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Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrnginvdv 15801 Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
Unit       /r

Theoremrngidpropd 15802* The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)

Theoremdvdsrpropd 15803* The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
r r

Theoremunitpropd 15804* The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
Unit Unit

Theoreminvrpropd 15805* The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)

Theoremisirred 15806* An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       Irred

Theoremisnirred 15807* The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       Irred

Theoremisirred2 15808* Expand out the set differences from isirred 15806. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       Irred

Theoremopprirred 15809 Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
oppr       Irred       Irred

Theoremirredn0 15810 The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirredcl 15811 An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirrednu 15812 An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred       Unit

Theoremirredn1 15813 The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirredrmul 15814 The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred       Unit

Theoremirredlmul 15815 The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred       Unit

Theoremirredmul 15816 If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred              Unit

Theoremirredneg 15817 The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

Theoremirrednegb 15818 An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred

10.4.5  Ring homomorphisms

Syntaxcrh 15819 Ring homomorphisms.
RingHom

Syntaxcrs 15820 Ring isomorphisms.
RingIso

Definitiondf-rnghom 15821* Define the set of ring homomorphisms from to . (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Definitiondf-rngiso 15822* Define the set of ring isomorphisms from to . (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingIso RingHom RingHom

Theoremdfrhm2 15823* The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom mulGrp MndHom mulGrp

Theoremrhmrcl1 15824 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Theoremrhmrcl2 15825 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Theoremisrhm 15826 A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
mulGrp       mulGrp       RingHom MndHom

Theoremrhmmhm 15827 A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
mulGrp       mulGrp       RingHom MndHom

Theoremrhmghm 15828 A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom

Theoremrhmf 15829 A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhmmul 15830 A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom

Theoremisrhm2d 15831* Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
RingHom

Theoremisrhmd 15832* Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhm1 15833 Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
RingHom

Theoremrhmco 15834 The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom RingHom RingHom

Theorempwsco1rhm 15835* Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                                           RingHom

Theorempwsco2rhm 15836* Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                      RingHom        RingHom

10.5  Division rings and fields

10.5.1  Definition and basic properties

Syntaxcdr 15837 Extend class notation with class of all division rings.

Syntaxcfield 15838 Class of fields.
Field

Definitiondf-drng 15839 Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
Unit

Definitiondf-field 15840 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field

Theoremisdrng 15841 The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngunit 15842 Elementhood in the set of units when is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngui 15843 The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremdrngrng 15844 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)

Theoremdrnggrp 15845 A division ring is a group. (Contributed by NM, 8-Sep-2011.)

Theoremisfld 15846 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field

Theoremisdrng2 15847 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
mulGrps

Theoremdrngprop 15848 If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)

Theoremdrngmgp 15849 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
mulGrps

Theoremdrngmcl 15850 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)

Theoremdrngid 15851 A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
mulGrps

Theoremdrngunz 15852 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)

Theoremdrngid2 15853 Properties showing that an element is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremdrnginvrcl 15854 Closure of the multiplicative inverse in a division ring. (reccl 9687 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrn0 15855 The multiplicative inverse in a division ring is nonzero. (recne0 9693 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrl 15856 Property of the multiplicative inverse in a division ring. (recid2 9695 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrr 15857 Property of the multiplicative inverse in a division ring. (recid 9694 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrngmul0or 15858 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)

Theoremdrngmulne0 15859 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)

Theoremdrngmuleq0 15860 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)

Theoremopprdrng 15861 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
oppr

Theoremisdrngd 15862* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." (Contributed by NM, 2-Aug-2013.)

Theoremisdrngrd 15863* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." This version of isdrngd 15862 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)

Theoremdrngpropd 15864* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)

Theoremfldpropd 15865* If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
Field Field

10.5.2  Subrings of a ring

Syntaxcsubrg 15866 Extend class notation with all subrings of a ring.
SubRing

Syntaxcrgspn 15867 Extend class notation with span of a set of elements over a ring.
RingSpan

Definitiondf-subrg 15868* Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity.

The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset of (where multiplication is component-wise) contains the false identity which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

SubRing s

Definitiondf-rgspn 15869* The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan SubRing

Theoremissubrg 15870 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing s

Theoremsubrgss 15871 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgid 15872 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing

Theoremsubrgrng 15873 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrgcrng 15874 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        SubRing

Theoremsubrgrcl 15875 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing

Theoremsubrgsubg 15876 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremsubrg0 15877 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrg1cl 15878 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgbas 15879 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrg1 15880 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrgacl 15881 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgmcl 15882 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgsubm 15883 A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
mulGrp       SubRing SubMnd

Theoremsubrgdvds 15884 If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        r       r       SubRing

Theoremsubrguss 15885 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       SubRing

Theoremsubrginv 15886 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s               Unit              SubRing

Theoremsubrgdv 15887 A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        /r       Unit       /r       SubRing

Theoremsubrgunit 15888 An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit              SubRing

Theoremsubrgugrp 15889 The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       mulGrps        SubRing SubGrp

Theoremissubrg2 15890* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremopprsubrg 15891 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubRing SubRing

Theoremsubrgint 15892 The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
SubRing SubRing

Theoremsubrgin 15893 The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
SubRing SubRing SubRing

Theoremsubrgmre 15894 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
SubRing Moore

Theoremissubdrg 15895* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
s                      SubRing

Theoremsubsubrg 15896 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        SubRing SubRing SubRing

Theoremsubsubrg2 15897 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
s        SubRing SubRing SubRing

Theoremissubrg3 15898 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
mulGrp       SubRing SubGrp SubMnd

Theoremresrhm 15899 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        RingHom SubRing RingHom

Theoremrhmeql 15900 The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
RingHom RingHom SubRing

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