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Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem317prm 13001 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 3 1 7  e. 
 Prime
 
Theorem631prm 13002 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 6 3 1  e. 
 Prime
 
6.2.15  Very large primes
 
Theorem1259lem1 13003 Lemma for 1259prm 13008. Calculate a power mod. In decimal, we calculate  2 ^ 1 6  =  5 2 N  +  6 8  ==  6 8 and  2 ^ 1 7  ==  6 8  x.  2  =  1 3 6 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 1 7 )  mod  N )  =  (;; 1 3 6  mod  N )
 
Theorem1259lem2 13004 Lemma for 1259prm 13008. Calculate a power mod. In decimal, we calculate  2 ^ 3 4  =  ( 2 ^ 1 7 ) ^ 2  ==  1
3 6 ^ 2  ==  1 4 N  +  8 7 0. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 3 4 )  mod  N )  =  (;; 8 7 0  mod  N )
 
Theorem1259lem3 13005 Lemma for 1259prm 13008. Calculate a power mod. In decimal, we calculate  2 ^ 3 8  =  2 ^ 3 4  x.  2 ^ 4  ==  8
7 0  x.  1 6  =  1 1 N  +  7 1 and  2 ^ 7 6  =  ( 2 ^ 3 4 ) ^ 2  ==  7
1 ^ 2  =  4 N  +  5  ==  5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 7 6 )  mod  N )  =  ( 5  mod 
 N )
 
Theorem1259lem4 13006 Lemma for 1259prm 13008. Calculate a power mod. In decimal, we calculate  2 ^ 3 0 6  =  ( 2 ^ 7 6 ) ^ 4  x.  4  ==  5 ^ 4  x.  4  =  2 N  -  1 8,  2 ^ 6 1 2  =  ( 2 ^ 3 0 6 ) ^ 2  ==  1 8 ^ 2  =  3 2 4,  2 ^ 6 2 9  =  2 ^ 6 1 2  x.  2 ^ 1 7  ==  3 2 4  x.  1 3 6  =  3 5 N  -  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 6 2 9 ) ^ 2  ==  1 ^ 2  =  1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^
 ( N  -  1
 ) )  mod  N )  =  ( 1  mod  N )
 
Theorem1259lem5 13007 Lemma for 1259prm 13008. Calculate the GCD of  2 ^ 3 4  -  1  ==  8 6 9 with  N  =  1 2 5 9. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( ( 2 ^; 3 4 )  -  1 )  gcd  N )  =  1
 
Theorem1259prm 13008 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  N  e.  Prime
 
Theorem2503lem1 13009 Lemma for 2503prm 13012. Calculate a power mod. In decimal, we calculate  2 ^ 1 8  =  5 1 2 ^ 2  =  1 0 4 N  +  1 8 3 2  ==  1 8 3 2. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  ( ( 2 ^; 1 8 )  mod  N )  =  (;;; 1 8 3 2 
 mod  N )
 
Theorem2503lem2 13010 Lemma for 2503prm 13012. Calculate a power mod. We calculate  2 ^ 1 9  =  2 ^ 1 8  x.  2  ==  1 8 3 2  x.  2  =  N  +  1 1 6 1,  2 ^ 3 8  =  ( 2 ^ 1 9 ) ^ 2  ==  1
1 6 1 ^ 2  =  5 3 8 N  +  1 3 0 7,  2 ^ 3 9  =  2 ^ 3 8  x.  2  ==  1 3 0 7  x.  2  =  N  +  1 1 1,  2 ^ 7 8  =  ( 2 ^ 3 9 ) ^ 2  ==  1
1 1 ^ 2  =  5 N  - 
1 9 4,  2 ^ 1 5 6  =  ( 2 ^ 7 8 ) ^ 2  ==  1 9 4 ^ 2  =  1 5 N  +  9 1,  2 ^ 3 1 2  =  ( 2 ^ 1 5 6 ) ^ 2  ==  9 1 ^ 2  =  3 N  +  7 7 2,  2 ^ 6 2 4  =  ( 2 ^ 3 1 2 ) ^ 2  ==  7 7 2 ^ 2  =  2 3 8 N  + 
2 7 0,  2 ^ 1 2 4 8  =  ( 2 ^ 6 2 4 ) ^
2  ==  2 7 0 ^ 2  =  2 9 N  + 
3 1 3,  2 ^ 1 2 5 1  =  2 ^ 1 2 4 8  x.  8  ==  3 1 3  x.  8  =  N  +  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 1 2 5 1 ) ^ 2  ==  1 ^ 2  =  1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  ( ( 2 ^
 ( N  -  1
 ) )  mod  N )  =  ( 1  mod  N )
 
Theorem2503lem3 13011 Lemma for 2503prm 13012. Calculate the GCD of  2 ^ 1 8  -  1  ==  1 8 3 1 with  N  =  2 5 0 3. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  ( ( ( 2 ^; 1 8 )  -  1 )  gcd  N )  =  1
 
Theorem2503prm 13012 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  N  e.  Prime
 
