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Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgrpoidinvlem1 20701 Lemma for grpoidinv 20705. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem2 20702 Lemma for grpoidinv 20705. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem3 20703* Lemma for grpoidinv 20705. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem4 20704* Lemma for grpoidinv 20705. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinv 20705* 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 20706* 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 20707 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theorem0ngrp 20708 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)

Theoremgrporn 20709 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 20710* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremfngid 20711 GId is a function. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrposn 20712 The group operation for the singleton group. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremgrpoidval 20713* Lemma for grpoidcl 20714 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoidcl 20714 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 20715* 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 20716 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 20717 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 20718 Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)

Theoremgrpoinveu 20719* 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 20720 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 20721* 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 20722* 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 20723 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 20724 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 20725 The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrporinv 20726 The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoinvid1 20727 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 20728 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 20729 The inverse of the identity element of a group. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremgrpolcan 20730 Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Theoremgrpo2grp 20731 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 20732* 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 20733* 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 20734 An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

Theoremgrpoasscan2 20735 An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgrpo2inv 20736 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 20737 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 20738 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 20739* Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivval 20740 Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivinv 20741 Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpoinvdiv 20742 Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivf 20743 Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivcl 20744 Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivdiv 20745 Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpomuldivass 20746 Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivid 20747 Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
GId

Theoremgrpopncan 20748 Cancellation law for group division. (pncan 8937 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponpcan 20749 Cancellation law for group division. (npcan 8940 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpopnpcan2 20750 Cancellation law for mixed addition and group division. (pnpcan2 8967 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponnncan2 20751 Cancellation law for group division. (nnncan2 8964 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponpncan 20752 Cancellation law for group division. (npncan 8949 analog.) (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpodiveq 20753 Relationship between group division and group multiplication. (Contributed by Mario Carneiro, 11-Jul-2014.) (New usage is discouraged.)

Theoremgxfval 20754* 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 20755 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 20756 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 20757 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 20758 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 20759 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 20760 A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 20763 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxnn0suc 20761 Induction on group power (lemma with nonnegative exponent - use gxsuc 20769 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxcl 20762 Closure of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxneg 20763 A negative group power is the inverse of the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxneg2 20764 The inverse of a negative group power is the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxm1 20765 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 20766 The group power operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxinv 20767 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxinv2 20768 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxsuc 20769 Induction on group power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxid 20770 The identity element of a group to any power remains unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremgxnn0add 20771 The group power of a sum is the group product of the powers (lemma with nonnegative exponent - use gxadd 20772 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxadd 20772 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 20773 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 20774 The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 20775 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxmul 20775 The group power of a product is the composition of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxmodid 20776 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 20777 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.)

15.1.2  Definition and basic properties of Abelian groups

Syntaxcablo 20778 Extend class notation with the class of all Abelian group operations.

Definitiondf-ablo 20779* Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremisablo 20780* The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablogrpo 20781 An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablocom 20782 An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablo32 20783 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremablo4 20784 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremisabloi 20785* Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremablomuldiv 20786 Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremablodivdiv 20787 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablodivdiv4 20788 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablodiv32 20789 Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablonnncan 20790 Cancellation law for group division. (nnncan 8962 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablonncan 20791 Cancellation law for group division. (nncan 8956 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremablonnncan1 20792 Cancellation law for group division. (nnncan1 8963 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremgxdi 20793 Distribution of group power over group operation for abelian groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremisgrpda 20794* Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)

Theoremisgrpod 20795* Properties that determine a group operation. (Renamed from isgrpd 14342 to isgrpod 20795 to prevent naming conflict. -NM 5-Jun-2013) (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)

Theoremisabloda 20796* Properties that determine an Abelian group operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.)

Theoremisablod 20797* Properties that determine an Abelian group operation. (Changed label from isabld 14937 to isablod 20797-NM 6-Aug-2013.) (Contributed by Jeff Madsen, 5-Dec-2009.) (New usage is discouraged.)

15.1.3  Subgroups

Syntaxcsubgo 20798 Extend class notation to include the class of subgroups.

Definitiondf-subgo 20799 Define the set of subgroups of . (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)

Theoremissubgo 20800 The predicate "is a subgroup of ." (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 12-Jul-2014.) (New usage is discouraged.)

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