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Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempolcon2N 30401 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theorempolcon2bN 30402 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theorempclss2polN 30403 The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

Theorempcl0N 30404 The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

Theorempcl0bN 30405 The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

TheorempmaplubN 30406 The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)

TheoremsspmaplubN 30407 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 30408 Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

TheorempaddunN 30409 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 5691.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)

Theorempoldmj1N 30410 De Morgan's law for polarity of projective sum. (oldmj1 29704 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)

Theorempmapj2N 30411 The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)

TheorempmapocjN 30412 The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)

TheorempolatN 30413 The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorem2polatN 30414 Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

TheorempnonsingN 30415 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 30416 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.

Definitiondf-psubclN 30417* 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 30418* The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheoremispsubclN 30419 The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempsubcliN 30420 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

Theorempsubcli2N 30421 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempsubclsubN 30422 A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

TheorempsubclssatN 30423 A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

TheorempmapidclN 30424 Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Theorem0psubclN 30425 The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

Theorem1psubclN 30426 The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

TheorematpsubclN 30427 A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

TheorempmapsubclN 30428 A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theoremispsubcl2N 30429* Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

TheorempsubclinN 30430 The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

TheorempaddatclN 30431 The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

TheorempclfinclN 30432 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 30382 and also pclcmpatN 30383. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

TheoremlinepsubclN 30433 A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

TheorempolsubclN 30434 A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Theorempoml4N 30435 Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

Theorempoml5N 30436 Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theorempoml6N 30437 Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem1N 30438 Lemma for osumclN 30449. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem2N 30439 Lemma for osumclN 30449. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem3N 30440 Lemma for osumclN 30449. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem4N 30441 Lemma for osumclN 30449. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem5N 30442 Lemma for osumclN 30449. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem6N 30443 Lemma for osumclN 30449. Use atom exchange hlatexch1 29877 to swap and . (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem7N 30444* Lemma for osumclN 30449. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem8N 30445 Lemma for osumclN 30449. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem9N 30446 Lemma for osumclN 30449. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem10N 30447 Lemma for osumclN 30449. Contradict osumcllem9N 30446. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem11N 30448 Lemma for osumclN 30449. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

TheoremosumclN 30449 Closure of orthogonal sum. If and are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

TheorempmapojoinN 30450 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 30334 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)

TheorempexmidN 30451 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 30435. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 30449. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Theorempexmidlem1N 30452 Lemma for pexmidN 30451. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem2N 30453 Lemma for pexmidN 30451. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem3N 30454 Lemma for pexmidN 30451. Use atom exchange hlatexch1 29877 to swap and . (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem4N 30455* Lemma for pexmidN 30451. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem5N 30456 Lemma for pexmidN 30451. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem6N 30457 Lemma for pexmidN 30451. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem7N 30458 Lemma for pexmidN 30451. Contradict pexmidlem6N 30457. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem8N 30459 Lemma for pexmidN 30451. The contradiction of pexmidlem6N 30457 and pexmidlem7N 30458 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

TheorempexmidALTN 30460 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 30435. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Theorempl42lem1N 30461 Lemma for pl42N 30465. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42lem2N 30462 Lemma for pl42N 30465. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42lem3N 30463 Lemma for pl42N 30465. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42lem4N 30464 Lemma for pl42N 30465. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42N 30465 Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Syntaxclh 30466 Extend class notation with set of all co-atoms (lattice hyperplanes).

Syntaxclaut 30467 Extend class notation with set of all lattice automorphisms.

SyntaxcwpointsN 30468 Extend class notation with W points.

SyntaxcpautN 30469 Extend class notation with set of all projective automorphisms.

Definitiondf-lhyp 30470* Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e. all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)

Definitiondf-laut 30471* Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)

Definitiondf-watsN 30472* Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom" . These are all atoms not in the polarity of , which is the hyperplane determined by . Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)

Definitiondf-pautN 30473* Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)

TheoremwatfvalN 30474* The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremwatvalN 30475 Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremiswatN 30476 The predicate "is a W atom" (corresponding to fiducial atom ). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Theoremlhpset 30477* The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)

Theoremislhp 30478 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)

Theoremislhp2 30479 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)

Theoremlhpbase 30480 A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)

Theoremlhp1cvr 30481 The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)

Theoremlhplt 30482 An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Theoremlhp2lt 30483 The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)

Theoremlhpexlt 30484* There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Theoremlhp0lt 30485 A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Theoremlhpn0 30486 A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)

Theoremlhpexle 30487* There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)

Theoremlhpexnle 30488* There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)

Theoremlhpexle1lem 30489* Lemma for lhpexle1 30490 and others that eliminates restrictions on . (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle1 30490* There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle2lem 30491* Lemma for lhpexle2 30492. (Contributed by NM, 19-Jun-2013.)

Theoremlhpexle2 30492* There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle3lem 30493* There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant*,simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)

Theoremlhpexle3 30494* There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)

Theoremlhpex2leN 30495* There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)

Theoremlhpoc 30496 The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)

Theoremlhpoc2N 30497 The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Theoremlhpocnle 30498 The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)

Theoremlhpocat 30499 The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)

Theoremlhpocnel 30500 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)

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