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Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremressbas2 13201 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 13202 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 13203 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 13204 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 13205 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  X  ->  ( Ws  B )  =  W )
 
Theoremressinbas 13206 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 13207 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 13208 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( A  e.  X  /\  B  C_  A )  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
 
Theoremwunress 13209 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 13210 Extend class notation with group (addition) operation.
 class  +g
 
Syntaxcmulr 13211 Extend class notation with ring multiplication.
 class  .r
 
Syntaxcstv 13212 Extend class notation with involution.
 class  * r
 
Syntaxcsca 13213 Extend class notation with scalar field.
 class Scalar
 
Syntaxcvsca 13214 Extend class notation with scalar product.
 class  .s
 
Syntaxcip 13215 Extend class notation with Hermitian form (inner product).
 class  .i
 
Syntaxcts 13216 Extend class notation with the topology component of a topological space.
 class TopSet
 
Syntaxcple 13217 Extend class notation with less-than-or-equal for posets.
 class  le
 
Syntaxcoc 13218 Extend class notation with the class of orthocomplementation extractors.
 class  oc
 
Syntaxcds 13219 Extend class notation with the metric space distance function.
 class  dist
 
Syntaxcunif 13220 Extend class notation with the uniform structure.
 class  Unif
 
Syntaxchom 13221 Extend class notation with the hom-set structure.
 class  Hom
 
Syntaxcco 13222 Extend class notation with the composition operation.
 class comp
 
Definitiondf-plusg 13223 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 +g  = Slot  2
 
Definitiondf-mulr 13224 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .r  = Slot  3
 
Definitiondf-starv 13225 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 13226 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 13227 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .s  = Slot  6
 
Definitiondf-ip 13228 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .i  = Slot  8
 
Definitiondf-tset 13229 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 13230 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 13231 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 13232 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 13233 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 Unif  = Slot ; 1 3
 
Definitiondf-hom 13234 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Hom  = Slot ; 1 4
 
Definitiondf-cco 13235 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- comp  = Slot ; 1
 5
 
Theoremstrlemor0 13236 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 13237 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 13238 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 13239 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 13240 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 13241 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   =>    |- 
 { <. A ,  X >. } Struct  <. I ,  I >.
 
Theoremstrle2 13242 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 13243 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 13244 Index value of the df-plusg 13223 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( +g  `  ndx )  =  2
 
Theoremplusgid 13245 Utility theorem: index-independent form of df-plusg 13223. (Contributed by NM, 20-Oct-2012.)
 |- 
 +g  = Slot  ( +g  ` 
 ndx )
 
Theorem2strstr 13246 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 13247 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 13248 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 13249 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 13250 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 13251 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 13252  +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  H  =  ( Gs  A )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( A  e.  V  ->  .+  =  ( +g  `  H ) )
 
Theoremgrpbasex 13253 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 13250 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |-  B  =  ( Base `  G )
 
Theoremgrpplusgx 13254 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 13254 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |- 
 .+  =  ( +g  `  G )
 
Theoremmulrndx 13255 Index value of the df-mulr 13224 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .r `  ndx )  =  3
 
Theoremmulrid 13256 Utility theorem: index-independent form of df-mulr 13224. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |- 
 .r  = Slot  ( .r ` 
 ndx )
 
Theoremrngstr 13257 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 13258 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 13259 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 13260 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 13261 Index value of the df-starv 13225 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( * r `  ndx )  =  4
 
Theoremstarvid 13262 Utility theorem: index-independent form of df-starv 13225. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  * r  = Slot  ( * r `  ndx )
 
Theoremressmulr 13263  .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 13264  * r is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
 |-  S  =  ( Rs  A )   &    |-  .*  =  ( * r `  R )   =>    |-  ( A  e.  V  ->  .*  =  ( * r `  S ) )
 
Theoremsrngfn 13265 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 13266 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 13267 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 13268 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 13269 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 13270 Index value of the df-sca 13226 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (Scalar `  ndx )  =  5
 
Theoremscaid 13271 Utility theorem: index-independent form of scalar df-sca 13226. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |- Scalar  = Slot  (Scalar `  ndx )
 
Theoremvscandx 13272 Index value of the df-vsca 13227 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .s `  ndx )  =  6
 
Theoremvscaid 13273 Utility theorem: index-independent form of scalar product df-vsca 13227. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .s  = Slot  ( .s ` 
 ndx )
 
Theoremlmodstr 13274 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 13275 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 ) )
 
Theoremlmodplusg 13276 The additive operation 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.  >. } )   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  W )
 )
 
Theoremlmodsca 13277 The set of scalars 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.  >. } )   =>    |-  ( F  e.  X  ->  F  =  (Scalar `  W ) )
 
Theoremlmodvsca 13278 The scalar product operation 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.  >. } )   =>    |-  (  .x.  e.  X  ->  .x.  =  ( .s `  W ) )
 
Theoremalgstr 13279 Lemma to shorten proofs of algbase 13280 through algvsca 13284. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  A Struct  <. 1 ,  6
 >.
 
Theoremalgbase 13280 The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( B  e.  V  ->  B  =  ( Base `  A ) )
 
Theoremalgaddg 13281 The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  A ) )
 
Theoremalgmulr 13282 The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .X.  e.  V  -> 
 .X.  =  ( .r `  A ) )
 
Theoremalgsca 13283 The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( S  e.  V  ->  S  =  (Scalar `  A ) )
 
Theoremalgvsca 13284 The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .s
 `  A ) )
 
Theoremresssca 13285 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  H  =  ( Gs  A )   &    |-  F  =  (Scalar `  G )   =>    |-  ( A  e.  V  ->  F  =  (Scalar `  H ) )
 
Theoremressvsca 13286  .s is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  H  =  ( Gs  A )   &    |-  .x.  =  ( .s `  G )   =>    |-  ( A  e.  V  ->  .x.  =  ( .s `  H ) )
 
Theoremipndx 13287 Index value of the df-ip 13228 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .i `  ndx )  =  8
 
Theoremipid 13288 Utility theorem: index-independent form of df-ip 13228. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |- 
 .i  = Slot  ( .i ` 
 ndx )
 
Theoremphlstr 13289 A constructed pre-Hilbert space is a structure. Starting from lmodstr 13274 (which has 4 members), we chain strleun 13240 once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  H Struct  <. 1 ,  8 >.
 
Theoremphlbase 13290 The base set of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  ( B  e.  X  ->  B  =  (
 Base `  H ) )
 
Theoremphlplusg 13291 The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  H ) )
 
Theoremphlsca 13292 The ring of scalars of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  ( T  e.  X  ->  T  =  (Scalar `  H ) )
 
Theoremphlvsca 13293 The scalar product operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  (  .x.  e.  X  ->  .x.  =  ( .s `  H ) )
 
Theoremphlip 13294 The inner product (Hermitian form) operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  (  .,  e.  X  ->  .,  =  ( .i `  H ) )
 
Theoremtsetndx 13295 Index value of the df-tset 13229 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (TopSet `  ndx )  =  9
 
Theoremtsetid 13296 Utility theorem: index-independent form of df-tset 13229. (Contributed by NM, 20-Oct-2012.)
 |- TopSet  = Slot  (TopSet `  ndx )
 
Theoremtopgrpstr 13297 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  W Struct  <. 1 ,  9
 >.
 
Theoremtopgrpbas 13298 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  ( B  e.  X  ->  B  =  ( Base `  W ) )
 
Theoremtopgrpplusg 13299 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  W ) )
 
Theoremtopgrptset 13300 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  ( J  e.  X  ->  J  =  (TopSet `  W ) )
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