P. 134 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13301-13400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	topnpropgd 13301	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 13302	Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
Syntax	cpt 13303	Extend class notation with a function whose value is a product topology.
class Xt_
Syntax	c0g 13304	Extend class notation with group identity element.
class 0g
Syntax	cgsu 13305	Extend class notation to include finitely supported group sums.
class gsumg
Definition	df-0g 13306*	Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-igsum 13307. 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 13307*	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 13308*	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 13309*	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 13310*	The topology generated by a basis. See also tgval2 14740 and tgval3 14747. (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 13311	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 13312	Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
|- ( F e. V -> ( Xt_ ` F ) e. _V )
Syntax	cprds 13313	The function constructing structure products.
class X_s
Syntax	cpws 13314	The function constructing structure powers.
class ^s
Definition	df-prds 13315*	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.) (Revised by Zhi Wang, 18-Aug-2024.)
|- 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. ( ( 2nd ` a ) h c ) , 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 13316	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 13317	Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
|- ( ( S e. V /\ R e. W ) -> ( S X_s R ) e. _V )
Theorem	imasvalstrd 13318	An image structure value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- U = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , L >. , <. ( dist ` ndx ) , D >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Z )   &    |- ( ph -> ., e. P )   &    |- ( ph -> O e. Q )   &    |- ( ph -> L e. R )   &    |- ( ph -> D e. A )   =>    |- ( ph -> U Struct <. 1 , ; 1 2 >. )
Theorem	prdsvalstrd 13319	Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
|- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Z )   &    |- ( ph -> ., e. P )   &    |- ( ph -> O e. Q )   &    |- ( ph -> L e. R )   &    |- ( ph -> D e. A )   &    |- ( ph -> H e. T )   &    |- ( ph -> .xb e. U )   =>    |- ( ph -> ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) u. ( { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , L >. , <. ( dist ` ndx ) , D >. } u. { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .xb >. } ) ) Struct <. 1 , ; 1 5 >. )
Theorem	prdsvallem 13320*	Lemma for prdsval 13321. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 13321, dependency on df-hom 13149 removed. (Revised by AV, 13-Oct-2024.)
|- ( f e. v , g e. v |-> X_ x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) ) e. _V
Theorem	prdsval 13321*	Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- K = ( Base ` S )   &    |- ( ph -> dom R = I )   &    |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )   &    |- ( ph -> .+ = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) )   &    |- ( ph -> .X. = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) )   &    |- ( ph -> .x. = ( f e. K , g e. B |-> ( x e. I |-> ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) ) )   &    |- ( ph -> ., = ( f e. B , g e. B |-> ( S gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) )   &    |- ( ph -> O = ( Xt_ ` ( TopOpen o. R ) ) )   &    |- ( ph -> .<_ = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } )   &    |- ( ph -> D = ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) )   &    |- ( ph -> H = ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) )   &    |- ( ph -> .xb = ( a e. ( B X. B ) , c e. B |-> ( d e. ( ( 2nd ` a ) H c ) , e e. ( H ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) )   &    |- ( ph -> S e. W )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> P = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) u. ( { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } u. { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .xb >. } ) ) )
Theorem	prdsbaslemss 13322	Lemma for prdsbas 13324 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- A = ( E ` P )   &    |- E = Slot ( E ` ndx )   &    |- ( E ` ndx ) e. NN   &    |- ( ph -> T e. X )   &    |- ( ph -> { <. ( E ` ndx ) , T >. } C_ P )   =>    |- ( ph -> A = T )
Theorem	prdssca 13323	Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   =>    |- ( ph -> S = (Scalar ` P ) )
Theorem	prdsbas 13324*	Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- B = ( Base ` P )   &    |- ( ph -> dom R = I )   =>    |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )
Theorem	prdsplusg 13325*	Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- B = ( Base ` P )   &    |- ( ph -> dom R = I )   &    |- .+ = ( +g ` P )   =>    |- ( ph -> .+ = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) )
Theorem	prdsmulr 13326*	Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- B = ( Base ` P )   &    |- ( ph -> dom R = I )   &    |- .x. = ( .r ` P )   =>    |- ( ph -> .x. = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) )
Theorem	prdsbas2 13327*	The base set of a structure product is an indexed set product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   =>    |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )
Theorem	prdsbasmpt 13328*	A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   =>    |- ( ph -> ( ( x e. I |-> U ) e. B <-> A. x e. I U e. ( Base ` ( R ` x ) ) ) )
Theorem	prdsbasfn 13329	Points in the structure product are functions; use this with dffn5im 5681 to establish equalities. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   &    |- ( ph -> T e. B )   =>    |- ( ph -> T Fn I )
Theorem	prdsbasprj 13330	Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   &    |- ( ph -> T e. B )   &    |- ( ph -> J e. I )   =>    |- ( ph -> ( T ` J ) e. ( Base ` ( R ` J ) ) )
Theorem	prdsplusgval 13331*	Value of a componentwise sum in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .+ = ( +g ` Y )   =>    |- ( ph -> ( F .+ G ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) )
Theorem	prdsplusgfval 13332	Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> J e. I )   =>    |- ( ph -> ( ( F .+ G ) ` J ) = ( ( F ` J ) ( +g ` ( R ` J ) ) ( G ` J ) ) )
Theorem	prdsmulrval 13333*	Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .x. = ( .r ` Y )   =>    |- ( ph -> ( F .x. G ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) )
Theorem	prdsmulrfval 13334	Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R Fn I )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .x. = ( .r ` Y )   &    |- ( ph -> J e. I )   =>    |- ( ph -> ( ( F .x. G ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) )
Theorem	prdsbas3 13335*	The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.)
