P. 138 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13701-13800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sgrpcl 13701	Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)
|- B = ( Base ` G )   &    |- .o. = ( +g ` G )   =>    |- ( ( G e. Smgrp /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B )
Theorem	sgrp0 13702	Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
|- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Smgrp )
Theorem	sgrp1 13703	The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.)
|- M = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }   =>    |- ( I e. V -> M e. Smgrp )
Theorem	issgrpd 13704*	Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> G e. V )   =>    |- ( ph -> G e. Smgrp )
Theorem	sgrppropd 13705*	If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.)
|- ( ph -> K e. V )   &    |- ( ph -> L e. W )   &    |- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( K e. Smgrp <-> L e. Smgrp ) )
7.1.5  Definition and basic properties of monoids According to Wikipedia ("Monoid", https://en.wikipedia.org/wiki/Monoid, 6-Feb-2020,) "In abstract algebra [...] a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity.". In the following, monoids are defined in the second way (as semigroups with identity), see df-mnd 13707, whereas many authors define magmas in the first way (as algebraic structure with a single associative binary operation and an identity element, i.e. without the need of a definition for/knowledge about semigroups), see ismnd 13709. See, for example, the definition in [Lang] p. 3: "A monoid is a set G, with a law of composition which is associative, and having a unit element".
Syntax	cmnd 13706	Extend class notation with class of all monoids.
class Mnd
Definition	df-mnd 13707*	A monoid is a semigroup, which has a two-sided neutral element. Definition 2 in [BourbakiAlg1] p. 12. In other words (according to the definition in [Lang] p. 3), a monoid is a set equipped with an everywhere defined internal operation (see mndcl 13713), whose operation is associative (see mndass 13714) and has a two-sided neutral element (see mndid 13715), see also ismnd 13709. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
|- Mnd = { g e. Smgrp | [. ( Base ` g ) / b ]. [. ( +g ` g ) / p ]. E. e e. b A. x e. b ( ( e p x ) = x /\ ( x p e ) = x ) }
Theorem	ismnddef 13708*	The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
Theorem	ismnd 13709*	The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 13713), whose operation is associative (so, a semigroup, see also mndass 13714) and has a two-sided neutral element (see mndid 13715). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Mnd <-> ( A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
Theorem	sgrpidmndm 13710*	A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Smgrp /\ E. e e. B ( E. w w e. e /\ e = .0. ) ) -> G e. Mnd )
Theorem	mndsgrp 13711	A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
|- ( G e. Mnd -> G e. Smgrp )
Theorem	mndmgm 13712	A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
|- ( M e. Mnd -> M e. Mgm )
Theorem	mndcl 13713	Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Proof shortened by AV, 8-Feb-2020.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
Theorem	mndass 13714	A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
Theorem	mndid 13715*	A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Mnd -> E. u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )
Theorem	mndideu 13716*	The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Mnd -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )
Theorem	mnd32g 13717	Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> ( Y .+ Z ) = ( Z .+ Y ) )   =>    |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )
Theorem	mnd12g 13718	Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> ( X .+ Y ) = ( Y .+ X ) )   =>    |- ( ph -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )
Theorem	mnd4g 13719	Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> W e. B )   &    |- ( ph -> ( Y .+ Z ) = ( Z .+ Y ) )   =>    |- ( ph -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )
Theorem	mndidcl 13720	The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Mnd -> .0. e. B )
Theorem	mndbn0 13721	The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 13720). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
|- B = ( Base ` G )   =>    |- ( G e. Mnd -> B =/= (/) )
Theorem	hashfinmndnn 13722	A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> (♯ ` B ) e. NN )
Theorem	mndplusf 13723	The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.)
|- B = ( Base ` G )   &    |- .+^ = ( +f ` G )   =>    |- ( G e. Mnd -> .+^ : ( B X. B ) --> B )
Theorem	mndlrid 13724	A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )
Theorem	mndlid 13725	The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ X e. B ) -> ( .0. .+ X ) = X )
Theorem	mndrid 13726	The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ X e. B ) -> ( X .+ .0. ) = X )
Theorem	ismndd 13727*	Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> .0. e. B )   &    |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )   &    |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )   =>    |- ( ph -> G e. Mnd )
Theorem	mndpfo 13728	The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.)
