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**Theorem List for Metamath Proof Explorer -** 35201-35300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	4atexlempw 35201	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$

Theorem	4atexlemp 35202	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemq 35203	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlems 35204	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemt 35205	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemutvt 35206	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempnq 35207	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemnslpq 35208	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemkl 35209	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemkc 35210	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemwb 35211	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempsb 35212	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlemqtb 35213	Lemma for 4atexlem7 35227. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempns 35214	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemswapqr 35215	Lemma for 4atexlem7 35227. Swap and , so that theorems involving can be reused for . Note that must be expanded because it involves . (Contributed by NM, 25-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemu 35216	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemv 35217	Lemma for 4atexlem7 35227. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemunv 35218	Lemma for 4atexlem7 35227. (Contributed by NM, 21-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemtlw 35219	Lemma for 4atexlem7 35227. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemntlpq 35220	Lemma for 4atexlem7 35227. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemc 35221	Lemma for 4atexlem7 35227. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemnclw 35222	Lemma for 4atexlem7 35227. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemex2 35223*	Lemma for 4atexlem7 35227. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemcnd 35224	Lemma for 4atexlem7 35227. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	4atexlemex4 35225*	Lemma for 4atexlem7 35227. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemex6 35226*	Lemma for 4atexlem7 35227. (Contributed by NM, 25-Nov-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atexlem7 35227*	Whenever there are at least 4 atoms under $.\/$ (specifically, , , , and $.\/$ $./\$ ), there are also at least 4 atoms under $.\/$ . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p $\/$ q/0 and hence p $\/$ s/0 contains at least four atoms..." Note that by cvlsupr2 34496, our $.\/$ $.\/$ is a shorter way to express $/\$ $/\$ $.\/$ . With a longer proof, the condition $.\/$ could be eliminated (see 4atex 35228), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex 35228*	Whenever there are at least 4 atoms under $.\/$ (specifically, , , , and $.\/$ $./\$ ), there are also at least 4 atoms under $.\/$ . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p $\/$ q/0 and hence p $\/$ s/0 contains at least four atoms..." Note that by cvlsupr2 34496, our $.\/$ $.\/$ is a shorter way to express $/\$ $/\$ $.\/$ . (Contributed by NM, 27-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2 35229*	More general version of 4atex 35228 for a line $.\/$ not necessarily connected to $.\/$ . (Contributed by NM, 27-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0aOLDN 35230*	Same as 4atex2 35229 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0bOLDN 35231*	Same as 4atex2 35229 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0cOLDN 35232*	Same as 4atex2 35229 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 35230 or 4atex2-0bOLDN 35231? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex3 35233*	More general version of 4atex 35228 for a line $.\/$ not necessarily connected to $.\/$ . (Contributed by NM, 29-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lautset 35234*	The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
$/\$

Theorem	islaut 35235*	The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)
$/\$

Theorem	lautle 35236	Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
$/\$ $/\$ $/\$

Theorem	laut1o 35237	A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)
$/\$

Theorem	laut11 35238	One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcl 35239	A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	lautcnvclN 35240	Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lautcnvle 35241	Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcnv 35242	The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
$/\$

Theorem	lautlt 35243	Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcvr 35244	Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautj 35245	Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lautm 35246	Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	lauteq 35247*	A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)
$/\$ $/\$ $/\$

Theorem	idlaut 35248	The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)


Theorem	lautco 35249	The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$

Theorem	pautsetN 35250*	The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispautN 35251*	The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$

Syntax	cldil 35252	Extend class notation with set of all lattice dilations.


Syntax	cltrn 35253	Extend class notation with set of all lattice translations.


Syntax	cdilN 35254	Extend class notation with set of all dilations.


Syntax	ctrnN 35255	Extend class notation with set of all translations.


Definition	df-ldil 35256*	Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)


Definition	df-ltrn 35257*	Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$/\$

Definition	df-dilN 35258*	Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)


Definition	df-trnN 35259*	Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)


Theorem	ldilfset 35260*	The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)


Theorem	ldilset 35261*	The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)
$/\$

Theorem	isldil 35262*	The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$/\$ $/\$

Theorem	ldillaut 35263	A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ldil1o 35264	A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$

Theorem	ldilval 35265	Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	idldil 35266	The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)
$/\$

Theorem	ldilcnv 35267	The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)
$/\$ $/\$

Theorem	ldilco 35268	The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$ $/\$

Theorem	ltrnfset 35269*	The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	ltrnset 35270*	The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	isltrn 35271*	The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	isltrn2N 35272*	The predicate "is a lattice translation". Version of isltrn 35271 that considers only different and . TODO: Can this eliminate some separate proofs for the case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	ltrnu 35273	Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom . Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	ltrnldil 35274	A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrnlaut 35275	A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrn1o 35276	A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrncl 35277	Closure of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrn11 35278	One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncnvnid 35279	If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	ltrncoidN 35280	Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analog of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	ltrnle 35281	Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncnvleN 35282	Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrnm 35283	Lattice translation of a meet. (Contributed by NM, 20-May-2012.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	ltrnj 35284	Lattice translation of a meet. TODO: change antecedent to (Contributed by NM, 25-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	ltrncvr 35285	Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrnval1 35286	Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrnid 35287*	A lattice translation is the identity function iff all atoms not under the fiducial co-atom are equal to their values. (Contributed by NM, 24-May-2012.)
$/\$ $/\$

Theorem	ltrnnid 35288*	If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom and not equal to its translation. (Contributed by NM, 24-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrnatb 35289	The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrncnvatb 35290	The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	ltrnel 35291	The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrnat 35292	The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 35291 uses. (Contributed by NM, 25-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrncnvat 35293	The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)
$/\$ $/\$ $/\$

Theorem	ltrncnvel 35294	The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrncoelN 35295	Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 35291 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrncoat 35296	Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 35291, ltrnat 35292 uses. (Contributed by NM, 1-May-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncoval 35297	Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncnv 35298	The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)
$/\$ $/\$

Theorem	ltrn11at 35299	Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrneq2 35300*	The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)
$/\$ $/\$ $/\$

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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 37201-37279

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