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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdprd2db 15601* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd DProd DProd

Theoremdprd2d2 15602* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp       DProd        DProd DProd        DProd DProd DProd DProd

Theoremdmdprdsplit2lem 15603 Lemma for dmdprdsplit 15605. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp                     Cntz              DProd        DProd        DProd DProd        DProd DProd        mrClsSubGrp

Theoremdmdprdsplit2 15604 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp                     Cntz              DProd        DProd        DProd DProd        DProd DProd        DProd

Theoremdmdprdsplit 15605 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp                     Cntz              DProd DProd DProd DProd DProd DProd DProd

Theoremdprdsplit 15606 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp                            DProd        DProd DProd DProd

Theoremdmdprdpr 15607 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz              SubGrp       SubGrp       DProd

Theoremdprdpr 15608 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 26-Apr-2016.)
Cntz              SubGrp       SubGrp                            DProd

Theoremdpjlem 15609 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      DProd

Theoremdpjcntz 15610 The two subgroups that appear in dpjval 15614 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      Cntz       DProd

Theoremdpjdisj 15611 The two subgroups that appear in dpjval 15614 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd

Theoremdpjlsm 15612 The two subgroups that appear in dpjval 15614 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd DProd

Theoremdpjfval 15613* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              DProd

Theoremdpjval 15614 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj                     DProd

Theoremdpjf 15615 The -th index projection is a function from the direct product to the -th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              DProd

Theoremdpjidcl 15616* The key property of projections: the sum of all the projections of is . (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd                      g

Theoremdpjeq 15617* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd                             g

Theoremdpjid 15618* The key property of projections: the sum of all the projections of is . (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd        g

Theoremdpjlid 15619 The -th index projection acts as the identity on elements of the -th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjrid 15620 The -th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjghm 15621 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              s DProd

Theoremdpjghm2 15622 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              s DProd s

10.3.5  The Fundamental Theorem of Abelian Groups

Theoremablfacrplem 15623* Lemma for ablfacrp2 15625. (Contributed by Mario Carneiro, 19-Apr-2016.)

Theoremablfacrp 15624* A finite abelian group whose order factors into relatively prime integers, itself "factors" into two subgroups that have trivial intersection and whose product is the whole group. Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 19-Apr-2016.)

Theoremablfacrp2 15625* The factors of ablfacrp 15624 have the expected orders (which allows for repeated application to decompose into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremablfac1lem 15626* Lemma for ablfac1b 15628. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremablfac1a 15627* The factors of ablfac1b 15628 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremablfac1b 15628* Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd

Theoremablfac1c 15629* The factors of ablfac1b 15628 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd

Theoremablfac1eulem 15630* Lemma for ablfac1eu 15631. (Contributed by Mario Carneiro, 27-Apr-2016.)
DProd DProd                                           DProd

Theoremablfac1eu 15631* The factorization of ablfac1b 15628 is unique, in that any other factorization into prime power factors (even if the exponents are different) must be equal to . (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd DProd

Theorempgpfac1lem1 15632* Lemma for pgpfac1 15638. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp

Theorempgpfac1lem2 15633* Lemma for pgpfac1 15638. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g

Theorempgpfac1lem3a 15634* Lemma for pgpfac1 15638. (Contributed by Mario Carneiro, 4-Jun-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g

Theorempgpfac1lem3 15635* Lemma for pgpfac1 15638. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g                            SubGrp

Theorempgpfac1lem4 15636* Lemma for pgpfac1 15638. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g       SubGrp

Theorempgpfac1lem5 15637* Lemma for pgpfac1 15638 (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp SubGrp        SubGrp

Theorempgpfac1 15638* Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                                    SubGrp

Theorempgpfaclem1 15639* Lemma for pgpfac 15642. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               SubGrp       SubGrp Word DProd DProd        s        mrClsSubGrp              gEx                                          SubGrp                     Word        DProd        DProd        concat        Word DProd DProd

