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

Theoremisdrngd 15801* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." (Contributed by NM, 2-Aug-2013.)

Theoremisdrngrd 15802* Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." This version of isdrngd 15801 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)

Theoremdrngpropd 15803* 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 15804* 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 15805 Extend class notation with all subrings of a ring.
SubRing

Syntaxcrgspn 15806 Extend class notation with span of a set of elements over a ring.
RingSpan

Definitiondf-subrg 15807* 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 15808* 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 15809 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing s

Theoremsubrgss 15810 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgid 15811 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing

Theoremsubrgrng 15812 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrgcrng 15813 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        SubRing

Theoremsubrgrcl 15814 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing

Theoremsubrgsubg 15815 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremsubrg0 15816 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrg1cl 15817 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
SubRing

Theoremsubrgbas 15818 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s        SubRing

Theoremsubrg1 15819 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s               SubRing

Theoremsubrgacl 15820 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgmcl 15821 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubRing

Theoremsubrgsubm 15822 A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
mulGrp       SubRing SubMnd

Theoremsubrgdvds 15823 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 15824 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        Unit       Unit       SubRing

Theoremsubrginv 15825 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
s               Unit              SubRing

Theoremsubrgdv 15826 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 15827 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 15828 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 15829* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
SubRing SubGrp

Theoremopprsubrg 15830 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubRing SubRing

Theoremsubrgint 15831 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 15832 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 15833 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
SubRing Moore

Theoremissubdrg 15834* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
s                      SubRing

Theoremsubsubrg 15835 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
s        SubRing SubRing SubRing

Theoremsubsubrg2 15836 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
s        SubRing SubRing SubRing

Theoremissubrg3 15837 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
mulGrp       SubRing SubGrp SubMnd

Theoremresrhm 15838 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        RingHom SubRing RingHom

Theoremrhmeql 15839 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 15840 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 15841 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 15842* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
s                      RingHom

Theoremsubrgpropd 15843* 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 15844* 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 15845 The set of absolute values on a ring.
AbsVal

Definitiondf-abv 15846* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 11982 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfval 15847* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabv 15848* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabvd 15849* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
AbsVal

Theoremabvrcl 15850 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfge0 15851 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvf 15852 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvcl 15853 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvge0 15854 The absolute value of a number is greater or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabveq0 15855 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvne0 15856 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvgt0 15857 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvmul 15858 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvtri 15859 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv0 15860 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1z 15861 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1 15862 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvneg 15863 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvsubtri 15864 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal

Theoremabvrec 15865 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal

Theoremabvdiv 15866 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal                     /r

Theoremabvdom 15867 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 15868 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 15869* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvtriv 15870* 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 15867 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvpropd 15871* 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 15872 Extend class notation with the functionalization of the *-ring involution.

Syntaxcsr 15873 Extend class notation with class of all *-rings.

Definitiondf-staf 15874* 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 15875* 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 15876* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstafval 15877 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstaffn 15878 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 15879 The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrhm 15880 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrng 15881 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcnv 15882 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngf1o 15883 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcl 15884 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngnvl 15885 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngadd 15886 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngmul 15887 The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrng1 15888 The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 15890.) (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrng0 15889 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremissrngd 15890* Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)

10.6  Left modules

10.6.1  Definition and basic properties

Syntaxclmod 15891 Extend class notation with class of all left modules.

Syntaxcscaf 15892 The functionalization of the scalar multiplication operation.

Definitiondf-lmod 15893* Define the class of all left modules, which are generalizations of left vector spaces. A left module over a ring is an (Abelian) group (vectors) together with a ring (scalars) and a left scalar product connecting them. (Contributed by NM, 4-Nov-2013.)
Scalar

Definitiondf-scaf 15894* Define the functionalization of the operator. This restricts the value of to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremislmod 15895* The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodlema 15896 Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremislmodd 15897* Properties that determine a left module. See note in isgrpd2 14769 regarding the on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremlmodgrp 15898 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)

Theoremlmodrng 15899 The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodfgrp 15900 The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

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