P. 344 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 34301-34400 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	cdlemg24 34301*	Combine cdlemg16z 34272 and cdlemg22 34300. TODO: Fix comment. (Contributed by NM, 24-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg37 34302*	Use cdlemg8 34244 to eliminate the $.\/$ condition of cdlemg24 34301. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg25zz 34303	cdlemg16zz 34273 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg26zz 34304	cdlemg16zz 34273 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg27a 34305	For use with case when $.\/$ $./\$ $.\/$ or $.\/$ $./\$ $.\/$ is zero, letting us establish $/\$ $.\/$ via 4atex 33687. TODO: Fix comment. (Contributed by NM, 28-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdlemg28a 34306	Part of proof of Lemma G of [Crawley] p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg31b0N 34307	TODO: Fix comment. (Contributed by NM, 30-May-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Theorem	cdlemg31b0a 34308	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Theorem	cdlemg27b 34309	TODO: Fix comment. (Contributed by NM, 28-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdlemg31a 34310	TODO: fix comment. (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg31b 34311	TODO: fix comment. (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg31c 34312	Show that when is an atom, it is not under . TODO: Is there a shorter direct proof? TODO: should we eliminate here? (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg31d 34313	Eliminate from cdlemg31c 34312. TODO: Prove directly. TODO: do we need to eliminate ? It might be better to do this all at once at the end. See also cdlemg29 34318 vs. cdlemg28 34317. (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg33b0 34314*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdlemg33c0 34315*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$

Theorem	cdlemg28b 34316*	Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that is redundant here (but simplifies cdlemg28 34317.) (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg28 34317*	Part of proof of Lemma G of [Crawley] p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 34318 (and maybe leading up to this too)? (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg29 34318*	Eliminate and from cdlemg28 34317. TODO: would it be better to do this later? (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg33a 34319*	TODO: Fix comment. (Contributed by NM, 29-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg33b 34320*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg33c 34321*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg33d 34322*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg33e 34323*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg33 34324*	Combine cdlemg33b 34320, cdlemg33c 34321, cdlemg33d 34322, cdlemg33e 34323. TODO: Fix comment. (Contributed by NM, 30-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg34 34325*	Use cdlemg33 to eliminate from cdlemg29 34318. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg35 34326*	TODO: Fix comment. TODO: should we have a more general version of hlsupr 32997 to avoid the conditions? (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg36 34327*	Use cdlemg35 to eliminate from cdlemg34 34325. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg38 34328	Use cdlemg37 34302 to eliminate from cdlemg36 34327. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg39 34329	Eliminate conditions from cdlemg38 34328. TODO: Would this better be done at cdlemg35 34326? TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg40 34330	Eliminate conditions from cdlemg39 34329. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg41 34331	Convert cdlemg40 34330 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	ltrnco 34332	The composition of two translations is a translation. Part of proof of Lemma G of [Crawley] p. 116, line 15 on p. 117. (Contributed by NM, 31-May-2013.)
$/\$ $/\$ $/\$

Theorem	trlcocnv 34333	Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)
$/\$ $/\$ $/\$

Theorem	trlcoabs 34334	Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trlcoabs2N 34335	Absorption of the trace of a composition. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trlcoat 34336	The trace of a composition of two translations is an atom if their traces are different. (Contributed by NM, 15-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlcocnvat 34337	Commonly used special case of trlcoat 34336. (Contributed by NM, 1-Jul-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlconid 34338	The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlcolem 34339	Lemma for trlco 34340. (Contributed by NM, 1-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	trlco 34340	The trace of a composition of translations is less than or equal to the join of their traces. Part of proof of Lemma G of [Crawley] p. 116, second paragraph on p. 117. (Contributed by NM, 2-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	trlcone 34341	If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg42 34342	Part of proof of Lemma G of [Crawley] p. 116, first line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemg43 34343	Part of proof of Lemma G of [Crawley] p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdlemg44a 34344	Part of proof of Lemma G of [Crawley] p. 116, fourth line of third paragraph on p. 117: "so fg(p) = gf(p)." (Contributed by NM, 3-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg44b 34345	Eliminate , from cdlemg44a 34344. (Contributed by NM, 3-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg44 34346	Part of proof of Lemma G of [Crawley] p. 116, fifth line of third paragraph on p. 117: "and hence fg = gf." (Contributed by NM, 3-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	cdlemg47a 34347	TODO: fix comment. TODO: Use this above in place of antecedents? (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	cdlemg46 34348*	Part of proof of Lemma G of [Crawley] p. 116, seventh line of third paragraph on p. 117: "hf and f have different traces." (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg47 34349*	Part of proof of Lemma G of [Crawley] p. 116, ninth line of third paragraph on p. 117: "we conclude that gf = fg." (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg48 34350	Elmininate from cdlemg47 34349. (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrncom 34351	Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116. (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	ltrnco4 34352	Rearrange a composition of 4 translations, analogous to an4 838. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	trljco 34353	Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trljco2 34354	Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Syntax	ctgrp 34355	Extend class notation with translation group.


