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**Theorem List for Metamath Proof Explorer -** 33501-33600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	paddass 33501	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.)
$/\$ $/\$ $/\$

Theorem	padd12N 33502	Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	padd4N 33503	Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	paddidm 33504	Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
$/\$

Theorem	paddclN 33505	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.)
$/\$ $/\$

Theorem	paddssw1 33506	Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	paddssw2 33507	Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	paddss 33508	Subset law for projective subspace sum. (unss 3545 analog.) (Contributed by NM, 7-Mar-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	pmodlem1 33509*	Lemma for pmod1i 33511. (Contributed by NM, 9-Mar-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pmodlem2 33510	Lemma for pmod1i 33511. (Contributed by NM, 9-Mar-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pmod1i 33511	The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
$/\$ $/\$ $/\$

Theorem	pmod2iN 33512	Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pmodN 33513	The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pmodl42N 33514	Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	pmapjoin 33515	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.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapjat1 33516	The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapjat2 33517	The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapjlln1 33518	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.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	hlmod1i 33519	A version of the modular law pmod1i 33511 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	atmod1i1 33520	Version of modular law pmod1i 33511 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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	atmod1i1m 33521	Version of modular law pmod1i 33511 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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$

Theorem	atmod1i2 33522	Version of modular law pmod1i 33511 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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	llnmod1i2 33523	Version of modular law pmod1i 33511 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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	atmod2i1 33524	Version of modular law pmod2iN 33512 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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	atmod2i2 33525	Version of modular law pmod2iN 33512 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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	llnmod2i2 33526	Version of modular law pmod1i 33511 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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	atmod3i1 33527	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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $.\/$

Theorem	atmod3i2 33528	Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $.\/$

Theorem	atmod4i1 33529	Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$ $./\$

Theorem	atmod4i2 33530	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-Mar-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$ $./\$

Theorem	llnexchb2lem 33531	Lemma for llnexchb2 33532. (Contributed by NM, 17-Nov-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$

Theorem	llnexchb2 33532	Line exchange property (compare cvlatexchb2 32999 for atoms). (Contributed by NM, 17-Nov-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$ $./\$

Theorem	llnexch2N 33533	Line exchange property (compare cvlatexch2 33001 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$

Theorem	dalawlem1 33534	Lemma for dalaw 33549. Special case of dath2 33400, where is replaced by $.\/$ $./\$ $.\/$ . The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 33400. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem2 33535	Lemma for dalaw 33549. Utility lemma that breaks $.\/$ $./\$ $.\/$ into a join of two pieces. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$

Theorem	dalawlem3 33536	Lemma for dalaw 33549. First piece of dalawlem5 33538. (Contributed by NM, 4-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem4 33537	Lemma for dalaw 33549. Second piece of dalawlem5 33538. (Contributed by NM, 4-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem5 33538	Lemma for dalaw 33549. Special case to eliminate the requirement $.\/$ $./\$ $.\/$ $.\/$ in dalawlem1 33534. (Contributed by NM, 4-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem6 33539	Lemma for dalaw 33549. First piece of dalawlem8 33541. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem7 33540	Lemma for dalaw 33549. Second piece of dalawlem8 33541. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem8 33541	Lemma for dalaw 33549. Special case to eliminate the requirement $.\/$ $./\$ $.\/$ $.\/$ in dalawlem1 33534. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem9 33542	Lemma for dalaw 33549. Special case to eliminate the requirement $.\/$ $./\$ $.\/$ $.\/$ in dalawlem1 33534. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem10 33543	Lemma for dalaw 33549. Combine dalawlem5 33538, dalawlem8 33541, and dalawlem9 . (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem11 33544	Lemma for dalaw 33549. First part of dalawlem13 33546. (Contributed by NM, 17-Sep-2012.)
$.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem12 33545	Lemma for dalaw 33549. Second part of dalawlem13 33546. (Contributed by NM, 17-Sep-2012.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem13 33546	Lemma for dalaw 33549. Special case to eliminate the requirement $.\/$ $.\/$ in dalawlem1 33534. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem14 33547	Lemma for dalaw 33549. Combine dalawlem10 33543 and dalawlem13 33546. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalawlem15 33548	Lemma for dalaw 33549. Swap variable triples and in dalawlem14 33547, to obtain the elimination of the remaining conditions in dalawlem1 33534. (Contributed by NM, 6-Oct-2012.)
$.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	dalaw 33549	Desargues' law, derived from Desargues' theorem dath 33399 and with no conditions on the atoms. If triples and are centrally perspective, i.e. $.\/$ $./\$ $.\/$ $.\/$ , then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Syntax	cpclN 33550	Extend class notation with projective subspace closure.


Definition	df-pclN 33551*	Projective subspace closure, which is the smallest projective subspace containing an arbitrary set of atoms. The subspace closure of the union of a set of projective subspaces is their supremum in . Related to an analogous definition of closure used in Lemma 3.1.4 of [PtakPulmannova] p. 68. (Note that this closure is not necessarily one of the closed projective subspaces of df-psubclN 33598.) (Contributed by NM, 7-Sep-2013.)


Theorem	pclfvalN 33552*	The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)


Theorem	pclvalN 33553*	Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pclclN 33554	Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	elpclN 33555*	Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	elpcliN 33556	Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pclssN 33557	Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pclssidN 33558	A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pclidN 33559	The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pclbtwnN 33560	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.)
$/\$ $/\$ $/\$

Theorem	pclunN 33561	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.)
$/\$ $/\$

Theorem	pclun2N 33562	The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pclfinN 33563*	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 33613. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pclcmpatN 33564*	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.)
$/\$ $/\$ $/\$

Syntax	cpolN 33565	Extend class notation with polarity of projective subspace $m$.


Definition	df-polarityN 33566*	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 ensures it is defined when . (Contributed by NM, 23-Oct-2011.)


Theorem	polfvalN 33567*	The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)


Theorem	polvalN 33568*	Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
$/\$

Theorem	polval2N 33569	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.)
$/\$

Theorem	polsubN 33570	The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	polssatN 33571	The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pol0N 33572	The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)


Theorem	pol1N 33573	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.)


Theorem	2pol0N 33574	The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)


Theorem	polpmapN 33575	The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polpmapN 33576	Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polvalN 33577	Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polssN 33578	A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	3polN 33579	Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$

Theorem	polcon3N 33580	Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	2polcon4bN 33581	Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	polcon2N 33582	Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	polcon2bN 33583	Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pclss2polN 33584	The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pcl0N 33585	The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)


Theorem	pcl0bN 33586	The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pmaplubN 33587	The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
$/\$

Theorem	sspmaplubN 33588	A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$

Theorem	2pmaplubN 33589	Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$

Theorem	paddunN 33590	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 5710.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	poldmj1N 33591	De Morgan's law for polarity of projective sum. (oldmj1 32885 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pmapj2N 33592	The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapocjN 33593	The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $.\/$

Theorem	polatN 33594	The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polatN 33595	Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pnonsingN 33596	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.)
$/\$

Syntax	cpscN 33597	Extend class notation with set of all closed projective subspaces for a Hilbert lattice.


Definition	df-psubclN 33598*	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.)
$/\$

Theorem	psubclsetN 33599*	The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispsubclN 33600	The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

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7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35636

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