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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtrlcoat 29601 The trace of a composition of two translations is an atom if their traces are different. (Contributed by NM, 15-Jun-2013.)

Theoremtrlcocnvat 29602 Commonly used special case of trlcoat 29601. (Contributed by NM, 1-Jul-2013.)

Theoremtrlconid 29603 The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013.)

Theoremtrlcolem 29604 Lemma for trlco 29605. (Contributed by NM, 1-Jun-2013.)

Theoremtrlco 29605 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 29606 If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)

Theoremcdlemg42 29607 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 29608 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 29609 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 29610 Eliminate , from cdlemg44a 29609. (Contributed by NM, 3-Jun-2013.)

Theoremcdlemg44 29611 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 29612 TODO: fix comment. TODO: Use this above in place of antecedents? (Contributed by NM, 5-Jun-2013.)

Theoremcdlemg46 29613* 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 29614* 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 29615 Elmininate from cdlemg47 29614. (Contributed by NM, 5-Jun-2013.)

Theoremltrncom 29616 Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116 (Contributed by NM, 5-Jun-2013.)

Theoremltrnco4 29617 Rearrange a composition of 4 translations, analogous to an4 800. (Contributed by NM, 10-Jun-2013.)

Theoremtrljco 29618 Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)

Theoremtrljco2 29619 Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)

Syntaxctgrp 29620 Extend class notation with translation group.

Definitiondf-tgrp 29621* 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 29622* The translation group maps for a lattice . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpset 29623* The translation group for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpbase 29624 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 29625* The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpov 29626 The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpgrplem 29627 Lemma for tgrpgrp 29628. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpgrp 29628 The translation group is a group. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpabl 29629 The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)

Syntaxctendo 29630 Extend class notation with translation group endomorphisms.

Syntaxcedring 29631 Extend class notation with division ring on trace-preserving endomorphisms.

Syntaxcedring-rN 29632 Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove theorems if not used.

Definitiondf-tendo 29633* Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)

Definitiondf-edring-rN 29634* 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 29635* 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 29636* The set of all trace-preserving endomorphisms on the set of translations for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremtendoset 29637* The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom . (Contributed by NM, 8-Jun-2013.)

Theoremistendo 29638* 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 29639 Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremistendod 29640* Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)

Theoremtendof 29641 Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoeq1 29642* Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)

Theoremtendovalco 29643 Value of composition of translations in a a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocoval 29644 Value of composition of endomorphisms in a a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocl 29645 Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoco2 29646 Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)

Theoremtendoidcl 29647 The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)

Theoremtendo1mul 29648 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendo1mulr 29649 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

Theoremtendococl 29650 The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoid 29651 The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)

Theoremtendoeq2 29652* Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 29702, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)

Theoremtendoplcbv 29653* Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)

Theoremtendopl 29654* Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendopl2 29655* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcl2 29656* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplco2 29657* Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)

Theoremtendopltp 29658* Trace-preserving property of endomorphism sum operation , based on theorem trlco 29605. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 29605) 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 29659* Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)

Theoremtendoplcom 29660* The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)

Theoremtendoplass 29661* The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)

Theoremtendodi1 29662* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendodi2 29663* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)

Theoremtendo0cbv 29664* Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)

Theoremtendo02 29665* Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0co2 29666* The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 29899? (Contributed by NM, 11-Jun-2013.)

Theoremtendo0tp 29667* Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)

Theoremtendo0cl 29668* The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0pl 29669* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendo0plr 29670* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)

Theoremtendoicbv 29671* Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi 29672* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoi2 29673* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoicl 29674* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl 29675* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Theoremtendoipl2 29676* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)

Theoremerngfset 29677* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremerngset 29678* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremerngbase 29679 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 29680* Ring addition operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngplus 29681* Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngplus2 29682 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfmul 29683* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)

Theoremerngmul 29684 Ring addition operation. (Contributed by NM, 10-Jun-2013.)

Theoremerngfset-rN 29685* The division rings on trace-preserving endomorphisms for a lattice . (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)

Theoremerngset-rN 29686* The division ring on trace-preserving endomorphisms for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)

Theoremerngbase-rN 29687 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 29688* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngplus-rN 29689* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngplus2-rN 29690 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremerngfmul-rN 29691* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)

Theoremerngmul-rN 29692 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)

Theoremcdlemh1 29693 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemh2 29694 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)

Theoremcdlemh 29695 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

Theoremcdlemi1 29696 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)

Theoremcdlemi2 29697 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)

Theoremcdlemi 29698 Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)

Theoremcdlemj1 29699 Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)

Theoremcdlemj2 29700 Part of proof of Lemma J of [Crawley] p. 118. Eliminate . (Contributed by NM, 20-Jun-2013.)

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