|- B = ( Base ` G )   &    |- .+^ = ( +f ` G )   =>    |- ( G e. Mnd -> .+^ : ( B X. B ) -onto-> B )
Theorem	mndfo 13729	The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ .+ Fn ( B X. B ) ) -> .+ : ( B X. B ) -onto-> B )
Theorem	mndpropd 13730*	If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( K e. Mnd <-> L e. Mnd ) )
Theorem	mndprop 13731	If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   =>    |- ( K e. Mnd <-> L e. Mnd )
Theorem	issubmnd 13732*	Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- H = ( G ↾s S )   =>    |- ( ( G e. Mnd /\ S C_ B /\ .0. e. S ) -> ( H e. Mnd <-> A. x e. S A. y e. S ( x .+ y ) e. S ) )
Theorem	ress0g 13733	0g is unaffected by restriction. This is a bit more generic than submnd0 13734. (Contributed by Thierry Arnoux, 23-Oct-2017.)
|- S = ( R ↾s A )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. = ( 0g ` S ) )
Theorem	submnd0 13734	The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- H = ( G ↾s S )   =>    |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> .0. = ( 0g ` H ) )
Theorem	mndinvmod 13735*	Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> A e. B )   =>    |- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )
Theorem	imasmnd2 13736*	The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )   &    |- ( ph -> R e. W )   &    |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V )   &    |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( F ` ( ( x .+ y ) .+ z ) ) = ( F ` ( x .+ ( y .+ z ) ) ) )   &    |- ( ph -> .0. e. V )   &    |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )   &    |- ( ( ph /\ x e. V ) -> ( F ` ( x .+ .0. ) ) = ( F ` x ) )   =>    |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )
Theorem	imasmnd 13737*	The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )   &    |- ( ph -> R e. Mnd )   &    |- .0. = ( 0g ` R )   =>    |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )
Theorem	imasmndf1 13738	The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- U = ( F "s R )   &    |- V = ( Base ` R )   =>    |- ( ( F : V -1-1-> B /\ R e. Mnd ) -> U e. Mnd )
Theorem	mnd1 13739	The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.)
|- M = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }   =>    |- ( I e. V -> M e. Mnd )
Theorem	mnd1id 13740	The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.)
|- M = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }   =>    |- ( I e. V -> ( 0g ` M ) = I )
7.1.6  Monoid homomorphisms and submonoids
Syntax	cmhm 13741	Hom-set generator class for monoids.
class MndHom
Syntax	csubmnd 13742	Class function taking a monoid to its lattice of submonoids.
class SubMnd
Definition	df-mhm 13743*	A monoid homomorphism is a function on the base sets which preserves the binary operation and the identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- MndHom = ( s e. Mnd , t e. Mnd |-> { f e. ( ( Base ` t ) ^m ( Base ` s ) ) | ( A. x e. ( Base ` s ) A. y e. ( Base ` s ) ( f ` ( x ( +g ` s ) y ) ) = ( ( f ` x ) ( +g ` t ) ( f ` y ) ) /\ ( f ` ( 0g ` s ) ) = ( 0g ` t ) ) } )
Definition	df-submnd 13744*	A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- SubMnd = ( s e. Mnd |-> { t e. ~P ( Base ` s ) | ( ( 0g ` s ) e. t /\ A. x e. t A. y e. t ( x ( +g ` s ) y ) e. t ) } )
Theorem	ismhm 13745*	Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- B = ( Base ` S )   &    |- C = ( Base ` T )   &    |- .+ = ( +g ` S )   &    |- .+^ = ( +g ` T )   &    |- .0. = ( 0g ` S )   &    |- Y = ( 0g ` T )   =>    |- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` .0. ) = Y ) ) )
Theorem	mhmex 13746	The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( S MndHom T ) e. _V )
Theorem	mhmrcl1 13747	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( F e. ( S MndHom T ) -> S e. Mnd )
Theorem	mhmrcl2 13748	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( F e. ( S MndHom T ) -> T e. Mnd )
Theorem	mhmf 13749	A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- B = ( Base ` S )   &    |- C = ( Base ` T )   =>    |- ( F e. ( S MndHom T ) -> F : B --> C )
Theorem	mhmpropd 13750*	Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
|- ( ph -> B = ( Base ` J ) )   &    |- ( ph -> C = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> C = ( Base ` M ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )   =>    |- ( ph -> ( J MndHom K ) = ( L MndHom M ) )
Theorem	mhmlin 13751	A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- .+^ = ( +g ` T )   =>    |- ( ( F e. ( S MndHom T ) /\ X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )
Theorem	mhm0 13752	A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- .0. = ( 0g ` S )   &    |- Y = ( 0g ` T )   =>    |- ( F e. ( S MndHom T ) -> ( F ` .0. ) = Y )
Theorem	idmhm 13753	The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)
|- B = ( Base ` M )   =>    |- ( M e. Mnd -> ( _I |` B ) e. ( M MndHom M ) )
Theorem	mhmf1o 13754	A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.)
