P. 336 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 33501-33600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ltrnid 33501*	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 33502*	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 33503	The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrncnvatb 33504	The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	ltrnel 33505	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 33506	The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 33505 uses. (Contributed by NM, 25-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrncnvat 33507	The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)
$/\$ $/\$ $/\$

Theorem	ltrncnvel 33508	The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrncoelN 33509	Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 33505 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrncoat 33510	Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 33505, ltrnat 33506 uses. (Contributed by NM, 1-May-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncoval 33511	Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltrncnv 33512	The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)
$/\$ $/\$

Theorem	ltrn11at 33513	Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrneq2 33514*	The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)
$/\$ $/\$ $/\$

Theorem	ltrneq 33515*	The equality of two translations is determined by their equality at atoms not under co-atom . (Contributed by NM, 20-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	idltrn 33516	The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)
$/\$

Theorem	ltrnmw 33517	Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	dilfsetN 33518*	The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)


Theorem	dilsetN 33519*	The set of dilations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
$/\$

Theorem	isdilN 33520*	The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	trnfsetN 33521*	The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)


Theorem	trnsetN 33522*	The set of translations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
$/\$

Theorem	istrnN 33523*	The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$

Syntax	ctrl 33524	Extend class notation with set of all traces of lattice translations.


Definition	df-trl 33525*	Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)


Theorem	trlfset 33526*	The set of all traces of lattice translations for a lattice . (Contributed by NM, 20-May-2012.)
$.\/$ $./\$ $.\/$ $./\$

Theorem	trlset 33527*	The set of traces of lattice translations for a fiducial co-atom . (Contributed by NM, 20-May-2012.)
$.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	trlval 33528*	The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	trlval2 33529	The value of the trace of a lattice translation, given any atom not under the fiducial co-atom . Note: this requires only the weaker assumption ; we use for convenience. (Contributed by NM, 20-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	trlcl 33530	Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.)
$/\$ $/\$

Theorem	trlcnv 33531	The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)
$/\$ $/\$

Theorem	trljat1 33532	The value of a translation of an atom not under the fiducial co-atom , joined with trace. Equation above Lemma C in [Crawley] p. 112. Todo: shorten with atmod3i1 33230? (Contributed by NM, 22-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trljat2 33533	The value of a translation of an atom not under the fiducial co-atom , joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trljat3 33534	The value of a translation of an atom not under the fiducial co-atom , joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trlat 33535	If an atom differs from its translation, the trace is an atom. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	trl0 33536	If an atom not under the fiducial co-atom equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	trlator0 33537	The trace of a lattice translation is an atom or zero. (Contributed by NM, 5-May-2013.)
$/\$ $/\$ $\/$

Theorem	trlatn0 33538	The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.)
$/\$ $/\$

Theorem	trlnidat 33539	The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
$/\$ $/\$ $/\$

Theorem	ltrnnidn 33540	If a lattice translation is not the identity, then the translation of any atom not under the fiducial co-atom is different from the atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 24-May-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrnideq 33541	Property of the identity lattice translation. (Contributed by NM, 27-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlid0 33542	The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)
$/\$

Theorem	trlnidatb 33543	A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 33539? Why do both this and ltrnideq 33541 need trlnidat 33539? (Contributed by NM, 4-Jun-2013.)
$/\$ $/\$

Theorem	trlid0b 33544	A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.)
$/\$ $/\$

Theorem	trlnid 33545	Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrn2ateq 33546	Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrnateq 33547	If any atom (under ) is not equal to its translation, so is any other atom. (Contributed by NM, 6-May-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrnatneq 33548	If any atom (under ) is not equal to its translation, so is any other atom. TODO: isn't needed to prove this. Will removing it shorten (and not lengthen) proofs using it? (Contributed by NM, 6-May-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrnatlw 33549	If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	trlle 33550	The trace of a lattice translation is less than the fiducial co-atom . (Contributed by NM, 25-May-2012.)
$/\$ $/\$

Theorem	trlne 33551	The trace of a lattice translation is not equal to any atom not under the fiducial co-atom . Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlnle 33552	The atom not under the fiducial co-atom is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlval3 33553	The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	trlval4 33554	The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	trlval5 33555	The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	arglem1N 33556	Lemma for Desargues' law. Theorem 13.3 of [Crawley] p. 110, 3rd and 4th lines from bottom. In these lemmas, , , , , , , , , , , and represent Crawley's a₀, a₁, a₂, b₀, b₁, b₂, c, z₀, z₁, z₂, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemc1 33557	Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 33230? (Contributed by NM, 29-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemc2 33558	Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$

Theorem	cdlemc3 33559	Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdlemc4 33560	Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$

Theorem	cdlemc5 33561	Lemma for cdlemc 33563. (Contributed by NM, 26-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdlemc6 33562	Lemma for cdlemc 33563. (Contributed by NM, 26-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdlemc 33563	Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdlemd1 33564	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdlemd2 33565	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$

Theorem	cdlemd3 33566	Part of proof of Lemma D in [Crawley] p. 113. The requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdlemd4 33567	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$

Theorem	cdlemd5 33568	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemd6 33569	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Theorem	cdlemd7 33570	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemd8 33571	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemd9 33572	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemd 33573	If two translations agree at any atom not under the fiducial co-atom , then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrneq3 33574	Two translations agree at any atom not under the fiducial co-atom iff they are equal. (Contributed by NM, 25-Jul-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme00a 33575	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme0aa 33576	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	cdleme0a 33577	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme0b 33578	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme0c 33579	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme0cp 33580	Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 33963- swap consequent equality; make antecedent use df-3an 962. (Contributed by NM, 13-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme0cq 33581	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme0dN 33582	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme0e 33583	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme0fN 33584	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme0gN 33585	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemeulpq 33586	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme01N 33587	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Theorem	cdleme02N 33588	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$

Theorem	cdleme0ex1N 33589*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Theorem	cdleme0ex2N 33590*	Part of proof of Lemma E in [Crawley] p. 113. Note that $.\/$ $.\/$ is a shorter way to express $/\$ $/\$ $.\/$ . (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$

Theorem	cdleme0moN 33591*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $\/$

Theorem	cdleme1b 33592	Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing is a lattice element. represents their f(r). (Contributed by NM, 6-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme1 33593	Part of proof of Lemma E in [Crawley] p. 113. represents their f(r). Here we show r $\/$ f(r) = r $\/$ u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme2 33594	Part of proof of Lemma E in [Crawley] p. 113. represents f(r). is the fiducial co-atom (hyperplane) w. Here we show that (r $\/$ f(r)) $/\$ w = u in their notation (4th line from bottom on p. 113). (Contributed by NM, 5-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	cdleme3b 33595	Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 33602 and cdleme3 33603. (Contributed by NM, 6-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme3c 33596	Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 33602 and cdleme3 33603. (Contributed by NM, 6-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme3d 33597	Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 33602 and cdleme3 33603. (Contributed by NM, 6-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$

Theorem	cdleme3e 33598	Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 33602 and cdleme3 33603. (Contributed by NM, 6-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme3fN 33599	Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 33602 and cdleme3 33603. TODO: Delete - duplicates cdleme0e 33583. (Contributed by NM, 6-Jun-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme3g 33600	Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 33602 and cdleme3 33603. (Contributed by NM, 7-Jun-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

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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 35301-35339

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