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Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnninfdclemlt 12501* Lemma for nninfdc 12503. The function from nninfdclemf 12499 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
 |-  ( ph  ->  A  C_ 
 NN )   &    |-  ( ph  ->  A. x  e.  NN DECID  x  e.  A )   &    |-  ( ph  ->  A. m  e.  NN  E. n  e.  A  m  <  n )   &    |-  ( ph  ->  ( J  e.  A  /\  1  <  J ) )   &    |-  F  =  seq 1
 ( ( y  e. 
 NN ,  z  e. 
 NN  |-> inf ( ( A  i^i  ( ZZ>= `  (
 y  +  1 ) ) ) ,  RR ,  <  ) ) ,  ( i  e.  NN  |->  J ) )   &    |-  ( ph  ->  U  e.  NN )   &    |-  ( ph  ->  V  e.  NN )   &    |-  ( ph  ->  U  <  V )   =>    |-  ( ph  ->  ( F `  U )  <  ( F `  V ) )
 
Theoremnninfdclemf1 12502* Lemma for nninfdc 12503. The function from nninfdclemf 12499 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
 |-  ( ph  ->  A  C_ 
 NN )   &    |-  ( ph  ->  A. x  e.  NN DECID  x  e.  A )   &    |-  ( ph  ->  A. m  e.  NN  E. n  e.  A  m  <  n )   &    |-  ( ph  ->  ( J  e.  A  /\  1  <  J ) )   &    |-  F  =  seq 1
 ( ( y  e. 
 NN ,  z  e. 
 NN  |-> inf ( ( A  i^i  ( ZZ>= `  (
 y  +  1 ) ) ) ,  RR ,  <  ) ) ,  ( i  e.  NN  |->  J ) )   =>    |-  ( ph  ->  F : NN -1-1-> A )
 
Theoremnninfdc 12503* An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
 |-  ( ( A  C_  NN  /\  A. x  e. 
 NN DECID  x  e.  A  /\  A. m  e.  NN  E. n  e.  A  m  <  n )  ->  om  ~<_  A )
 
Theoremunbendc 12504* An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.)
 |-  ( ( A  C_  NN  /\  A. x  e. 
 NN DECID  x  e.  A  /\  A. m  e.  NN  E. n  e.  A  m  <  n )  ->  A  ~~ 
 NN )
 
Theoremprminf 12505 There are an infinite number of primes. Theorem 1.7 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 28-Nov-2012.)
 |- 
 Prime  ~~  NN
 
Theoreminfpn2 12506* There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 12392, so by unbendc 12504 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)
 |-  S  =  { n  e.  NN  |  ( 1  <  n  /\  A. m  e.  NN  (
 ( n  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  n ) ) ) }   =>    |-  S  ~~  NN
 
PART 6  BASIC STRUCTURES
 
6.1  Extensible structures
 
6.1.1  Basic definitions

An "extensible structure" (or "structure" in short, at least in this section) is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, 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 extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit.

An extensible structure is implemented as a function (a set of ordered pairs) on a finite (and not necessarily sequential) subset of  NN. 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). By convention, we normally avoid direct reference to the hard-coded numeric index and instead use structure component extractors such as ndxid 12535 and strslfv 12556. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. See the comment of basendx 12566 for more details on numeric indices versus the structure component extractors.

There are many other possible ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we are aware of, and (2) is easier to do proofs with. We cannot use an approach that uses "hidden" arguments; Metamath does not support hidden arguments, and in any case we want nothing hidden. It would be possible to use a categorical approach (e.g., something vaguely similar to Lean's mathlib). However, instances (the chain of proofs that an  X is a  Y via a bunch of forgetful functors) can cause serious performance problems for automated tooling, and the resulting proofs would be painful to look at directly (in the case of Lean, they are long past the level where people would find it acceptable to look at them directly). Metamath is working under much stricter conditions than this, and it has still managed to achieve about the same level of flexibility through this "extensible structure" approach.

To create a substructure of a given extensible structure, you can simply use the multifunction restriction operator for extensible structures ↾s as defined in df-iress 12519. This 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, defining a function using the base set and applying that, or explicitly truncating the slot before use.

Extensible structures only work well when they represent concrete categories, where there is a "base set", morphisms are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, ↾s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization.

