P. 130 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	unifndxnn 12901	The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
|- ( UnifSet ` ndx ) e. NN
Theorem	basendxltunifndx 12902	The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( Base ` ndx ) < ( UnifSet ` ndx )
Theorem	unifndxnbasendx 12903	The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- ( UnifSet ` ndx ) =/= ( Base ` ndx )
Theorem	unifndxntsetndx 12904	The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( UnifSet ` ndx ) =/= (TopSet ` ndx )
Theorem	slotsdifunifndx 12905	The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
|- ( ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) ) /\ ( ( le ` ndx ) =/= ( UnifSet ` ndx ) /\ ( dist ` ndx ) =/= ( UnifSet ` ndx ) ) )
Theorem	homid 12906	Utility theorem: index-independent form of df-hom 12779. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- Hom = Slot ( Hom ` ndx )
Theorem	homslid 12907	Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- ( Hom = Slot ( Hom ` ndx ) /\ ( Hom ` ndx ) e. NN )
Theorem	ccoid 12908	Utility theorem: index-independent form of df-cco 12780. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- comp = Slot (comp ` ndx )
Theorem	ccoslid 12909	Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- (comp = Slot (comp ` ndx ) /\ (comp ` ndx ) e. NN )
6.1.3  Definition of the structure product
Syntax	crest 12910	Extend class notation with the function returning a subspace topology.
class ↾t
Syntax	ctopn 12911	Extend class notation with the topology extractor function.
class TopOpen
Definition	df-rest 12912*	Function returning the subspace topology induced by the topology y and the set x. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
|- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
Definition	df-topn 12913	Define the topology extractor function. This differs from df-tset 12774 when a structure has been restricted using df-iress 12686; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
Theorem	restfn 12914	The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
|- ↾t Fn ( _V X. _V )
Theorem	topnfn 12915	The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- TopOpen Fn _V
Theorem	restval 12916*	The subspace topology induced by the topology J on the set A. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
Theorem	elrest 12917*	The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|- ( ( J e. V /\ B e. W ) -> ( A e. ( J ↾t B ) <-> E. x e. J A = ( x i^i B ) ) )
Theorem	elrestr 12918	Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|- ( ( J e. V /\ S e. W /\ A e. J ) -> ( A i^i S ) e. ( J ↾t S ) )
Theorem	restid2 12919	The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )
Theorem	restsspw 12920	The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J ↾t A ) C_ ~P A
Theorem	restid 12921	The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- X = U. J   =>    |- ( J e. V -> ( J ↾t X ) = J )
Theorem	topnvalg 12922	Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
|- B = ( Base ` W )   &    |- J = (TopSet ` W )   =>    |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )
Theorem	topnidg 12923	Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- B = ( Base ` W )   &    |- J = (TopSet ` W )   =>    |- ( ( W e. V /\ J C_ ~P B ) -> J = ( TopOpen ` W ) )
Theorem	topnpropgd 12924	The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
|- ( ph -> ( Base ` K ) = ( Base ` L ) )   &    |- ( ph -> (TopSet ` K ) = (TopSet ` L ) )   &    |- ( ph -> K e. V )   &    |- ( ph -> L e. W )   =>    |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
Syntax	ctg 12925	Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
Syntax	cpt 12926	Extend class notation with a function whose value is a product topology.
class Xt_
Syntax	c0g 12927	Extend class notation with group identity element.
class 0g
Syntax	cgsu 12928	Extend class notation to include finitely supported group sums.
class gsumg
Definition	df-0g 12929*	Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-igsum 12930. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.)
|- 0g = ( g e. _V |-> ( iota e ( e e. ( Base ` g ) /\ A. x e. ( Base ` g ) ( ( e ( +g ` g ) x ) = x /\ ( x ( +g ` g ) e ) = x ) ) ) )
Definition	df-igsum 12930*	Define a finite group sum (also called "iterated sum") of a structure. Given G gsumg F where F : A --> ( Base ` G ), the set of indices is A and the values are given by F at each index. A group sum over a multiplicative group may be viewed as a product. The definition is meaningful in different contexts, depending on the size of the index set A and each demanding different properties of G. 1. If A = (/) and G has an identity element, then the sum equals this identity. 2. If A = ( M ... N ) and G is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., ( ( F ` 1 ) + ( F ` 2 ) ) + ( F ` 3 ), etc. 3. This definition does not handle other cases. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)
|- gsumg = ( w e. _V , f e. _V |-> ( iota x ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) ) )
Definition	df-topgen 12931*	Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.)
