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	prdsval 13301*	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 13302	Lemma for prdsbas 13304 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 13303	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 13304*	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 13305*	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 13306*	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 13307*	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 13308*	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 13309	Points in the structure product are functions; use this with dffn5im 5678 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 13310	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 13311*	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 13312	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 13313*	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 13314	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 13315*	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 13316*	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 13317*	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 13318*	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 13319	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 13320	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 13321	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 13322	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 13323	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 13324	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 13325	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 13326*	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 13327	Image structure function.
class "s
Syntax	cqus 13328	Quotient structure function.
class /.s
Syntax	cxps 13329	Binary product structure function.
class X.s
Definition	df-iimas 13330*	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 4731, 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 4730) and/or R ( with df-iress 13035) 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 13331*	Define a quotient ring (or quotient group), which is a special case of an image structure df-iimas 13330 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 13332*	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 13333	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 13334*	Value of an image structure. The is a lemma for the theorems imasbas 13335, imasplusg 13336, and imasmulr 13337 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 13335	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 13336*	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 13337*	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 13338	Lemma for f1ocpbl 13339. (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 13339	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 13340	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 13341	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 13342*	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 13343*	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 13344*	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 13345*	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 13346*	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 13347*	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 13348*	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 13349*	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 13350*	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 13351*	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 13352*	The function in qusval 13351 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 13353	Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )
Theorem	qusin 13354	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 13355	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 13356*	Value of the function in qusval 13351. (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 13357*	Value of the function in qusval 13351. (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 13358*	Lemma for ercpbl 13359. (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 13359*	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 13360*	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 13361*	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 13362*	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 13363*	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 13364*	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 13365*	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 13366*	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 13367	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 13368	Biconditional version of fnpr2o 13367. (Contributed by Jim Kingdon, 27-Sep-2023.)
|- ( ( A e. _V /\ B e. _V ) <-> { <. (/) , A >. , <. 1o , B >. } Fn 2o )
Theorem	fvpr0o 13369	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 13370	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 13371	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 13372*	Elementhood in the target space of the function F appearing in xpsval 13380. (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 13373	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 13374*	Elementhood in the target space of the function F appearing in xpsval 13380. (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 13375	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 13376*	The value of the function appearing in xpsval 13380. (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 13377*	The function appearing in xpsval 13380 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 13378*	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 13379*	The function appearing in xpsval 13380 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 13380*	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 6005, binary operations are defined by a rule, and with df-ov 6003, 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 6003 (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 6003). The definition of magmas (Mgm, see df-mgm 13384) 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 13381	Extend class notation with group addition as a function.
class +f
Syntax	cmgm 13382	Extend class notation with class of all magmas.
class Mgm
Definition	df-plusf 13383*	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 13394), while +g only has closure (mgmcl 13387). (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 13384*	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 13385*	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 ) )
Theorem	ismgmn0 13386*	The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( A e. B -> ( M e. Mgm <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )
Theorem	mgmcl 13387	Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( ( M e. Mgm /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B )
Theorem	isnmgm 13388	A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.)
|- B = ( Base ` M )   &    |- .o. = ( +g ` M )   =>    |- ( ( X e. B /\ Y e. B /\ ( X .o. Y ) e/ B ) -> M e/ Mgm )
Theorem	mgmsscl 13389	If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. (Contributed by AV, 17-Feb-2024.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   =>    |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X ( +g ` G ) Y ) e. S )
Theorem	plusffvalg 13390*	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +f ` G )   =>    |- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )
Theorem	plusfvalg 13391	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +f ` G )   =>    |- ( ( G e. V /\ X e. B /\ Y e. B ) -> ( X .+^ Y ) = ( X .+ Y ) )
Theorem	plusfeqg 13392	If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +f ` G )   =>    |- ( ( G e. V /\ .+ Fn ( B X. B ) ) -> .+^ = .+ )
Theorem	plusffng 13393	The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
|- B = ( Base ` G )   &    |- .+^ = ( +f ` G )   =>    |- ( G e. V -> .+^ Fn ( B X. B ) )
Theorem	mgmplusf 13394	The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.)
|- B = ( Base ` M )   &    |- .+^ = ( +f ` M )   =>    |- ( M e. Mgm -> .+^ : ( B X. B ) --> B )
Theorem	intopsn 13395	The internal operation for a set is the trivial operation iff the set is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
|- ( ( .o. : ( B X. B ) --> B /\ Z e. B ) -> ( B = { Z } <-> .o. = { <. <. Z , Z >. , Z >. } ) )
Theorem	mgmb1mgm1 13396	The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.)
|- B = ( Base ` M )   &    |- .+ = ( +g ` M )   =>    |- ( ( M e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
Theorem	mgm0 13397	Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
|- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Mgm )
Theorem	mgm1 13398	The structure with one element and the only closed internal operation for a singleton is a magma. (Contributed by AV, 10-Feb-2020.)
|- M = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }   =>    |- ( I e. V -> M e. Mgm )
Theorem	opifismgmdc 13399*	A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)
|- B = ( Base ` M )   &    |- ( +g ` M ) = ( x e. B , y e. B |-> if ( ps , C , D ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> DECID ps )   &    |- ( ph -> E. x x e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> C e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> D e. B )   =>    |- ( ph -> M e. Mgm )
7.1.2  Identity elements According to Wikipedia ("Identity element", 7-Feb-2020, https://en.wikipedia.org/wiki/Identity_element): "In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it.". Or in more detail "... an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity." We concentrate on two-sided identities in the following. The existence of an identity (an identity is unique if it exists, see mgmidmo 13400) is an important property of monoids, and therefore also for groups, but also for magmas not required to be associative. Magmas with an identity element are called "unital magmas" (see Definition 2 in [BourbakiAlg1] p. 12) or, if the magmas are cancellative, "loops" (see definition in [Bruck] p. 15). In the context of extensible structures, the identity element (of any magma M) is defined as "group identity element" ( 0g ` M ), see df-0g 13286. Related theorems which are already valid for magmas are provided in the following.
Theorem	mgmidmo 13400*	A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)
|- E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x )