P. 167 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mnd12g 16601	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 16602	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 16603	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	mndplusf 16604	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 16605	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 16606	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 16607	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 16608*	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 16609	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 16610	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 16611*	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 16612	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 16613*	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 16614	0g is unaffected by restriction. This is a bit more generic than submnd0 16615. (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 16615	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	prdsplusgcl 16616	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 16617*	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 16618	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 16619	Zero 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 16620	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 16621	Zero in a product of monoids. (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 16622*	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 16623*	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 16624	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	xpsmnd 16625	The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- T = ( R X.s S )   =>    |- ( ( R e. Mnd /\ S e. Mnd ) -> T e. Mnd )
Theorem	mnd1 16626	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	mnd1OLD 16627	The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) Obsolete version of mnd1 16626 as of 11-Feb-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- M = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }   =>    |- ( I e. V -> M e. Mnd )
Theorem	mnd1id 16628	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 )
10.1.6  Monoid homomorphisms and submonoids
Syntax	cmhm 16629	Hom-set generator class for monoids.
class MndHom
Syntax	csubmnd 16630	Class function taking a monoid to its lattice of submonoids.
class SubMnd
Definition	df-mhm 16631*	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 16632*	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 16633*	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	mhmrcl1 16634	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( F e. ( S MndHom T ) -> S e. Mnd )
Theorem	mhmrcl2 16635	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( F e. ( S MndHom T ) -> T e. Mnd )
Theorem	mhmf 16636	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 16637*	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 16638	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 16639	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 16640	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 16641	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 16642	Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- ( S e. (SubMnd ` M ) -> M e. Mnd )
Theorem	issubm 16643*	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 16644	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 16645*	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	submss 16646	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 16647	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 16648	Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- .0. = ( 0g ` M )   =>    |- ( S e. (SubMnd ` M ) -> .0. e. S )
Theorem	submcl 16649	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 16650	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 16651	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 16652	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 16653	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	0mhm 16654	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 16655	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 16656	One direction of resmhm2b 16657. (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 16657	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 16658	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 16659	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 16660	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 ) )
Theorem	submacs 16661	Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
|- B = ( Base ` G )   =>    |- ( G e. Mnd -> (SubMnd ` G ) e. (ACS ` B ) )
Theorem	mrcmndind 16662*	(( From SO's determinants branch )). TODO: Appropriate description to be added! (Contributed by SO, 14-Jul-2018.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( x = ( y .+ z ) -> ( ps <-> th ) )   &    |- ( x = .0. -> ( ps <-> ta ) )   &    |- ( x = A -> ( ps <-> et ) )   &    |- .0. = ( 0g ` M )   &    |- .+ = ( +g ` M )   &    |- B = ( Base ` M )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> G C_ B )   &    |- ( ph -> B = ( (mrCls ` (SubMnd ` M ) ) ` G ) )   &    |- ( ph -> ta )   &    |- ( ( ( ph /\ y e. B /\ z e. G ) /\ ch ) -> th )   &    |- ( ph -> A e. B )   =>    |- ( ph -> et )
Theorem	prdspjmhm 16663*	A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> I e. V )   &    |- ( ph -> S e. X )   &    |- ( ph -> R : I --> Mnd )   &    |- ( ph -> A e. I )   =>    |- ( ph -> ( x e. B |-> ( x ` A ) ) e. ( Y MndHom ( R ` A ) ) )
Theorem	pwspjmhm 16664*	A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   =>    |- ( ( R e. Mnd /\ I e. V /\ A e. I ) -> ( x e. B |-> ( x ` A ) ) e. ( Y MndHom R ) )
Theorem	pwsdiagmhm 16665*	Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   &    |- F = ( x e. B |-> ( I X. { x } ) )   =>    |- ( ( R e. Mnd /\ I e. W ) -> F e. ( R MndHom Y ) )
Theorem	pwsco1mhm 16666*	Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- Y = ( R ^s A )   &    |- Z = ( R ^s B )   &    |- C = ( Base ` Z )   &    |- ( ph -> R e. Mnd )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( g e. C |-> ( g o. F ) ) e. ( Z MndHom Y ) )
Theorem	pwsco2mhm 16667*	Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- Y = ( R ^s A )   &    |- Z = ( S ^s A )   &    |- B = ( Base ` Y )   &    |- ( ph -> A e. V )   &    |- ( ph -> F e. ( R MndHom S ) )   =>    |- ( ph -> ( g e. B |-> ( F o. g ) ) e. ( Y MndHom Z ) )
10.1.7  Ordered 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-gsum 15390. If order is not significant, it is simpler to use families instead.
