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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdlemg17j 30801* TODO: fix comment. (Contributed by NM, 11-May-2013.)

Theoremcdlemg17pq 30802* Utility theorem for swapping and . TODO: fix comment. (Contributed by NM, 11-May-2013.)

Theoremcdlemg17bq 30803* cdlemg17b 30792 with and swapped. Antecedent is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b 30792 also? (Contributed by NM, 13-May-2013.)

Theoremcdlemg17iqN 30804* cdlemg17i 30799 with and swapped. (Contributed by NM, 13-May-2013.) (New usage is discouraged.)

Theoremcdlemg17irq 30805* cdlemg17ir 30800 with and swapped. (Contributed by NM, 13-May-2013.)

Theoremcdlemg17jq 30806* cdlemg17j 30801 with and swapped. (Contributed by NM, 13-May-2013.)

Theoremcdlemg17 30807* Part of Lemma G of [Crawley] p. 117, lines 7 and 8. We show an argument whose value at equals itself. TODO: fix comment. (Contributed by NM, 12-May-2013.)

Theoremcdlemg18a 30808 Show two lines are different. TODO: fix comment. (Contributed by NM, 14-May-2013.)

Theoremcdlemg18b 30809 Lemma for cdlemg18c 30810. TODO: fix comment. (Contributed by NM, 15-May-2013.)

Theoremcdlemg18c 30810 Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)

Theoremcdlemg18d 30811* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)

Theoremcdlemg18 30812* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)

Theoremcdlemg19a 30813* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)

Theoremcdlemg19 30814* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)

Theoremcdlemg20 30815* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 23-May-2013.)

Theoremcdlemg21 30816* Version of cdlemg19 with instead of as a condition. (Contributed by NM, 23-May-2013.)

Theoremcdlemg22 30817* cdlemg21 30816 with condition removed. (Contributed by NM, 23-May-2013.)

Theoremcdlemg24 30818* Combine cdlemg16z 30789 and cdlemg22 30817. TODO: Fix comment. (Contributed by NM, 24-May-2013.)

Theoremcdlemg37 30819* Use cdlemg8 30761 to eliminate the condition of cdlemg24 30818. (Contributed by NM, 31-May-2013.)

Theoremcdlemg25zz 30820 cdlemg16zz 30790 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)

Theoremcdlemg26zz 30821 cdlemg16zz 30790 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)

Theoremcdlemg27a 30822 For use with case when or is zero, letting us establish via 4atex 30206. TODO: Fix comment. (Contributed by NM, 28-May-2013.)

Theoremcdlemg28a 30823 Part of proof of Lemma G of [Crawley] p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-May-2013.)

Theoremcdlemg31b0N 30824 TODO: Fix comment. (Contributed by NM, 30-May-2013.) (New usage is discouraged.)

Theoremcdlemg31b0a 30825 TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg27b 30826 TODO: Fix comment. (Contributed by NM, 28-May-2013.)

Theoremcdlemg31a 30827 TODO: fix comment. (Contributed by NM, 29-May-2013.)

Theoremcdlemg31b 30828 TODO: fix comment. (Contributed by NM, 29-May-2013.)

Theoremcdlemg31c 30829 Show that when is an atom, it is not under . TODO: Is there a shorter direct proof? Todo: should we eliminate here? (Contributed by NM, 29-May-2013.)

Theoremcdlemg31d 30830 Eliminate from cdlemg31c 30829. TODO: Prove directly. Todo: do we need to eliminate ? It might be better to do this all at once at the end. See also cdlemg29 30835 vs. cdlemg28 30834. (Contributed by NM, 29-May-2013.)

Theoremcdlemg33b0 30831* TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg33c0 30832* TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg28b 30833* Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that is redundant here (but simplifies cdlemg28 30834.) (Contributed by NM, 29-May-2013.)

Theoremcdlemg28 30834* Part of proof of Lemma G of [Crawley] p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 30835 (and maybe leading up to this too)? (Contributed by NM, 29-May-2013.)

Theoremcdlemg29 30835* Eliminate and from cdlemg28 30834. TODO: would it be better to do this later? (Contributed by NM, 29-May-2013.)

Theoremcdlemg33a 30836* TODO: Fix comment. (Contributed by NM, 29-May-2013.)

