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Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdlemk43N 34601* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 31-Jul-2013.) (New usage is discouraged.)

Theoremcdlemk35u 34602* Substitution version of cdlemk35 34550. (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk55u1 34603* Lemma for cdlemk55u 34604. (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk55u 34604* Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120. , stand for g, h. represents tau. (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk39u1 34605* Lemma for cdlemk39u 34606. (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk39u 34606* Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by . (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk19u1 34607* cdlemk19 34507 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk19u 34608* Part of Lemma K of [Crawley] p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with , , . (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk56 34609* Part of Lemma K of [Crawley] p. 118. Line 11, p. 120, "tau is in Delta" i.e. is a trace-preserving endormorphism. (Contributed by NM, 31-Jul-2013.)

Theoremcdlemk19w 34610* Use a fixed element to eliminate in cdlemk19u 34608. (Contributed by NM, 1-Aug-2013.)

Theoremcdlemk56w 34611* Use a fixed element to eliminate in cdlemk56 34609. (Contributed by NM, 1-Aug-2013.)

Theoremcdlemk 34612* Lemma K of [Crawley] p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use , , and to represent f, k, and tau. (Contributed by NM, 1-Aug-2013.)

Theoremtendoex 34613* Generalization of Lemma K of [Crawley] p. 118, cdlemk 34612. TODO: can this be used to shorten uses of cdlemk 34612? (Contributed by NM, 15-Oct-2013.)

Theoremcdleml1N 34614 Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)

Theoremcdleml2N 34615* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)

Theoremcdleml3N 34616* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)

Theoremcdleml4N 34617* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)

Theoremcdleml5N 34618* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)

Theoremcdleml6 34619* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)

Theoremcdleml7 34620* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)

Theoremcdleml8 34621* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)

Theoremcdleml9 34622* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)

Theoremdva1dim 34623* Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) whose trace is rather than itself; exists by cdlemf 34201. is the division ring base by erngdv 34631, and is the scalar product by dvavsca 34655. must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)

Theoremdvhb1dimN 34624* Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)

Theoremerng1lem 34625 Value of the endomorphism division ring unit. (Contributed by NM, 12-Oct-2013.)

Theoremerngdvlem1 34626* Lemma for eringring 34630. (Contributed by NM, 4-Aug-2013.)

Theoremerngdvlem2N 34627* Lemma for eringring 34630. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem3 34628* Lemma for eringring 34630. (Contributed by NM, 6-Aug-2013.)

Theoremerngdvlem4 34629* Lemma for erngdv 34631. (Contributed by NM, 11-Aug-2013.)

Theoremeringring 34630 An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.)

Theoremerngdv 34631 An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)

Theoremerng0g 34632* The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremerng1r 34633 The division ring unit of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremerngdvlem1-rN 34634* Lemma for eringring 34630. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem2-rN 34635* Lemma for eringring 34630. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem3-rN 34636* Lemma for eringring 34630. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem4-rN 34637* Lemma for erngdv 34631. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)

Theoremerngring-rN 34638 An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)

Theoremerngdv-rN 34639 An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)

Syntaxcdveca 34640 Extend class notation with constructed vector space A.

Definitiondf-dveca 34641* Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013.)
Scalar

Theoremdvafset 34642* The constructed partial vector space A for a lattice . (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvaset 34643* The constructed partial vector space A for a lattice . (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvasca 34644 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom ). (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvabase 34645 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom ). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvafplusg 34646* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvaplusg 34647* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar

Theoremdvaplusgv 34648 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar

Theoremdvafmulr 34649* Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvamulr 34650 Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar

Theoremdvavbase 34651 The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom ). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvafvadd 34652* The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvavadd 34653 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)

Theoremdvafvsca 34654* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvavsca 34655 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)

Theoremtendospid 34656 Identity property of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendospcl 34657 Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendospass 34658 Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendospdi1 34659 Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendocnv 34660 Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)

Theoremtendospdi2 34661* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

TheoremtendospcanN 34662* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdvaabl 34663 The constructed partial vector space A for a lattice is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvalveclem 34664 Lemma for dvalvec 34665. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvalvec 34665 The constructed partial vector space A for a lattice is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdva0g 34666 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)

Syntaxcdia 34667 Extend class notation with partial isomorphism A.

Definitiondf-disoa 34668* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)

Theoremdiaffval 34669* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiafval 34670* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiaval 34671* The partial isomorphism A for a lattice . Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)

Theoremdiaelval 34672 Member of the partial isomorphism A for a lattice . (Contributed by NM, 3-Dec-2013.)

Theoremdiafn 34673* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)

Theoremdiadm 34674* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)

Theoremdiaeldm 34675 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)

TheoremdiadmclN 34676 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiadmleN 34677 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdian0 34678 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)

Theoremdia0eldmN 34679 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdia1eldmN 34680 The fiducial hyperplane (the largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdiass 34681 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)

Theoremdiael 34682 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)

Theoremdiatrl 34683 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)

TheoremdiaelrnN 34684 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Theoremdialss 34685 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdiaord 34686 The partial isomorphism A for a lattice is order-preserving in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)

Theoremdia11N 34687 The partial isomorphism A for a lattice is one-to-one in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)

Theoremdiaf11N 34688 The partial isomorphism A for a lattice is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiaclN 34689 Closure of partial isomorphism A for a lattice . (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiacnvclN 34690 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdia0 34691 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)

Theoremdia1N 34692 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Theoremdia1elN 34693 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiaglbN 34694* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiameetN 34695 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiainN 34696 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaintclN 34697 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiasslssN 34698 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdiassdvaN 34699 The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)

Theoremdia1dim 34700* Two expressions for the 1-dimensional subspaces of partial vector space A (when is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

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