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Theorem List for Intuitionistic Logic Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstrsetsid 12001 Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  S Struct  <. M ,  N >. )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  ( E ` 
 ndx )  e.  dom  S )   =>    |-  ( ph  ->  S  =  ( S sSet  <. ( E `
  ndx ) ,  ( E `  S ) >. ) )
 
Theoremfvsetsid 12002 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
 |-  ( ( F  e.  V  /\  X  e.  W  /\  Y  e.  U ) 
 ->  ( ( F sSet  <. X ,  Y >. ) `  X )  =  Y )
 
Theoremsetsfun 12003 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
 |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( I  e.  U  /\  E  e.  W ) )  ->  Fun  ( G sSet  <. I ,  E >. ) )
 
Theoremsetsfun0 12004 A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 12003 is useful for proofs based on isstruct2r 11979 which requires  Fun  ( F 
\  { (/) } ) for 
F to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
 |-  ( ( ( G  e.  V  /\  Fun  ( G  \  { (/) } )
 )  /\  ( I  e.  U  /\  E  e.  W ) )  ->  Fun  ( ( G sSet  <. I ,  E >. )  \  { (/)
 } ) )
 
Theoremsetsn0fun 12005 The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
 |-  ( ph  ->  S Struct  X )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  E  e.  W )   =>    |-  ( ph  ->  Fun  (
 ( S sSet  <. I ,  E >. )  \  { (/)
 } ) )
 
Theoremsetsresg 12006 The structure replacement function does not affect the value of  S away from  A. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
 |-  ( ( S  e.  V  /\  A  e.  W  /\  B  e.  X ) 
 ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } )
 )  =  ( S  |`  ( _V  \  { A } ) ) )
 
Theoremsetsabsd 12007 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
 |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  (
 ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. ) )
 
Theoremsetscom 12008 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( ( S  e.  V  /\  A  =/=  B )  /\  ( C  e.  W  /\  D  e.  X )
 )  ->  ( ( S sSet  <. A ,  C >. ) sSet  <. B ,  D >. )  =  ( ( S sSet  <. B ,  D >. ) sSet  <. A ,  C >. ) )
 
Theoremstrslfvd 12009 Deduction version of strslfv 12012. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  <. ( E `  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrslfv2d 12010 Deduction version of strslfv 12012. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  `' `' S )   &    |-  ( ph  ->  <.
 ( E `  ndx ) ,  C >.  e.  S )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrslfv2 12011 A variation on strslfv 12012 to avoid asserting that  S itself is a function, which involves sethood of all the ordered pair components of  S. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  S  e.  _V   &    |-  Fun  `' `' S   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( C  e.  V  ->  C  =  ( E `
  S ) )
 
Theoremstrslfv 12012 Extract a structure component  C (such as the base set) from a structure  S with a component extractor  E (such as the base set extractor df-base 11974). By virtue of ndxslid 11993, this can be done without having to refer to the hard-coded numeric index of  E. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  S Struct  X   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   =>    |-  ( C  e.  V  ->  C  =  ( E `  S ) )
 
Theoremstrslfv3 12013 Variant on strslfv 12012 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  ( ph  ->  U  =  S )   &    |-  S Struct  X   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   &    |-  ( ph  ->  C  e.  V )   &    |-  A  =  ( E `
  U )   =>    |-  ( ph  ->  A  =  C )
 
Theoremstrslssd 12014 Deduction version of strslss 12015. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  Fun  T )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ph  ->  <. ( E `
  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  ( E `  T )  =  ( E `  S ) )
 
Theoremstrslss 12015 Propagate component extraction to a structure  T from a subset structure  S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)
 |-  T  e.  _V   &    |-  Fun  T   &    |-  S  C_  T   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( E `  T )  =  ( E `  S )
 
Theoremstrsl0 12016 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   =>    |-  (/)  =  ( E `  (/) )
 
Theorembase0 12017 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  (/)  =  ( Base `  (/) )
 
Theoremsetsslid 12018 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   =>    |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  =  ( E `  ( W sSet  <. ( E `  ndx ) ,  C >. ) ) )
 
Theoremsetsslnid 12019 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( E `  ndx )  =/=  D   &    |-  D  e.  NN   =>    |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
 