Theoremcdlemg33b 30837* TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg33c 30838* TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg33d 30839* TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg33e 30840* TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg33 30841* Combine cdlemg33b 30837, cdlemg33c 30838, cdlemg33d 30839, cdlemg33e 30840. TODO: Fix comment. (Contributed by NM, 30-May-2013.)

Theoremcdlemg34 30842* Use cdlemg33 to eliminate from cdlemg29 30835. TODO: Fix comment. (Contributed by NM, 31-May-2013.)

Theoremcdlemg35 30843* TODO: Fix comment. TODO: should we have a more general version of hlsupr 29516 to avoid the conditions? (Contributed by NM, 31-May-2013.)

Theoremcdlemg36 30844* Use cdlemg35 to eliminate from cdlemg34 30842. TODO: Fix comment. (Contributed by NM, 31-May-2013.)

Theoremcdlemg38 30845 Use cdlemg37 30819 to eliminate from cdlemg36 30844. TODO: Fix comment. (Contributed by NM, 31-May-2013.)

Theoremcdlemg39 30846 Eliminate conditions from cdlemg38 30845. TODO: Would this better be done at cdlemg35 30843? TODO: Fix comment. (Contributed by NM, 31-May-2013.)

Theoremcdlemg40 30847 Eliminate conditions from cdlemg39 30846. TODO: Fix comment. (Contributed by NM, 31-May-2013.)

Theoremcdlemg41 30848 Convert cdlemg40 30847 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.)

Theoremltrnco 30849 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 30850 Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)

Theoremtrlcoabs 30851 Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.)

Theoremtrlcoabs2N 30852 Absorption of the trace of a composition. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

Theoremtrlcoat 30853 The trace of a composition of two translations is an atom if their traces are different. (Contributed by NM, 15-Jun-2013.)

Theoremtrlcocnvat 30854 Commonly used special case of trlcoat 30853. (Contributed by NM, 1-Jul-2013.)

Theoremtrlconid 30855 The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013.)

Theoremtrlcolem 30856 Lemma for trlco 30857. (Contributed by NM, 1-Jun-2013.)

Theoremtrlco 30857 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 30858 If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)

Theoremcdlemg42 30859 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 30860 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 30861 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 30862 Eliminate , from cdlemg44a 30861. (Contributed by NM, 3-Jun-2013.)

Theoremcdlemg44 30863 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 30864 TODO: fix comment. TODO: Use this above in place of antecedents? (Contributed by NM, 5-Jun-2013.)

Theoremcdlemg46 30865* 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 30866* 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 30867 Elmininate from cdlemg47 30866. (Contributed by NM, 5-Jun-2013.)

Theoremltrncom 30868 Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116 (Contributed by NM, 5-Jun-2013.)

Theoremltrnco4 30869 Rearrange a composition of 4 translations, analogous to an4 798. (Contributed by NM, 10-Jun-2013.)

Theoremtrljco 30870 Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)

Theoremtrljco2 30871 Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)

Syntaxctgrp 30872 Extend class notation with translation group.

Definitiondf-tgrp 30873* 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 30874* The translation group maps for a lattice . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpset 30875* The translation group for a fiducial co-atom . (Contributed by NM, 5-Jun-2013.)

Theoremtgrpbase 30876 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 30877* The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpov 30878 The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)

Theoremtgrpgrplem 30879 Lemma for tgrpgrp 30880. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpgrp 30880 The translation group is a group. (Contributed by NM, 6-Jun-2013.)

Theoremtgrpabl 30881 The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)

Syntaxctendo 30882 Extend class notation with translation group endomorphisms.

Syntaxcedring 30883 Extend class notation with division ring on trace-preserving endomorphisms.

Syntaxcedring-rN 30884 Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove theorems if not used.

Definitiondf-tendo 30885* Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)

Definitiondf-edring-rN 30886* 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 30887* 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 30888* The set of all trace-preserving endomorphisms on the set of translations for a lattice . (Contributed by NM, 8-Jun-2013.)

Theoremtendoset 30889* The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom . (Contributed by NM, 8-Jun-2013.)

Theoremistendo 30890* 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 30891 Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremistendod 30892* Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)

Theoremtendof 30893 Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoeq1 30894* Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)

Theoremtendovalco 30895 Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocoval 30896 Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendocl 30897 Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Theoremtendoco2 30898 Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)

Theoremtendoidcl 30899 The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)

Theoremtendo1mul 30900 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)

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