem58 29919 Lemma for dath 29925. Analog of dalem57 29918 for . (Contributed by NM, 10-Aug-2012.)

Theoremdalem59 29920 Lemma for dath 29925. Analog of dalem57 29918 for . (Contributed by NM, 10-Aug-2012.)

Theoremdalem60 29921 Lemma for dath 29925. is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)

Theoremdalem61 29922 Lemma for dath 29925. Show that atoms , , and lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms and . (Contributed by NM, 11-Aug-2012.)

Theoremdalem62 29923 Lemma for dath 29925. Eliminate the condition containing dummy variables and . (Contributed by NM, 11-Aug-2012.)

Theoremdalem63 29924 Lemma for dath 29925. Combine the cases where and are different planes with the case where and are the same plane. (Contributed by NM, 11-Aug-2012.)

Theoremdath 29925 Desargues' Theorem of projective geometry (proved for a Hilbert lattice). Assume each triple of atoms (points) and forms a triangle (i.e. determines a plane). Assume that lines , , and meet at a "center of perspectivity" . (We also assume that is not on any of the 6 lines forming the two triangles.) Then the atoms , , are colinear, forming an "axis of perspectivity".

Our proof roughly follows Theorem 2.7.1, p. 78 in Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press (1988). Unlike them, we don't assume is an atom to make this theorem slightly more general for easier future use. However, we prove that must be an atom in dalemcea 29849.

For a visual demonstration, see the "Desargue's Theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html. The points I, J, and K there define the axis of perspectivity.

See theorem dalaw 30075 for Desargues Law, which eliminates all of the preconditions on the atoms except for central perspectivity. (Contributed by NM, 20-Aug-2012.)

Theoremdath2 29926 Version of Desargues' Theorem dath 29925 with a different variable ordering. (Contributed by NM, 7-Oct-2012.)

Theoremlineset 29927* The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)

Theoremisline 29928* The predicate "is a line". (Contributed by NM, 19-Sep-2011.)

Theoremislinei 29929* Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)

TheorempointsetN 29930* The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)

TheoremispointN 29931* The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)

TheorematpointN 29932 The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempsubspset 29933* The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)

Theoremispsubsp 29934* The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)

Theoremispsubsp2 29935* The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)

Theorempsubspi 29936* Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)

Theorempsubspi2N 29937 Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)

Theorem0psubN 29938 The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)

TheoremsnatpsubN 29939 The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)

TheorempointpsubN 29940 A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)

TheoremlinepsubN 29941 A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

TheorematpsubN 29942 The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)

Theorempsubssat 29943 A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)

TheorempsubatN 29944 A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)

Theorempmapfval 29945* The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)

Theorempmapval 29946* Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)

Theoremelpmap 29947 Member of a projective map. (Contributed by NM, 27-Jan-2012.)

Theorempmapssat 29948 The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)

TheorempmapssbaN 29949 A weakening of pmapssat 29948 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)

Theorempmaple 29950 The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)

Theorempmap11 29951 The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)

Theorempmapat 29952 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)

Theoremelpmapat 29953 Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)

Theorempmap0 29954 Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)

Theorempmapeq0 29955 A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)

Theorempmap1N 29956 Value of the projective map of a Hilbert lattice at lattice unit. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)

Theorempmapsub 29957 The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)

Theorempmapglbx 29958* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 29959, where we read as . Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Theorempmapglb 29959* The projective map of the GLB of a set of lattice elements . Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Theorempmapglb2N 29960* The projective map of the GLB of a set of lattice elements . Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows . (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)

Theorempmapglb2xN 29961* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 29960, where we read as . Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows . (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)

Theorempmapmeet 29962 The projective map of a meet. (Contributed by NM, 25-Jan-2012.)

Theoremisline2 29963* Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)

Theoremlinepmap 29964 A line described with a projective map. (Contributed by NM, 3-Feb-2012.)

Theoremisline3 29965* Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)

Theoremisline4N 29966* Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)

Theoremlneq2at 29967 A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)

TheoremlnatexN 29968* There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

TheoremlnjatN 29969* Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

TheoremlncvrelatN 29970 A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Theoremlncvrat 29971 A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)

Theoremlncmp 29972 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)

Theorem2lnat 29973 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)

Theorem2atm2atN 29974 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Theorem2llnma1b 29975 Generalization of 2llnma1 29976. (Contributed by NM, 26-Apr-2013.)

Theorem2llnma1 29976 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)

Theorem2llnma3r 29977 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)

Theorem2llnma2 29978 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)

Theorem2llnma2rN 29979 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013.) (New usage is discouraged.)

18.27.10  Construction of a vector space from a Hilbert lattice

Theoremcdlema1N 29980 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)

Theoremcdlema2N 29981 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)

Theoremcdlemblem 29982 Lemma for cdlemb 29983. (Contributed by NM, 8-May-2012.)

Theoremcdlemb 29983* Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)

Syntaxcpadd 29984 Extend class notation with projective subspace sum.

Definitiondf-padd 29985* Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)

Theorempaddfval 29986* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)

Theorempaddval 29987* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)

Theoremelpadd 29988* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)

Theoremelpaddn0 29989* Member of projective subspace sum of non-empty sets. (Contributed by NM, 3-Jan-2012.)

Theorempaddvaln0N 29990* Projective subspace sum operation value for non-empty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)

Theoremelpaddri 29991 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)

TheoremelpaddatriN 29992 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)

Theoremelpaddat 29993* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)

TheoremelpaddatiN 29994* Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theoremelpadd2at 29995 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)

Theoremelpadd2at2 29996 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)

TheorempaddunssN 29997 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)

Theoremelpadd0 29998 Member of projective subspace sum with at least one empty set.. (Contributed by NM, 29-Dec-2011.)

Theorempaddval0 29999 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)

Theorempadd01 30000 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

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