|- topGen = ( x e. _V |-> { y | y C_ U. ( x i^i ~P y ) } )
Definition	df-pt 12932*	Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
|- Xt_ = ( f e. _V |-> ( topGen ` { x | E. g ( ( g Fn dom f /\ A. y e. dom f ( g ` y ) e. ( f ` y ) /\ E. z e. Fin A. y e. ( dom f \ z ) ( g ` y ) = U. ( f ` y ) ) /\ x = X_ y e. dom f ( g ` y ) ) } ) )
Theorem	tgval 12933*	The topology generated by a basis. See also tgval2 14287 and tgval3 14294. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )
Theorem	tgvalex 12934	The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
|- ( B e. V -> ( topGen ` B ) e. _V )
Theorem	ptex 12935	Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
|- ( F e. V -> ( Xt_ ` F ) e. _V )
Syntax	cprds 12936	The function constructing structure products.
class X_s
Syntax	cpws 12937	The function constructing structure powers.
class ^s
Definition	df-prds 12938*	Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
|- X_s = ( s e. _V , r e. _V |-> [_ X_ x e. dom r ( Base ` ( r ` x ) ) / v ]_ [_ ( f e. v , g e. v |-> X_ x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) ) / h ]_ ( ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( +g ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .r ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( .r ` ( r ` x ) ) ( g ` x ) ) ) ) >. } u. { <. (Scalar ` ndx ) , s >. , <. ( .s ` ndx ) , ( f e. ( Base ` s ) , g e. v |-> ( x e. dom r |-> ( f ( .s ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .i ` ndx ) , ( f e. v , g e. v |-> ( s gsumg ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) >. } ) u. ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. r ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ v /\ A. x e. dom r ( f ` x ) ( le ` ( r ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. v , g e. v |-> sup ( ( ran ( x e. dom r |-> ( ( f ` x ) ( dist ` ( r ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( a e. ( v X. v ) , c e. v |-> ( d e. ( c h ( 2nd ` a ) ) , e e. ( h ` a ) |-> ( x e. dom r |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( r ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) ) )
Theorem	reldmprds 12939	The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
|- Rel dom X_s
Theorem	prdsex 12940	Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
|- ( ( S e. V /\ R e. W ) -> ( S X_s R ) e. _V )
Definition	df-pws 12941*	Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- ^s = ( r e. _V , i e. _V |-> ( (Scalar ` r ) X_s ( i X. { r } ) ) )
6.1.4  Definition of the structure quotient
Syntax	cimas 12942	Image structure function.
class "s
Syntax	cqus 12943	Quotient structure function.
class /.s
Syntax	cxps 12944	Binary product structure function.
class X.s
Definition	df-iimas 12945*	Define an image structure, which takes a structure and a function on the base set, and maps all the operations via the function. For this to work properly f must either be injective or satisfy the well-definedness condition f ( a ) = f ( c ) /\ f ( b ) = f ( d ) -> f ( a + b ) = f ( c + d ) for each relevant operation. Note that although we call this an "image" by association to df-ima 4676, in order to keep the definition simple we consider only the case when the domain of F is equal to the base set of R. Other cases can be achieved by restricting F (with df-res 4675) and/or R ( with df-iress 12686) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by AV, 6-Oct-2020.)