 
Syntaxcstr 12507 Extend class notation with the class of structures with components numbered below  A.
 class Struct
 
Syntaxcnx 12508 Extend class notation with the structure component index extractor.
 class  ndx
 
Syntaxcsts 12509 Set components of a structure.
 class sSet
 
Syntaxcslot 12510 Extend class notation with the slot function.
 class Slot  A
 
Syntaxcbs 12511 Extend class notation with the class of all base set extractors.
 class  Base
 
Syntaxcress 12512 Extend class notation with the extensible structure builder restriction operator.
 classs
 
Definitiondf-struct 12513* 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.

As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set 
(/) to be extensible structures. Because of 0nelfun 5253, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 12524:  F Struct  X  ->  Fun  ( F  \  { (/)
} ).

Allowing an extensible structure to contain the empty set ensures that expressions like  { <. A ,  B >. ,  <. C ,  D >. } are structures without asserting or implying that  A,  B,  C and  D are sets (if  A or  B is a proper class, then  <. A ,  B >.  =  (/), see opprc 3814). (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 12514 Define the structure component index extractor. See Theorem ndxarg 12534 to understand its purpose. The restriction to  NN ensures that  ndx is a set. The restriction to some set is necessary 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 12517) another set such as the set of finite ordinals 
om (df-iom 4608). (Contributed by NM, 4-Sep-2011.)
 |- 
 ndx  =  (  _I  |` 
 NN )
 
Definitiondf-slot 12515* Define the slot extractor for extensible structures. The class Slot  A is a function whose argument can be any set, although it is meaningful only if that set is a member of an extensible structure (such as a partially ordered set or a group).

Note that Slot  A is implemented as "evaluation at  A". That is,  (Slot  A `  S ) is defined to be  ( S `  A ), where  A will typically be a small nonzero natural number. Each extensible structure  S is a function defined on specific natural number "slots", and this function extracts the value at a particular slot.

The special "structure"  ndx, defined as the identity function restricted to  NN, can be used to extract the number  A from a slot, since  (Slot  A `  ndx )  =  A (see ndxarg 12534). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression  ( Base `  ndx ) in theorems and proofs instead of its value 1).

The class Slot cannot be defined as  ( x  e.  _V  |->  ( f  e. 
_V  |->  ( f `  x ) ) ) because each Slot  A is a function on the proper class  _V so is itself a proper class, and the values of functions are sets (fvex 5554). It is necessary to allow proper classes as values of Slot  A since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)

 |- Slot  A  =  ( x  e.  _V  |->  ( x `  A ) )
 
Theoremsloteq 12516 Equality theorem for the Slot construction. The converse holds if  A (or  B) is a set. (Contributed by BJ, 27-Dec-2021.)
 |-  ( A  =  B  -> Slot 
 A  = Slot  B )
 
Definitiondf-base 12517 Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 Base  = Slot  1
 
Definitiondf-sets 12518* Set a component of an extensible structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-iress 12519 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- sSet  =  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
 { e } )
 )
 
Definitiondf-iress 12519* 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, defining a function using the base set and applying that, or explicitly truncating the slot before use.

(Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.)

(Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.)

 |-s  =  ( w  e.  _V ,  x  e.  _V  |->  ( w sSet  <. ( Base ` 
 ndx ) ,  ( x  i^i  ( Base `  w ) ) >. ) )
 
Theorembrstruct 12520 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |- 
 Rel Struct
 
Theoremisstruct2im 12521 The property of being a structure with components in  ( 1st `  X
) ... ( 2nd `  X
). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
 |-  ( F Struct  X  ->  ( X  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( ... `  X ) ) )
 
Theoremisstruct2r 12522 The property of being a structure with components in  ( 1st `  X
) ... ( 2nd `  X
). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
 |-  ( ( ( X  e.  (  <_  i^i  ( NN  X.  NN )
 )  /\  Fun  ( F 
 \  { (/) } )
 )  /\  ( F  e.  V  /\  dom  F  C_  ( ... `  X ) ) )  ->  F Struct  X )
 
Theoremstructex 12523 A structure is a set. (Contributed by AV, 10-Nov-2021.)
 |-  ( G Struct  X  ->  G  e.  _V )
 
Theoremstructn0fun 12524 A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
 |-  ( F Struct  X  ->  Fun  ( F  \  { (/)
 } ) )
 
Theoremisstructim 12525 The property of being a structure with components in  M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
 |-  ( F Struct  <. M ,  N >.  ->  ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( M ... N ) ) )
 
