P. 340 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 33901-34000 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	pclun2N 33901	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 33902*	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 33952. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pclcmpatN 33903*	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 33904	Extend class notation with polarity of projective subspace $m$.


Definition	df-polarityN 33905*	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 33906*	The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)


Theorem	polvalN 33907*	Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
$/\$

Theorem	polval2N 33908	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 33909	The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	polssatN 33910	The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pol0N 33911	The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)


Theorem	pol1N 33912	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 33913	The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)


Theorem	polpmapN 33914	The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polpmapN 33915	Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polvalN 33916	Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polssN 33917	A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	3polN 33918	Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$

Theorem	polcon3N 33919	Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	2polcon4bN 33920	Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	polcon2N 33921	Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	polcon2bN 33922	Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pclss2polN 33923	The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pcl0N 33924	The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)


Theorem	pcl0bN 33925	The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
$/\$

Theorem	pmaplubN 33926	The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
$/\$

Theorem	sspmaplubN 33927	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 33928	Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$

Theorem	paddunN 33929	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 5806.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	poldmj1N 33930	De Morgan's law for polarity of projective sum. (oldmj1 33224 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pmapj2N 33931	The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapocjN 33932	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 33933	The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	2polatN 33934	Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pnonsingN 33935	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 33936	Extend class notation with set of all closed projective subspaces for a Hilbert lattice.


Definition	df-psubclN 33937*	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 33938*	The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispsubclN 33939	The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	psubcliN 33940	Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	psubcli2N 33941	Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	psubclsubN 33942	A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	psubclssatN 33943	A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pmapidclN 33944	Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$/\$

Theorem	0psubclN 33945	The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)


Theorem	1psubclN 33946	The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)


Theorem	atpsubclN 33947	A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	pmapsubclN 33948	A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispsubcl2N 33949*	Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)


Theorem	psubclinN 33950	The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	paddatclN 33951	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.)
$/\$ $/\$

Theorem	pclfinclN 33952	The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 33902 and also pclcmpatN 33903. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
$/\$ $/\$

Theorem	linepsubclN 33953	A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	polsubclN 33954	A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	poml4N 33955	Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	poml5N 33956	Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	poml6N 33957	Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	osumcllem1N 33958	Lemma for osumclN 33969. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	osumcllem2N 33959	Lemma for osumclN 33969. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	osumcllem3N 33960	Lemma for osumclN 33969. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$

Theorem	osumcllem4N 33961	Lemma for osumclN 33969. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem5N 33962	Lemma for osumclN 33969. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	osumcllem6N 33963	Lemma for osumclN 33969. Use atom exchange hlatexch1 33397 to swap and . (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	osumcllem7N 33964*	Lemma for osumclN 33969. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem8N 33965	Lemma for osumclN 33969. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem9N 33966	Lemma for osumclN 33969. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem10N 33967	Lemma for osumclN 33969. Contradict osumcllem9N 33966. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem11N 33968	Lemma for osumclN 33969. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	osumclN 33969	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.)
$/\$ $/\$ $/\$

Theorem	pmapojoinN 33970	For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 33854 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidN 33971	Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 33955. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 33969. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pexmidlem1N 33972	Lemma for pexmidN 33971. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	pexmidlem2N 33973	Lemma for pexmidN 33971. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem3N 33974	Lemma for pexmidN 33971. Use atom exchange hlatexch1 33397 to swap and . (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem4N 33975*	Lemma for pexmidN 33971. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem5N 33976	Lemma for pexmidN 33971. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem6N 33977	Lemma for pexmidN 33971. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem7N 33978	Lemma for pexmidN 33971. Contradict pexmidlem6N 33977. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem8N 33979	Lemma for pexmidN 33971. The contradiction of pexmidlem6N 33977 and pexmidlem7N 33978 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.)
$/\$ $/\$ $/\$

Theorem	pexmidALTN 33980	Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 33955. 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.)
$/\$ $/\$

Theorem	pl42lem1N 33981	Lemma for pl42N 33985. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	pl42lem2N 33982	Lemma for pl42N 33985. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42lem3N 33983	Lemma for pl42N 33985. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pl42lem4N 33984	Lemma for pl42N 33985. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42N 33985	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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Syntax	clh 33986	Extend class notation with set of all co-atoms (lattice hyperplanes).


Syntax	claut 33987	Extend class notation with set of all lattice automorphisms.


Syntax	cwpointsN 33988	Extend class notation with W points.


Syntax	cpautN 33989	Extend class notation with set of all projective automorphisms.


Definition	df-lhyp 33990*	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.)


Definition	df-laut 33991*	Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)
$/\$

Definition	df-watsN 33992*	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.)
$\$

Definition	df-pautN 33993*	Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
$/\$

Theorem	watfvalN 33994*	The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$\$

Theorem	watvalN 33995	Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $\$

Theorem	iswatN 33996	The predicate "is a W atom" (corresponding to fiducial atom ). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lhpset 33997*	The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)


Theorem	islhp 33998	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)
$/\$

Theorem	islhp2 33999	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhpbase 34000	A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)

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