|- Y = ( S X_s ( x e. I |-> R ) )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> A. x e. I R e. X )   &    |- K = ( Base ` R )   =>    |- ( ph -> B = X_ x e. I K )
Theorem	prdsbasmpt2 13336*	A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015.) (Revised by Mario Carneiro, 13-Sep-2015.)
|- Y = ( S X_s ( x e. I |-> R ) )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> A. x e. I R e. X )   &    |- K = ( Base ` R )   =>    |- ( ph -> ( ( x e. I |-> U ) e. B <-> A. x e. I U e. K ) )
Theorem	prdsbascl 13337*	An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.)
|- Y = ( S X_s ( x e. I |-> R ) )   &    |- B = ( Base ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> A. x e. I R e. X )   &    |- K = ( Base ` R )   &    |- ( ph -> F e. B )   =>    |- ( ph -> A. x e. I ( F ` x ) e. K )
Definition	df-pws 13338*	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 } ) ) )
Theorem	pwsval 13339	Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- F = (Scalar ` R )   =>    |- ( ( R e. V /\ I e. W ) -> Y = ( F X_s ( I X. { R } ) ) )
Theorem	pwsbas 13340	Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   =>    |- ( ( R e. V /\ I e. W ) -> ( B ^m I ) = ( Base ` Y ) )
Theorem	pwselbasb 13341	Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   &    |- V = ( Base ` Y )   =>    |- ( ( R e. W /\ I e. Z ) -> ( X e. V <-> X : I --> B ) )
Theorem	pwselbas 13342	An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   &    |- V = ( Base ` Y )   &    |- ( ph -> R e. W )   &    |- ( ph -> I e. Z )   &    |- ( ph -> X e. V )   =>    |- ( ph -> X : I --> B )
Theorem	pwsplusgval 13343	Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- ( ph -> R e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` Y )   =>    |- ( ph -> ( F .+b G ) = ( F oF .+ G ) )
Theorem	pwsmulrval 13344	Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- ( ph -> R e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` Y )   =>    |- ( ph -> ( F .xb G ) = ( F oF .x. G ) )
Theorem	pwsdiagel 13345	Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` R )   &    |- C = ( Base ` Y )   =>    |- ( ( ( R e. V /\ I e. W ) /\ A e. B ) -> ( I X. { A } ) e. C )
Theorem	pwssnf1o 13346*	Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- Y = ( R ^s { I } )   &    |- B = ( Base ` R )   &    |- F = ( x e. B |-> ( { I } X. { x } ) )   &    |- C = ( Base ` Y )   =>    |- ( ( R e. V /\ I e. W ) -> F : B -1-1-onto-> C )
6.1.4  Definition of the structure quotient
Syntax	cimas 13347	Image structure function.
class "s
Syntax	cqus 13348	Quotient structure function.
class /.s
Syntax	cxps 13349	Binary product structure function.
class X.s
Definition	df-iimas 13350*	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 4732, 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 4731) and/or R ( with df-iress 13055) 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 13351*	Define a quotient ring (or quotient group), which is a special case of an image structure df-iimas 13350 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 13352*	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 13353	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 13354*	Value of an image structure. The is a lemma for the theorems imasbas 13355, imasplusg 13356, and imasmulr 13357 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 13355	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 13356*	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 13357*	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 13358	Lemma for f1ocpbl 13359. (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 13359	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 13360	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 13361	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 13362*	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 13363*	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 13364*	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 13365*	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 13366*	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 13367*	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 13368*	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 13369*	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 13370*	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 13371*	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 13372*	The function in qusval 13371 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 13373	Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )
Theorem	qusin 13374	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 13375	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 13376*	Value of the function in qusval 13371. (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 13377*	Value of the function in qusval 13371. (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 13378*	Lemma for ercpbl 13379. (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 13379*	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 13380*	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 13381*	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 13382*	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 13383*	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 13384*	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 13385*	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 13386*	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 13387	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 13388	Biconditional version of fnpr2o 13387. (Contributed by Jim Kingdon, 27-Sep-2023.)
|- ( ( A e. _V /\ B e. _V ) <-> { <. (/) , A >. , <. 1o , B >. } Fn 2o )
Theorem	fvpr0o 13389	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 13390	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 13391	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 13392*	Elementhood in the target space of the function F appearing in xpsval 13400. (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 13393	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 13394*	Elementhood in the target space of the function F appearing in xpsval 13400. (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 13395	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 13396*	The value of the function appearing in xpsval 13400. (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 13397*	The function appearing in xpsval 13400 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 13398*	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 13399*	The function appearing in xpsval 13400 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 13400*	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 ) )