Theoremisstructr 12526 The property of being a structure with components in  M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
 |-  ( ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  ( Fun  ( F  \  { (/)
 } )  /\  F  e.  V  /\  dom  F  C_  ( M ... N ) ) )  ->  F Struct 
 <. M ,  N >. )
 
Theoremstructcnvcnv 12527 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  X  ->  `' `' F  =  ( F  \  { (/) } )
 )
 
Theoremstructfung 12528 The converse of the converse of a structure is a function. Closed form of structfun 12529. (Contributed by AV, 12-Nov-2021.)
 |-  ( F Struct  X  ->  Fun  `' `' F )
 
Theoremstructfun 12529 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)
 |-  F Struct  X   =>    |- 
 Fun  `' `' F
 
Theoremstructfn 12530 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 ) )
 
Theoremstrnfvnd 12531 Deduction version of strnfvn 12532. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
 |-  E  = Slot  N   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( E `  S )  =  ( S `  N ) )
 
Theoremstrnfvn 12532 Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 12517) and  N is a fixed integer such as  1.  S is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12556. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

 |-  S  e.  _V   &    |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( E `  S )  =  ( S `  N )
 
Theoremstrfvssn 12533 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.) (Revised by Jim Kingdon, 19-Jan-2023.)
 |-  E  = Slot  N   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( E `  S )  C_  U.
 ran  S )
 
Theoremndxarg 12534 Get the numeric argument from a defined structure component extractor such as df-base 12517. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( E `  ndx )  =  N
 
Theoremndxid 12535 A structure component extractor is defined by its own index. This theorem, together with strslfv 12556 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 12517, making it easier to change should the need arise.

(Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.)

 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  E  = Slot  ( E `
  ndx )
 
Theoremndxslid 12536 A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 12556. (Contributed by Jim Kingdon, 29-Jan-2023.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
 
Theoremslotslfn 12537 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   =>    |-  E  Fn  _V
 
Theoremslotex 12538 Existence of slot value. A corollary of slotslfn 12537. (Contributed by Jim Kingdon, 12-Feb-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   =>    |-  ( A  e.  V  ->  ( E `  A )  e.  _V )
 
Theoremstrndxid 12539 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
 |-  ( ph  ->  S  e.  V )   &    |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  ( E `  S ) )
 
Theoremreldmsets 12540 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |- 
 Rel  dom sSet
 
Theoremsetsvalg 12541 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 }
 ) )
 
Theoremsetsvala 12542 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.)
 |-  ( ( S  e.  V  /\  A  e.  X  /\  B  e.  W ) 
 ->  ( S sSet  <. A ,  B >. )  =  ( ( S  |`  ( _V  \  { A } )
 )  u.  { <. A ,  B >. } )
 )
 
Theoremsetsex 12543 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
 |-  ( ( S  e.  V  /\  A  e.  X  /\  B  e.  W ) 
 ->  ( S sSet  <. A ,  B >. )  e.  _V )
 
Theoremstrsetsid 12544 Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  S Struct  <. M ,  N >. )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  ( E ` 
 ndx )  e.  dom  S )   =>    |-  ( ph  ->  S  =  ( S sSet  <. ( E `
  ndx ) ,  ( E `  S ) >. ) )
 
Theoremfvsetsid 12545 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
 |-  ( ( F  e.  V  /\  X  e.  W  /\  Y  e.  U ) 
 ->  ( ( F sSet  <. X ,  Y >. ) `  X )  =  Y )
 
Theoremsetsfun 12546 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
 |-  ( ( ( G  e.  V  /\  Fun  G )  /\  ( I  e.  U  /\  E  e.  W ) )  ->  Fun  ( G sSet  <. I ,  E >. ) )
 
Theoremsetsfun0 12547 A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 12546 is useful for proofs based on isstruct2r 12522 which requires  Fun  ( F 
\  { (/) } ) for 
F to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
 |-  ( ( ( G  e.  V  /\  Fun  ( G  \  { (/) } )
 )  /\  ( I  e.  U  /\  E  e.  W ) )  ->  Fun  ( ( G sSet  <. I ,  E >. )  \  { (/)
 } ) )
 
Theoremsetsn0fun 12548 The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
 |-  ( ph  ->  S Struct  X )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  E  e.  W )   =>    |-  ( ph  ->  Fun  (
 ( S sSet  <. I ,  E >. )  \  { (/)
 } ) )
 
