P. 363 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 36201-36300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	cdlemk36 36201*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 18-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk37 36202*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 18-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk38 36203*	Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. TODO: derive more directly with r19.23 3022? (Contributed by NM, 19-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk39 36204*	Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of tau, represented by . (Contributed by NM, 19-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk40 36205*	TODO: fix comment. (Contributed by NM, 31-Jul-2013.)


Theorem	cdlemk40t 36206*	TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
$/\$

Theorem	cdlemk40f 36207*	TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
$/\$

Theorem	cdlemk41 36208*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemkfid1N 36209	Lemma for cdlemkfid3N 36213. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemkid1 36210	Lemma for cdlemkid 36224. (Contributed by NM, 24-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdlemkfid2N 36211	Lemma for cdlemkfid3N 36213. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkid2 36212*	Lemma for cdlemkid 36224. (Contributed by NM, 24-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkfid3N 36213*	TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemky 36214*	Part of proof of Lemma K of [Crawley] p. 118. TODO: clean up stuff. represents in cdlemk31 36184. (Contributed by NM, 21-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkyu 36215*	Convert between function and explicit forms. represents in cdlemkuu 36183. TODO: Clean all this up. (Contributed by NM, 21-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkyuu 36216*	cdlemkyu 36215 with some hypotheses eliminated. TODO: Clean all this up. (Contributed by NM, 21-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk11ta 36217*	Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119. , stand for g, h. TODO: fix comment. (Contributed by NM, 21-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk19ylem 36218*	Lemma for cdlemk19y 36220. (Contributed by NM, 30-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk11tb 36219*	Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119. , stand for g, h. cdlemk11ta 36217 with hypotheses removed. TODO: Can this be proved directly with no quantification? (Contributed by NM, 21-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk19y 36220*	cdlemk19 36157 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkid3N 36221*	Lemma for cdlemkid 36224. (Contributed by NM, 25-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkid4 36222*	Lemma for cdlemkid 36224. (Contributed by NM, 25-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkid5 36223*	Lemma for cdlemkid 36224. (Contributed by NM, 25-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkid 36224*	The value of the tau function (in Lemma K of [Crawley] p. 118) on the identity relation. (Contributed by NM, 25-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk35s 36225*	Substitution version of cdlemk35 36200. (Contributed by NM, 22-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk35s-id 36226*	Substitution version of cdlemk35 36200. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk39s 36227*	Substitution version of cdlemk39 36204. TODO: Can any commonality with cdlemk35s 36225 be exploited? (Contributed by NM, 23-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk39s-id 36228*	Substitution version of cdlemk39 36204 with non-identity requirement on removed. TODO: Can any commonality with cdlemk35s 36225 be exploited? (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk42 36229*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk19xlem 36230*	Lemma for cdlemk19x 36231. (Contributed by NM, 30-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk19x 36231*	cdlemk19 36157 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk42yN 36232*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk11tc 36233*	Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119. , stand for g, h. TODO: fix comment. (Contributed by NM, 21-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk11t 36234*	Part of proof of Lemma K of [Crawley] p. 118. Eq. 5, line 36, p. 119. , stand for g, h. represents tau. (Contributed by NM, 21-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk45 36235*	Part of proof of Lemma K of [Crawley] p. 118. Line 37, p. 119. , stand for g, h. represents tau. They do not explicitly mention the requirement . (Contributed by NM, 22-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk46 36236*	Part of proof of Lemma K of [Crawley] p. 118. Line 38 (last line), p. 119. , stand for g, h. represents tau. (Contributed by NM, 22-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk47 36237*	Part of proof of Lemma K of [Crawley] p. 118. Line 2, p. 120. , stand for g, h. represents tau. (Contributed by NM, 22-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk48 36238*	Part of proof of Lemma K of [Crawley] p. 118. Line 4, p. 120. , stand for g, h. represents tau. (Contributed by NM, 22-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk49 36239*	Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 120. , stand for g, h. represents tau. (Contributed by NM, 23-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemk50 36240*	Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. , stand for g, h. represents tau. TODO: Combine into cdlemk52 36242? (Contributed by NM, 23-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemk51 36241*	Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. , stand for g, h. represents tau. TODO: Combine into cdlemk52 36242? (Contributed by NM, 23-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdlemk52 36242*	Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. , stand for g, h. represents tau. (Contributed by NM, 23-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk53a 36243*	Lemma for cdlemk53 36245. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk53b 36244*	Lemma for cdlemk53 36245. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk53 36245*	Part of proof of Lemma K of [Crawley] p. 118. Line 7, p. 120. , stand for g, h. represents tau. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk54 36246*	Part of proof of Lemma K of [Crawley] p. 118. Line 10, p. 120. , stand for g, h. represents tau. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk55a 36247*	Lemma for cdlemk55 36249. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk55b 36248*	Lemma for cdlemk55 36249. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk55 36249*	Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120. , stand for g, h. represents tau. (Contributed by NM, 26-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemkyyN 36250*	Part of proof of Lemma K of [Crawley] p. 118. TODO: clean up stuff. (Contributed by NM, 21-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk43N 36251*	Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 31-Jul-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk35u 36252*	Substitution version of cdlemk35 36200. (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk55u1 36253*	Lemma for cdlemk55u 36254. (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk55u 36254*	Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120. , stand for g, h. represents tau. (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk39u1 36255*	Lemma for cdlemk39u 36256. (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk39u 36256*	Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by . (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk19u1 36257*	cdlemk19 36157 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk19u 36258*	Part of Lemma K of [Crawley] p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with , , . (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk56 36259*	Part of Lemma K of [Crawley] p. 118. Line 11, p. 120, "tau is in Delta" i.e. is a trace-preserving endormorphism. (Contributed by NM, 31-Jul-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk19w 36260*	Use a fixed element to eliminate in cdlemk19u 36258. (Contributed by NM, 1-Aug-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk56w 36261*	Use a fixed element to eliminate in cdlemk56 36259. (Contributed by NM, 1-Aug-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemk 36262*	Lemma K of [Crawley] p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use , , and to represent f, k, and tau. (Contributed by NM, 1-Aug-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendoex 36263*	Generalization of Lemma K of [Crawley] p. 118, cdlemk 36262. TODO: can this be used to shorten uses of cdlemk 36262? (Contributed by NM, 15-Oct-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	cdleml1N 36264	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleml2N 36265*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleml3N 36266*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleml4N 36267*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleml5N 36268*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	cdleml6 36269*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleml7 36270*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleml8 36271*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleml9 36272*	Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	dva1dim 36273*	Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) whose trace is rather than itself; exists by cdlemf 35851. is the division ring base by erngdv 36281, and is the scalar product by dvavsca 36305. must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)
$/\$ $/\$

