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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrmdplusg 14801 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd                     concat

Theoremfrmdadd 14802 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd                     concat

Theoremvrmdfval 14803* The canonical injection from the generating set to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdval 14804 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdf 14805 The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd       Word

Theoremfrmdmnd 14806 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd

Theoremfrmd0 14807 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd

Theoremfrmdsssubm 14808 The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd       Word SubMnd

Theoremfrmdgsum 14809 Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd       varFMnd       Word g

Theoremfrmdss2 14810 A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of is Word ". (Contributed by Mario Carneiro, 2-Oct-2015.)
freeMnd       varFMnd       SubMnd Word

Theoremfrmdup1 14811* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd              Word g                             MndHom

Theoremfrmdup2 14812* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd              Word g                             varFMnd

Theoremfrmdup3 14813* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd              varFMnd       MndHom

10.2  Groups

10.2.1  Definition and basic properties

Definitiondf-grp 14814* Define class of all groups. A group is a monoid (df-mnd 14692) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group is an algebraic structure formed from a base set of elements (notated per df-base 13476) and an internal group operation (notated per df-plusg 13544). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 14820), associativity (so for any a, b, c, see grpass 14821), identity (there must be an element such that for any a), and inverse (for each element a in the base set, there must be an element in the base set such that ). It can be proven that the identity element is unique (grpideu 14823). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 15417). Subgroups can often be formed from groups, see df-subg 14943. An example of an (Abelian) group is the set of complex numbers over the group operation (addition), as proven in cnaddablx 15483; an Abelian group is a group as proven in ablgrp 15419. Other structures include groups, including unital rings (df-rng 15665) and fields (df-field 15840). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Definitiondf-minusg 14815* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)

Definitiondf-sbg 14816* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)

Definitiondf-mulg 14817* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremisgrp 14818* The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpmnd 14819 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremgrpcl 14820 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)

Theoremgrpass 14821 A group operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremgrpinvex 14822* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpideu 14823* The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)

Theoremgrpplusf 14824 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrppropd 14825* If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremgrpprop 14826 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)

Theoremgrppropstr 14827 Generalize a specific 2-element group to show that any set with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpss 14828 Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring is a group before we know that it is also a ring. (Theorem rnggrp 15671, on the other hand, requires that we know in advance that is a ring.) (Contributed by NM, 11-Oct-2013.)

Theoremisgrpd2e 14829* Deduce a group from its properties. In this version of isgrpd2 14830, we don't assume there is an expression for the inverse of . (Contributed by NM, 10-Aug-2013.)

Theoremisgrpd2 14830* Deduce a group from its properties. (negative) is normally dependent on i.e. read it as . Note: normally we don't use a antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2438, but we make an exception for theorems such as isgrpd2 14830, ismndd 14721, and islmodd 15958 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)

Theoremisgrpde 14831* Deduce a group from its properties. In this version of isgrpd 14832, we don't assume there is an expression for the inverse of . (Contributed by NM, 6-Jan-2015.)

Theoremisgrpd 14832* Deduce a group from its properties. Unlike isgrpd2 14830, this one goes straight from the base properties rather than going through . (negative) is normally dependent on i.e. read it as . (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremisgrpi 14833* Properties that determine a group. (negative) is normally dependent on i.e. read it as . (Contributed by NM, 3-Sep-2011.)

Theoremisgrpix 14834* Properties that determine a group. Read as . Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)

Theoremgrpidcl 14835 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremgrpbn0 14836 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremgrplid 14837 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrprid 14838 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrpn0 14839 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrprcan 14840 Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Theoremgrpinveu 14841* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)

Theoremgrpid 14842 Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremisgrpid2 14843 Properties showing that an element is the identity element of a group. (Contributed by NM, 7-Aug-2013.)

Theoremgrpidd2 14844* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14832. (Contributed by Mario Carneiro, 14-Jun-2015.)

Theoremgrpinvfval 14845* The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvval 14846* The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvfn 14847 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremgrpinvfvi 14848 The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremgrpsubfval 14849* Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)

Theoremgrpsubval 14850 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)

Theoremgrpinvf 14851 The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)

Theoremgrpinvcl 14852 A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremgrplinv 14853 The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrprinv 14854 The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpinvid1 14855 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremgrpinvid2 14856 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremisgrpinv 14857* Properties showing that a function is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremgrpinvid 14858 The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)

Theoremgrplcan 14859 Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)

Theoremgrpinvinv 14860 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)

Theoremgrpinvcnv 14861 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrpinv11 14862 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)

Theoremgrpinvf1o 14863 The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Theoremgrpinvnz 14864 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremgrpinvnzcl 14865 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremgrpsubinv 14866 Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)

Theoremgrplmulf1o 14867* Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremgrpinvpropd 14868* If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)

Theoremgrpinvadd 14869 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.)

Theoremgrpsubf 14870 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theoremgrpsubcl 14871 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubrcan 14872 Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)

Theoremgrpinvsub 14873 Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)

Theoremgrpinvval2 14874 A df-neg 9296-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremgrpsubid 14875 Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubid1 14876 Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)

Theoremgrpsubeq0 14877 If the difference between two group elements is zero, they are equal. (subeq0 9329 analog.) (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubadd 14878 Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubsub 14879 Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrpaddsubass 14880 Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)

Theoremgrppncan 14881 Cancellation law for subtraction (pncan 9313 analog). (Contributed by NM, 16-Apr-2014.)

Theoremgrpnpcan 14882 Cancellation law for subtraction (npcan 9316 analog). . (Contributed by NM, 19-Apr-2014.)

Theoremgrpsubsub4 14883 Double group subtraction (subsub4 9336 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)

Theoremgrppnpcan2 14884 Cancellation law for mixed addition and subtraction. (pnpcan2 9343 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrpnpncan 14885 Cancellation law for group subtraction. (npncan 9325 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrpnnncan2 14886 Cancellation law for group subtraction. (nnncan2 9340 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrplactfval 14887* The left group action of element of group . (Contributed by Paul Chapman, 18-Mar-2008.)

Theoremgrplactval 14888* The value of the left group action of element of group at . (Contributed by Paul Chapman, 18-Mar-2008.)

Theoremgrplactcnv 14889* The left group action of element of group maps the underlying set of one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Theoremgrplactf1o 14890* The left group action of element of group maps the underlying set of one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

Theoremgrpsubpropd 14891 Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)

Theoremgrpsubpropd2 14892* Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremmulgfval 14893* Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgval 14894 Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgfn 14895 Functionality of the group multiple function. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
.g

Theoremmulgfvi 14896 The group multiple function is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
.g       .g

Theoremmulg0 14897 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnn 14898 Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulg1 14899 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnnp1 14900 Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

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