P. 127 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12601-12700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ctiunctal 12601*	Variation of ctiunct 12600 which allows x to be present in ph. (Contributed by Jim Kingdon, 5-May-2024.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- ( ph -> A. x e. A G : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
Theorem	unct 12602*	The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.)
|- ( ( E. f f : om -onto-> ( A ⊔ 1o ) /\ E. g g : om -onto-> ( B ⊔ 1o ) ) -> E. h h : om -onto-> ( ( A u. B ) ⊔ 1o ) )
Theorem	omctfn 12603*	Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ( ph /\ x e. om ) -> E. g g : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. f ( f Fn om /\ A. x e. om ( f ` x ) : om -onto-> ( B ⊔ 1o ) ) )
Theorem	omiunct 12604*	The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12600 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ( ph /\ x e. om ) -> E. g g : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. om B ⊔ 1o ) )
Theorem	ssomct 12605*	A decidable subset of om is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
|- ( ( A C_ om /\ A. x e. om DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	ssnnctlemct 12606*	Lemma for ssnnct 12607. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 1 )   =>    |- ( ( A C_ NN /\ A. x e. NN DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	ssnnct 12607*	A decidable subset of NN is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	nninfdclemcl 12608*	Lemma for nninfdc 12613. (Contributed by Jim Kingdon, 25-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 -> P e. A )   &    |- ( ph -> Q e. A )   =>    |- ( ph -> ( P ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) Q ) e. A )
Theorem	nninfdclemf 12609*	Lemma for nninfdc 12613. A function from the natural numbers into A. (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 --> A )
Theorem	nninfdclemp1 12610*	Lemma for nninfdc 12613. Each element of the sequence F is greater than the previous element. (Contributed by Jim Kingdon, 26-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 -> ( F ` U ) < ( F ` ( U + 1 ) ) )
Theorem	nninfdclemlt 12611*	Lemma for nninfdc 12613. The function from nninfdclemf 12609 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 ) )
Theorem	nninfdclemf1 12612*	Lemma for nninfdc 12613. The function from nninfdclemf 12609 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 )
Theorem	nninfdc 12613*	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 )
Theorem	unbendc 12614*	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 )
Theorem	prminf 12615	There are an infinite number of primes. Theorem 1.7 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 28-Nov-2012.)
|- Prime ~~ NN
Theorem	infpn2 12616*	There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 12502, so by unbendc 12614 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 12645 and strslfv 12666. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. See the comment of basendx 12676 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 12629. 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.
Syntax	cstr 12617	Extend class notation with the class of structures with components numbered below A.
class Struct
Syntax	cnx 12618	Extend class notation with the structure component index extractor.
class ndx
Syntax	csts 12619	Set components of a structure.
class sSet
Syntax	cslot 12620	Extend class notation with the slot function.
class Slot A
Syntax	cbs 12621	Extend class notation with the class of all base set extractors.
class Base
Syntax	cress 12622	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-struct 12623*	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 5273, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 12634: 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 3826). (Contributed by Mario Carneiro, 29-Aug-2015.)
|- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
Definition	df-ndx 12624	Define the structure component index extractor. See Theorem ndxarg 12644 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 12627) another set such as the set of finite ordinals om (df-iom 4624). (Contributed by NM, 4-Sep-2011.)
|- ndx = ( _I |` NN )
Definition	df-slot 12625*	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 12644). 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 5575). 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 ) )
Theorem	sloteq 12626	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 )
Definition	df-base 12627	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
Definition	df-sets 12628*	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 12629 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 } ) )
Definition	df-iress 12629*	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 ) ) >. ) )
Theorem	brstruct 12630	The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- Rel Struct
Theorem	isstruct2im 12631	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 ) ) )
Theorem	isstruct2r 12632	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 )
Theorem	structex 12633	A structure is a set. (Contributed by AV, 10-Nov-2021.)
|- ( G Struct X -> G e. _V )
Theorem	structn0fun 12634	A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
|- ( F Struct X -> Fun ( F \ { (/) } ) )
Theorem	isstructim 12635	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 ) ) )
Theorem	isstructr 12636	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 >. )
Theorem	structcnvcnv 12637	Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- ( F Struct X -> `' `' F = ( F \ { (/) } ) )
Theorem	structfung 12638	The converse of the converse of a structure is a function. Closed form of structfun 12639. (Contributed by AV, 12-Nov-2021.)
|- ( F Struct X -> Fun `' `' F )
Theorem	structfun 12639	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
Theorem	structfn 12640	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 ) )
Theorem	strnfvnd 12641	Deduction version of strnfvn 12642. (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 ) )
Theorem	strnfvn 12642	Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 12627) 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 12666. (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 )
Theorem	strfvssn 12643	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 )
Theorem	ndxarg 12644	Get the numeric argument from a defined structure component extractor such as df-base 12627. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- E = Slot N   &    |- N e. NN   =>    |- ( E ` ndx ) = N
Theorem	ndxid 12645	A structure component extractor is defined by its own index. This theorem, together with strslfv 12666 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12627, 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 )
Theorem	ndxslid 12646	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 12666. (Contributed by Jim Kingdon, 29-Jan-2023.)
