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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstructfun 13401 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  X   =>    |- 
 Fun  `' `' F
 
Theoremstructfn 13402 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  <. M ,  N >.   =>    |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... N ) )
 
Theoremslotfn 13403 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  E  = Slot  N   =>    |-  E  Fn  _V
 
Theoremstrfvnd 13404 Deduction version of strfvn 13406. (Contributed by Mario Carneiro, 15-Nov-2014.)
 |-  E  = Slot  N   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  ( E `  S )  =  ( S `  N ) )
 
Theoremwunndx 13405 Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   =>    |-  ( ph  ->  ndx 
 e.  U )
 
Theoremstrfvn 13406 Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 13394) and  N is a fixed integer such as  1.  S is a structure, i.e. a specific member of a class of structures such as  Poset (df-poset 14323) where  S  e.  Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 13421. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

 |-  S  e.  _V   &    |-  E  = Slot  N   =>    |-  ( E `  S )  =  ( S `  N )
 
Theoremstrfvss 13407 A structure component extractor produces a value which is contained in a set dependent on  S, but not  E. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  E  = Slot  N   =>    |-  ( E `  S )  C_  U. ran  S
 
Theoremwunstr 13408 Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  = Slot  N   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  S  e.  U )   =>    |-  ( ph  ->  ( E `  S )  e.  U )
 
Theoremndxarg 13409 Get the numeric argument from a defined structure component extractor such as df-base 13394. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( E `  ndx )  =  N
 
Theoremndxid 13410 A structure component extractor is defined by its own index. This theorem, together with strfv 13421 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 13394 and the  10 in df-ple 13469, making it easier to change should the need arise. For example, we can refer to a specific poset with base set  B and order relation  L using  { <. ( Base `  ndx ) ,  B >. ,  <. ( le `  ndx ) ,  L >. } rather than  { <. 1 ,  B >. ,  <. 10 ,  L >. }. The latter, while shorter to state, requires revision if we later change  10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  E  = Slot  ( E `
  ndx )
 
Theoremreldmsets 13411 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |- 
 Rel  dom sSet
 
Theoremsetsvalg 13412 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( S  e.  V  /\  A  e.  W )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
 ) )
 
Theoremsetsval 13413 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( S  e.  V  /\  B  e.  W )  ->  ( S sSet  <. A ,  B >. )  =  ( ( S  |`  ( _V  \  { A } )
 )  u.  { <. A ,  B >. } )
 )
 
Theoremwunsets 13414 Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  S  e.  U )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( S sSet  A )  e.  U )
 
Theoremsetsres 13415 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 13416 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 13417 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 13418 Deduction version of strfv 13421. (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 13419 Deduction version of strfv 13421. (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 13420 A variation on strfv 13421 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 13421 Extract a structure component  C (such as the base set) from a structure  S (such as a member of  Poset, df-poset 14323) with a component extractor  E (such as the base set extractor df-base 13394). By virtue of ndxid 13410, 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 13422 Variant on strfv 13421 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 13423 Deduction version of strss 13424. (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 13424 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 13425 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 13426 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  (/)  =  ( Base `  (/) )
 
Theoremstrfvi 13427 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 13428 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 13429 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 13430 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 13431 Utility theorem: index-independent form of df-base 13394. (Contributed by NM, 20-Oct-2012.)
 |- 
 Base  = Slot  ( Base `  ndx )
 
Theoremelbasfv 13432 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 13433 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 13434 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
 
Theoremreldmress 13435 The structure restriction is a proper operator, so it can be used with ovprc1 6041. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |- 
 Rel  doms
 
Theoremressval 13436 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 13437 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 13438 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 13439 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 13440 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 13441 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 13442 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 13443 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 13444 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  X  ->  ( Ws  B )  =  W )
 
Theoremressinbas 13445 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 13446 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 13447 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( A  e.  X  /\  B  C_  A )  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
 
Theoremwunress 13448 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 13449 Extend class notation with group (addition) operation.
 class  +g
 
Syntaxcmulr 13450 Extend class notation with ring multiplication.
 class  .r
 
Syntaxcstv 13451 Extend class notation with involution.
 class  * r
 
Syntaxcsca 13452 Extend class notation with scalar field.
 class Scalar
 
Syntaxcvsca 13453 Extend class notation with scalar product.
 class  .s
 
Syntaxcip 13454 Extend class notation with Hermitian form (inner product).
 class  .i
 
Syntaxcts 13455 Extend class notation with the topology component of a topological space.
 class TopSet
 
Syntaxcple 13456 Extend class notation with less-than-or-equal for posets.
 class  le
 
Syntaxcoc 13457 Extend class notation with the class of orthocomplementation extractors.
 class  oc
 
Syntaxcds 13458 Extend class notation with the metric space distance function.
 class  dist
 
Syntaxcunif 13459 Extend class notation with the uniform structure.
 class  UnifSet
 
Syntaxchom 13460 Extend class notation with the hom-set structure.
 class  Hom
 
Syntaxcco 13461 Extend class notation with the composition operation.
 class comp
 
Definitiondf-plusg 13462 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 +g  = Slot  2
 
Definitiondf-mulr 13463 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .r  = Slot  3
 
Definitiondf-starv 13464 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 13465 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 13466 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .s  = Slot  6
 
Definitiondf-ip 13467 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .i  = Slot  8
 
Definitiondf-tset 13468 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 13469 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 13470 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 13471 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 13472 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 UnifSet  = Slot ; 1 3
 
Definitiondf-hom 13473 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Hom  = Slot ; 1 4
 
Definitiondf-cco 13474 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- comp  = Slot ; 1
 5
 
Theoremstrlemor0 13475 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 13476 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 13477 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 13478 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 13479 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 13480 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   =>    |- 
 { <. A ,  X >. } Struct  <. I ,  I >.
 
Theoremstrle2 13481 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 13482 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 13483 Index value of the df-plusg 13462 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( +g  `  ndx )  =  2
 
Theoremplusgid 13484 Utility theorem: index-independent form of df-plusg 13462. (Contributed by NM, 20-Oct-2012.)
 |- 
 +g  = Slot  ( +g  ` 
 ndx )
 
Theorem2strstr 13485 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 13486 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 13487 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 13488 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 13489 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 13490 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 13491  +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  H  =  ( Gs  A )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( A  e.  V  ->  .+  =  ( +g  `  H ) )
 
Theoremgrpbasex 13492 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 13489 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |-  B  =  ( Base `  G )
 
Theoremgrpplusgx 13493 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 13493 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |- 
 .+  =  ( +g  `  G )
 
Theoremmulrndx 13494 Index value of the df-mulr 13463 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .r `  ndx )  =  3
 
Theoremmulrid 13495 Utility theorem: index-independent form of df-mulr 13463. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |- 
 .r  = Slot  ( .r ` 
 ndx )
 
Theoremrngstr 13496 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 13497 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 13498 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 13499 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 13500 Index value of the df-starv 13464 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( * r `  ndx )  =  4
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