Theorembaseval 12020 Value of the base set extractor. (Normally it is preferred to work with  ( Base `  ndx ) rather than the hard-coded  1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
 |-  K  e.  _V   =>    |-  ( Base `  K )  =  ( K `  1 )
 
Theorembaseid 12021 Utility theorem: index-independent form of df-base 11974. (Contributed by NM, 20-Oct-2012.)
 |- 
 Base  = Slot  ( Base `  ndx )
 
Theorembasendx 12022 Index value of the base set extractor. (Normally it is preferred to work with  ( Base `  ndx ) rather than the hard-coded  1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.)
 |-  ( Base `  ndx )  =  1
 
Theorembasendxnn 12023 The index value of the base set extractor 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, 23-Sep-2020.)
 |-  ( Base `  ndx )  e. 
 NN
 
Theorembaseslid 12024 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
 |-  ( Base  = Slot  ( Base ` 
 ndx )  /\  ( Base `  ndx )  e. 
 NN )
 
Theorembasfn 12025 The base set extractor is a function on  _V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |- 
 Base  Fn  _V
 
Theoremreldmress 12026 The structure restriction is a proper operator, so it can be used with ovprc1 5807. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |- 
 Rel  doms
 
Theoremressid2 12027 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)
 |-  R  =  ( Ws  A )   &    |-  B  =  (
 Base `  W )   =>    |-  ( ( B 
 C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  R  =  W )
 
Theoremressval2 12028 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  R  =  ( Ws  A )   &    |-  B  =  (
 Base `  W )   =>    |-  ( ( -.  B  C_  A  /\  W  e.  X  /\  A  e.  Y )  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. ) )
 
Theoremressid 12029 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  X  ->  ( Ws  B )  =  W )
 
6.1.2  Slot definitions
 
Syntaxcplusg 12030 Extend class notation with group (addition) operation.
 class  +g
 
Syntaxcmulr 12031 Extend class notation with ring multiplication.
 class  .r
 
Syntaxcstv 12032 Extend class notation with involution.
 class  *r
 
Syntaxcsca 12033 Extend class notation with scalar field.
 class Scalar
 
Syntaxcvsca 12034 Extend class notation with scalar product.
 class  .s
 
Syntaxcip 12035 Extend class notation with Hermitian form (inner product).
 class  .i
 
Syntaxcts 12036 Extend class notation with the topology component of a topological space.
 class TopSet
 
Syntaxcple 12037 Extend class notation with "less than or equal to" for posets.
 class  le
 
Syntaxcoc 12038 Extend class notation with the class of orthocomplementation extractors.
 class  oc
 
Syntaxcds 12039 Extend class notation with the metric space distance function.
 class  dist
 
Syntaxcunif 12040 Extend class notation with the uniform structure.
 class  UnifSet
 
Syntaxchom 12041 Extend class notation with the hom-set structure.
 class  Hom
 
Syntaxcco 12042 Extend class notation with the composition operation.
 class comp
 
Definitiondf-plusg 12043 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 +g  = Slot  2
 
Definitiondf-mulr 12044 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .r  = Slot  3
 
Definitiondf-starv 12045 Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  *r  = Slot  4
 
Definitiondf-sca 12046 Define scalar field component of a vector space  v. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- Scalar  = Slot  5
 
Definitiondf-vsca 12047 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .s  = Slot  6
 
Definitiondf-ip 12048 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .i  = Slot  8
 
Definitiondf-tset 12049 Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- TopSet  = Slot  9
 
Definitiondf-ple 12050 Define "less than or equal to" ordering extractor for posets and related structures. We use ; 1 0 for the index to avoid conflict with  1 through  9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
 |- 
 le  = Slot ; 1 0
 
Definitiondf-ocomp 12051 Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 oc  = Slot ; 1 1
 
Definitiondf-ds 12052 Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 dist  = Slot ; 1 2
 
Definitiondf-unif 12053 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 UnifSet  = Slot ; 1 3
 
Definitiondf-hom 12054 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Hom  = Slot ; 1 4
 
Definitiondf-cco 12055 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- comp  = Slot ; 1
 5
 
Theoremstrleund 12056 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
 |-  ( ph  ->  F Struct  <. A ,  B >. )   &    |-  ( ph  ->  G Struct  <. C ,  D >. )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  ( F  u.  G ) Struct  <. A ,  D >. )
 
Theoremstrleun 12057 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  <. A ,  B >.   &    |-  G Struct 
 <. C ,  D >.   &    |-  B  <  C   =>    |-  ( F  u.  G ) Struct 
 <. A ,  D >.
 
