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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremissubrg3 15901 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
mulGrp       SubRing SubGrp SubMnd

Theoremresrhm 15902 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        RingHom SubRing RingHom

Theoremrhmeql 15903 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 15904 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 15905 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 15906* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
s                      RingHom

Theoremsubrgpropd 15907* 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 15908* 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 15909 The set of absolute values on a ring.
AbsVal

Definitiondf-abv 15910* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 12046 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfval 15911* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabv 15912* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremisabvd 15913* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
AbsVal

Theoremabvrcl 15914 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvfge0 15915 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvf 15916 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvcl 15917 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvge0 15918 The absolute value of a number is greater or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabveq0 15919 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvne0 15920 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvgt0 15921 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvmul 15922 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvtri 15923 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv0 15924 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1z 15925 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabv1 15926 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvneg 15927 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theoremabvsubtri 15928 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal

Theoremabvrec 15929 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal

Theoremabvdiv 15930 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
AbsVal                     /r

Theoremabvdom 15931 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 15932 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 15933* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvtriv 15934* 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 15931 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
AbsVal

Theoremabvpropd 15935* 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 15936 Extend class notation with the functionalization of the *-ring involution.

Syntaxcsr 15937 Extend class notation with class of all *-rings.

Definitiondf-staf 15938* 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 15939* 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 15940* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstafval 15941 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremstaffn 15942 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 15943 The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrhm 15944 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
oppr              RingHom

Theoremsrngrng 15945 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcnv 15946 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngf1o 15947 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngcl 15948 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngnvl 15949 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngadd 15950 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrngmul 15951 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 15952 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 15954.) (Contributed by Mario Carneiro, 6-Oct-2015.)

Theoremsrng0 15953 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremissrngd 15954* 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 15955 Extend class notation with class of all left modules.

Syntaxcscaf 15956 The functionalization of the scalar multiplication operation.

Definitiondf-lmod 15957* 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 15958* 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 15959* The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodlema 15960 Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremislmodd 15961* Properties that determine a left module. See note in isgrpd2 14833 regarding the on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremlmodgrp 15962 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)

Theoremlmodrng 15963 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 15964 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

Theoremlmodbn0 15965 The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodacl 15966 Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodmcl 15967 Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodsn0 15968 The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvacl 15969 Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodass 15970 Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodlcan 15971 Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvscl 15972 Closure of scalar product for a left module. (hvmulcl 22521 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremscaffval 15973* The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscafval 15974 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscafeq 15975 If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremscaffn 15976 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremlmodscaf 15977 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar

Theoremlmodvsdi 15978 Distributive law for scalar product. (ax-hvdistr1 22516 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

Theoremlmodvsdir 15979 Distributive law for scalar product. (ax-hvdistr1 22516 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

Theoremlmodvsass 15980 Associative law for scalar product. (ax-hvmulass 22515 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar

Theoremlmod0cl 15981 The ring zero in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmod1cl 15982 The ring unit in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvs1 15983 Scalar product with ring unit. (ax-hvmulid 22514 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmod0vcl 15984 The zero vector is a vector. (ax-hv0cl 22511 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vlid 15985 Left identity law for the zero vector. (hvaddid2 22530 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vrid 15986 Right identity law for the zero vector. (ax-hvaddid 22512 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vid 15987 Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmod0vs 15988 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 22518 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvs0 15989 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 22531 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvnegcl 15990 Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvnegid 15991 Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvneg1 15992 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodvsneg 15993 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

Theoremlmodvsubcl 15994 Closure of vector subtraction. (hvsubcl 22525 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodcom 15995 Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)

Theoremlmodabl 15996 A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)

Theoremlmodcmn 15997 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
CMnd

Theoremlmodnegadd 15998 Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlmod4 15999 Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsubadd 16000 Relationship between vector subtraction and addition. (hvsubadd 22584 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

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