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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrnco 31201 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.)

Theoremtrlcocnv 31202 Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)

Theoremtrlcoabs 31203 Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.)

Theoremtrlcoabs2N 31204 Absorption of the trace of a composition. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

Theoremtrlcoat 31205 The trace of a composition of two translations is an atom if their traces are different. (Contributed by NM, 15-Jun-2013.)

Theoremtrlcocnvat 31206 Commonly used special case of trlcoat 31205. (Contributed by NM, 1-Jul-2013.)

Theoremtrlconid 31207 The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013.)

Theoremtrlcolem 31208 Lemma for trlco 31209. (Contributed by NM, 1-Jun-2013.)

Theoremtrlco 31209 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.)

Theoremtrlcone 31210 If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)

Theoremcdlemg42 31211 Part of proof of Lemma G of [Crawley] p. 116, first line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)

Theoremcdlemg43 31212 Part of proof of Lemma G of [Crawley] p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)

Theoremcdlemg44a 31213 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.)

Theoremcdlemg44b 31214 Eliminate , from cdlemg44a 31213. (Contributed by NM, 3-Jun-2013.)

Theoremcdlemg44 31215 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.)

Theoremcdlemg47a 31216 TODO: fix comment. TODO: Use this above in place of antecedents? (Contributed by NM, 5-Jun-2013.)

Theoremcdlemg46 31217* 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.)

Theoremcdlemg47 31218* 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.)

Theoremcdlemg48 31219 Elmininate from cdlemg47 31218. (Contributed by NM, 5-Jun-2013.)

Theoremltrncom 31220 Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116 (Contributed by NM, 5-Jun-2013.)

Theoremltrnco4 31221 Rearrange a composition of 4 translations, analogous to an4 798. (Contributed by NM, 10-Jun-2013.)

Theoremtrljco 31222 Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)

Theoremtrljco2 31223 Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)

Syntaxctgrp 31224 Extend class notation with translation group.

Definitiondf-tgrp 31225* 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 31226* The translation group maps for a lattice . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpset 31227* The translation group for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpbase 31228 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 31229* The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpov 31230 The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpgrplem 31231 Lemma for tgrpgrp 31232. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpgrp 31232 The translation group is a group. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpabl 31233 The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)

Syntaxctendo 31234 Extend class notation with translation group endomorphisms.

Syntaxcedring 31235 Extend class notation with division ring on trace-preserving endomorphisms.

Syntaxcedring-rN 31236 Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove theorems if not used.

Definitiondf-tendo 31237* Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)

Definitiondf-edring-rN 31238* 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 31239* 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 31240* The set of all trace-preserving endomorphisms on the set of translations for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremtendoset 31241* The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom . (Contributed by NM, 8-Jun-2013.)

Theoremistendo 31242* 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 31243 Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremistendod 31244* Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)

Theoremtendof 31245 Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoeq1 31246* Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)

Theoremtendovalco 31247 Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocoval 31248 Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocl 31249 Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoco2 31250 Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)

Theoremtendoidcl 31251 The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)

Theoremtendo1mul 31252 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendo1mulr 31253 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendococl 31254 The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoid 31255 The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)

Theoremtendoeq2 31256* Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 31306, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)

Theoremtendoplcbv 31257* Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)

Theoremtendopl 31258* Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendopl2 31259* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcl2 31260* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplco2 31261* Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)

Theoremtendopltp 31262* Trace-preserving property of endomorphism sum operation , based on theorem trlco 31209. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 31209) 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 31263* Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcom 31264* The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)

Theoremtendoplass 31265* The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)

Theoremtendodi1 31266* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendodi2 31267* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendo0cbv 31268* Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)

Theoremtendo02 31269* Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0co2 31270* The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 31503? (Contributed by NM, 11-Jun-2013.)

Theoremtendo0tp 31271* Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0cl 31272* The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0pl 31273* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0plr 31274* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)

Theoremtendoicbv 31275* Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi 31276* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi2 31277* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoicl 31278* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl 31279* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl2 31280* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)

Theoremerngfset 31281* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremerngset 31282* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremerngbase 31283 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 31284* Ring addition operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngplus 31285* Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngplus2 31286 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfmul 31287* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngmul 31288 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfset-rN 31289* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)

Theoremerngset-rN 31290* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)

Theoremerngbase-rN 31291 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 31292* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngplus-rN 31293* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngplus2-rN 31294 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngfmul-rN 31295* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngmul-rN 31296 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremcdlemh1 31297 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemh2 31298 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)

Theoremcdlemh 31299 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemi1 31300 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)

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