|- "s = ( f e. _V , r e. _V |-> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } )
Definition	df-qus 12946*	Define a quotient ring (or quotient group), which is a special case of an image structure df-iimas 12945 where the image function is x |-> [ x ] e. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
Definition	df-xps 12947*	Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
|- X.s = ( r e. _V , s e. _V |-> ( `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) ) )
Theorem	imasex 12948	Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
|- ( ( F e. V /\ R e. W ) -> ( F "s R ) e. _V )
Theorem	imasival 12949*	Value of an image structure. The is a lemma for the theorems imasbas 12950, imasplusg 12951, and imasmulr 12952 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- .X. = ( .r ` R )   &    |- .x. = ( .s ` R )   &    |- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )   &    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> U = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )
Theorem	imasbas 12950	The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> B = ( Base ` U ) )
Theorem	imasplusg 12951*	The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ph -> R e. Z )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` U )   =>    |- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
Theorem	imasmulr 12952*	The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ph -> R e. Z )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` U )   =>    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
Theorem	f1ocpbllem 12953	Lemma for f1ocpbl 12954. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> F : V -1-1-onto-> X )   =>    |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) <-> ( A = C /\ B = D ) ) )
Theorem	f1ocpbl 12954	An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> F : V -1-1-onto-> X )   =>    |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) ) )
Theorem	f1ovscpbl 12955	An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)
|- ( ph -> F : V -1-1-onto-> X )   =>    |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) -> ( F ` ( A .+ B ) ) = ( F ` ( A .+ C ) ) ) )
Theorem	f1olecpbl 12956	An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> F : V -1-1-onto-> X )   =>    |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( A N B <-> C N D ) ) )
Theorem	imasaddfnlemg 12957*	The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )   &    |- ( ph -> V e. W )   &    |- ( ph -> .x. e. C )   =>    |- ( ph -> .xb Fn ( B X. B ) )
Theorem	imasaddvallemg 12958*	The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )   &    |- ( ph -> V e. W )   &    |- ( ph -> .x. e. C )   =>    |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
Theorem	imasaddflemg 12959*	The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )   &    |- ( ph -> V e. W )   &    |- ( ph -> .x. e. C )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   =>    |- ( ph -> .xb : ( B X. B ) --> B )
Theorem	imasaddfn 12960*	The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> R e. Z )   &    |- .x. = ( +g ` R )   &    |- .xb = ( +g ` U )   =>    |- ( ph -> .xb Fn ( B X. B ) )
Theorem	imasaddval 12961*	The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> R e. Z )   &    |- .x. = ( +g ` R )   &    |- .xb = ( +g ` U )   =>    |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
Theorem	imasaddf 12962*	The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> R e. Z )   &    |- .x. = ( +g ` R )   &    |- .xb = ( +g ` U )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   =>    |- ( ph -> .xb : ( B X. B ) --> B )
Theorem	imasmulfn 12963*	The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> R e. Z )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` U )   =>    |- ( ph -> .xb Fn ( B X. B ) )
Theorem	imasmulval 12964*	The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> R e. Z )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` U )   =>    |- ( ( ph /\ X e. V /\ Y e. V ) -> ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) )
Theorem	imasmulf 12965*	The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( 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 .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> R e. Z )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` U )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   =>    |- ( ph -> .xb : ( B X. B ) --> B )
Theorem	qusval 12966*	Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> .~ e. W )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> U = ( F "s R ) )
Theorem	quslem 12967*	The function in qusval 12966 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> .~ e. W )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> F : V -onto-> ( V /. .~ ) )
Theorem	qusex 12968	Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )
Theorem	qusin 12969	Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ e. W )   &    |- ( ph -> R e. Z )   &    |- ( ph -> ( .~ " V ) C_ V )   =>    |- ( ph -> U = ( R /.s ( .~ i^i ( V X. V ) ) ) )
Theorem	qusbas 12970	Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ e. W )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> ( V /. .~ ) = ( Base ` U ) )
Theorem	divsfval 12971*	Value of the function in qusval 12966. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   =>    |- ( ph -> ( F ` A ) = [ A ] .~ )
Theorem	divsfvalg 12972*	Value of the function in qusval 12966. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> A e. V )   =>    |- ( ph -> ( F ` A ) = [ A ] .~ )
Theorem	ercpbllemg 12973*	Lemma for ercpbl 12974. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. V )   =>    |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> A .~ B ) )
Theorem	ercpbl 12974*	Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) ) -> ( a .+ b ) e. V )   &    |- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) ) )   =>    |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) ) )
Theorem	erlecpbl 12975*	Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A N B <-> C N D ) ) )   =>    |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( A N B <-> C N D ) ) )
Theorem	qusaddvallemg 12976*	Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> R e. Z )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )   &    |- ( ph -> .x. e. W )   =>    |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )
Theorem	qusaddflemg 12977*	The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> R e. Z )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )   &    |- ( ph -> .x. e. W )   =>    |- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )
Theorem	qusaddval 12978*	The addition in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> R e. Z )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   &    |- .x. = ( +g ` R )   &    |- .xb = ( +g ` U )   =>    |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )
Theorem	qusaddf 12979*	The addition in a quotient structure as a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> R e. Z )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   &    |- .x. = ( +g ` R )   &    |- .xb = ( +g ` U )   =>    |- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )
Theorem	qusmulval 12980*	The multiplication in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> R e. Z )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` U )   =>    |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )
Theorem	qusmulf 12981*	The multiplication in a quotient structure as a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> R e. Z )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )   &    |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` U )   =>    |- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )
Theorem	fnpr2o 12982	Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
|- ( ( A e. V /\ B e. W ) -> { <. (/) , A >. , <. 1o , B >. } Fn 2o )
Theorem	fnpr2ob 12983	Biconditional version of fnpr2o 12982. (Contributed by Jim Kingdon, 27-Sep-2023.)
