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Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdalem51 28601 Lemma for dath 28614. 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 28602 Lemma for dath 28614. Lines and intersect at an atom. (Contributed by NM, 8-Aug-2012.)

Theoremdalem53 28603 Lemma for dath 28614. The auxliary axis of perspectivity is a line (analogous to the actual axis of perspectivity in dalem15 28556. (Contributed by NM, 8-Aug-2012.)

Theoremdalem54 28604 Lemma for dath 28614. Line intersects the auxiliary axis of perspectivity . (Contributed by NM, 8-Aug-2012.)

Theoremdalem55 28605 Lemma for dath 28614. Lines and intersect at the auxiliary line (later shown to be an axis of perspectivity; see dalem60 28610). (Contributed by NM, 8-Aug-2012.)

Theoremdalem56 28606 Lemma for dath 28614. Analog of dalem55 28605 for line . (Contributed by NM, 8-Aug-2012.)

Theoremdalem57 28607 Lemma for dath 28614. Axis of perspectivity point is on the auxiliary line . (Contributed by NM, 9-Aug-2012.)

Theoremdalem58 28608 Lemma for dath 28614. Analog of dalem57 28607 for . (Contributed by NM, 10-Aug-2012.)

Theoremdalem59 28609 Lemma for dath 28614. Analog of dalem57 28607 for . (Contributed by NM, 10-Aug-2012.)

Theoremdalem60 28610 Lemma for dath 28614. is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)

Theoremdalem61 28611 Lemma for dath 28614. 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 28612 Lemma for dath 28614. Eliminate the condition containing dummy variables and . (Contributed by NM, 11-Aug-2012.)

Theoremdalem63 28613 Lemma for dath 28614. Combine the cases where and are different planes with the case where and are the same plane. (Contributed by NM, 11-Aug-2012.)

Theoremdath 28614 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 28538.

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 28764 for Desargues Law, which eliminates all of the preconditions on the atoms except for central perspectivity. (Contributed by NM, 20-Aug-2012.)

Theoremdath2 28615 Version of Desargues' Theorem dath 28614 with a different variable ordering. (Contributed by NM, 7-Oct-2012.)

Theoremlineset 28616* The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)

Theoremisline 28617* The predicate "is a line". (Contributed by NM, 19-Sep-2011.)

Theoremislinei 28618* Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)

TheorempointsetN 28619* The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)

TheoremispointN 28620* The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)

TheorematpointN 28621 The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempsubspset 28622* The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)

Theoremispsubsp 28623* The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)

Theoremispsubsp2 28624* The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)

Theorempsubspi 28625* Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)

Theorempsubspi2N 28626 Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)

Theorem0psubN 28627 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 28628 The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)

TheorempointpsubN 28629 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 28630 A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

TheorematpsubN 28631 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 28632 A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)

TheorempsubatN 28633 A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)

Theorempmapfval 28634* The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)

Theorempmapval 28635* 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 28636 Member of a projective map. (Contributed by NM, 27-Jan-2012.)

Theorempmapssat 28637 The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)

TheorempmapssbaN 28638 A weakening of pmapssat 28637 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)

Theorempmaple 28639 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 28640 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 28641 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)

Theoremelpmapat 28642 Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)

Theorempmap0 28643 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 28644 A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)

Theorempmap1N 28645 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 28646 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 28647* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 28648, where we read as . Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Theorempmapglb 28648* 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 28649* 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 28650* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 28649, 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 28651 The projective map of a meet. (Contributed by NM, 25-Jan-2012.)

Theoremisline2 28652* Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)

Theoremlinepmap 28653 A line described with a projective map. (Contributed by NM, 3-Feb-2012.)

Theoremisline3 28654* Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)

Theoremisline4N 28655* Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)

Theoremlneq2at 28656 A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)

TheoremlnatexN 28657* There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

TheoremlnjatN 28658* 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 28659 A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Theoremlncvrat 28660 A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)

Theoremlncmp 28661 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)

Theorem2lnat 28662 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)

Theorem2atm2atN 28663 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Theorem2llnma1b 28664 Generalization of 2llnma1 28665. (Contributed by NM, 26-Apr-2013.)

Theorem2llnma1 28665 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)

Theorem2llnma3r 28666 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)

Theorem2llnma2 28667 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)

Theorem2llnma2rN 28668 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.)

16.22.10  Construction of a vector space from a Hilbert lattice

Theoremcdlema1N 28669 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)

Theoremcdlema2N 28670 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)

Theoremcdlemblem 28671 Lemma for cdlemb 28672. (Contributed by NM, 8-May-2012.)

Theoremcdlemb 28672* 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 28673 Extend class notation with projective subspace sum.

Definitiondf-padd 28674* 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 28675* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)

Theorempaddval 28676* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)

Theoremelpadd 28677* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)

Theoremelpaddn0 28678* Member of projective subspace sum of non-empty sets. (Contributed by NM, 3-Jan-2012.)

Theorempaddvaln0N 28679* Projective subspace sum operation value for non-empty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)

Theoremelpaddri 28680 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)

TheoremelpaddatriN 28681 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)

Theoremelpaddat 28682* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)

TheoremelpaddatiN 28683* Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theoremelpadd2at 28684 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)

Theoremelpadd2at2 28685 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)

TheorempaddunssN 28686 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)

Theoremelpadd0 28687 Member of projective subspace sum with at least one empty set.. (Contributed by NM, 29-Dec-2011.)

Theorempaddval0 28688 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)

Theorempadd01 28689 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

Theorempadd02 28690 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

Theorempaddcom 28691 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)

Theorempaddssat 28692 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)

Theoremsspadd1 28693 A projective subspace sum is a superset of its first summand. (ssun1 3248 analog.) (Contributed by NM, 3-Jan-2012.)

Theoremsspadd2 28694 A projective subspace sum is a superset of its second summand. (ssun2 3249 analog.) (Contributed by NM, 3-Jan-2012.)

Theorempaddss1 28695 Subset law for projective subspace sum. (unss1 3254 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddss2 28696 Subset law for projective subspace sum. (unss2 3256 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddss12 28697 Subset law for projective subspace sum. (unss12 3257 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddasslem1 28698 Lemma for paddass 28716. (Contributed by NM, 8-Jan-2012.)

Theorempaddasslem2 28699 Lemma for paddass 28716. (Contributed by NM, 8-Jan-2012.)

Theorempaddasslem3 28700* Lemma for paddass 28716. Restate projective space axiom ps-2 28356. (Contributed by NM, 8-Jan-2012.)

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