P. 143 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14201-14300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gsumvalfi 14201	Value of the finite group sum over an unordered finite set. (Contributed by Jim Kingdon, 24-Mar-2026.)
|- B = ( Base ` W )   &    |- ( ph -> W e. CMnd )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> G : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )   =>    |- ( ph -> ( W gsumg F ) = ( W gzsumgz ( F o. G ) ) )
Theorem	gzsumgsum1 14202	On an integer range starting at one, gzsumgz and gsumg agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> F : ( 1 ... N ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) = ( G gsumg F ) )
Theorem	gsum0cmn 14203	An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
|- ( G e. CMnd -> ( G gsumg (/) ) = ( 0g ` G ) )
Theorem	gzsumgsum 14204	On an integer range, gzsumgz and gsumg agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) = ( G gsumg F ) )
Theorem	gsumsncmn 14205*	Group sum of a singleton. (Contributed by Jim Kingdon, 2-Apr-2026.)
|- B = ( Base ` G )   &    |- ( k = M -> A = C )   =>    |- ( ( G e. CMnd /\ M e. V /\ C e. B ) -> ( G gsumg ( k e. { M } |-> A ) ) = C )
Theorem	gsump1 14206	Splitting off one element from a finite group sum. This would typically used in a proof by induction. (Contributed by Jim Kingdon, 3-Apr-2026.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> F : ( Y u. { Z } ) --> B )   &    |- ( ph -> Y e. Fin )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. Z e. Y )   =>    |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` Y ) ) .+ ( F ` Z ) ) )
Theorem	gsumzfi 14207*	Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. CMnd /\ A e. Fin ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
Theorem	gsumclfi 14208	Closure of a finite group sum. (Contributed by Jim Kingdon, 8-Apr-2026.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( G gsumg F ) e. B )
Theorem	gsumf1ofi 14209	Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> H : C -1-1-onto-> A )   =>    |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
Theorem	gsummptfidmadd 14210*	The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ x e. A ) -> C e. B )   &    |- ( ( ph /\ x e. A ) -> D e. B )   &    |- F = ( x e. A |-> C )   &    |- H = ( x e. A |-> D )   =>    |- ( ph -> ( G gsumg ( x e. A |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
Theorem	gsummptfidmadd2 14211*	The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ x e. A ) -> C e. B )   &    |- ( ( ph /\ x e. A ) -> D e. B )   &    |- F = ( x e. A |-> C )   &    |- H = ( x e. A |-> D )   =>    |- ( ph -> ( G gsumg ( F oF .+ H ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
Theorem	gsumsubmclfi 14212	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
|- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> F : A --> S )   =>    |- ( ph -> ( G gsumg F ) e. S )
Theorem	gsummhmfi 14213	Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> K e. ( G MndHom H ) )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
Theorem	gsummhm2fi 14214*	Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )   &    |- ( ( ph /\ k e. A ) -> X e. B )   &    |- ( x = X -> C = D )   &    |- ( x = ( G gsumg ( k e. A |-> X ) ) -> C = E )   =>    |- ( ph -> ( H gsumg ( k e. A |-> D ) ) = E )
Theorem	gsumconstcmn 14215*	Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. CMnd /\ A e. Fin /\ X e. B ) -> ( G gsumg ( k e. A |-> X ) ) = ( (♯ ` A ) .x. X ) )
Theorem	gsumressfi 14216*	The group sum in a substructure is the same as the group sum in the original structure. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- H = ( G ↾s S )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> S C_ B )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> .0. e. S )   &    |- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )   =>    |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )
Theorem	gsumsubmfi 14217	Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> F : A --> S )   &    |- H = ( G ↾s S )   =>    |- ( ph -> ( G gsumg F ) = ( H gsumg F ) )
7.2.7  Structure product
Syntax	cprds 14218	The function constructing structure products.
class X_s
Definition	df-prds 14219*	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 14220	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 14221	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	prdsval 14222*	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 14223	Lemma for prdsbas 14225 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 14224	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 14225*	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 14226*	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 14227*	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 14228*	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 14229*	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 14230	Points in the structure product are functions; use this with dffn5im 5748 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 14231	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 14232*	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 14233	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 14234*	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 14235	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 14236*	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 14237*	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 14238*	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 )
Theorem	prdsplusgsgrpcl 14239	Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I -->Smgrp )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   =>    |- ( ph -> ( F .+ G ) e. B )
Theorem	prdssgrpd 14240	The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I -->Smgrp )   =>    |- ( ph -> Y e. Smgrp )
Theorem	prdsplusgcl 14241	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 14242*	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 14243	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 14244	The identity in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Mnd )   =>    |- ( ph -> ( 0g o. R ) = ( 0g ` Y ) )
Theorem	prdsinvlem 14245*	Characterization of inverses 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 --> Grp )   &    |- ( ph -> F e. B )   &    |- .0. = ( 0g o. R )   &    |- N = ( y e. I |-> ( ( invg ` ( R ` y ) ) ` ( F ` y ) ) )   =>    |- ( ph -> ( N e. B /\ ( N .+ F ) = .0. ) )
Theorem	prdsgrpd 14246	The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Grp )   =>    |- ( ph -> Y e. Grp )
Theorem	prdsinvgd 14247*	Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> Grp )   &    |- B = ( Base ` Y )   &    |- N = ( invg ` Y )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N ` X ) = ( x e. I |-> ( ( invg ` ( R ` x ) ) ` ( X ` x ) ) ) )
7.2.8  Binary product on structures
Syntax	cxps 14248	Binary product structure function.
