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**Theorem List for Metamath Proof Explorer -** 34701-34800 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	lhpmcvr6N 34701*	Specialization of lhpmcvr2 34697. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$

Theorem	lhpm0atN 34702	If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	lhpmat 34703	An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	lhpmatb 34704	An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)
$./\$ $/\$ $/\$ $./\$

Theorem	lhp2at0 34705	Join and meet with different atoms under co-atom . (Contributed by NM, 15-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhp2atnle 34706	Inequality for 2 different atoms under co-atom . (Contributed by NM, 17-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhp2atne 34707	Inequality for joins with 2 different atoms under co-atom . (Contributed by NM, 22-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lhp2at0nle 34708	Inequality for 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhp2at0ne 34709	Inequality for joins with 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lhpelim 34710	Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 34703 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$

Theorem	lhpmod2i2 34711	Modular law for hyperplanes analogous to atmod2i2 34535 for atoms. (Contributed by NM, 9-Feb-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	lhpmod6i1 34712	Modular law for hyperplanes analogous to complement of atmod2i1 34534 for atoms. (Contributed by NM, 1-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	lhprelat3N 34713*	The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 34085. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	cdlemb2 34714*	Given two atoms not under the fiducial (reference) co-atom , there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhple 34715	Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat 34716	Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat4N 34717	Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat2 34718	Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpat3 34719	There is only one atom under both $.\/$ and co-atom . (Contributed by NM, 21-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	4atexlemk 34720	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemw 34721	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempw 34722	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$

Theorem	4atexlemp 34723	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemq 34724	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlems 34725	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemt 34726	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemutvt 34727	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempnq 34728	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemnslpq 34729	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemkl 34730	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemkc 34731	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemwb 34732	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempsb 34733	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlemqtb 34734	Lemma for 4atexlem7 34748. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempns 34735	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemswapqr 34736	Lemma for 4atexlem7 34748. 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 34737	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemv 34738	Lemma for 4atexlem7 34748. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemunv 34739	Lemma for 4atexlem7 34748. (Contributed by NM, 21-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemtlw 34740	Lemma for 4atexlem7 34748. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemntlpq 34741	Lemma for 4atexlem7 34748. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemc 34742	Lemma for 4atexlem7 34748. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemnclw 34743	Lemma for 4atexlem7 34748. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemex2 34744*	Lemma for 4atexlem7 34748. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemcnd 34745	Lemma for 4atexlem7 34748. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	4atexlemex4 34746*	Lemma for 4atexlem7 34748. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemex6 34747*	Lemma for 4atexlem7 34748. (Contributed by NM, 25-Nov-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atexlem7 34748*	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 34017, our $.\/$ $.\/$ is a shorter way to express $/\$ $/\$ $.\/$ . With a longer proof, the condition $.\/$ could be eliminated (see 4atex 34749), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex 34749*	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 34017, our $.\/$ $.\/$ is a shorter way to express $/\$ $/\$ $.\/$ . (Contributed by NM, 27-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2 34750*	More general version of 4atex 34749 for a line $.\/$ not necessarily connected to $.\/$ . (Contributed by NM, 27-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0aOLDN 34751*	Same as 4atex2 34750 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0bOLDN 34752*	Same as 4atex2 34750 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0cOLDN 34753*	Same as 4atex2 34750 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 34751 or 4atex2-0bOLDN 34752? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex3 34754*	More general version of 4atex 34749 for a line $.\/$ not necessarily connected to $.\/$ . (Contributed by NM, 29-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lautset 34755*	The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
$/\$

Theorem	islaut 34756*	The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)
$/\$

Theorem	lautle 34757	Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
$/\$ $/\$ $/\$

Theorem	laut1o 34758	A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)
$/\$

Theorem	laut11 34759	One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcl 34760	A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	lautcnvclN 34761	Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lautcnvle 34762	Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcnv 34763	The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
$/\$

Theorem	lautlt 34764	Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcvr 34765	Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautj 34766	Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lautm 34767	Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	lauteq 34768*	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 34769	The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)


Theorem	lautco 34770	The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$

Theorem	pautsetN 34771*	The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispautN 34772*	The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$

Syntax	cldil 34773	Extend class notation with set of all lattice dilations.


Syntax	cltrn 34774	Extend class notation with set of all lattice translations.


Syntax	cdilN 34775	Extend class notation with set of all dilations.


Syntax	ctrnN 34776	Extend class notation with set of all translations.


Definition	df-ldil 34777*	Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)


Definition	df-ltrn 34778*	Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$/\$

Definition	df-dilN 34779*	Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)


Definition	df-trnN 34780*	Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)


Theorem	ldilfset 34781*	The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)


Theorem	ldilset 34782*	The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)
$/\$

Theorem	isldil 34783*	The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$/\$ $/\$

Theorem	ldillaut 34784	A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ldil1o 34785	A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$

Theorem	ldilval 34786	Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	idldil 34787	The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)
$/\$

Theorem	ldilcnv 34788	The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)
$/\$ $/\$

Theorem	ldilco 34789	The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$ $/\$

Theorem	ltrnfset 34790*	The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	ltrnset 34791*	The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	isltrn 34792*	The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	isltrn2N 34793*	The predicate "is a lattice translation". Version of isltrn 34792 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 34794	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 34795	A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrnlaut 34796	A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrn1o 34797	A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrncl 34798	Closure of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrn11 34799	One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncnvnid 34800	If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-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

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