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**Theorem List for Metamath Proof Explorer -** 33301-33400 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	pmap1N 33301	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.)


Theorem	pmapsub 33302	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.)
$/\$

Theorem	pmapglbx 33303*	The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 33304, where we read as . Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
$/\$ $/\$

Theorem	pmapglb 33304*	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.)
$/\$ $/\$

Theorem	pmapglb2N 33305*	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.)
$/\$

Theorem	pmapglb2xN 33306*	The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 33305, 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.)
$/\$

Theorem	pmapmeet 33307	The projective map of a meet. (Contributed by NM, 25-Jan-2012.)
$./\$ $/\$ $/\$ $./\$

Theorem	isline2 33308*	Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
$.\/$ $/\$ $.\/$

Theorem	linepmap 33309	A line described with a projective map. (Contributed by NM, 3-Feb-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	isline3 33310*	Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	isline4N 33311*	Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
$/\$

Theorem	lneq2at 33312	A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lnatexN 33313*	There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	lnjatN 33314*	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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lncvrelatN 33315	A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	lncvrat 33316	A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	lncmp 33317	If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	2lnat 33318	Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
$./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	2atm2atN 33319	Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	2llnma1b 33320	Generalization of 2llnma1 33321. (Contributed by NM, 26-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	2llnma1 33321	Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	2llnma3r 33322	Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	2llnma2 33323	Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	2llnma2rN 33324	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

Theorem	cdlema1N 33325	A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $.\/$ $.\/$

Theorem	cdlema2N 33326	A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $./\$

Theorem	cdlemblem 33327	Lemma for cdlemb 33328. (Contributed by NM, 8-May-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdlemb 33328*	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.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Syntax	cpadd 33329	Extend class notation with projective subspace sum.


Definition	df-padd 33330*	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.)


Theorem	paddfval 33331*	Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
$.\/$ $.\/$

Theorem	paddval 33332*	Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	elpadd 33333*	Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)
$.\/$ $/\$ $/\$ $\/$ $\/$ $/\$ $.\/$

Theorem	elpaddn0 33334*	Member of projective subspace sum of nonempty sets. (Contributed by NM, 3-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	paddvaln0N 33335*	Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	elpaddri 33336	Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	elpaddatriN 33337	Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	elpaddat 33338*	Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	elpaddatiN 33339*	Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	elpadd2at 33340	Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	elpadd2at2 33341	Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	paddunssN 33342	Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	elpadd0 33343	Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.)
$/\$ $/\$ $/\$ $/\$ $\/$

Theorem	paddval0 33344	Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	padd01 33345	Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
$/\$

Theorem	padd02 33346	Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
$/\$

Theorem	paddcom 33347	Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
$/\$ $/\$

Theorem	paddssat 33348	A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)
$/\$ $/\$

Theorem	sspadd1 33349	A projective subspace sum is a superset of its first summand. (ssun1 3629 analog.) (Contributed by NM, 3-Jan-2012.)
$/\$ $/\$

Theorem	sspadd2 33350	A projective subspace sum is a superset of its second summand. (ssun2 3630 analog.) (Contributed by NM, 3-Jan-2012.)
$/\$ $/\$

Theorem	paddss1 33351	Subset law for projective subspace sum. (unss1 3635 analog.) (Contributed by NM, 7-Mar-2012.)
$/\$ $/\$

Theorem	paddss2 33352	Subset law for projective subspace sum. (unss2 3637 analog.) (Contributed by NM, 7-Mar-2012.)
$/\$ $/\$

Theorem	paddss12 33353	Subset law for projective subspace sum. (unss12 3638 analog.) (Contributed by NM, 7-Mar-2012.)
$/\$ $/\$ $/\$

Theorem	paddasslem1 33354	Lemma for paddass 33372. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	paddasslem2 33355	Lemma for paddass 33372. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	paddasslem3 33356*	Lemma for paddass 33372. Restate projective space axiom ps-2 33012. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$

Theorem	paddasslem4 33357*	Lemma for paddass 33372. Combine paddasslem1 33354, paddasslem2 33355, and paddasslem3 33356. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$

Theorem	paddasslem5 33358	Lemma for paddass 33372. Show by contradiction. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	paddasslem6 33359	Lemma for paddass 33372. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	paddasslem7 33360	Lemma for paddass 33372. Combine paddasslem5 33358 and paddasslem6 33359. (Contributed by NM, 9-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	paddasslem8 33361	Lemma for paddass 33372. (Contributed by NM, 8-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	paddasslem9 33362	Lemma for paddass 33372. Combine paddasslem7 33360 and paddasslem8 33361. (Contributed by NM, 9-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	paddasslem10 33363	Lemma for paddass 33372. Use paddasslem4 33357 to eliminate from paddasslem9 33362. (Contributed by NM, 9-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	paddasslem11 33364	Lemma for paddass 33372. The case when . (Contributed by NM, 11-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	paddasslem12 33365	Lemma for paddass 33372. The case when . (Contributed by NM, 11-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	paddasslem13 33366	Lemma for paddass 33372. The case when $.\/$ . (Unlike the proof in Maeda and Maeda, we don't need .) (Contributed by NM, 11-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	paddasslem14 33367	Lemma for paddass 33372. Remove , , and $.\/$ from antecedent of paddasslem10 33363, using paddasslem11 33364, paddasslem12 33365, and paddasslem13 33366. (Contributed by NM, 11-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	paddasslem15 33368	Lemma for paddass 33372. Use elpaddn0 33334 to eliminate and from paddasslem14 33367. (Contributed by NM, 11-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	paddasslem16 33369	Lemma for paddass 33372. Use elpaddn0 33334 to eliminate and from paddasslem15 33368. (Contributed by NM, 11-Jan-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	paddasslem17 33370	Lemma for paddass 33372. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	paddasslem18 33371	Lemma for paddass 33372. Combine paddasslem16 33369 and paddasslem17 33370. (Contributed by NM, 12-Jan-2012.)
$/\$ $/\$ $/\$

Theorem	paddass 33372	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 33373	Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	padd4N 33374	Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	paddidm 33375	Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
$/\$

Theorem	paddclN 33376	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 33377	Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	paddssw2 33378	Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	paddss 33379	Subset law for projective subspace sum. (unss 3640 analog.) (Contributed by NM, 7-Mar-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	pmodlem1 33380*	Lemma for pmod1i 33382. (Contributed by NM, 9-Mar-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pmodlem2 33381	Lemma for pmod1i 33382. (Contributed by NM, 9-Mar-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pmod1i 33382	The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
$/\$ $/\$ $/\$

Theorem	pmod2iN 33383	Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pmodN 33384	The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pmodl42N 33385	Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	pmapjoin 33386	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 33387	The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapjat2 33388	The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)
$.\/$ $/\$ $/\$ $.\/$

Theorem	pmapjlln1 33389	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 33390	A version of the modular law pmod1i 33382 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	atmod1i1 33391	Version of modular law pmod1i 33382 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 33392	Version of modular law pmod1i 33382 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 33393	Version of modular law pmod1i 33382 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 33394	Version of modular law pmod1i 33382 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 33395	Version of modular law pmod2iN 33383 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 33396	Version of modular law pmod2iN 33383 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 33397	Version of modular law pmod1i 33382 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 33398	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 33399	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 33400	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.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$ $./\$

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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 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40161

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