Theoremstrle1g 12058 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
 |-  I  e.  NN   &    |-  A  =  I   =>    |-  ( X  e.  V  ->  { <. A ,  X >. } Struct  <. I ,  I >. )
 
Theoremstrle2g 12059 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
 |-  I  e.  NN   &    |-  A  =  I   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  B  =  J   =>    |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  { <. A ,  X >. ,  <. B ,  Y >. } Struct  <. I ,  J >. )
 
Theoremstrle3g 12060 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  B  =  J   &    |-  J  <  K   &    |-  K  e.  NN   &    |-  C  =  K   =>    |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  P ) 
 ->  { <. A ,  X >. ,  <. B ,  Y >. ,  <. C ,  Z >. } Struct  <. I ,  K >. )
 
Theoremplusgndx 12061 Index value of the df-plusg 12043 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( +g  `  ndx )  =  2
 
Theoremplusgid 12062 Utility theorem: index-independent form of df-plusg 12043. (Contributed by NM, 20-Oct-2012.)
 |- 
 +g  = Slot  ( +g  ` 
 ndx )
 
Theoremplusgslid 12063 Slot property of  +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
 |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e. 
 NN )
 
Theoremopelstrsl 12064 The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( ph  ->  S Struct  X )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  <. ( E `  ndx ) ,  V >.  e.  S )   =>    |-  ( ph  ->  V  =  ( E `  S ) )
 
Theoremopelstrbas 12065 The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
 |-  ( ph  ->  S Struct  X )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  <. ( Base `  ndx ) ,  V >.  e.  S )   =>    |-  ( ph  ->  V  =  ( Base `  S )
 )
 
Theorem1strstrg 12066 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. }   =>    |-  ( B  e.  V  ->  G Struct  <. 1 ,  1
 >. )
 
Theorem1strbas 12067 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. }   =>    |-  ( B  e.  V  ->  B  =  ( Base `  G ) )
 
Theorem2strstrg 12068 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  (
 ( B  e.  V  /\  .+  e.  W ) 
 ->  G Struct  <. 1 ,  N >. )
 
Theorem2strbasg 12069 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  (
 ( B  e.  V  /\  .+  e.  W ) 
 ->  B  =  ( Base `  G ) )
 
Theorem2stropg 12070 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  (
 ( B  e.  V  /\  .+  e.  W ) 
 ->  .+  =  ( E `
  G ) )
 
Theorem2strstr1g 12071 A constructed two-slot structure. Version of 2strstrg 12068 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. N ,  .+  >. }   &    |-  ( Base `  ndx )  <  N   &    |-  N  e.  NN   =>    |-  (
 ( B  e.  V  /\  .+  e.  W ) 
 ->  G Struct  <. ( Base `  ndx ) ,  N >. )
 
Theorem2strbas1g 12072 The base set of a constructed two-slot structure. Version of 2strbasg 12069 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. N ,  .+  >. }   &    |-  ( Base `  ndx )  <  N   &    |-  N  e.  NN   =>    |-  (
 ( B  e.  V  /\  .+  e.  W ) 
 ->  B  =  ( Base `  G ) )
 
Theorem2strop1g 12073 The other slot of a constructed two-slot structure. Version of 2stropg 12070 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. N ,  .+  >. }   &    |-  ( Base `  ndx )  <  N   &    |-  N  e.  NN   &    |-  E  = Slot  N   =>    |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  =  ( E `  G ) )
 
Theorembasendxnplusgndx 12074 The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.)
 |-  ( Base `  ndx )  =/=  ( +g  `  ndx )
 
Theoremgrpstrg 12075 A constructed group is a structure on 
1 ... 2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  ( ( B  e.  V  /\  .+  e.  W )  ->  G Struct  <. 1 ,  2 >. )
 
Theoremgrpbaseg 12076 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  ( ( B  e.  V  /\  .+  e.  W )  ->  B  =  (
 Base `  G ) )
 
Theoremgrpplusgg 12077 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  =  ( +g  `  G ) )
 
Theoremmulrndx 12078 Index value of the df-mulr 12044 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .r `  ndx )  =  3
 