|- B = ( Base ` R )   &    |- C = ( Base ` S )   =>    |- ( F e. ( R MndHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S MndHom R ) ) )
Theorem	submrcl 13755	Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( S e. (SubMnd ` M ) -> M e. Mnd )
Theorem	issubm 13756*	Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- B = ( Base ` M )   &    |- .0. = ( 0g ` M )   &    |- .+ = ( +g ` M )   =>    |- ( M e. Mnd -> ( S e. (SubMnd ` M ) <-> ( S C_ B /\ .0. e. S /\ A. x e. S A. y e. S ( x .+ y ) e. S ) ) )
Theorem	issubm2 13757	Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- B = ( Base ` M )   &    |- .0. = ( 0g ` M )   &    |- H = ( M ↾s S )   =>    |- ( M e. Mnd -> ( S e. (SubMnd ` M ) <-> ( S C_ B /\ .0. e. S /\ H e. Mnd ) ) )
Theorem	issubmd 13758*	Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
|- B = ( Base ` M )   &    |- .+ = ( +g ` M )   &    |- .0. = ( 0g ` M )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> ch )   &    |- ( ( ph /\ ( ( x e. B /\ y e. B ) /\ ( th /\ ta ) ) ) -> et )   &    |- ( z = .0. -> ( ps <-> ch ) )   &    |- ( z = x -> ( ps <-> th ) )   &    |- ( z = y -> ( ps <-> ta ) )   &    |- ( z = ( x .+ y ) -> ( ps <-> et ) )   =>    |- ( ph -> { z e. B | ps } e. (SubMnd ` M ) )
Theorem	mndissubm 13759	If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ H e. Mnd ) -> ( ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubMnd ` G ) ) )
Theorem	submss 13760	Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- B = ( Base ` M )   =>    |- ( S e. (SubMnd ` M ) -> S C_ B )
Theorem	submid 13761	Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- B = ( Base ` M )   =>    |- ( M e. Mnd -> B e. (SubMnd ` M ) )
Theorem	subm0cl 13762	Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- .0. = ( 0g ` M )   =>    |- ( S e. (SubMnd ` M ) -> .0. e. S )
Theorem	submcl 13763	Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
|- .+ = ( +g ` M )   =>    |- ( ( S e. (SubMnd ` M ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )
Theorem	submmnd 13764	Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- H = ( M ↾s S )   =>    |- ( S e. (SubMnd ` M ) -> H e. Mnd )
Theorem	submbas 13765	The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
|- H = ( M ↾s S )   =>    |- ( S e. (SubMnd ` M ) -> S = ( Base ` H ) )
Theorem	subm0 13766	Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- H = ( M ↾s S )   &    |- .0. = ( 0g ` M )   =>    |- ( S e. (SubMnd ` M ) -> .0. = ( 0g ` H ) )
Theorem	subsubm 13767	A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
|- H = ( G ↾s S )   =>    |- ( S e. (SubMnd ` G ) -> ( A e. (SubMnd ` H ) <-> ( A e. (SubMnd ` G ) /\ A C_ S ) ) )
Theorem	0subm 13768	The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Mnd -> { .0. } e. (SubMnd ` G ) )
Theorem	insubm 13769	The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
|- ( ( A e. (SubMnd ` M ) /\ B e. (SubMnd ` M ) ) -> ( A i^i B ) e. (SubMnd ` M ) )
Theorem	0mhm 13770	The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- .0. = ( 0g ` N )   &    |- B = ( Base ` M )   =>    |- ( ( M e. Mnd /\ N e. Mnd ) -> ( B X. { .0. } ) e. ( M MndHom N ) )
Theorem	resmhm 13771	Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
|- U = ( S ↾s X )   =>    |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( F |` X ) e. ( U MndHom T ) )
Theorem	resmhm2 13772	One direction of resmhm2b 13773. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- U = ( T ↾s X )   =>    |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F e. ( S MndHom T ) )
Theorem	resmhm2b 13773	Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- U = ( T ↾s X )   =>    |- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )
Theorem	mhmco 13774	The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- ( ( F e. ( T MndHom U ) /\ G e. ( S MndHom T ) ) -> ( F o. G ) e. ( S MndHom U ) )
Theorem	mhmima 13775	The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
|- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) e. (SubMnd ` N ) )
Theorem	mhmeql 13776	The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
|- ( ( F e. ( S MndHom T ) /\ G e. ( S MndHom T ) ) -> dom ( F i^i G ) e. (SubMnd ` S ) )
7.1.7  Iterated sums in a monoid One important use of words is as formal composites in cases where order is significant, using the ordered general sum operator df-gzsum 13590. If order is not significant, it is simpler to use families instead.