Theorempgpfaclem2 15640* Lemma for pgpfac 15642. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               SubGrp       SubGrp Word DProd DProd        s        mrClsSubGrp              gEx                                          SubGrp                     Word DProd DProd

Theorempgpfaclem3 15641* Lemma for pgpfac 15642. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               SubGrp       SubGrp Word DProd DProd        Word DProd DProd

Theorempgpfac 15642* Full factorization of a finite abelian p-group, by iterating pgpfac1 15638. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               Word DProd DProd

Theoremablfaclem1 15643* Lemma for ablfac 15646. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                                           SubGrp Word DProd DProd        SubGrp Word DProd DProd

Theoremablfaclem2 15644* Lemma for ablfac 15646. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                                           SubGrp Word DProd DProd        Word                      ..^

Theoremablfaclem3 15645* Lemma for ablfac 15646. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                                           SubGrp Word DProd DProd

Theoremablfac 15646* The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                      Word DProd DProd

Theoremablfac2 15647* Choose generators for each cyclic group in ablfac 15646. (Contributed by Mario Carneiro, 28-Apr-2016.)
SubGrp s CycGrp pGrp                      .g              Word DProd DProd

10.4  Rings

10.4.1  Multiplicative Group

Syntaxcmgp 15648 Multiplicative group.
mulGrp

Definitiondf-mgp 15649 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 15772 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (rngmgp 15670) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8940). (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp sSet

Theoremfnmgp 15650 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp

Theoremmgpval 15651 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp              sSet

Theoremmgpplusg 15652 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp

Theoremmgplem 15653 Lemma for mgpbas 15654. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Slot

Theoremmgpbas 15654 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpsca 15655 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15724. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
mulGrp       Scalar       Scalar

Theoremmgptset 15656 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopSet TopSet

Theoremmgptopn 15657 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpds 15658 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpress 15659 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        mulGrp       s mulGrp

10.4.2  Definition and basic properties

Syntaxcrg 15660 Extend class notation with class of all (unital) rings.

Syntaxccrg 15661 Extend class notation with class of all (unital) commutative rings.

Syntaxcur 15662 Extend class notation with ring unit.

Definitiondf-rng 15663* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. So that the additive structure must be abelian (see rngcom 15692), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
mulGrp

Definitiondf-cring 15664 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp CMnd

Definitiondf-ur 15665 Define the multiplicative neutral element of a ring. This definition works by extracting the element, i.e. the neutral element in a group or monoid, and transfering it to the multiplicative monoid via the mulGrp function (df-mgp 15649). See also dfur2 15667, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
mulGrp

Theoremrngidval 15666 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
mulGrp

Theoremdfur2 15667* The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)

Theoremisrng 15668* The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
mulGrp

Theoremrnggrp 15669 A ring is a group. (Contributed by NM, 15-Sep-2011.)

Theoremrngmgp 15670 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
mulGrp

Theoremiscrng 15671 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp       CMnd

Theoremcrngmgp 15672 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp       CMnd

Theoremrngmnd 15673 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremcrngrng 15674 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremmgpf 15675 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp

Theoremrngi 15676 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngcl 15677 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremcrngcom 15678 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremiscrng2 15679* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremrngass 15680 Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngideu 15681* The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngdi 15682 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Theoremrngdir 15683 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Theoremrngidcl 15684 The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremrng0cl 15685 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)

Theoremrngidmlem 15686 Lemma for rnglidm 15687 and rngridm 15688. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremrnglidm 15687 The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)

Theoremrngridm 15688 The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)

Theoremisrngid 15689* Properties showing that an element is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)

Theoremrngidss 15690 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
mulGrps

Theoremrngacl 15691 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremrngcom 15692 Commutativity of the additive group of a ring. (See also lmodcom 15990.) (Contributed by Gérard Lang, 4-Dec-2014.)

Theoremrngabl 15693 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)

Theoremrngcmn 15694 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
CMnd

Theoremrngpropd 15695* If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremcrngpropd 15696* If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremrngprop 15697 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremisrngd 15698* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)

Theoremiscrngd 15699* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremrnglz 15700 The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.)

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