Theorem4001lem1 13013 Lemma for 4001prm 13017. Calculate a power mod. In decimal, we calculate  2 ^ 1 2  =  4 0 9 6  =  N  +  9 5,  2 ^ 2 4  =  ( 2 ^ 1 2 ) ^ 2  ==  9
5 ^ 2  =  2 N  +  1 0 2 3,  2 ^ 2 5  =  2 ^ 2 4  x.  2  ==  1 0 2 3  x.  2  =  2 0 4 6,  2 ^ 5 0  =  ( 2 ^ 2 5 ) ^ 2  ==  2
0 4 6 ^ 2  =  1 0 4 6 N  + 
1 0 7 0,  2 ^ 1 0 0  =  ( 2 ^ 5 0 ) ^ 2  ==  1 0 7 0 ^ 2  =  2 8 6 N  + 
6 1 4 and  2 ^ 2 0 0  =  ( 2 ^ 1 0 0 ) ^ 2  ==  6 1 4 ^ 2  =  9 4 N  +  9 0 2  ==  9 0 2. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( 2 ^;; 2 0 0 ) 
 mod  N )  =  (;; 9 0 2  mod 
 N )
 
Theorem4001lem2 13014 Lemma for 2503prm 13012. Calculate a power mod. In decimal, we calculate  2 ^ 4 0 0  =  ( 2 ^ 2 0 0 ) ^ 2  ==  9 0 2 ^ 2  =  2 0 3 N  + 
1 4 0 1 and  2 ^ 8 0 0  =  ( 2 ^ 4 0 0 ) ^ 2  ==  1 4 0 1 ^ 2  =  4 9 0 N  +  2 3 1 1  ==  2 3 1 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( 2 ^;; 8 0 0 ) 
 mod  N )  =  (;;; 2 3 1 1  mod 
 N )
 
Theorem4001lem3 13015 Lemma for 4001prm 13017. Calculate a power mod. In decimal, we calculate  2 ^ 1 0 0 0  =  2 ^ 8 0 0  x.  2 ^ 2 0 0  ==  2 3 1 1  x.  9 0 2  =  5 2 1 N  +  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 1 0 0 0 ) ^ 4  ==  1 ^ 4  =  1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( 2 ^
 ( N  -  1
 ) )  mod  N )  =  ( 1  mod  N )
 
Theorem4001lem4 13016 Lemma for 4001prm 13017. Calculate the GCD of  2 ^ 8 0 0  -  1  ==  2 3 1 0 with  N  =  4 0 0 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( ( 2 ^;; 8 0 0 )  -  1
 )  gcd  N )  =  1
 
Theorem4001prm 13017 4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  N  e.  Prime
 
PART 7  EXTENSIBLE STRUCTURES
 
7.1  Extensible structures
 
7.1.1  Basic definitions

An "extensible structure" is a function (set of ordered pairs) on a finite (and not necessarily sequential) subset of  NN, used to define a specific group, ring, poset, etc. The function's argument is the index of a structure component (such as  1 for the base set of a group), and its value is the component (such as the base set). A group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus every ring is also a group. This allows theorems from more general structures (groups) to be reused for more specialized structures (rings) without having to reprove them.

 
Syntaxcstr 13018 Extend class notation with the class of structures with components numbered below  A.
 class Struct
 
Syntaxcnx 13019 Extend class notation with the structure component index extractor.
 class  ndx
 
Syntaxcsts 13020 Set components of a structure.
 class sSet
 
Syntaxcslot 13021 Extend class notation with the slot function.
 class Slot  A
 
Syntaxcbs 13022 Extend class notation with the class of all base set extractors.
 class  Base
 
Syntaxcress 13023 Extend class notation with the extensible structure builder restriction operator.
 classs
 
Definitiondf-struct 13024* Define a structure with components in  M ... N. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN 
 X.  NN ) )  /\  Fun  ( f  \  { (/)
 } )  /\  dom  f  C_  ( ... `  x ) ) }
 
Definitiondf-ndx 13025 Define the structure component index extractor. See theorem ndxarg 13042 to understand its purpose. The restriction to  NN allows  ndx to exist as a set, since  _I is a proper class. In principle, we could have chosen  CC or (if we revise all structure component definitions such as df-base 13027) another set such as the natural ordinal numbers  om (df-om 4548). (Contributed by NM, 4-Sep-2011.)
 |- 
 ndx  =  (  _I  |` 
 NN )
 
Definitiondf-slot 13026* Define slot extractor for posets and related structures. Note that the function argument can be any set, although it is meaningful only if it is a member of  Poset (df-poset 13924) when used for posets or of  Grp (df-grp 14324) when used from groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |- Slot  A  =  ( x  e.  _V  |->  ( x `  A ) )
 
Definitiondf-base 13027 Define the base set extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 Base  = Slot  1
 
Definitiondf-sets 13028* Set one or more components of a structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 13029 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 15161, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the  +g slot instead of the  .r slot. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- sSet  =  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  {  e } ) )  u. 
 { e } )
 )
 
Definitiondf-ress 13029* Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the  Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like  Ring), defining a function using the base set and applying that (like  TopGrp), or explicitly truncating the slot before use (like  MetSp).