|- ( ( A e. _V /\ B e. _V ) <-> { <. (/) , A >. , <. 1o , B >. } Fn 2o )
Theorem	fvpr0o 12984	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
|- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
Theorem	fvpr1o 12985	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
|- ( B e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
Theorem	fvprif 12986	The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
Theorem	xpsfrnel 12987*	Elementhood in the target space of the function F appearing in xpsval 12995. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( G e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )
Theorem	xpsfeq 12988	A function on 2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
|- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } = G )
Theorem	xpsfrnel2 12989*	Elementhood in the target space of the function F appearing in xpsval 12995. (Contributed by Mario Carneiro, 15-Aug-2015.)
|- ( { <. (/) , X >. , <. 1o , Y >. } e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( X e. A /\ Y e. B ) )
Theorem	xpscf 12990	Equivalent condition for the pair function to be a proper function on A. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( { <. (/) , X >. , <. 1o , Y >. } : 2o --> A <-> ( X e. A /\ Y e. A ) )
Theorem	xpsfval 12991*	The value of the function appearing in xpsval 12995. (Contributed by Mario Carneiro, 15-Aug-2015.)
|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )   =>    |- ( ( X e. A /\ Y e. B ) -> ( X F Y ) = { <. (/) , X >. , <. 1o , Y >. } )
Theorem	xpsff1o 12992*	The function appearing in xpsval 12995 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = { (/) , 1o }. (Contributed by Mario Carneiro, 15-Aug-2015.)
|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )   =>    |- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )
Theorem	xpsfrn 12993*	A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.)
|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )   =>    |- ran F = X_ k e. 2o if ( k = (/) , A , B )
Theorem	xpsff1o2 12994*	The function appearing in xpsval 12995 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = { (/) , 1o }. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )   =>    |- F : ( A X. B ) -1-1-onto-> ran F
Theorem	xpsval 12995*	Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )   &    |- G = (Scalar ` R )   &    |- U = ( G X_s { <. (/) , R >. , <. 1o , S >. } )   =>    |- ( ph -> T = ( `' F "s U ) )
PART 7  BASIC ALGEBRAIC STRUCTURES
7.1  Monoids
7.1.1  Magmas According to Wikipedia ("Magma (algebra)", 08-Jan-2020, https://en.wikipedia.org/wiki/magma_(algebra)) "In abstract algebra, a magma [...] is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.". Since the concept of a "binary operation" is used in different variants, these differences are explained in more detail in the following: With df-mpo 5927, binary operations are defined by a rule, and with df-ov 5925, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation 5925 (19-Jan-2020), "... a binary operation on a set S is a mapping of the elements of the Cartesian product S X. S to S: f : S X. S --> S. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, binary operations mapping the elements of the Cartesian product S X. S are more precisely called internal binary operations. If, in addition, the result is also contained in the set S, the operation should be called closed internal binary operation. Therefore, a "binary operation on a set S" according to Wikipedia is a "closed internal binary operation" in a more precise terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations 5925). The definition of magmas (Mgm, see df-mgm 12999) concentrates on the closure property of the associated operation, and poses no additional restrictions on it. In this way, it is most general and flexible.
Syntax	cplusf 12996	Extend class notation with group addition as a function.
class +f
Syntax	cmgm 12997	Extend class notation with class of all magmas.
class Mgm
Definition	df-plusf 12998*	Define group addition function. Usually we will use +g directly instead of +f, and they have the same behavior in most cases. The main advantage of +f for any magma is that it is a guaranteed function (mgmplusf 13009), while +g only has closure (mgmcl 13002). (Contributed by Mario Carneiro, 14-Aug-2015.)
|- +f = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) y ) ) )
Definition	df-mgm 12999*	A magma is a set equipped with an everywhere defined internal operation. Definition 1 in [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
|- Mgm = { g | [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b ( x o y ) e. b }
Theorem	ismgm 13000*	The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( M e. V -> ( M e. Mgm <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )