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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsetsres 13101 The structure replacement function does not affect the value of  S away from  A. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( S  e.  V  ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } )
 )  =  ( S  |`  ( _V  \  { A } ) ) )
 
Theoremsetsabs 13102 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. ) )
 
Theoremsetscom 13103 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 >. ) )
 
Theoremstrfvd 13104 Deduction version of strfv 13107. (Contributed by Mario Carneiro, 15-Nov-2014.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  <. ( E `  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrfv2d 13105 Deduction version of strfv 13107. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  `' `' S )   &    |-  ( ph  ->  <.
 ( E `  ndx ) ,  C >.  e.  S )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrfv2 13106 A variation on strfv 13107 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.)
 |-  S  e.  _V   &    |-  Fun  `' `' S   &    |-  E  = Slot  ( E `  ndx )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( C  e.  V  ->  C  =  ( E `
  S ) )
 
Theoremstrfv 13107 Extract a structure component  C (such as the base set) from a structure  S (such as a member of  Poset, df-poset 14007) with a component extractor  E (such as the base set extractor df-base 13080). By virtue of ndxid 13096, 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 Mario Carneiro, 29-Aug-2015.)
 |-  S Struct  X   &    |-  E  = Slot  ( E `  ndx )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   =>    |-  ( C  e.  V  ->  C  =  ( E `  S ) )
 
Theoremstrfv3 13108 Variant on strfv 13107 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
 |-  ( ph  ->  U  =  S )   &    |-  S Struct  X   &    |-  E  = Slot  ( E `  ndx )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   &    |-  ( ph  ->  C  e.  V )   &    |-  A  =  ( E `
  U )   =>    |-  ( ph  ->  A  =  C )
 
Theoremstrssd 13109 Deduction version of strss 13110. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  Fun  T )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ph  ->  <. ( E `
  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  ( E `  T )  =  ( E `  S ) )
 
Theoremstrss 13110 Propagate component extraction to a structure  T from a subset structure  S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
 |-  T  e.  _V   &    |-  Fun  T   &    |-  S  C_  T   &    |-  E  = Slot  ( E `  ndx )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( E `  T )  =  ( E `  S )
 
Theoremstr0 13111 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
 |-  F  = Slot  I   =>    |-  (/)  =  ( F `
  (/) )
 
Theorembase0 13112 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  (/)  =  ( Base `  (/) )
 
Theoremstrfvi 13113 Structure slot extractors cannot distinguish between proper classes and  (/), so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  E  = Slot  N   &    |-  X  =  ( E `  S )   =>    |-  X  =  ( E `
  (  _I  `  S ) )
 
Theoremsetsid 13114 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  E  = Slot  ( E `
  ndx )   =>    |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  =  ( E `  ( W sSet  <. ( E `  ndx ) ,  C >. ) ) )
 
Theoremsetsnid 13115 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( E `  ndx )  =/=  D   =>    |-  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) )
 
Theorembaseval 13116 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.) (Contributed by NM, 4-Sep-2011.)
 |-  K  e.  _V   =>    |-  ( Base `  K )  =  ( K `  1 )
 
Theorembaseid 13117 Utility theorem: index-independent form of df-base 13080. (Contributed by NM, 20-Oct-2012.)
 |- 
 Base  = Slot  ( Base `  ndx )
 
Theoremelbasfv 13118 Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  S  =  ( F `
  Z )   &    |-  B  =  ( Base `  S )   =>    |-  ( X  e.  B  ->  Z  e.  _V )
 
Theoremelbasov 13119 Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |- 
 Rel  dom  O   &    |-  S  =  ( X O Y )   &    |-  B  =  ( Base `  S )   =>    |-  ( A  e.  B  ->  ( X  e.  _V  /\  Y  e.  _V )
 )
 
Theorembasendx 13120 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.) (Contributed by Mario Carneiro, 2-Aug-2013.)
 |-  ( Base `  ndx )  =  1
 
Theoremreldmress 13121 The structure restriction is a proper operator, so it can be used with ovprc1 5785. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |- 
 Rel  doms
 
Theoremressval 13122 Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  R  =  ( Ws  A )   &    |-  B  =  (
 Base `  W )   =>    |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <. ( Base ` 
 ndx ) ,  ( A  i^i  B ) >. ) ) )
 
Theoremressid2 13123 General behavior of trivial 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 )
 
Theoremressval2 13124 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 ) >. ) )
 
Theoremressbas 13125 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.)
 |-  R  =  ( Ws  A )   &    |-  B  =  (
 Base `  W )   =>    |-  ( A  e.  V  ->  ( A  i^i  B )  =  ( Base `  R ) )
 