class X.s
Definition	df-xps 14249*	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	xpsval 14250*	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 ) )
7.2.9  Structure power
Syntax	cpws 14251	The function constructing structure powers.
class ^s
Definition	df-pws 14252*	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 14253	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 14254	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 14255	Membership in the base set of a structure power. (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 14256	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 14257	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 14258	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 14259	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 14260*	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 )
Theorem	pwsmnd 14261	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 14262	The identity in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Mnd /\ I e. V ) -> ( I X. { .0. } ) = ( 0g ` Y ) )
Theorem	pwsgrp 14263	A structure power of a group is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   =>    |- ( ( R e. Grp /\ I e. V ) -> Y e. Grp )
Theorem	pwsinvg 14264	Negation in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- M = ( invg ` R )   &    |- N = ( invg ` Y )   =>    |- ( ( R e. Grp /\ I e. V /\ X e. B ) -> ( N ` X ) = ( M o. X ) )
Theorem	pwssub 14265	Subtraction in a structure power. (Contributed by Mario Carneiro, 12-Jan-2015.)
|- Y = ( R ^s I )   &    |- B = ( Base ` Y )   &    |- M = ( -g ` R )   &    |- .- = ( -g ` Y )   =>    |- ( ( ( R e. Grp /\ I e. V ) /\ ( F e. B /\ G e. B ) ) -> ( F .- G ) = ( F oF M G ) )
7.3  Rings
7.3.1  Multiplicative Group
Syntax	cmgp 14266	Multiplicative group.
class mulGrp
Definition	df-mgp 14267	Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 14312). (Contributed by Mario Carneiro, 21-Dec-2014.)
|- mulGrp = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )
Theorem	fnmgp 14268	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
|- mulGrp Fn _V
Theorem	mgpvalg 14269	Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. V -> M = ( R sSet <. ( +g ` ndx ) , .x. >. ) )
Theorem	mgpplusgg 14270	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. V -> .x. = ( +g ` M ) )
Theorem	mgpplusg 14271	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- .x. = ( +g ` M )
Theorem	mgpex 14272	Existence of the multiplication group. If R is known to be a semiring, see srgmgp 14321. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> M e. _V )
Theorem	mgpbasg 14273	Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. V -> B = ( Base ` M ) )
Theorem	mgpbas 14274	Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( Base ` R )   =>    |- B = ( Base ` M )
Theorem	mgpscag 14275	The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
|- M = (mulGrp ` R )   &    |- S = (Scalar ` R )   =>    |- ( R e. V -> S = (Scalar ` M ) )
Theorem	mgptsetg 14276	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> (TopSet ` R ) = (TopSet ` M ) )
Theorem	mgptopng 14277	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- J = ( TopOpen ` R )   =>    |- ( R e. V -> J = ( TopOpen ` M ) )
Theorem	mgpdsg 14278	Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( dist ` R )   =>    |- ( R e. V -> B = ( dist ` M ) )
Theorem	mgpress 14279	Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
|- S = ( R ↾s A )   &    |- M = (mulGrp ` R )   =>    |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )
7.3.2  Non-unital rings ("rngs") According to Wikipedia, "... in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 28-Mar-2025).
Syntax	crng 14280	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-rng 14281*	Define the class of all non-unital rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
|- Rng = { f e. Abel | ( (mulGrp ` f ) e. Smgrp /\ [. ( Base ` f ) / b ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. A. x e. b A. y e. b A. z e. b ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) ) }
Theorem	isrng 14282*	The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
|- B = ( Base ` R )   &    |- G = (mulGrp ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Rng <-> ( R e. Abel /\ G e. Smgrp /\ A. x e. B A. y e. B A. z e. B ( ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) /\ ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) ) ) )
Theorem	rngabl 14283	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
|- ( R e. Rng -> R e. Abel )
Theorem	rngmgp 14284	A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
|- G = (mulGrp ` R )   =>    |- ( R e. Rng -> G e. Smgrp )
Theorem	rngmgpf 14285	Restricted functionality of the multiplicative group on non-unital rings (mgpf 14364 analog). (Contributed by AV, 22-Feb-2025.)
|- (mulGrp |` Rng ) :Rng -->Smgrp
Theorem	rnggrp 14286	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
|- ( R e. Rng -> R e. Grp )
Theorem	rngass 14287	Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
Theorem	rngdi 14288	Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )
Theorem	rngdir 14289	Distributive law for the multiplication operation of a non-unital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) )
Theorem	rngacl 14290	Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   =>    |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
Theorem	rng0cl 14291	The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Rng -> .0. e. B )
Theorem	rngcl 14292	Closure of the multiplication operation of a non-unital ring. (Contributed by AV, 17-Apr-2020.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
Theorem	rnglz 14293	The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 14397. (Revised by AV, 17-Apr-2020.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = .0. )
Theorem	rngrz 14294	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14398. (Revised by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) = .0. )
Theorem	rngmneg1 14295	Negation of a product in a non-unital ring (mulneg1 8723 analog). In contrast to ringmneg1 14407, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )
Theorem	rngmneg2 14296	Negation of a product in a non-unital ring (mulneg2 8724 analog). In contrast to ringmneg2 14408, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )
Theorem	rngm2neg 14297	Double negation of a product in a non-unital ring (mul2neg 8726 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 14409. (Revised by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) )
Theorem	rngansg 14298	Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
|- ( R e. Rng -> (NrmSGrp ` R ) = (SubGrp ` R ) )
Theorem	rngsubdi 14299	Ring multiplication distributes over subtraction. (subdi 8713 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14410. (Revised by AV, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )
Theorem	rngsubdir 14300	Ring multiplication distributes over subtraction. (subdir 8714 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14411. (Revised by AV, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )