P. 361 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 36001-36100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	cdlemg35 36001*	TODO: Fix comment. TODO: should we have a more general version of hlsupr 34672 to avoid the conditions? (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg36 36002*	Use cdlemg35 to eliminate from cdlemg34 36000. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg38 36003	Use cdlemg37 35977 to eliminate from cdlemg36 36002. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg39 36004	Eliminate conditions from cdlemg38 36003. TODO: Would this better be done at cdlemg35 36001? TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg40 36005	Eliminate conditions from cdlemg39 36004. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemg41 36006	Convert cdlemg40 36005 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	ltrnco 36007	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 36008	Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)
$/\$ $/\$ $/\$

Theorem	trlcoabs 36009	Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trlcoabs2N 36010	Absorption of the trace of a composition. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trlcoat 36011	The trace of a composition of two translations is an atom if their traces are different. (Contributed by NM, 15-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlcocnvat 36012	Commonly used special case of trlcoat 36011. (Contributed by NM, 1-Jul-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlconid 36013	The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	trlcolem 36014	Lemma for trlco 36015. (Contributed by NM, 1-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	trlco 36015	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 36016	If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg42 36017	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 36018	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 36019	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 36020	Eliminate , from cdlemg44a 36019. (Contributed by NM, 3-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemg44 36021	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 36022	TODO: fix comment. TODO: Use this above in place of antecedents? (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	cdlemg46 36023*	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 36024*	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 36025	Elmininate from cdlemg47 36024. (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ltrncom 36026	Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116. (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	ltrnco4 36027	Rearrange a composition of 4 translations, analogous to an4 865. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	trljco 36028	Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	trljco2 36029	Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Syntax	ctgrp 36030	Extend class notation with translation group.


Definition	df-tgrp 36031*	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 36032*	The translation group maps for a lattice . (Contributed by NM, 5-Jun-2013.)


Theorem	tgrpset 36033*	The translation group for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)
$/\$

Theorem	tgrpbase 36034	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 36035*	The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)
$/\$

Theorem	tgrpov 36036	The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tgrpgrplem 36037	Lemma for tgrpgrp 36038. (Contributed by NM, 6-Jun-2013.)
$/\$

Theorem	tgrpgrp 36038	The translation group is a group. (Contributed by NM, 6-Jun-2013.)
$/\$

Theorem	tgrpabl 36039	The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)
$/\$

Syntax	ctendo 36040	Extend class notation with translation group endomorphisms.


Syntax	cedring 36041	Extend class notation with division ring on trace-preserving endomorphisms.


Syntax	cedring-rN 36042	Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove theorems if not used.


Definition	df-tendo 36043*	Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)
$/\$ $/\$

Definition	df-edring-rN 36044*	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 36045*	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 36046*	The set of all trace-preserving endomorphisms on the set of translations for a lattice . (Contributed by NM, 8-Jun-2013.)
$/\$ $/\$

Theorem	tendoset 36047*	The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom . (Contributed by NM, 8-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	istendo 36048*	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 36049	Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	istendod 36050*	Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendof 36051	Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$

Theorem	tendoeq1 36052*	Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendovalco 36053	Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendocoval 36054	Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendocl 36055	Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendoco2 36056	Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	tendoidcl 36057	The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)
$/\$

Theorem	tendo1mul 36058	Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
$/\$ $/\$

Theorem	tendo1mulr 36059	Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
$/\$ $/\$

Theorem	tendococl 36060	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 36061	The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)
$/\$ $/\$

Theorem	tendoeq2 36062*	Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 36112, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendoplcbv 36063*	Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)


Theorem	tendopl 36064*	Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
$/\$

Theorem	tendopl2 36065*	Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$

Theorem	tendoplcl2 36066*	Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendoplco2 36067*	Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	tendopltp 36068*	Trace-preserving property of endomorphism sum operation , based on theorem trlco 36015. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 36015) 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 36069*	Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendoplcom 36070*	The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendoplass 36071*	The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendodi1 36072*	Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendodi2 36073*	Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	tendo0cbv 36074*	Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)


Theorem	tendo02 36075*	Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)


Theorem	tendo0co2 36076*	The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 36309? (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	tendo0tp 36077*	Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)
$/\$ $/\$

Theorem	tendo0cl 36078*	The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-Jun-2013.)
$/\$

Theorem	tendo0pl 36079*	Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)
$/\$ $/\$

Theorem	tendo0plr 36080*	Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)
$/\$ $/\$

Theorem	tendoicbv 36081*	Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)


Theorem	tendoi 36082*	Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)


Theorem	tendoi2 36083*	Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
$/\$

Theorem	tendoicl 36084*	Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
$/\$ $/\$

Theorem	tendoipl 36085*	Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
$/\$ $/\$

Theorem	tendoipl2 36086*	Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)
$/\$ $/\$

Theorem	erngfset 36087*	The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.)


Theorem	erngset 36088*	The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)
$/\$

Theorem	erngbase 36089	The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom ). TODO: the .t hypothesis isn't used. (Also look at others.) (Contributed by NM, 9-Jun-2013.)
$/\$

Theorem	erngfplus 36090*	Ring addition operation. (Contributed by NM, 9-Jun-2013.)
$/\$

Theorem	erngplus 36091*	Ring addition operation. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	erngplus2 36092	Ring addition operation. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	erngfmul 36093*	Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)
$/\$

Theorem	erngmul 36094	Ring addition operation. (Contributed by NM, 10-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	erngfset-rN 36095*	The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)


Theorem	erngset-rN 36096*	The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)
$/\$

Theorem	erngbase-rN 36097	The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom ). (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
$/\$

Theorem	erngfplus-rN 36098*	Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
$/\$

Theorem	erngplus-rN 36099*	Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	erngplus2-rN 36100	Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

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