Theorem	dvhb1dimN 36274*	Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	erng1lem 36275	Value of the endomorphism division ring unit. (Contributed by NM, 12-Oct-2013.)
$/\$ $/\$

Theorem	erngdvlem1 36276*	Lemma for eringring 36280. (Contributed by NM, 4-Aug-2013.)
$/\$

Theorem	erngdvlem2N 36277*	Lemma for eringring 36280. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
$/\$

Theorem	erngdvlem3 36278*	Lemma for eringring 36280. (Contributed by NM, 6-Aug-2013.)
$/\$

Theorem	erngdvlem4 36279*	Lemma for erngdv 36281. (Contributed by NM, 11-Aug-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	eringring 36280	An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.)
$/\$

Theorem	erngdv 36281	An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
$/\$

Theorem	erng0g 36282*	The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
$/\$

Theorem	erng1r 36283	The division ring unit of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
$/\$

Theorem	erngdvlem1-rN 36284*	Lemma for eringring 36280. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
$/\$

Theorem	erngdvlem2-rN 36285*	Lemma for eringring 36280. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
$/\$

Theorem	erngdvlem3-rN 36286*	Lemma for eringring 36280. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
$/\$

Theorem	erngdvlem4-rN 36287*	Lemma for erngdv 36281. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	erngring-rN 36288	An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
$/\$

Theorem	erngdv-rN 36289	An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
$/\$

Syntax	cdveca 36290	Extend class notation with constructed vector space A.


Definition	df-dveca 36291*	Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013.)
Scalar

Theorem	dvafset 36292*	The constructed partial vector space A for a lattice . (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theorem	dvaset 36293*	The constructed partial vector space A for a lattice . (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
$/\$ Scalar

Theorem	dvasca 36294	The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom ). (Contributed by NM, 22-Jun-2014.)
Scalar $/\$

Theorem	dvabase 36295	The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom ). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar $/\$

Theorem	dvafplusg 36296*	Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar $/\$

Theorem	dvaplusg 36297*	Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar $/\$ $/\$ $/\$

Theorem	dvaplusgv 36298	Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar $/\$ $/\$ $/\$ $/\$

Theorem	dvafmulr 36299*	Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar $/\$

Theorem	dvamulr 36300	Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar $/\$ $/\$ $/\$

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