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Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrnco4 31001 Rearrange a composition of 4 translations, analogous to an4 797. (Contributed by NM, 10-Jun-2013.)

Theoremtrljco 31002 Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)

Theoremtrljco2 31003 Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)

Syntaxctgrp 31004 Extend class notation with translation group.

Definitiondf-tgrp 31005* 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.)

Theoremtgrpfset 31006* The translation group maps for a lattice . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpset 31007* The translation group for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpbase 31008 The base set of the translation group is the set of all translations (for a fiducial co-atom ). (Contributed by NM, 5-Jun-2013.)

Theoremtgrpopr 31009* The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpov 31010 The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpgrplem 31011 Lemma for tgrpgrp 31012. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpgrp 31012 The translation group is a group. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpabl 31013 The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)

Syntaxctendo 31014 Extend class notation with translation group endomorphisms.

Syntaxcedring 31015 Extend class notation with division ring on trace-preserving endomorphisms.

Syntaxcedring-rN 31016 Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove theorems if not used.

Definitiondf-tendo 31017* Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)

Definitiondf-edring-rN 31018* Define division ring on trace-preserving endomorphisms. Definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)

Definitiondf-edring 31019* 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.)

Theoremtendofset 31020* The set of all trace-preserving endomorphisms on the set of translations for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremtendoset 31021* The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom . (Contributed by NM, 8-Jun-2013.)

Theoremistendo 31022* 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.)

Theoremtendotp 31023 Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremistendod 31024* Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)

Theoremtendof 31025 Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoeq1 31026* Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)

Theoremtendovalco 31027 Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocoval 31028 Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocl 31029 Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoco2 31030 Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)

Theoremtendoidcl 31031 The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)

Theoremtendo1mul 31032 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendo1mulr 31033 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendococl 31034 The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoid 31035 The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)

Theoremtendoeq2 31036* Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 31086, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)

Theoremtendoplcbv 31037* Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)

Theoremtendopl 31038* Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendopl2 31039* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcl2 31040* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplco2 31041* Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)

Theoremtendopltp 31042* Trace-preserving property of endomorphism sum operation , based on theorem trlco 30989. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 30989) 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.)

Theoremtendoplcl 31043* Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcom 31044* The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)

Theoremtendoplass 31045* The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)

Theoremtendodi1 31046* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendodi2 31047* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendo0cbv 31048* Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)

Theoremtendo02 31049* Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0co2 31050* The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 31283? (Contributed by NM, 11-Jun-2013.)

Theoremtendo0tp 31051* Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0cl 31052* The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0pl 31053* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0plr 31054* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)

Theoremtendoicbv 31055* Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi 31056* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi2 31057* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoicl 31058* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl 31059* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl2 31060* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)

Theoremerngfset 31061* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremerngset 31062* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremerngbase 31063 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.)

Theoremerngfplus 31064* Ring addition operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngplus 31065* Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngplus2 31066 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfmul 31067* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngmul 31068 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfset-rN 31069* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)

Theoremerngset-rN 31070* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)

Theoremerngbase-rN 31071 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.)

Theoremerngfplus-rN 31072* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngplus-rN 31073* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngplus2-rN 31074 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngfmul-rN 31075* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngmul-rN 31076 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremcdlemh1 31077 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemh2 31078 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)

Theoremcdlemh 31079 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemi1 31080 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)

Theoremcdlemi2 31081 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)

Theoremcdlemi 31082 Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)

Theoremcdlemj1 31083 Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)

Theoremcdlemj2 31084 Part of proof of Lemma J of [Crawley] p. 118. Eliminate . (Contributed by NM, 20-Jun-2013.)

Theoremcdlemj3 31085 Part of proof of Lemma J of [Crawley] p. 118. Eliminate . (Contributed by NM, 20-Jun-2013.)

Theoremtendocan 31086 Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of [Crawley] p. 118. (Contributed by NM, 21-Jun-2013.)

Theoremtendoid0 31087* A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (Contributed by NM, 1-Aug-2013.)

Theoremtendo0mul 31088* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)

Theoremtendo0mulr 31089* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)

Theoremtendo1ne0 31090* The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.)

Theoremtendoconid 31091* The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)

Theoremtendotr 31092* The trace of the value of a non-zero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)

Theoremcdlemk1 31093 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)

Theoremcdlemk2 31094 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)

Theoremcdlemk3 31095 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)

Theoremcdlemk4 31096 Part of proof of Lemma K of [Crawley] p. 118, last line. We use for their h, since is already used. (Contributed by NM, 24-Jun-2013.)

Theoremcdlemk5a 31097 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)

Theoremcdlemk5 31098 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 25-Jun-2013.)

Theoremcdlemk6 31099 Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 30148. (Contributed by NM, 25-Jun-2013.)

Theoremcdlemk8 31100 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 26-Jun-2013.)

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