Theoremsetsresg 12549 The structure replacement function does not affect the value of  S away from  A. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
 |-  ( ( S  e.  V  /\  A  e.  W  /\  B  e.  X ) 
 ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } )
 )  =  ( S  |`  ( _V  \  { A } ) ) )
 
Theoremsetsabsd 12550 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
 |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  (
 ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. ) )
 
Theoremsetscom 12551 Different components can be set in any order. (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 >. ) )
 
Theoremsetscomd 12552 Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
 |-  ( ph  ->  A  e.  Y )   &    |-  ( ph  ->  B  e.  Z )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  D  e.  X )   =>    |-  ( ph  ->  (
 ( S sSet  <. A ,  C >. ) sSet  <. B ,  D >. )  =  ( ( S sSet  <. B ,  D >. ) sSet  <. A ,  C >. ) )
 
Theoremstrslfvd 12553 Deduction version of strslfv 12556. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  <. ( E `  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrslfv2d 12554 Deduction version of strslfv 12556. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  `' `' S )   &    |-  ( ph  ->  <.
 ( E `  ndx ) ,  C >.  e.  S )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrslfv2 12555 A variation on strslfv 12556 to avoid asserting that  S itself is a function, which involves sethood of all the ordered pair components of  S. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  S  e.  _V   &    |-  Fun  `' `' S   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( C  e.  V  ->  C  =  ( E `
  S ) )
 
Theoremstrslfv 12556 Extract a structure component  C (such as the base set) from a structure  S with a component extractor  E (such as the base set extractor df-base 12517). By virtue of ndxslid 12536, this can be done without having to refer to the hard-coded numeric index of  E. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  S Struct  X   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   =>    |-  ( C  e.  V  ->  C  =  ( E `  S ) )
 
Theoremstrslfv3 12557 Variant on strslfv 12556 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
 |-  ( ph  ->  U  =  S )   &    |-  S Struct  X   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   &    |-  ( ph  ->  C  e.  V )   &    |-  A  =  ( E `
  U )   =>    |-  ( ph  ->  A  =  C )
 
Theoremstrslssd 12558 Deduction version of strslss 12559. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  Fun  T )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ph  ->  <. ( E `
  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  ( E `  T )  =  ( E `  S ) )
 
Theoremstrslss 12559 Propagate component extraction to a structure  T from a subset structure  S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)
 |-  T  e.  _V   &    |-  Fun  T   &    |-  S  C_  T   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( E `  T )  =  ( E `  S )
 
Theoremstrsl0 12560 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   =>    |-  (/)  =  ( E `  (/) )
 
Theorembase0 12561 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  (/)  =  ( Base `  (/) )
 
Theoremsetsslid 12562 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   =>    |-  ( ( W  e.  A  /\  C  e.  V )  ->  C  =  ( E `  ( W sSet  <. ( E `  ndx ) ,  C >. ) ) )
 
Theoremsetsslnid 12563 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
 |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( E `  ndx )  =/=  D   &    |-  D  e.  NN   =>    |-  ( ( W  e.  A  /\  C  e.  V )  ->  ( E `  W )  =  ( E `  ( W sSet  <. D ,  C >. ) ) )
 
Theorembaseval 12564 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 12565 Utility theorem: index-independent form of df-base 12517. (Contributed by NM, 20-Oct-2012.)
 |- 
 Base  = Slot  ( Base `  ndx )
 
Theorembasendx 12566 Index value of the base set extractor.

Use of this theorem is discouraged since the particular value  1 for the index is an implementation detail. It is generally sufficient to work with  ( Base `  ndx ) and use theorems such as baseid 12565 and basendxnn 12567.

The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint, in proofs such as lmodstrd 12672. Although we have a few theorems such as basendxnplusgndx 12633, we do not intend to add such theorems for every pair of indices (which would be quadradically many in the number of indices).

