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Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempmap0 33401 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 33402 A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)

Theorempmap1N 33403 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 33404 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 33405* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 33406, where we read as . Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Theorempmapglb 33406* 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 33407* 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 33408* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 33407, 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 33409 The projective map of a meet. (Contributed by NM, 25-Jan-2012.)

Theoremisline2 33410* Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)

Theoremlinepmap 33411 A line described with a projective map. (Contributed by NM, 3-Feb-2012.)

Theoremisline3 33412* Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)

Theoremisline4N 33413* Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)

Theoremlneq2at 33414 A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)

TheoremlnatexN 33415* There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

TheoremlnjatN 33416* 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 33417 A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Theoremlncvrat 33418 A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)

Theoremlncmp 33419 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)

Theorem2lnat 33420 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)

Theorem2atm2atN 33421 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Theorem2llnma1b 33422 Generalization of 2llnma1 33423. (Contributed by NM, 26-Apr-2013.)

Theorem2llnma1 33423 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)

Theorem2llnma3r 33424 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)

Theorem2llnma2 33425 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)

Theorem2llnma2rN 33426 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.)

21.21.13  Construction of a vector space from a Hilbert lattice

Theoremcdlema1N 33427 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)

Theoremcdlema2N 33428 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)

Theoremcdlemblem 33429 Lemma for cdlemb 33430. (Contributed by NM, 8-May-2012.)

Theoremcdlemb 33430* 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 33431 Extend class notation with projective subspace sum.

Definitiondf-padd 33432* 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 33433* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)

Theorempaddval 33434* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)

Theoremelpadd 33435* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)

Theoremelpaddn0 33436* Member of projective subspace sum of nonempty sets. (Contributed by NM, 3-Jan-2012.)

Theorempaddvaln0N 33437* Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)

Theoremelpaddri 33438 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)

TheoremelpaddatriN 33439 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)

Theoremelpaddat 33440* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)

TheoremelpaddatiN 33441* Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theoremelpadd2at 33442 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)

Theoremelpadd2at2 33443 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)

TheorempaddunssN 33444 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)

Theoremelpadd0 33445 Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.)

Theorempaddval0 33446 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)

Theorempadd01 33447 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

Theorempadd02 33448 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

Theorempaddcom 33449 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)

Theorempaddssat 33450 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)

Theoremsspadd1 33451 A projective subspace sum is a superset of its first summand. (ssun1 3588 analog.) (Contributed by NM, 3-Jan-2012.)

Theoremsspadd2 33452 A projective subspace sum is a superset of its second summand. (ssun2 3589 analog.) (Contributed by NM, 3-Jan-2012.)

Theorempaddss1 33453 Subset law for projective subspace sum. (unss1 3594 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddss2 33454 Subset law for projective subspace sum. (unss2 3596 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddss12 33455 Subset law for projective subspace sum. (unss12 3597 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddasslem3 33458* Lemma for paddass 33474. Restate projective space axiom ps-2 33114. (Contributed by NM, 8-Jan-2012.)

Theorempaddasslem11 33466 Lemma for paddass 33474. The case when . (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem12 33467 Lemma for paddass 33474. The case when . (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem13 33468 Lemma for paddass 33474. The case when . (Unlike the proof in Maeda and Maeda, we don't need .) (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem17 33472 Lemma for paddass 33474. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)

Theorempaddass 33474 Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (Contributed by NM, 29-Dec-2011.)

Theorempadd12N 33475 Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempadd4N 33476 Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempaddidm 33477 Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)

TheorempaddclN 33478 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 33479 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddssw2 33480 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddss 33481 Subset law for projective subspace sum. (unss 3599 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempmodlem1 33482* Lemma for pmod1i 33484. (Contributed by NM, 9-Mar-2012.)

Theorempmodlem2 33483 Lemma for pmod1i 33484. (Contributed by NM, 9-Mar-2012.)

Theorempmod1i 33484 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)

Theorempmod2iN 33485 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

TheorempmodN 33486 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)

Theorempmodl42N 33487 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempmapjoin 33488 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 33489 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)

Theorempmapjat2 33490 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)

Theorempmapjlln1 33491 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 33492 A version of the modular law pmod1i 33484 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)

Theorematmod1i1 33493 Version of modular law pmod1i 33484 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 33494 Version of modular law pmod1i 33484 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 33495 Version of modular law pmod1i 33484 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 33496 Version of modular law pmod1i 33484 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 33497 Version of modular law pmod2iN 33485 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 33498 Version of modular law pmod2iN 33485 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 33499 Version of modular law pmod1i 33484 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 33500 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.)

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