(Credit for this operator goes to Mario Carneiro).

See ressbas 13072 for the altered base set, and resslem 13075 (subrg0 15387, ressplusg 13124, subrg1 15390, ressmulr 13135) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

 |-s  =  ( w  e.  _V ,  x  e.  _V  |->  if ( ( Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base ` 
 ndx ) ,  ( x  i^i  ( Base `  w ) ) >. ) ) )
 
Theorembrstruct 13030 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |- 
 Rel Struct
 
Theoremisstruct2 13031 The property of being a structure with components in  ( 1st `  X
) ... ( 2nd `  X
). (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  X  <->  ( X  e.  (  <_  i^i  ( NN  X. 
 NN ) )  /\  Fun  ( F  \  { (/)
 } )  /\  dom  F 
 C_  ( ... `  X ) ) )
 
Theoremisstruct 13032 The property of being a structure with components in  M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  <. M ,  N >. 
 <->  ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  Fun  ( F  \  { (/) } )  /\  dom 
 F  C_  ( M ... N ) ) )
 
Theoremstructcnvcnv 13033 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  X  ->  `' `' F  =  ( F  \  { (/) } )
 )
 
Theoremstructfun 13034 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  X   =>    |- 
 Fun  `' `' F
 
Theoremstructfn 13035 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 13036 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  E  = Slot  N   =>    |-  E  Fn  _V
 
Theoremstrfvnd 13037 Deduction version of strfvn 13039. (Contributed by Mario Carneiro, 15-Nov-2014.)
 |-  E  = Slot  N   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  ( E `  S )  =  ( S `  N ) )
 
Theoremwunndx 13038 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 13039 Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 13027) 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 13924) 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 13054. (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 13040 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 13041 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 13042 Get the numeric argument from a defined structure component extractor such as df-base 13027. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( E `  ndx )  =  N
 
Theoremndxid 13043 A structure component extractor is defined by its own index. This theorem, together with strfv 13054 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 13027 and the  10 in df-ple 13102, 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 13044 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |- 
 Rel  dom sSet
 
Theoremsetsvalg 13045 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 13046 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 13047 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 13048 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 13049 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 13050 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 13051 Deduction version of strfv 13054. (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 13052 Deduction version of strfv 13054. (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 13053 A variation on strfv 13054 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 13054 Extract a structure component  C (such as the base set) from a structure  S (such as a member of  Poset, df-poset 13924) with a component extractor  E (such as the base set extractor df-base 13027). By virtue of ndxid 13043, 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 13055 Variant on strfv 13054 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 13056 Deduction version of strss 13057. (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 13057 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 13058 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 13059 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  (/)  =  ( Base `  (/) )
 
Theoremstrfvi 13060 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 13061 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 13062 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 13063 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 13064 Utility theorem: index-independent form of df-base 13027. (Contributed by NM, 20-Oct-2012.)
 |- 
 Base  = Slot  ( Base `  ndx )
 
Theoremelbasfv 13065 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 13066 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 13067 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 13068 The structure restriction is a proper operator, so it can be used with ovprc1 5738. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |- 
 Rel  doms
 
Theoremressval 13069 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 13070 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 13071 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 13072 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 13073 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 13074 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 13075 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 13076 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 13077 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  X  ->  ( Ws  B )  =  W )
 
Theoremressinbas 13078 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 13079 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 13080 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( A  e.  X  /\  B  C_  A )  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
 
Theoremwunress 13081 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 13082 Extend class notation with group (addition) operation.
 class  +g
 
Syntaxcmulr 13083 Extend class notation with ring multiplication.
 class  .r
 
Syntaxcstv 13084 Extend class notation with involution.
 class  * r
 
Syntaxcsca 13085 Extend class notation with scalar field.
 class Scalar
 
Syntaxcvsca 13086 Extend class notation with scalar product.
 class  .s
 
Syntaxcip 13087 Extend class notation with Hermitian form (inner product).
 class  .i
 
Syntaxcts 13088 Extend class notation with the topology component of a topological space.
 class TopSet
 
Syntaxcple 13089 Extend class notation with less-than-or-equal for posets.
 class  le
 
Syntaxcoc 13090 Extend class notation with the class of orthocomplementation extractors.
 class  oc
 
Syntaxcds 13091 Extend class notation with the metric space distance function.
 class  dist
 
Syntaxcunif 13092 Extend class notation with the uniform structure.
 class  Unif
 
Syntaxchom 13093 Extend class notation with the hom-set structure.
 class  Hom
 
Syntaxcco 13094 Extend class notation with the composition operation.
 class comp
 
Definitiondf-plusg 13095 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 +g  = Slot  2
 
Definitiondf-mulr 13096 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .r  = Slot  3
 
Definitiondf-starv 13097 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 13098 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 13099 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .s  = Slot  6
 
Definitiondf-ip 13100 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .i  = Slot  8
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