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Theorem List for Intuitionistic Logic Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtsetid 12701 Utility theorem: index-independent form of df-tset 12611. (Contributed by NM, 20-Oct-2012.)
 |- TopSet  = Slot  (TopSet `  ndx )
 
Theoremtsetslid 12702 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
 
Theoremtsetndxnn 12703 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
 |-  (TopSet `  ndx )  e. 
 NN
 
Theorembasendxlttsetndx 12704 The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
 |-  ( Base `  ndx )  < 
 (TopSet `  ndx )
 
Theoremtsetndxnbasendx 12705 The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
 |-  (TopSet `  ndx )  =/=  ( Base `  ndx )
 
Theoremtsetndxnplusgndx 12706 The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
 |-  (TopSet `  ndx )  =/=  ( +g  `  ndx )
 
Theoremtsetndxnmulrndx 12707 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
 |-  (TopSet `  ndx )  =/=  ( .r `  ndx )
 
Theoremtsetndxnstarvndx 12708 The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
 |-  (TopSet `  ndx )  =/=  ( *r `  ndx )
 
Theoremslotstnscsi 12709 The slots Scalar,  .s and  .i are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
 |-  ( (TopSet `  ndx )  =/=  (Scalar `  ndx )  /\  (TopSet `  ndx )  =/=  ( .s `  ndx )  /\  (TopSet `  ndx )  =/=  ( .i `  ndx ) )
 
Theoremtopgrpstrd 12710 A constructed topological group is a structure. (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  ->  W Struct  <.
 1 ,  9 >.
 )
 
Theoremtopgrpbasd 12711 The base set 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  ->  B  =  ( Base `  W )
 )
 
Theoremtopgrpplusgd 12712 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 )
 )
 
Theoremtopgrptsetd 12713 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 )
 )
 
Theoremplendx 12714 Index value of the df-ple 12612 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
 |-  ( le `  ndx )  = ; 1 0
 
Theorempleid 12715 Utility theorem: self-referencing, index-independent form of df-ple 12612. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theorempleslid 12716 Slot property of  le. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  ( le  = Slot  ( le `  ndx )  /\  ( le `  ndx )  e.  NN )
 
Theoremplendxnn 12717 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
 
Theorembasendxltplendx 12718 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 )
 
Theoremplendxnbasendx 12719 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 )
 
Theoremplendxnplusgndx 12720 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 )
 
Theoremplendxnmulrndx 12721 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 )
 
Theoremplendxnscandx 12722 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 )
 
Theoremplendxnvscandx 12723 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 )
 
Theoremslotsdifplendx 12724 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 ) )
 
Theoremdsndx 12725 Index value of the df-ds 12614 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 12726 Utility theorem: index-independent form of df-ds 12614. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremdsslid 12727 Slot property of  dist. (Contributed by Jim Kingdon, 6-May-2023.)
 |-  ( dist  = Slot  ( dist ` 
 ndx )  /\  ( dist `  ndx )  e. 
 NN )
 
Theoremdsndxnn 12728 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
 
Theorembasendxltdsndx 12729 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 )
 
Theoremdsndxnbasendx 12730 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 )
 
Theoremdsndxnplusgndx 12731 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 )
 
Theoremdsndxnmulrndx 12732 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 )
 
Theoremslotsdnscsi 12733 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 ) )
 
Theoremdsndxntsetndx 12734 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 )
 
Theoremslotsdifdsndx 12735 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 ) )
 
Theoremunifndx 12736 Index value of the df-unif 12615 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
 |-  ( UnifSet `  ndx )  = ; 1
 3
 
Theoremunifid 12737 Utility theorem: index-independent form of df-unif 12615. (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |- 
 UnifSet  = Slot  ( UnifSet `  ndx )
 
Theoremunifndxnn 12738 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
 
Theorembasendxltunifndx 12739 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 )
 
Theoremunifndxnbasendx 12740 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 )
 