Theorem	gsumvallem2 13777*	Lemma for properties of the set of identities of G. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }   =>    |- ( G e. Mnd -> O = { .0. } )
Theorem	gzsumwsubmcl 13778	Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
|- ( ( S e. (SubMnd ` G ) /\ W e. Word S ) -> ( G gzsumgz W ) e. S )
Theorem	gzsumwcl 13779	Closure of the composite of a word in a structure G. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- B = ( Base ` G )   =>    |- ( ( G e. Mnd /\ W e. Word B ) -> ( G gzsumgz W ) e. B )
Theorem	gzsumwmhm 13780	Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
|- B = ( Base ` M )   =>    |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H ` ( M gzsumgz W ) ) = ( N gzsumgz ( H o. W ) ) )
Theorem	gzsumcl 13781	Closure of an ordered group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 16-Aug-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) e. B )
7.2  Groups
7.2.1  Definition and basic properties
Syntax	cgrp 13782	Extend class notation with class of all groups.
class Grp
Syntax	cminusg 13783	Extend class notation with inverse of group element.
class invg
Syntax	csg 13784	Extend class notation with group subtraction (or division) operation.
class -g
Definition	df-grp 13785*	Define class of all groups. A group is a monoid (df-mnd 13707) 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 G is an algebraic structure formed from a base set of elements (notated ( Base ` G ) per df-base 13336) and an internal group operation (notated ( +g ` G ) per df-plusg 13421). 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 13790), associativity (so ( ( a +g b ) +g c ) = ( a +g ( b +g c ) ) for any a, b, c, see grpass 13791), identity (there must be an element e = ( 0g ` G ) such that e +g a = a +g e = a for any a), and inverse (for each element a in the base set, there must be an element b = invg a in the base set such that a +g b = b +g a = e). It can be proven that the identity element is unique (grpideu 13793). Groups need not be commutative; a commutative group is an Abelian group. Subgroups can often be formed from groups. An example of an (Abelian) group is the set of complex numbers CC over the group operation + (addition). Other structures include groups, including unital rings and fields. (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- Grp = { g e. Mnd | A. a e. ( Base ` g ) E. m e. ( Base ` g ) ( m ( +g ` g ) a ) = ( 0g ` g ) }
Definition	df-minusg 13786*	Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
|- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ w e. ( Base ` g ) ( w ( +g ` g ) x ) = ( 0g ` g ) ) ) )
Definition	df-sbg 13787*	Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
|- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
Theorem	isgrp 13788*	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.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Grp <-> ( G e. Mnd /\ A. a e. B E. m e. B ( m .+ a ) = .0. ) )
Theorem	grpmnd 13789	A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. Grp -> G e. Mnd )
Theorem	grpcl 13790	Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
Theorem	grpass 13791	A group operation is associative. (Contributed by NM, 14-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
Theorem	grpinvex 13792*	Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
Theorem	grpideu 13793*	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.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> E! u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) )
Theorem	grpassd 13794	A group operation is associative. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
Theorem	grpmndd 13795	A group is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mnd )
Theorem	grpcld 13796	Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+ Y ) e. B )
Theorem	grpplusf 13797	The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- F = ( +f ` G )   =>    |- ( G e. Grp -> F : ( B X. B ) --> B )
Theorem	grpplusfo 13798	The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.)
|- B = ( Base ` G )   &    |- F = ( +f ` G )   =>    |- ( G e. Grp -> F : ( B X. B ) -onto-> B )
Theorem	grppropd 13799*	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.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( K e. Grp <-> L e. Grp ) )
Theorem	grpprop 13800	If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   =>    |- ( K e. Grp <-> L e. Grp )