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

Theoremdrngprop 15801 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 15802 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
mulGrps

Theoremdrngmcl 15803 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)

Theoremdrngid 15804 A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
mulGrps

Theoremdrngunz 15805 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)

Theoremdrngid2 15806 Properties showing that an element is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremdrnginvrcl 15807 Closure of the multiplicative inverse in a division ring. (reccl 9641 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrn0 15808 The multiplicative inverse in a division ring is nonzero. (recne0 9647 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrl 15809 Property of the multiplicative inverse in a division ring. (recid2 9649 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrnginvrr 15810 Property of the multiplicative inverse in a division ring. (recid 9648 analog.) (Contributed by NM, 19-Apr-2014.)

Theoremdrngmul0or 15811 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)

Theoremdrngmulne0 15812 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)

Theoremdrngmuleq0 15813 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)

Theoremopprdrng 15814 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
oppr

Theoremisdrngd 15815* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." (Contributed by NM, 2-Aug-2013.)

Theoremisdrngrd 15816* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." This version of isdrngd 15815 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)

Theoremdrngpropd 15817* 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 15818* 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 15819 Extend class notation with all subrings of a ring.
SubRing

Syntaxcrgspn 15820 Extend class notation with span of a set of elements over a ring.
RingSpan

Definitiondf-subrg 15821* 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 15822* 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 15823 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing s

Theoremsubrgss 15824 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgid 15825 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing

Theoremsubrgrng 15826 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrgcrng 15827 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        SubRing

Theoremsubrgrcl 15828 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing

Theoremsubrgsubg 15829 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremsubrg0 15830 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrg1cl 15831 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgbas 15832 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrg1 15833 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrgacl 15834 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgmcl 15835 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgsubm 15836 A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
mulGrp       SubRing SubMnd

Theoremsubrgdvds 15837 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 15838 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       SubRing

Theoremsubrginv 15839 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s               Unit              SubRing

Theoremsubrgdv 15840 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 15841 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 15842 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 15843* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremopprsubrg 15844 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubRing SubRing

Theoremsubrgint 15845 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 15846 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 15847 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
SubRing Moore

Theoremissubdrg 15848* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
s                      SubRing

Theoremsubsubrg 15849 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        SubRing SubRing SubRing

Theoremsubsubrg2 15850 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
s        SubRing SubRing SubRing

Theoremissubrg3 15851 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
mulGrp       SubRing SubGrp SubMnd

Theoremresrhm 15852 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        RingHom SubRing RingHom

Theoremrhmeql 15853 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

Theoremrhmima 15854 The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
RingHom SubRing SubRing

Theoremcntzsubr 15855 Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
mulGrp       Cntz       SubRing

Theorempwsdiagrhm 15856* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
s                      RingHom

Theoremsubrgpropd 15857* If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)
SubRing SubRing

Theoremrhmpropd 15858* Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
RingHom RingHom

10.5.3  Absolute value (abstract algebra)

Syntaxcabv 15859 The set of absolute values on a ring.
AbsVal

Definitiondf-abv 15860* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 11996 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfval 15861* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabv 15862* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabvd 15863* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
AbsVal

Theoremabvrcl 15864 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfge0 15865 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvf 15866 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvcl 15867 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvge0 15868 The absolute value of a number is greater or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabveq0 15869 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvne0 15870 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvgt0 15871 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvmul 15872 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvtri 15873 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv0 15874 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1z 15875 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1 15876 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvneg 15877 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvsubtri 15878 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal

Theoremabvrec 15879 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal

Theoremabvdiv 15880 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal                     /r

Theoremabvdom 15881 Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal

Theoremabvres 15882 The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
AbsVal       s        AbsVal       SubRing

Theoremabvtrivd 15883* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvtriv 15884* The trivial absolute value. (This theorem is true as long as is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 15881 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvpropd 15885* If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal AbsVal

10.5.4  Star rings

Syntaxcstf 15886 Extend class notation with the functionalization of the *-ring involution.

Syntaxcsr 15887 Extend class notation with class of all *-rings.

Definitiondf-staf 15888* Define the functionalization of the involution in a star ring. This is not strictly necessary but by having as an actual function we can state the principal properties of an involution much more cleanly. (Contributed by Mario Carneiro, 6-Oct-2015.)

Definitiondf-srng 15889* Define class of all star rings. A star ring is a ring with an involution (conjugation) function. Involution (unlike say the ring zero) is not unique and therefore must be added as a new component to the ring. For example, two possible involutions for complex numbers are the identity function and complex conjugation. Definition of involution in [Holland95] p. 204. (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
RingHom oppr

Theoremstaffval 15890* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstafval 15891 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstaffn 15892 The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremissrng 15893 The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrhm 15894 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrng 15895 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcnv 15896 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngf1o 15897 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcl 15898 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngnvl 15899 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngadd 15900 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)

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