Theoremmulrid 12079 Utility theorem: index-independent form of df-mulr 12044. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |- 
 .r  = Slot  ( .r ` 
 ndx )
 
Theoremmulrslid 12080 Slot property of  .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
 |-  ( .r  = Slot  ( .r `  ndx )  /\  ( .r `  ndx )  e.  NN )
 
Theoremplusgndxnmulrndx 12081 The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
 |-  ( +g  `  ndx )  =/=  ( .r `  ndx )
 
Theorembasendxnmulrndx 12082 The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
 |-  ( Base `  ndx )  =/=  ( .r `  ndx )
 
Theoremrngstrg 12083 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  ( ( B  e.  V  /\  .+  e.  W  /\  .x.  e.  X )  ->  R Struct  <. 1 ,  3 >. )
 
Theoremrngbaseg 12084 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  ( ( B  e.  V  /\  .+  e.  W  /\  .x.  e.  X )  ->  B  =  ( Base `  R )
 )
 
Theoremrngplusgg 12085 The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  ( ( B  e.  V  /\  .+  e.  W  /\  .x.  e.  X )  ->  .+  =  ( +g  `  R )
 )
 
Theoremrngmulrg 12086 The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  ( ( B  e.  V  /\  .+  e.  W  /\  .x.  e.  X )  ->  .x.  =  ( .r `  R ) )
 
Theoremstarvndx 12087 Index value of the df-starv 12045 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( *r `  ndx )  =  4
 
Theoremstarvid 12088 Utility theorem: index-independent form of df-starv 12045. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  *r  = Slot  ( *r `  ndx )
 
Theoremstarvslid 12089 Slot property of  *r. (Contributed by Jim Kingdon, 4-Feb-2023.)
 |-  ( *r  = Slot 
 ( *r `  ndx )  /\  ( *r `  ndx )  e.  NN )
 
Theoremsrngstrd 12090 A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( *r `  ndx ) ,  .*  >. } )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  .+  e.  W )   &    |-  ( ph  ->  .x.  e.  X )   &    |-  ( ph  ->  .*  e.  Y )   =>    |-  ( ph  ->  R Struct  <. 1 ,  4 >.
 )
 
Theoremsrngbased 12091 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( *r `  ndx ) ,  .*  >. } )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  .+  e.  W )   &    |-  ( ph  ->  .x.  e.  X )   &    |-  ( ph  ->  .*  e.  Y )   =>    |-  ( ph  ->  B  =  ( Base `  R ) )
 
Theoremsrngplusgd 12092 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( *r `  ndx ) ,  .*  >. } )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  .+  e.  W )   &    |-  ( ph  ->  .x.  e.  X )   &    |-  ( ph  ->  .*  e.  Y )   =>    |-  ( ph  ->  .+  =  ( +g  `  R ) )
 
Theoremsrngmulrd 12093 The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( *r `  ndx ) ,  .*  >. } )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  .+  e.  W )   &    |-  ( ph  ->  .x.  e.  X )   &    |-  ( ph  ->  .*  e.  Y )   =>    |-  ( ph  ->  .x. 
 =  ( .r `  R ) )
 
Theoremsrnginvld 12094 The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( *r `  ndx ) ,  .*  >. } )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  .+  e.  W )   &    |-  ( ph  ->  .x.  e.  X )   &    |-  ( ph  ->  .*  e.  Y )   =>    |-  ( ph  ->  .*  =  ( *r `
  R ) )
 
Theoremscandx 12095 Index value of the df-sca 12046 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (Scalar `  ndx )  =  5
 
Theoremscaid 12096 Utility theorem: index-independent form of scalar df-sca 12046. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |- Scalar  = Slot  (Scalar `  ndx )
 
Theoremscaslid 12097 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  (Scalar  = Slot  (Scalar `  ndx )  /\  (Scalar `  ndx )  e.  NN )
 
Theoremvscandx 12098 Index value of the df-vsca 12047 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .s `  ndx )  =  6
 
Theoremvscaid 12099 Utility theorem: index-independent form of scalar product df-vsca 12047. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .s  = Slot  ( .s ` 
 ndx )
 
Theoremvscaslid 12100 Slot property of  .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  ( .s  = Slot  ( .s `  ndx )  /\  ( .s `  ndx )  e.  NN )
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