Theorem	gsumvallem2 16668*	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 16669	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	gsumz 16670*	Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
Theorem	gsumwsubmcl 16671	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	gsumws1 16672	A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
|- B = ( Base ` G )   =>    |- ( S e. B -> ( G gsumg <" S "> ) = S )
Theorem	gsumwcl 16673	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	gsumccat 16674	Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) )
Theorem	gsumws2 16675	Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ S e. B /\ T e. B ) -> ( G gsumg <" S T "> ) = ( S .+ T ) )
Theorem	gsumccatsn 16676	Homomorphic property of composites with a singleton. (Contributed by AV, 20-Jan-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Mnd /\ W e. Word B /\ Z e. B ) -> ( G gsumg ( W ++ <" Z "> ) ) = ( ( G gsumg W ) .+ Z ) )
Theorem	gsumspl 16677	The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- B = ( Base ` M )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> S e. Word B )   &    |- ( ph -> F e. ( 0 ... T ) )   &    |- ( ph -> T e. ( 0 ... ( # ` S ) ) )   &    |- ( ph -> X e. Word B )   &    |- ( ph -> Y e. Word B )   &    |- ( ph -> ( M gsumg X ) = ( M gsumg Y ) )   =>    |- ( ph -> ( M gsumg ( S splice <. F , T , X >. ) ) = ( M gsumg ( S splice <. F , T , Y >. ) ) )
Theorem	gsumwmhm 16678	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 ) ) )
Theorem	gsumwspan 16679*	The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)
|- B = ( Base ` M )   &    |- K = (mrCls ` (SubMnd ` M ) )   =>    |- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) = ran ( w e. Word G |-> ( M gsumg w ) ) )
10.1.8  Free monoids
Syntax	cfrmd 16680	Extend class definition with the free monoid construction.
class freeMnd
Syntax	cvrmd 16681	Extend class notation with free monoid injection.
class varFMnd
Definition	df-frmd 16682	Define a free monoid over a set i of generators, defined as the set of finite strings on I with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- freeMnd = ( i e. _V |-> { <. ( Base ` ndx ) , Word i >. , <. ( +g ` ndx ) , ( ++ |` (Word i X. Word i ) ) >. } )
Definition	df-vrmd 16683*	Define a free monoid over a set i of generators, defined as the set of finite strings on I with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- varFMnd = ( i e. _V |-> ( j e. i |-> <" j "> ) )
Theorem	frmdval 16684	Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- M = (freeMnd ` I )   &    |- ( I e. V -> B = Word I )   &    |- .+ = ( ++ |` ( B X. B ) )   =>    |- ( I e. V -> M = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. } )
Theorem	frmdbas 16685	The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` M )   =>    |- ( I e. V -> B = Word I )
Theorem	frmdelbas 16686	An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` M )   =>    |- ( X e. B -> X e. Word I )
Theorem	frmdplusg 16687	The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` M )   &    |- .+ = ( +g ` M )   =>    |- .+ = ( ++ |` ( B X. B ) )
Theorem	frmdadd 16688	Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` M )   &    |- .+ = ( +g ` M )   =>    |- ( ( X e. B /\ Y e. B ) -> ( X .+ Y ) = ( X ++ Y ) )
Theorem	vrmdfval 16689*	The canonical injection from the generating set I to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- U = (varFMnd ` I )   =>    |- ( I e. V -> U = ( j e. I |-> <" j "> ) )
Theorem	vrmdval 16690	The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- U = (varFMnd ` I )   =>    |- ( ( I e. V /\ A e. I ) -> ( U ` A ) = <" A "> )
Theorem	vrmdf 16691	The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
|- U = (varFMnd ` I )   =>    |- ( I e. V -> U : I -->Word I )
Theorem	frmdmnd 16692	A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- M = (freeMnd ` I )   =>    |- ( I e. V -> M e. Mnd )
Theorem	frmd0 16693	The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- M = (freeMnd ` I )   =>    |- (/) = ( 0g ` M )
Theorem	frmdsssubm 16694	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.)
|- M = (freeMnd ` I )   =>    |- ( ( I e. V /\ J C_ I ) -> Word J e. (SubMnd ` M ) )
Theorem	frmdgsum 16695	Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- M = (freeMnd ` I )   &    |- U = (varFMnd ` I )   =>    |- ( ( I e. V /\ W e. Word I ) -> ( M gsumg ( U o. W ) ) = W )
Theorem	frmdss2 16696	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 J is Word J". (Contributed by Mario Carneiro, 2-Oct-2015.)
|- M = (freeMnd ` I )   &    |- U = (varFMnd ` I )   =>    |- ( ( I e. V /\ J C_ I /\ A e. (SubMnd ` M ) ) -> ( ( U " J ) C_ A <-> Word J C_ A ) )
Theorem	frmdup1 16697*	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.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` G )   &    |- E = ( x e. Word I |-> ( G gsumg ( A o. x ) ) )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> I e. X )   &    |- ( ph -> A : I --> B )   =>    |- ( ph -> E e. ( M MndHom G ) )
Theorem	frmdup2 16698*	The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` G )   &    |- E = ( x e. Word I |-> ( G gsumg ( A o. x ) ) )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> I e. X )   &    |- ( ph -> A : I --> B )   &    |- U = (varFMnd ` I )   &    |- ( ph -> Y e. I )   =>    |- ( ph -> ( E ` ( U ` Y ) ) = ( A ` Y ) )
Theorem	frmdup3lem 16699*	Lemma for frmdup3 16700. (Contributed by Mario Carneiro, 18-Jul-2016.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` G )   &    |- U = (varFMnd ` I )   =>    |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ ( F e. ( M MndHom G ) /\ ( F o. U ) = A ) ) -> F = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) )
Theorem	frmdup3 16700*	Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.)
|- M = (freeMnd ` I )   &    |- B = ( Base ` G )   &    |- U = (varFMnd ` I )   =>    |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> E! m e. ( M MndHom G ) ( m o. U ) = A )