Theoremressbas2 13126 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  R  =  ( Ws  A )   &    |-  B  =  (
 Base `  W )   =>    |-  ( A  C_  B  ->  A  =  (
 Base `  R ) )
 
Theoremressbasss 13127 The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  R  =  ( Ws  A )   &    |-  B  =  (
 Base `  W )   =>    |-  ( Base `  R )  C_  B
 
Theoremresslem 13128 Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  R  =  ( Ws  A )   &    |-  C  =  ( E `  W )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  1  <  N   =>    |-  ( A  e.  V  ->  C  =  ( E `
  R ) )
 
Theoremress0 13129 All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( (/)s  A )  =  (/)
 
Theoremressid 13130 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  X  ->  ( Ws  B )  =  W )
 
Theoremressinbas 13131 Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  B  =  ( Base `  W )   =>    |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B ) ) )
 
Theoremressress 13132 Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
 |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
 
Theoremressabs 13133 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( A  e.  X  /\  B  C_  A )  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
 
Theoremwunress 13134 Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  W  e.  U )   =>    |-  ( ph  ->  ( Ws  A )  e.  U )
 
7.1.2  Slot definitions
 
Syntaxcplusg 13135 Extend class notation with group (addition) operation.
 class  +g
 
Syntaxcmulr 13136 Extend class notation with ring multiplication.
 class  .r
 
Syntaxcstv 13137 Extend class notation with involution.
 class  * r
 
Syntaxcsca 13138 Extend class notation with scalar field.
 class Scalar
 
Syntaxcvsca 13139 Extend class notation with scalar product.
 class  .s
 
Syntaxcip 13140 Extend class notation with Hermitian form (inner product).
 class  .i
 
Syntaxcts 13141 Extend class notation with the topology component of a topological space.
 class TopSet
 
Syntaxcple 13142 Extend class notation with less-than-or-equal for posets.
 class  le
 
Syntaxcoc 13143 Extend class notation with the class of orthocomplementation extractors.
 class  oc
 
Syntaxcds 13144 Extend class notation with the metric space distance function.
 class  dist
 
Syntaxcunif 13145 Extend class notation with the uniform structure.
 class  Unif
 
Syntaxchom 13146 Extend class notation with the hom-set structure.
 class  Hom
 
Syntaxcco 13147 Extend class notation with the composition operation.
 class comp
 
Definitiondf-plusg 13148 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 +g  = Slot  2
 
Definitiondf-mulr 13149 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .r  = Slot  3
 
Definitiondf-starv 13150 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 13151 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 13152 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .s  = Slot  6
 
Definitiondf-ip 13153 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .i  = Slot  8
 
Definitiondf-tset 13154 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 13155 Define less-than-or-equal ordering extractor for posets and related structures. We use  10 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.)
 |- 
 le  = Slot  10
 
Definitiondf-ocomp 13156 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 13157 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 13158 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 Unif  = Slot ; 1 3
 
Definitiondf-hom 13159 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Hom  = Slot ; 1 4
 
Definitiondf-cco 13160 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- comp  = Slot ; 1
 5
 
Theoremstrlemor0 13161 Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  ( Fun  `' `' (/)  /\  dom  (/)  C_  ( 1 ... 0 ) )
 
Theoremstrlemor1 13162 Add one element to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... I ) )   &    |-  I  e.  NN0   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  A  =  J   &    |-  G  =  ( F  u.  { <. A ,  X >. } )   =>    |-  ( Fun  `' `' G  /\  dom  G  C_  (
 1 ... J ) )
 
Theoremstrlemor2 13163 Add two elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... I ) )   &    |-  I  e.  NN0   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  A  =  J   &    |-  J  <  K   &    |-  K  e.  NN   &    |-  B  =  K   &    |-  G  =  ( F  u.  { <. A ,  X >. ,  <. B ,  Y >. } )   =>    |-  ( Fun  `' `' G  /\  dom  G  C_  ( 1 ... K ) )
 
Theoremstrlemor3 13164 Add three elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... I ) )   &    |-  I  e.  NN0   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  A  =  J   &    |-  J  <  K   &    |-  K  e.  NN   &    |-  B  =  K   &    |-  K  <  L   &    |-  L  e.  NN   &    |-  C  =  L   &    |-  G  =  ( F  u.  { <. A ,  X >. , 
 <. B ,  Y >. , 
 <. C ,  Z >. } )   =>    |-  ( Fun  `' `' G  /\  dom  G  C_  (
 1 ... L ) )
 
Theoremstrleun 13165 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 >.
 
Theoremstrle1 13166 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   =>    |- 
 { <. A ,  X >. } Struct  <. I ,  I >.
 