Theoremunifndxntsetndx 12741 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 )
 
Theoremslotsdifunifndx 12742 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 ) ) )
 
Theoremhomid 12743 Utility theorem: index-independent form of df-hom 12616. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- 
 Hom  = Slot  ( Hom  `  ndx )
 
Theoremhomslid 12744 Slot property of  Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
 |-  ( Hom  = Slot  ( Hom  `  ndx )  /\  ( Hom  `  ndx )  e. 
 NN )
 
Theoremccoid 12745 Utility theorem: index-independent form of df-cco 12617. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- comp  = Slot  (comp `  ndx )
 
Theoremccoslid 12746 Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
 |-  (comp  = Slot  (comp `  ndx )  /\  (comp `  ndx )  e.  NN )
 
6.1.3  Definition of the structure product
 
Syntaxcrest 12747 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 12748 Extend class notation with the topology extractor function.
 class  TopOpen
 
Definitiondf-rest 12749* 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 ) ) )
 
Definitiondf-topn 12750 Define the topology extractor function. This differs from df-tset 12611 when a structure has been restricted using df-iress 12523; 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 )
 ) )
 
Theoremrestfn 12751 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 12752 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 12753* 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 )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
 
Theoremelrest 12754* 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.  ( Jt  B )  <->  E. x  e.  J  A  =  ( x  i^i  B ) ) )
 
Theoremelrestr 12755 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.  ( Jt  S ) )
 
Theoremrestid2 12756 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 )  ->  ( Jt  A )  =  J )
 
Theoremrestsspw 12757 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( Jt  A )  C_  ~P A
 
Theoremrestid 12758 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  ->  ( Jt  X )  =  J )
 
Theoremtopnvalg 12759 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  ->  ( Jt  B )  =  (
 TopOpen `  W ) )
 
Theoremtopnidg 12760 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 ) )
 
Theoremtopnpropgd 12761 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 ) )
 
Syntaxctg 12762 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 12763 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Syntaxc0g 12764 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 12765 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-0g 12766* Define group identity element. Remark: this definition is required here because the symbol  0g is already used in df-igsum 12767. 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 ) ) ) )
 
Definitiondf-igsum 12767* Define a finite group sum (also called "iterated sum") of a structure.

Given  G  gsumg  F where  F : A --> ( Base `  G ), the set of indices is  A and the values are given by  F at each index. A group sum over a multiplicative group may be viewed as a product. The definition is meaningful in different contexts, depending on the size of the index set  A and each demanding different properties of  G.

1. If  A  =  (/) and  G has an identity element, then the sum equals this identity.

2. If  A  =  ( M ... N ) and 
G is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e.,  ( ( F `  1 )  +  ( F ` 
2 ) )  +  ( F `  3
), etc.

3. This definition does not handle other cases.

(Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)

 |- 
 gsumg  =  ( w  e.  _V ,  f  e.  _V  |->  ( iota x ( ( dom  f  =  (/)  /\  x  =  ( 0g
 `  w ) )  \/  E. m E. n  e.  ( ZZ>= `  m ) ( dom  f  =  ( m
 ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  f
 ) `  n )
 ) ) ) )
 
Definitiondf-topgen 12768* 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 ) } )
 
Definitiondf-pt 12769* 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
 ) ) } )
 )
 
Theoremtgval 12770* The topology generated by a basis. See also tgval2 14028 and tgval3 14035. (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 ) } )
 
Theoremtgvalex 12771 The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
 |-  ( B  e.  V  ->  ( topGen `  B )  e.  _V )
 
Theoremptex 12772 Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
 |-  ( F  e.  V  ->  ( Xt_ `  F )  e.  _V )
 
