P. 137 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13601-13700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	topgrpplusgd 13601	The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> .+ = ( +g ` W ) )
Theorem	topgrptsetd 13602	The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> J = (TopSet ` W ) )
Theorem	plendx 13603	Index value of the df-ple 13500 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
|- ( le ` ndx ) = ; 1 0
Theorem	pleid 13604	Utility theorem: self-referencing, index-independent form of df-ple 13500. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
|- le = Slot ( le ` ndx )
Theorem	pleslid 13605	Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
|- ( le = Slot ( le ` ndx ) /\ ( le ` ndx ) e. NN )
Theorem	plendxnn 13606	The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
|- ( le ` ndx ) e. NN
Theorem	basendxltplendx 13607	The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
|- ( Base ` ndx ) < ( le ` ndx )
Theorem	plendxnbasendx 13608	The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
|- ( le ` ndx ) =/= ( Base ` ndx )
Theorem	plendxnplusgndx 13609	The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( le ` ndx ) =/= ( +g ` ndx )
Theorem	plendxnmulrndx 13610	The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= ( .r ` ndx )
Theorem	plendxnscandx 13611	The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= (Scalar ` ndx )
Theorem	plendxnvscandx 13612	The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= ( .s ` ndx )
Theorem	slotsdifplendx 13613	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
|- ( ( *r ` ndx ) =/= ( le ` ndx ) /\ (TopSet ` ndx ) =/= ( le ` ndx ) )
Theorem	ocndx 13614	Index value of the df-ocomp 13501 slot. (Contributed by Mario Carneiro, 25-Oct-2015.) (New usage is discouraged.)
|- ( oc ` ndx ) = ; 1 1
Theorem	ocid 13615	Utility theorem: index-independent form of df-ocomp 13501. (Contributed by Mario Carneiro, 25-Oct-2015.)
|- oc = Slot ( oc ` ndx )
Theorem	basendxnocndx 13616	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- ( Base ` ndx ) =/= ( oc ` ndx )
Theorem	plendxnocndx 13617	The slot for the orthocomplementation is not the slot for the order in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- ( le ` ndx ) =/= ( oc ` ndx )
Theorem	dsndx 13618	Index value of the df-ds 13502 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( dist ` ndx ) = ; 1 2
Theorem	dsid 13619	Utility theorem: index-independent form of df-ds 13502. (Contributed by Mario Carneiro, 23-Dec-2013.)
|- dist = Slot ( dist ` ndx )
Theorem	dsslid 13620	Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
|- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
Theorem	dsndxnn 13621	The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
|- ( dist ` ndx ) e. NN
Theorem	basendxltdsndx 13622	The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( Base ` ndx ) < ( dist ` ndx )
Theorem	dsndxnbasendx 13623	The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
|- ( dist ` ndx ) =/= ( Base ` ndx )
Theorem	dsndxnplusgndx 13624	The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( dist ` ndx ) =/= ( +g ` ndx )
Theorem	dsndxnmulrndx 13625	The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- ( dist ` ndx ) =/= ( .r ` ndx )
Theorem	slotsdnscsi 13626	The slots Scalar, .s and .i are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
|- ( ( dist ` ndx ) =/= (Scalar ` ndx ) /\ ( dist ` ndx ) =/= ( .s ` ndx ) /\ ( dist ` ndx ) =/= ( .i ` ndx ) )
Theorem	dsndxntsetndx 13627	The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( dist ` ndx ) =/= (TopSet ` ndx )
Theorem	slotsdifdsndx 13628	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
|- ( ( *r ` ndx ) =/= ( dist ` ndx ) /\ ( le ` ndx ) =/= ( dist ` ndx ) )
Theorem	unifndx 13629	Index value of the df-unif 13503 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
|- ( UnifSet ` ndx ) = ; 1 3
Theorem	unifid 13630	Utility theorem: index-independent form of df-unif 13503. (Contributed by Thierry Arnoux, 17-Dec-2017.)
|- UnifSet = Slot ( UnifSet ` ndx )
Theorem	unifndxnn 13631	The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
|- ( UnifSet ` ndx ) e. NN
Theorem	basendxltunifndx 13632	The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( Base ` ndx ) < ( UnifSet ` ndx )
Theorem	unifndxnbasendx 13633	The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- ( UnifSet ` ndx ) =/= ( Base ` ndx )
Theorem	unifndxntsetndx 13634	The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( UnifSet ` ndx ) =/= (TopSet ` ndx )
Theorem	slotsdifunifndx 13635	The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
|- ( ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) ) /\ ( ( le ` ndx ) =/= ( UnifSet ` ndx ) /\ ( dist ` ndx ) =/= ( UnifSet ` ndx ) ) )
Theorem	homndx 13636	Index value of the df-hom 13504 slot. (Contributed by Mario Carneiro, 7-Jan-2017.) (New usage is discouraged.)