Theoremstrle2 13167 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  B  =  J   =>    |-  { <. A ,  X >. ,  <. B ,  Y >. } Struct  <. I ,  J >.
 
Theoremstrle3 13168 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   =>    |- 
 { <. A ,  X >. ,  <. B ,  Y >. ,  <. C ,  Z >. } Struct  <. I ,  K >.
 
Theoremplusgndx 13169 Index value of the df-plusg 13148 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( +g  `  ndx )  =  2
 
Theoremplusgid 13170 Utility theorem: index-independent form of df-plusg 13148. (Contributed by NM, 20-Oct-2012.)
 |- 
 +g  = Slot  ( +g  ` 
 ndx )
 
Theorem2strstr 13171 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  G Struct  <.
 1 ,  N >.
 
Theorem2strbas 13172 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  ( B  e.  V  ->  B  =  ( Base `  G ) )
 
Theorem2strop 13173 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  (  .+  e.  V  ->  .+  =  ( E `  G ) )
 
Theoremgrpstr 13174 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 ) ,  .+  >. }   =>    |-  G Struct  <. 1 ,  2
 >.
 
Theoremgrpbase 13175 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  ->  B  =  ( Base `  G ) )
 
Theoremgrpplusg 13176 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 ) ,  .+  >. }   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  G ) )
 
Theoremressplusg 13177  +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  H  =  ( Gs  A )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( A  e.  V  ->  .+  =  ( +g  `  H ) )
 
Theoremgrpbasex 13178 The base of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpbase 13175 instead. (Contributed by NM, 17-Oct-2012.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |-  B  =  ( Base `  G )
 
Theoremgrpplusgx 13179 The operation of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpplusgx 13179 instead. (Contributed by NM, 17-Oct-2012.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |- 
 .+  =  ( +g  `  G )
 
Theoremmulrndx 13180 Index value of the df-mulr 13149 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .r `  ndx )  =  3
 
Theoremmulrid 13181 Utility theorem: index-independent form of df-mulr 13149. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |- 
 .r  = Slot  ( .r ` 
 ndx )
 
Theoremrngstr 13182 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  R Struct  <. 1 ,  3 >.
 
Theoremrngbase 13183 The base set 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  ->  B  =  (
 Base `  R ) )
 
Theoremrngplusg 13184 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.  >. }   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  R ) )
 
Theoremrngmulr 13185 The muliplicative 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.  >. }   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .r `  R ) )
 
Theoremstarvndx 13186 Index value of the df-starv 13150 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( * r `  ndx )  =  4
 
Theoremstarvid 13187 Utility theorem: index-independent form of df-starv 13150. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  * r  = Slot  ( * r `  ndx )
 
Theoremressmulr 13188  .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( A  e.  V  ->  .x.  =  ( .r `  S ) )
 
Theoremressstarv 13189  * r is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
 |-  S  =  ( Rs  A )   &    |-  .*  =  ( * r `  R )   =>    |-  ( A  e.  V  ->  .*  =  ( * r `  S ) )
 
Theoremsrngfn 13190 A constructed star ring is a function with domain contained in  1 thru  4. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( * r `  ndx ) ,  .*  >. } )   =>    |-  R Struct  <. 1 ,  4
 >.
 
Theoremsrngbase 13191 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( * r `  ndx ) ,  .*  >. } )   =>    |-  ( B  e.  X  ->  B  =  ( Base `  R ) )
 
Theoremsrngplusg 13192 The addition 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 ) ,  .*  >. } )   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  R ) )
 
Theoremsrngmulr 13193 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 ) ,  .*  >. } )   =>    |-  (  .x.  e.  X  ->  .x.  =  ( .r
 `  R ) )
 
Theoremsrnginvl 13194 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 ) ,  .*  >. } )   =>    |-  (  .*  e.  X  ->  .*  =  ( * r `  R ) )
 
Theoremscandx 13195 Index value of the df-sca 13151 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (Scalar `  ndx )  =  5
 
Theoremscaid 13196 Utility theorem: index-independent form of scalar df-sca 13151. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |- Scalar  = Slot  (Scalar `  ndx )
 
Theoremvscandx 13197 Index value of the df-vsca 13152 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .s `  ndx )  =  6
 
Theoremvscaid 13198 Utility theorem: index-independent form of scalar product df-vsca 13152. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .s  = Slot  ( .s ` 
 ndx )
 
Theoremlmodstr 13199 A constructed left module or left vector space is a function. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  F >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. } )   =>    |-  W Struct  <.
 1 ,  6 >.
 
Theoremlmodbase 13200 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  F >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. } )   =>    |-  ( B  e.  X  ->  B  =  ( Base `  W ) )
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