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Theorem List for Metamath Proof Explorer - 34401-34500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtendof 34401 Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoeq1 34402* Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)

Theoremtendovalco 34403 Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocoval 34404 Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocl 34405 Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoco2 34406 Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)

Theoremtendoidcl 34407 The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)

Theoremtendo1mul 34408 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendo1mulr 34409 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendococl 34410 The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoid 34411 The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)

Theoremtendoeq2 34412* Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 34462, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)

Theoremtendoplcbv 34413* Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)

Theoremtendopl 34414* Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendopl2 34415* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcl2 34416* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplco2 34417* Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)

Theoremtendopltp 34418* Trace-preserving property of endomorphism sum operation , based on theorem trlco 34365. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 34365) 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 34419* Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcom 34420* The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)

Theoremtendoplass 34421* The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)

Theoremtendodi1 34422* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendodi2 34423* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendo0cbv 34424* Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)

Theoremtendo02 34425* Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0co2 34426* The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 34659? (Contributed by NM, 11-Jun-2013.)

Theoremtendo0tp 34427* Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0cl 34428* The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0pl 34429* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0plr 34430* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)

Theoremtendoicbv 34431* Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi 34432* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi2 34433* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoicl 34434* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl 34435* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl2 34436* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)

Theoremerngfset 34437* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremerngset 34438* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremerngbase 34439 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 34440* Ring addition operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngplus 34441* Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngplus2 34442 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfmul 34443* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngmul 34444 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfset-rN 34445* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)

Theoremerngset-rN 34446* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)

Theoremerngbase-rN 34447 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 34448* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngplus-rN 34449* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngplus2-rN 34450 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngfmul-rN 34451* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngmul-rN 34452 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremcdlemh1 34453 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemh2 34454 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)

Theoremcdlemh 34455 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemi1 34456 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)

Theoremcdlemi2 34457 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)

Theoremcdlemi 34458 Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)

Theoremcdlemj1 34459 Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)

Theoremcdlemj2 34460 Part of proof of Lemma J of [Crawley] p. 118. Eliminate . (Contributed by NM, 20-Jun-2013.)

Theoremcdlemj3 34461 Part of proof of Lemma J of [Crawley] p. 118. Eliminate . (Contributed by NM, 20-Jun-2013.)

Theoremtendocan 34462 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 34463* 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 34464* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)

Theoremtendo0mulr 34465* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)

Theoremtendo1ne0 34466* The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.)

Theoremtendoconid 34467* The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)

Theoremtendotr 34468* 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 34469 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)

Theoremcdlemk2 34470 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)

Theoremcdlemk3 34471 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)

Theoremcdlemk4 34472 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 34473 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)

Theoremcdlemk5 34474 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 25-Jun-2013.)

Theoremcdlemk6 34475 Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 33522. (Contributed by NM, 25-Jun-2013.)

Theoremcdlemk8 34476 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 26-Jun-2013.)

Theoremcdlemk9 34477 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)

Theoremcdlemk9bN 34478 Part of proof of Lemma K of [Crawley] p. 118. TODO: is this needed? If so, shorten with cdlemk9 34477 if that one is also needed. (Contributed by NM, 28-Jun-2013.) (New usage is discouraged.)

Theoremcdlemki 34479* Part of proof of Lemma K of [Crawley] p. 118. TODO: Eliminate and put into cdlemksel 34483. (Contributed by NM, 25-Jun-2013.)

Theoremcdlemkvcl 34480 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)

Theoremcdlemk10 34481 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)

Theoremcdlemksv 34482* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)

Theoremcdlemksel 34483* Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 34479? (Contributed by NM, 26-Jun-2013.)

Theoremcdlemksat 34484* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)

Theoremcdlemksv2 34485* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function at the fixed parameter. (Contributed by NM, 26-Jun-2013.)

Theoremcdlemk7 34486* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013.)

Theoremcdlemk11 34487* Part of proof of Lemma K of [Crawley] p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-Jun-2013.)

Theoremcdlemk12 34488* Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-Jun-2013.)

Theoremcdlemkoatnle 34489* Utility lemma. (Contributed by NM, 2-Jul-2013.)

Theoremcdlemk13 34490* Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. , are k1, f1. (Contributed by NM, 1-Jul-2013.)

Theoremcdlemkole 34491* Utility lemma. (Contributed by NM, 2-Jul-2013.)

Theoremcdlemk14 34492* Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. , are k1, f1. (Contributed by NM, 1-Jul-2013.)

Theoremcdlemk15 34493* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. , are k1, f1. (Contributed by NM, 1-Jul-2013.)

Theoremcdlemk16a 34494* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)

Theoremcdlemk16 34495* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.)

Theoremcdlemk17 34496* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. , are k1, f1. (Contributed by NM, 1-Jul-2013.)

Theoremcdlemk1u 34497* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)

Theoremcdlemk5auN 34498* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) (New usage is discouraged.)

Theoremcdlemk5u 34499* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)

Theoremcdlemk6u 34500* Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 33522. (Contributed by NM, 4-Jul-2013.)

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