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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-gid 26001* Define a function that maps a group operation to the group's identity element. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Definitiondf-ginv 26002* Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
GId

Definitiondf-gdiv 26003* Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-gx 26004* Define a function that maps a group operation to the group's power operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremisgrpo 26005* The predicate "is a group operation." Note that is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremisgrpo2 26006* The predicate "is a group operation." (Contributed by NM, 23-Oct-2012.) (New usage is discouraged.)

Theoremisgrpoi 26007* Properties that determine a group operation. Read as . (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremgrpofo 26008 A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)

Theoremgrpocl 26009 Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpolidinv 26010* A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremgrpon0 26011 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)

Theoremgrpoass 26012 A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem1 26013 Lemma for grpoidinv 26017. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem2 26014 Lemma for grpoidinv 26017. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem3 26015* Lemma for grpoidinv 26017. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem4 26016* Lemma for grpoidinv 26017. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinv 26017* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Theoremgrpoideu 26018* The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Theoremgrporndm 26019 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theorem0ngrp 26020 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)

Theoremgrporn 26021 The range of a group operation. Useful for satisfying group base set hypotheses of the form . (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremgidval 26022* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremfngid 26023 GId is a function. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrposn 26024 The group operation for the singleton group. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremgrpoidval 26025* Lemma for grpoidcl 26026 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoidcl 26026 The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoidinv2 26027* A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpolid 26028 The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrporid 26029 The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrporcan 26030 Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)

Theoremgrpoinveu 26031* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoid 26032 Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GId

Theoremgrpoinvfval 26033* The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoinvval 26034* The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoinvcl 26035 A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpoinv 26036 The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpolinv 26037 The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrporinv 26038 The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoinvid1 26039 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoinvid2 26040 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoinvid 26041 The inverse of the identity element of a group. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremgrpolcan 26042 Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Theoremgrpo2grp 26043 Convert a group operation to a group structure. (Contributed by NM, 25-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)

Theoremisgrp2d 26044* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremisgrp2i 26045* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremgrpoasscan1 26046 An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

Theoremgrpoasscan2 26047 An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgrpo2inv 26048 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Theoremgrpoinvf 26049 Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpoinvop 26050 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Theoremgrpodivfval 26051* Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivval 26052 Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivinv 26053 Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpoinvdiv 26054 Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivf 26055 Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivcl 26056 Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivdiv 26057 Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpomuldivass 26058 Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivid 26059 Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
GId

Theoremgrpopncan 26060 Cancellation law for group division. (pncan 9901 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponpcan 26061 Cancellation law for group division. (npcan 9904 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpopnpcan2 26062 Cancellation law for mixed addition and group division. (pnpcan2 9934 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponnncan2 26063 Cancellation law for group division. (nnncan2 9931 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponpncan 26064 Cancellation law for group division. (npncan 9915 analog.) (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpodiveq 26065 Relationship between group division and group multiplication. (Contributed by Mario Carneiro, 11-Jul-2014.) (New usage is discouraged.)

Theoremgxfval 26066* The value of the group power operator function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgxval 26067 The result of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgxpval 26068 The result of the group power operator when the exponent is positive. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxnval 26069 The result of the group power operator when the exponent is negative. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgx0 26070 The result of the group power operator when the exponent is zero. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgx1 26071 The result of the group power operator when the exponent is one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxnn0neg 26072 A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 26075 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxnn0suc 26073 Induction on group power (lemma with nonnegative exponent - use gxsuc 26081 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxcl 26074 Closure of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxneg 26075 A negative group power is the inverse of the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxneg2 26076 The inverse of a negative group power is the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxm1 26077 The result of the group power operator when the exponent is minus one. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxcom 26078 The group power operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxinv 26079 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxinv2 26080 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxsuc 26081 Induction on group power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxid 26082 The identity element of a group to any power remains unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremgxnn0add 26083 The group power of a sum is the group product of the powers (lemma with nonnegative exponent - use gxadd 26084 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxadd 26084 The group power of a sum is the group product of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxsub 26085 The group power of a difference is the group quotient of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxnn0mul 26086 The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 26087 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxmul 26087 The group power of a product is the composition of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxmodid 26088 Casting out powers of the identity element leaves the group power unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremresgrprn 26089 The underlying set of a group operation which is a restriction of a mapping. (Contributed by Paul Chapman, 25-Mar-2008.) (New usage is discouraged.)

18.1.2  Definition and basic properties of Abelian groups

Syntaxcablo 26090 Extend class notation with the class of all Abelian group operations.

Definitiondf-ablo 26091* Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremisablo 26092* The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablogrpo 26093 An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablocom 26094 An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablo32 26095 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremablo4 26096 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremisabloi 26097* Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremablomuldiv 26098 Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremablodivdiv 26099 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablodivdiv4 26100 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

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