|- ( Hom ` ndx ) = ; 1 4
Theorem	homid 13637	Utility theorem: index-independent form of df-hom 13504. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- Hom = Slot ( Hom ` ndx )
Theorem	homslid 13638	Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- ( Hom = Slot ( Hom ` ndx ) /\ ( Hom ` ndx ) e. NN )
Theorem	ccondx 13639	Index value of the df-cco 13505 slot. (Contributed by Mario Carneiro, 7-Jan-2017.) (New usage is discouraged.)
|- (comp ` ndx ) = ; 1 5
Theorem	ccoid 13640	Utility theorem: index-independent form of df-cco 13505. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- comp = Slot (comp ` ndx )
Theorem	ccoslid 13641	Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- (comp = Slot (comp ` ndx ) /\ (comp ` ndx ) e. NN )
6.1.3  Various definitions used by the structure product
Syntax	crest 13642	Extend class notation with the function returning a subspace topology.
class ↾t
Syntax	ctopn 13643	Extend class notation with the topology extractor function.
class TopOpen
Definition	df-rest 13644*	Function returning the subspace topology induced by the topology y and the set x. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
|- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
Definition	df-topn 13645	Define the topology extractor function. This differs from df-tset 13499 when a structure has been restricted using df-iress 13409; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
Theorem	restfn 13646	The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
|- ↾t Fn ( _V X. _V )
Theorem	topnfn 13647	The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- TopOpen Fn _V
Theorem	restval 13648*	The subspace topology induced by the topology J on the set A. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
Theorem	elrest 13649*	The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|- ( ( J e. V /\ B e. W ) -> ( A e. ( J ↾t B ) <-> E. x e. J A = ( x i^i B ) ) )
Theorem	elrestr 13650	Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|- ( ( J e. V /\ S e. W /\ A e. J ) -> ( A i^i S ) e. ( J ↾t S ) )
Theorem	restid2 13651	The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )
Theorem	restsspw 13652	The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J ↾t A ) C_ ~P A
Theorem	restid 13653	The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- X = U. J   =>    |- ( J e. V -> ( J ↾t X ) = J )
Theorem	topnvalg 13654	Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
|- B = ( Base ` W )   &    |- J = (TopSet ` W )   =>    |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )
Theorem	topnidg 13655	Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- B = ( Base ` W )   &    |- J = (TopSet ` W )   =>    |- ( ( W e. V /\ J C_ ~P B ) -> J = ( TopOpen ` W ) )
Theorem	topnpropgd 13656	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 13657	Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
Syntax	cpt 13658	Extend class notation with a function whose value is a product topology.
class Xt_
Syntax	c0g 13659	Extend class notation with group identity element.
class 0g
Syntax	cgzsu 13660	Extend class notation to include group sums over integer ranges.
class gzsumgz
Definition	df-0g 13661*	Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gzsum 13662. 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-gzsum 13662*	Define a finite group sum (also called "iterated sum") of a structure. Given G gzsumgz 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. But see df-gsumfi 14200 for the case where A is a finite set (which need not specify an order) and G is a commutative monoid. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)
|- gzsumgz = ( 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 13663*	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 13664*	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 13665*	The topology generated by a basis. See also tgval2 15201 and tgval3 15208. (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 13666	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 13667	Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
|- ( F e. V -> ( Xt_ ` F ) e. _V )
Theorem	imasvalstrd 13668	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 13669	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 13670*	Lemma for prdsval 14222. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 14222, dependency on df-hom 13504 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
6.1.4  Definition of the structure quotient
Syntax	cimas 13671	Image structure function.
class "s
Syntax	cqus 13672	Quotient structure function.
class /.s
Definition	df-iimas 13673*	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 4787, 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 4786) and/or R ( with df-iress 13409) 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 13674*	Define a quotient ring (or quotient group), which is a special case of an image structure df-iimas 13673 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 ) )
Theorem	imasex 13675	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 13676*	Value of an image structure. The is a lemma for the theorems imasbas 13677, imasplusg 13678, and imasmulr 13679 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 13677	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 13678*	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 13679*	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 13680	Lemma for f1ocpbl 13681. (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 13681	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 13682	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 13683	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 13684*	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 13685*	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 13686*	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 13687*	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 13688*	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 13689*	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 13690*	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 13691*	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 13692*	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 13693*	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 13694*	The function in qusval 13693 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 13695	Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )
Theorem	qusin 13696	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 13697	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 13698*	Value of the function in qusval 13693. (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 13699*	Value of the function in qusval 13693. (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 13700*	Lemma for ercpbl 13701. (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 ) )