Syntaxcprds 12773 The function constructing structure products.
 class  X_s
 
Syntaxcpws 12774 The function constructing structure powers.
 class  ^s
 
Definitiondf-prds 12775* Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
 |-  X_s  =  ( s  e.  _V ,  r  e.  _V  |->  [_ X_ x  e.  dom  r ( Base `  (
 r `  x )
 )  /  v ]_ [_ ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
  x ) ( Hom  `  ( r `  x ) ) ( g `  x ) ) )  /  h ]_ ( ( { <. (
 Base `  ndx ) ,  v >. ,  <. ( +g  ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x ) ) ( g `
  x ) ) ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .r `  ( r `
  x ) ) ( g `  x ) ) ) )
 >. }  u.  { <. (Scalar `  ndx ) ,  s >. ,  <. ( .s `  ndx ) ,  ( f  e.  ( Base `  s
 ) ,  g  e.  v  |->  ( x  e. 
 dom  r  |->  ( f ( .s `  (
 r `  x )
 ) ( g `  x ) ) ) ) >. ,  <. ( .i
 `  ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .i `  ( r `
  x ) ) ( g `  x ) ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  r )
 ) >. ,  <. ( le ` 
 ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  ( r `  x ) ) ( g `
  x ) ) } >. ,  <. ( dist ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  sup ( ( ran  ( x  e.  dom  r  |->  ( ( f `  x ) ( dist `  (
 r `  x )
 ) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  h >. , 
 <. (comp `  ndx ) ,  ( a  e.  (
 v  X.  v ) ,  c  e.  v  |->  ( d  e.  (
 c h ( 2nd `  a ) ) ,  e  e.  ( h `
  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x ) ( <. ( ( 1st `  a
 ) `  x ) ,  ( ( 2nd `  a
 ) `  x ) >. (comp `  ( r `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } )
 ) )
 
Theoremreldmprds 12776 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
 
Theoremprdsex 12777 Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
 |-  ( ( S  e.  V  /\  R  e.  W )  ->  ( S X_s R )  e.  _V )
 
Definitiondf-pws 12778* Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |- 
 ^s  =  ( r  e. 
 _V ,  i  e. 
 _V  |->  ( (Scalar `  r
 ) X_s ( i  X.  {
 r } ) ) )
 
6.1.4  Definition of the structure quotient
 
Syntaxcimas 12779 Image structure function.
 class  "s
 
Syntaxcqus 12780 Quotient structure function.
 class  /.s
 
Syntaxcxps 12781 Binary product structure function.
 class  X.s
 
Definitiondf-iimas 12782* 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 4657, 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 4656) and/or  R ( with df-iress 12523) 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 ) ) >. }
 >. } )
 
Definitiondf-qus 12783* Define a quotient ring (or quotient group), which is a special case of an image structure df-iimas 12782 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 ) )
 
Definitiondf-xps 12784* 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 >. } ) ) )
 
Theoremimasex 12785 Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
 |-  ( ( F  e.  V  /\  R  e.  W )  ->  ( F  "s  R )  e.  _V )
 
Theoremimasival 12786* Value of an image structure. The is a lemma for the theorems imasbas 12787, imasplusg 12788, and imasmulr 12789 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  >. } )
 
Theoremimasbas 12787 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 ) )
 
Theoremimasplusg 12788* 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 ) ) >. } )
 
Theoremimasmulr 12789* 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 ) ) >. } )
 
Theoremf1ocpbllem 12790 Lemma for f1ocpbl 12791. (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 ) ) )
 
Theoremf1ocpbl 12791 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 ) ) ) )
 
Theoremf1ovscpbl 12792 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 ) ) ) )
 
Theoremf1olecpbl 12793 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 ) ) )
 
Theoremimasaddfnlemg 12794* 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 ) )
 
Theoremimasaddvallemg 12795* 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 ) ) )
 
Theoremimasaddflemg 12796* 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 )
 
Theoremimasaddfn 12797* 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 ) )
 
Theoremimasaddval 12798* 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 ) ) )
 
Theoremimasaddf 12799* 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 )
 
Theoremimasmulfn 12800* 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 ) )
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