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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pclclN 33501 | Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.) |
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Theorem | elpclN 33502* | Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
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Theorem | elpcliN 33503 | Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
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Theorem | pclssN 33504 | Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.) |
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Theorem | pclssidN 33505 | A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
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Theorem | pclidN 33506 | The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.) |
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Theorem | pclbtwnN 33507 | 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.) |
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Theorem | pclunN 33508 | 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.) |
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Theorem | pclun2N 33509 | The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
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Theorem | pclfinN 33510* | 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 33560. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.) |
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Theorem | pclcmpatN 33511* | 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.) |
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Syntax | cpolN 33512 | Extend class notation with polarity of projective subspace $m$. |
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Definition | df-polarityN 33513* |
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 ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | polfvalN 33514* | The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.) |
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Theorem | polvalN 33515* | Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.) |
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Theorem | polval2N 33516 | 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.) |
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Theorem | polsubN 33517 | The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
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Theorem | polssatN 33518 | The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
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Theorem | pol0N 33519 | The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.) |
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Theorem | pol1N 33520 | 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.) |
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Theorem | 2pol0N 33521 | The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
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Theorem | polpmapN 33522 | The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
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Theorem | 2polpmapN 33523 | Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
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Theorem | 2polvalN 33524 | Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | 2polssN 33525 | A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.) |
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Theorem | 3polN 33526 | Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
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Theorem | polcon3N 33527 | Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
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Theorem | 2polcon4bN 33528 | Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
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Theorem | polcon2N 33529 | Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
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Theorem | polcon2bN 33530 | Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
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Theorem | pclss2polN 33531 | The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
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Theorem | pcl0N 33532 | The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
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Theorem | pcl0bN 33533 | The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
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Theorem | pmaplubN 33534 | The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.) |
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Theorem | sspmaplubN 33535 | A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
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Theorem | 2pmaplubN 33536 | Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
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Theorem | paddunN 33537 | 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 5892.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
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Theorem | poldmj1N 33538 | De Morgan's law for polarity of projective sum. (oldmj1 32832 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.) |
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Theorem | pmapj2N 33539 | The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.) |
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Theorem | pmapocjN 33540 | The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.) |
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Theorem | polatN 33541 | The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) |
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Theorem | 2polatN 33542 | Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | pnonsingN 33543 | 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.) |
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Syntax | cpscN 33544 | Extend class notation with set of all closed projective subspaces for a Hilbert lattice. |
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Definition | df-psubclN 33545* | 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.) |
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Theorem | psubclsetN 33546* | The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
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Theorem | ispsubclN 33547 | The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
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Theorem | psubcliN 33548 | Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
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Theorem | psubcli2N 33549 | Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
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Theorem | psubclsubN 33550 | A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
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Theorem | psubclssatN 33551 | A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | pmapidclN 33552 | Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.) |
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Theorem | 0psubclN 33553 | The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | 1psubclN 33554 | The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | atpsubclN 33555 | A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | pmapsubclN 33556 | A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
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Theorem | ispsubcl2N 33557* | Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
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Theorem | psubclinN 33558 | The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
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Theorem | paddatclN 33559 | 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.) |
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Theorem | pclfinclN 33560 | The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 33510 and also pclcmpatN 33511. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
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Theorem | linepsubclN 33561 | A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | polsubclN 33562 | A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
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Theorem | poml4N 33563 | Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
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Theorem | poml5N 33564 | Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
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Theorem | poml6N 33565 | Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem1N 33566 | Lemma for osumclN 33577. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem2N 33567 | Lemma for osumclN 33577. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem3N 33568 | Lemma for osumclN 33577. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem4N 33569 | Lemma for osumclN 33577. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem5N 33570 | Lemma for osumclN 33577. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem6N 33571 |
Lemma for osumclN 33577. Use atom exchange hlatexch1 33005 to swap ![]() ![]() |
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Theorem | osumcllem7N 33572* | Lemma for osumclN 33577. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem8N 33573 | Lemma for osumclN 33577. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem9N 33574 | Lemma for osumclN 33577. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem10N 33575 | Lemma for osumclN 33577. Contradict osumcllem9N 33574. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumcllem11N 33576 | Lemma for osumclN 33577. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
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Theorem | osumclN 33577 |
Closure of orthogonal sum. If ![]() ![]() |
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Theorem | pmapojoinN 33578 | For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 33462 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.) |
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Theorem | pexmidN 33579 | Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 33563. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 33577. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
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Theorem | pexmidlem1N 33580 |
Lemma for pexmidN 33579. Holland's proof implicitly requires ![]() ![]() ![]() |
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Theorem | pexmidlem2N 33581 | Lemma for pexmidN 33579. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
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Theorem | pexmidlem3N 33582 |
Lemma for pexmidN 33579. Use atom exchange hlatexch1 33005 to swap ![]() ![]() |
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Theorem | pexmidlem4N 33583* | Lemma for pexmidN 33579. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
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Theorem | pexmidlem5N 33584 | Lemma for pexmidN 33579. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
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Theorem | pexmidlem6N 33585 | Lemma for pexmidN 33579. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.) |
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Theorem | pexmidlem7N 33586 | Lemma for pexmidN 33579. Contradict pexmidlem6N 33585. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.) |
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Theorem | pexmidlem8N 33587 |
Lemma for pexmidN 33579. The contradiction of pexmidlem6N 33585 and
pexmidlem7N 33586 shows that there can be no atom ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | pexmidALTN 33588 |
Excluded middle law for closed projective subspaces, which is equivalent
to (and derived from) the orthomodular law poml4N 33563. Lemma 3.3(2) in
[Holland95] p. 215. In our proof, we
use the variables ![]() ![]() ![]() ![]() ![]() |
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Theorem | pl42lem1N 33589 | Lemma for pl42N 33593. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.) |
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Theorem | pl42lem2N 33590 | Lemma for pl42N 33593. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.) |
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Theorem | pl42lem3N 33591 | Lemma for pl42N 33593. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.) |
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Theorem | pl42lem4N 33592 | Lemma for pl42N 33593. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.) |
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Theorem | pl42N 33593 | 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.) |
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Syntax | clh 33594 | Extend class notation with set of all co-atoms (lattice hyperplanes). |
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Syntax | claut 33595 | Extend class notation with set of all lattice automorphisms. |
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Syntax | cwpointsN 33596 | Extend class notation with W points. |
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Syntax | cpautN 33597 | Extend class notation with set of all projective automorphisms. |
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Definition | df-lhyp 33598* | 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.) |
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Definition | df-laut 33599* | Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.) |
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Definition | df-watsN 33600* |
Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference)
atom" ![]() ![]() ![]() ![]() ![]() ![]() |
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