P. 132 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13101-13200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gsumprval 13101	Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N = ( M + 1 ) )   &    |- ( ph -> F : { M , N } --> B )   =>    |- ( ph -> ( G gsumg F ) = ( ( F ` M ) .+ ( F ` N ) ) )
Theorem	gsumpr12val 13102	Value of the group sum operation over the pair { 1 , 2 }. (Contributed by AV, 14-Dec-2018.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> F : { 1 , 2 } --> B )   =>    |- ( ph -> ( G gsumg F ) = ( ( F ` 1 ) .+ ( F ` 2 ) ) )
7.1.4  Semigroups A semigroup (Smgrp, see df-sgrp 13104) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup 13104). In other words, a semigroup is an associative magma. The notion of semigroup is a generalization of that of group where the existence of an identity or inverses is not required.
Syntax	csgrp 13103	Extend class notation with class of all semigroups.
class Smgrp
Definition	df-sgrp 13104*	A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 13058), whose operation is associative. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4 . (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
|- Smgrp = { g e. Mgm | [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) }
Theorem	issgrp 13105*	The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( M e. Smgrp <-> ( M e. Mgm /\ A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
Theorem	issgrpv 13106*	The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( M e. V -> ( M e. Smgrp <-> A. x e. B A. y e. B ( ( x .o. y ) e. B /\ A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) ) )
Theorem	issgrpn0 13107*	The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( A e. B -> ( M e. Smgrp <-> A. x e. B A. y e. B ( ( x .o. y ) e. B /\ A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) ) )
Theorem	isnsgrp 13108	A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( ( ( X .o. Y ) .o. Z ) =/= ( X .o. ( Y .o. Z ) ) -> M e/ Smgrp ) )
Theorem	sgrpmgm 13109	A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
|- ( M e. Smgrp -> M e. Mgm )
Theorem	sgrpass 13110	A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
|- B = ( Base ` G )   &    |- .o. = ( +g ` G )   =>    |- ( ( G e. Smgrp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .o. Y ) .o. Z ) = ( X .o. ( Y .o. Z ) ) )
Theorem	sgrpcl 13111	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 13112	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 13113	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 13114*	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 13115*	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 ) )
Theorem	prdsplusgsgrpcl 13116	Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I -->Smgrp )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   =>    |- ( ph -> ( F .+ G ) e. B )
Theorem	prdssgrpd 13117	The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I -->Smgrp )   =>    |- ( ph -> Y 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 13119, 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 13121. 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 13118	Extend class notation with class of all monoids.
class Mnd
Definition	df-mnd 13119*	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 13125), whose operation is associative (see mndass 13126) and has a two-sided neutral element (see mndid 13127), see also ismnd 13121. (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 13120*	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 13121*	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 13125), whose operation is associative (so, a semigroup, see also mndass 13126) and has a two-sided neutral element (see mndid 13127). (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 13122*	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 13123	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 13124	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 13125	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 13126	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 13127*	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 13128*	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 13129	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 13130	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 13131	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 13132	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 13133	The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 13132). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
|- B = ( Base ` G )   =>    |- ( G e. Mnd -> B =/= (/) )
Theorem	hashfinmndnn 13134	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 13135	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 13136	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 13137	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 13138	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 13139*	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 13140	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 13141	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 13142*	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 13143	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 13144*	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 13145	0g is unaffected by restriction. This is a bit more generic than submnd0 13146. (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 13146	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 13147*	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	prdsplusgcl 13148	Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I --> Mnd )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   =>    |- ( ph -> ( F .+ G ) e. B )
Theorem	prdsidlem 13149*	Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I --> Mnd )   &    |- .0. = ( 0g o. R )   =>    |- ( ph -> ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
Theorem	prdsmndd 13150	The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Mnd )   =>    |- ( ph -> Y e. Mnd )
Theorem	prds0g 13151	The identity in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Mnd )   =>    |- ( ph -> ( 0g o. R ) = ( 0g ` Y ) )
Theorem	pwsmnd 13152	The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   =>    |- ( ( R e. Mnd /\ I e. V ) -> Y e. Mnd )
Theorem	pws0g 13153	The identity in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Mnd /\ I e. V ) -> ( I X. { .0. } ) = ( 0g ` Y ) )
Theorem	imasmnd2 13154*	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 13155*	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 13156	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 13157	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 13158	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 13159	Hom-set generator class for monoids.
class MndHom
Syntax	csubmnd 13160	Class function taking a monoid to its lattice of submonoids.
class SubMnd
Definition	df-mhm 13161*	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 13162*	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 13163*	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 13164	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 13165	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( F e. ( S MndHom T ) -> S e. Mnd )
Theorem	mhmrcl2 13166	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( F e. ( S MndHom T ) -> T e. Mnd )
Theorem	mhmf 13167	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 13168*	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 13169	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 13170	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 13171	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 13172	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 13173	Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( S e. (SubMnd ` M ) -> M e. Mnd )
Theorem	issubm 13174*	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 13175	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 13176*	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 13177	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 13178	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 13179	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 13180	Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- .0. = ( 0g ` M )   =>    |- ( S e. (SubMnd ` M ) -> .0. e. S )
Theorem	submcl 13181	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 13182	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 13183	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 13184	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 13185	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 13186	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 13187	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 13188	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 13189	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 13190	One direction of resmhm2b 13191. (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 13191	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 13192	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 13193	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 13194	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 general sum operator df-igsum 12961. If order is not significant, it is simpler to use families instead.
Theorem	gsumvallem2 13195*	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	gsumsubm 13196	Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> F : A --> S )   &    |- H = ( G ↾s S )   =>    |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )
Theorem	gsumfzz 13197*	Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 15-Aug-2025.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ M e. ZZ /\ N e. ZZ ) -> ( G gsumg ( k e. ( M ... N ) |-> .0. ) ) = .0. )
Theorem	gsumwsubmcl 13198	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 gsumg W ) e. S )
Theorem	gsumwcl 13199	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 gsumg W ) e. B )
Theorem	gsumwmhm 13200	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 gsumg W ) ) = ( N gsumg ( H o. W ) ) )