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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrmdss2 14501 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 14502* 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 14503* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd              Word g                             varFMnd

Theoremfrmdup3 14504* 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 14505* Define class of all groups. A group is a monoid (df-mnd 14383) 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 13169) and an internal group operation (notated per df-plusg 13237). 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 14511), associativity (so for any a, b, c, see grpass 14512), 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 14514). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 15108). Subgroups can often be formed from groups, see df-subg 14634. An example of an (Abelian) group is the set of complex numbers over the group operation (addition), as proven in cnaddablx 15174; an Abelian group is a group as proven in ablgrp 15110. Other structures include groups, including unital rings (df-rng 15356) and fields (df-field 15531). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Definitiondf-minusg 14506* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)

Definitiondf-sbg 14507* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)

Definitiondf-mulg 14508* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremisgrp 14509* 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 14510 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremgrpcl 14511 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)

Theoremgrpass 14512 A group operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremgrpinvex 14513* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpideu 14514* 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 14515 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrppropd 14516* 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 14517 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 14518 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 14519 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 15362, on the other hand, requires that we know in advance that is a ring.) (Contributed by NM, 11-Oct-2013.)

Theoremisgrpd2e 14520* Deduce a group from its properties. In this version of isgrpd2 14521, we don't assume there is an expression for the inverse of . (Contributed by NM, 10-Aug-2013.)

Theoremisgrpd2 14521* 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 2296, but we make an exception for theorems such as isgrpd2 14521, ismndd 14412, and islmodd 15649 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)

Theoremisgrpde 14522* Deduce a group from its properties. In this version of isgrpd 14523, we don't assume there is an expression for the inverse of . (Contributed by NM, 6-Jan-2015.)

Theoremisgrpd 14523* Deduce a group from its properties. Unlike isgrpd2 14521, 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 14524* Properties that determine a group. (negative) is normally dependent on i.e. read it as . (Contributed by NM, 3-Sep-2011.)

Theoremisgrpix 14525* 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 14526 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremgrpbn0 14527 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremgrplid 14528 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrprid 14529 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrpn0 14530 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrprcan 14531 Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Theoremgrpinveu 14532* 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 14533 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 14534 Properties showing that an element is the identity element of a group. (Contributed by NM, 7-Aug-2013.)

Theoremgrpidd2 14535* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14523. (Contributed by Mario Carneiro, 14-Jun-2015.)

Theoremgrpinvfval 14536* The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvval 14537* The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvfn 14538 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremgrpinvfvi 14539 The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremgrpsubfval 14540* Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)

Theoremgrpsubval 14541 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)

Theoremgrpinvf 14542 The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)

Theoremgrpinvcl 14543 A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremgrplinv 14544 The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrprinv 14545 The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpinvid1 14546 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremgrpinvid2 14547 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremisgrpinv 14548* 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 14549 The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)

Theoremgrplcan 14550 Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)

Theoremgrpinvinv 14551 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)

Theoremgrpinvcnv 14552 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrpinv11 14553 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)

Theoremgrpinvf1o 14554 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 14555 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremgrpinvnzcl 14556 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremgrpsubinv 14557 Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)

Theoremgrplmulf1o 14558* Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremgrpinvpropd 14559* 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 14560 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 14561 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theoremgrpsubcl 14562 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubrcan 14563 Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)

Theoremgrpinvsub 14564 Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)

Theoremgrpinvval2 14565 A df-neg 9056-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremgrpsubid 14566 Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubid1 14567 Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)

Theoremgrpsubeq0 14568 If the difference between two group elements is zero, they are equal. (subeq0 9089 analog.) (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubadd 14569 Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)

Theoremgrpsubsub 14570 Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrpaddsubass 14571 Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)

Theoremgrppncan 14572 Cancellation law for subtraction (pncan 9073 analog). (Contributed by NM, 16-Apr-2014.)

Theoremgrpnpcan 14573 Cancellation law for subtraction (npcan 9076 analog). . (Contributed by NM, 19-Apr-2014.)

Theoremgrpsubsub4 14574 Double group subtraction (subsub4 9096 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)

Theoremgrppnpcan2 14575 Cancellation law for mixed addition and subtraction. (pnpcan2 9103 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrpnpncan 14576 Cancellation law for group subtraction. (npncan 9085 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrpnnncan2 14577 Cancellation law for group subtraction. (nnncan2 9100 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrplactfval 14578* The left group action of element of group . (Contributed by Paul Chapman, 18-Mar-2008.)

Theoremgrplactval 14579* The value of the left group action of element of group at . (Contributed by Paul Chapman, 18-Mar-2008.)

Theoremgrplactcnv 14580* 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 14581* 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 14582 Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)

Theoremgrpsubpropd2 14583* Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremmulgfval 14584* Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgval 14585 Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgfn 14586 Functionality of the group multiple function. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
.g

Theoremmulgfvi 14587 The group multiple function is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
.g       .g

Theoremmulg0 14588 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnn 14589 Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulg1 14590 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnnp1 14591 Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulg2 14592 Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
.g

Theoremmulgnegnn 14593 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnn0p1 14594 Group multiple (exponentiation) operation at a successor, extended to . (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnnsubcl 14595* Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
.g

Theoremmulgnn0subcl 14596* Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
.g

Theoremmulgsubcl 14597* Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
.g

Theoremmulgnncl 14598 Closure of the group multiple (exponentiation) operation. TODO: This can be generalized to a magma if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgnn0cl 14599 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremmulgcl 14600 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

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