P. 334 of Theorem List - Metamath Proof Explorer

Home	Metamath Proof Explorer Theorem List (p. 334 of 393)	< Previous Next >
		Browser slow? Try the Unicode version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List

Color key:

Metamath Proof Explorer
(1-26397)

Hilbert Space Explorer
(26398-27920)

Users' Mathboxes
(27921-39291)

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

Theorem	lhpelim 33301	Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 33294 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$

Theorem	lhpmod2i2 33302	Modular law for hyperplanes analogous to atmod2i2 33126 for atoms. (Contributed by NM, 9-Feb-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	lhpmod6i1 33303	Modular law for hyperplanes analogous to complement of atmod2i1 33125 for atoms. (Contributed by NM, 1-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	lhprelat3N 33304*	The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 32676. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	cdlemb2 33305*	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 33306	Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat 33307	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 33308	Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat2 33309	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 33310	There is only one atom under both $.\/$ and co-atom . (Contributed by NM, 21-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	4atexlemk 33311	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemw 33312	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempw 33313	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$

Theorem	4atexlemp 33314	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemq 33315	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlems 33316	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemt 33317	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemutvt 33318	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempnq 33319	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemnslpq 33320	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemkl 33321	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemkc 33322	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemwb 33323	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempsb 33324	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlemqtb 33325	Lemma for 4atexlem7 33339. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempns 33326	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemswapqr 33327	Lemma for 4atexlem7 33339. 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 33328	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemv 33329	Lemma for 4atexlem7 33339. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemunv 33330	Lemma for 4atexlem7 33339. (Contributed by NM, 21-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemtlw 33331	Lemma for 4atexlem7 33339. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemntlpq 33332	Lemma for 4atexlem7 33339. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemc 33333	Lemma for 4atexlem7 33339. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemnclw 33334	Lemma for 4atexlem7 33339. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	4atexlemex2 33335*	Lemma for 4atexlem7 33339. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemcnd 33336	Lemma for 4atexlem7 33339. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	4atexlemex4 33337*	Lemma for 4atexlem7 33339. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemex6 33338*	Lemma for 4atexlem7 33339. (Contributed by NM, 25-Nov-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atexlem7 33339*	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 32608, our $.\/$ $.\/$ is a shorter way to express $/\$ $/\$ $.\/$ . With a longer proof, the condition $.\/$ could be eliminated (see 4atex 33340), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex 33340*	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 32608, our $.\/$ $.\/$ is a shorter way to express $/\$ $/\$ $.\/$ . (Contributed by NM, 27-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2 33341*	More general version of 4atex 33340 for a line $.\/$ not necessarily connected to $.\/$ . (Contributed by NM, 27-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0aOLDN 33342*	Same as 4atex2 33341 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0bOLDN 33343*	Same as 4atex2 33341 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex2-0cOLDN 33344*	Same as 4atex2 33341 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 33342 or 4atex2-0bOLDN 33343? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	4atex3 33345*	More general version of 4atex 33340 for a line $.\/$ not necessarily connected to $.\/$ . (Contributed by NM, 29-May-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lautset 33346*	The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
$/\$

Theorem	islaut 33347*	The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)
$/\$

Theorem	lautle 33348	Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
$/\$ $/\$ $/\$

Theorem	laut1o 33349	A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)
$/\$

Theorem	laut11 33350	One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcl 33351	A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	lautcnvclN 33352	Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lautcnvle 33353	Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcnv 33354	The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
$/\$

Theorem	lautlt 33355	Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautcvr 33356	Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	lautj 33357	Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lautm 33358	Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	lauteq 33359*	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 33360	The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)


Theorem	lautco 33361	The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$

Theorem	pautsetN 33362*	The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$

Theorem	ispautN 33363*	The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$

Syntax	cldil 33364	Extend class notation with set of all lattice dilations.


Syntax	cltrn 33365	Extend class notation with set of all lattice translations.


Syntax	cdilN 33366	Extend class notation with set of all dilations.


Syntax	ctrnN 33367	Extend class notation with set of all translations.


Definition	df-ldil 33368*	Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)


Definition	df-ltrn 33369*	Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$/\$

Definition	df-dilN 33370*	Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)


Definition	df-trnN 33371*	Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)


Theorem	ldilfset 33372*	The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)


Theorem	ldilset 33373*	The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)
$/\$

Theorem	isldil 33374*	The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$/\$ $/\$

Theorem	ldillaut 33375	A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ldil1o 33376	A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$

Theorem	ldilval 33377	Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	idldil 33378	The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)
$/\$

Theorem	ldilcnv 33379	The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)
$/\$ $/\$

Theorem	ldilco 33380	The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
$/\$ $/\$ $/\$

Theorem	ltrnfset 33381*	The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	ltrnset 33382*	The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	isltrn 33383*	The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	isltrn2N 33384*	The predicate "is a lattice translation". Version of isltrn 33383 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 33385	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 33386	A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrnlaut 33387	A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrn1o 33388	A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)
$/\$ $/\$

Theorem	ltrncl 33389	Closure of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrn11 33390	One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncnvnid 33391	If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	ltrncoidN 33392	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 33393	Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncnvleN 33394	Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrnm 33395	Lattice translation of a meet. (Contributed by NM, 20-May-2012.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Theorem	ltrnj 33396	Lattice translation of a meet. TODO: change antecedent to (Contributed by NM, 25-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	ltrncvr 33397	Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrnval1 33398	Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrnid 33399*	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 33400*	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.)
$/\$ $/\$ $/\$ $/\$

< Previous Next >

Page List Jump to page: Contents 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 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 39201-39291

< Previous Next >