(New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.)

 |-  ( Base `  ndx )  =  1
 
Theorembasendxnn 12567 The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.)
 |-  ( Base `  ndx )  e. 
 NN
 
Theorembaseslid 12568 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
 |-  ( Base  = Slot  ( Base ` 
 ndx )  /\  ( Base `  ndx )  e. 
 NN )
 
Theorembasfn 12569 The base set extractor is a function on  _V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |- 
 Base  Fn  _V
 
Theorembasmex 12570 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
 |-  B  =  ( Base `  G )   =>    |-  ( A  e.  B  ->  G  e.  _V )
 
Theorembasmexd 12571 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  G  e.  _V )
 
Theoremreldmress 12572 The structure restriction is a proper operator, so it can be used with ovprc1 5931. (Contributed by Stefan O'Rear, 29-Nov-2014.)
 |- 
 Rel  doms
 
Theoremressvalsets 12573 Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
 |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  =  ( W sSet  <. ( Base ` 
 ndx ) ,  ( A  i^i  ( Base `  W ) ) >. ) )
 
Theoremressex 12574 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
 |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Ws  A )  e.  _V )
 
Theoremressval2 12575 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 ) >. ) )
 
Theoremressbasd 12576 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
 |-  ( ph  ->  R  =  ( Ws  A ) )   &    |-  ( ph  ->  B  =  (
 Base `  W ) )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( A  i^i  B )  =  ( Base `  R ) )
 
Theoremressbas2d 12577 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  ( ph  ->  R  =  ( Ws  A ) )   &    |-  ( ph  ->  B  =  (
 Base `  W ) )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  A  =  ( Base `  R ) )
 
Theoremressbasssd 12578 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.)
 |-  ( ph  ->  R  =  ( Ws  A ) )   &    |-  ( ph  ->  B  =  (
 Base `  W ) )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  (
 Base `  R )  C_  B )
 
Theoremressbasid 12579 The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  V  ->  ( Base `  ( Ws  B ) )  =  B )
 
Theoremstrressid 12580 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
 |-  ( ph  ->  B  =  ( Base `  W )
 )   &    |-  ( ph  ->  W Struct  <. M ,  N >. )   &    |-  ( ph  ->  Fun  W )   &    |-  ( ph  ->  ( Base ` 
 ndx )  e.  dom  W )   =>    |-  ( ph  ->  ( Ws  B )  =  W )
 
Theoremressval3d 12581 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
 |-  R  =  ( Ss  A )   &    |-  B  =  (
 Base `  S )   &    |-  E  =  ( Base `  ndx )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  E  e.  dom  S )   &    |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  R  =  ( S sSet  <. E ,  A >. ) )
 
Theoremresseqnbasd 12582 The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
 |-  R  =  ( Ws  A )   &    |-  C  =  ( E `  W )   &    |-  ( E  = Slot  ( E `
  ndx )  /\  ( E `  ndx )  e. 
 NN )   &    |-  ( E `  ndx )  =/=  ( Base `  ndx )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  C  =  ( E `  R ) )
 
Theoremressinbasd 12583 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.)
 |-  ( ph  ->  B  =  ( Base `  W )
 )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  W  e.  V )   =>    |-  ( ph  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B ) ) )
 
Theoremressressg 12584 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  /\  W  e.  Z ) 
 ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
 
Theoremressabsg 12585 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( A  e.  X  /\  B  C_  A  /\  W  e.  Y ) 
 ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
 
6.1.2  Slot definitions
 
Syntaxcplusg 12586 Extend class notation with group (addition) operation.
 class  +g
 
Syntaxcmulr 12587 Extend class notation with ring multiplication.
 class  .r
 
Syntaxcstv 12588 Extend class notation with involution.
 class  *r
 
Syntaxcsca 12589 Extend class notation with scalar field.
 class Scalar
 
Syntaxcvsca 12590 Extend class notation with scalar product.
 class  .s
 
Syntaxcip 12591 Extend class notation with Hermitian form (inner product).
 class  .i
 
Syntaxcts 12592 Extend class notation with the topology component of a topological space.
 class TopSet
 
Syntaxcple 12593 Extend class notation with "less than or equal to" for posets.
 class  le
 
Syntaxcoc 12594 Extend class notation with the class of orthocomplementation extractors.
 class  oc
 
Syntaxcds 12595 Extend class notation with the metric space distance function.
 class  dist
 
Syntaxcunif 12596 Extend class notation with the uniform structure.
 class  UnifSet
 
Syntaxchom 12597 Extend class notation with the hom-set structure.
 class  Hom
 
Syntaxcco 12598 Extend class notation with the composition operation.
 class comp
 
Definitiondf-plusg 12599 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 +g  = Slot  2
 
Definitiondf-mulr 12600 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 .r  = Slot  3
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