Definition	df-tgrp 34356*	Define the class of all translation groups. is normally a member of . Each base set is the set of all lattice translations with respect to a hyperplane , and the operation is function composition. Similar to definition of G in [Crawley] p. 116, third paragraph (which defines this for geomodular lattices). (Contributed by NM, 5-Jun-2013.)


Theorem	tgrpfset 34357*	The translation group maps for a lattice . (Contributed by NM, 5-Jun-2013.)


Theorem	tgrpset 34358*	The translation group for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)
$/\$

Theorem	tgrpbase 34359	The base set of the translation group is the set of all translations (for a fiducial co-atom ). (Contributed by NM, 5-Jun-2013.)
$/\$

Theorem	tgrpopr 34360*	The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)
$/\$

Theorem	tgrpov 34361	The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tgrpgrplem 34362	Lemma for tgrpgrp 34363. (Contributed by NM, 6-Jun-2013.)
$/\$

Theorem	tgrpgrp 34363	The translation group is a group. (Contributed by NM, 6-Jun-2013.)
$/\$

Theorem	tgrpabl 34364	The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)
$/\$

Syntax	ctendo 34365	Extend class notation with translation group endomorphisms.


Syntax	cedring 34366	Extend class notation with division ring on trace-preserving endomorphisms.


Syntax	cedring-rN 34367	Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove theorems if not used.


Definition	df-tendo 34368*	Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)
$/\$ $/\$

Definition	df-edring-rN 34369*	Define division ring on trace-preserving endomorphisms. Definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)


Definition	df-edring 34370*	Define division ring on trace-preserving endomorphisms. The multiplication operation is reversed composition, per the definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)


Theorem	tendofset 34371*	The set of all trace-preserving endomorphisms on the set of translations for a lattice . (Contributed by NM, 8-Jun-2013.)
$/\$ $/\$

Theorem	tendoset 34372*	The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom . (Contributed by NM, 8-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	istendo 34373*	The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (Contributed by NM, 8-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendotp 34374	Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	istendod 34375*	Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendof 34376	Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$

Theorem	tendoeq1 34377*	Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendovalco 34378	Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendocoval 34379	Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendocl 34380	Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendoco2 34381	Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	tendoidcl 34382	The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)
$/\$

Theorem	tendo1mul 34383	Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
$/\$ $/\$

Theorem	tendo1mulr 34384	Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
$/\$ $/\$

Theorem	tendococl 34385	The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendoid 34386	The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)
$/\$ $/\$

Theorem	tendoeq2 34387*	Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 34437, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendoplcbv 34388*	Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)


Theorem	tendopl 34389*	Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
$/\$

Theorem	tendopl2 34390*	Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$

Theorem	tendoplcl2 34391*	Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendoplco2 34392*	Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	tendopltp 34393*	Trace-preserving property of endomorphism sum operation , based on theorem trlco 34340. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 34340) that Delta is a subring of E." (In our development, we will bypass their E and go directly to their Delta, whose base set is our .) (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendoplcl 34394*	Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendoplcom 34395*	The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendoplass 34396*	The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendodi1 34397*	Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendodi2 34398*	Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendo0cbv 34399*	Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)


Theorem	tendo02 34400*	Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 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 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-40909

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