|- E = Slot N   &    |- N e. NN   =>    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
Theorem	slotslfn 12647	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
Theorem	slotex 12648	Existence of slot value. A corollary of slotslfn 12647. (Contributed by Jim Kingdon, 12-Feb-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   =>    |- ( A e. V -> ( E ` A ) e. _V )
Theorem	strndxid 12649	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 ) )
Theorem	reldmsets 12650	The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
|- Rel dom sSet
Theorem	setsvalg 12651	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 } ) )
Theorem	setsvala 12652	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 >. } ) )
Theorem	setsex 12653	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 )
Theorem	strsetsid 12654	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 ) >. ) )
Theorem	fvsetsid 12655	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 )
Theorem	setsfun 12656	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 >. ) )
Theorem	setsfun0 12657	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 12656 is useful for proofs based on isstruct2r 12632 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 >. ) \ { (/) } ) )
Theorem	setsn0fun 12658	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 >. ) \ { (/) } ) )
Theorem	setsresg 12659	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 } ) ) )
Theorem	setsabsd 12660	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 >. ) )
Theorem	setscom 12661	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 >. ) )
Theorem	setscomd 12662	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 >. ) )
Theorem	strslfvd 12663	Deduction version of strslfv 12666. (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 ) )
Theorem	strslfv2d 12664	Deduction version of strslfv 12666. (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 ) )
Theorem	strslfv2 12665	A variation on strslfv 12666 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 ) )
Theorem	strslfv 12666	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 12627). By virtue of ndxslid 12646, 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 ) )
Theorem	strslfv3 12667	Variant on strslfv 12666 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 )
Theorem	strslssd 12668	Deduction version of strslss 12669. (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 ) )
Theorem	strslss 12669	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 )
Theorem	strsl0 12670	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 ` (/) )
Theorem	base0 12671	The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- (/) = ( Base ` (/) )
Theorem	setsslid 12672	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 >. ) ) )
Theorem	setsslnid 12673	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 >. ) ) )
Theorem	baseval 12674	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 )
Theorem	baseid 12675	Utility theorem: index-independent form of df-base 12627. (Contributed by NM, 20-Oct-2012.)
|- Base = Slot ( Base ` ndx )
Theorem	basendx 12676	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 12675 and basendxnn 12677. 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 12784. Although we have a few theorems such as basendxnplusgndx 12745, 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
Theorem	basendxnn 12677	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
Theorem	baseslid 12678	The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
|- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
Theorem	basfn 12679	The base set extractor is a function on _V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
|- Base Fn _V
Theorem	basmex 12680	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 )
Theorem	basmexd 12681	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 )
Theorem	basm 12682*	A structure whose base is inhabited is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
|- B = ( Base ` G )   =>    |- ( A e. B -> E. j j e. G )
Theorem	relelbasov 12683	Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)
|- Rel dom O   &    |- Rel O   &    |- S = ( X O Y )   &    |- B = ( Base ` S )   =>    |- ( A e. B -> ( X e. _V /\ Y e. _V ) )
Theorem	reldmress 12684	The structure restriction is a proper operator, so it can be used with ovprc1 5955. (Contributed by Stefan O'Rear, 29-Nov-2014.)
|- Rel dom ↾s
Theorem	ressvalsets 12685	Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
|- ( ( W e. X /\ A e. Y ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
Theorem	ressex 12686	Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
|- ( ( W e. X /\ A e. Y ) -> ( W ↾s A ) e. _V )
Theorem	ressval2 12687	Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
|- R = ( W ↾s A )   &    |- B = ( Base ` W )   =>    |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
Theorem	ressbasd 12688	Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
|- ( ph -> R = ( W ↾s A ) )   &    |- ( ph -> B = ( Base ` W ) )   &    |- ( ph -> W e. X )   &    |- ( ph -> A e. V )   =>    |- ( ph -> ( A i^i B ) = ( Base ` R ) )
Theorem	ressbas2d 12689	Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- ( ph -> R = ( W ↾s A ) )   &    |- ( ph -> B = ( Base ` W ) )   &    |- ( ph -> W e. X )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> A = ( Base ` R ) )
Theorem	ressbasssd 12690	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 = ( W ↾s A ) )   &    |- ( ph -> B = ( Base ` W ) )   &    |- ( ph -> W e. X )   &    |- ( ph -> A e. V )   =>    |- ( ph -> ( Base ` R ) C_ B )
Theorem	ressbasid 12691	The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
|- B = ( Base ` W )   =>    |- ( W e. V -> ( Base ` ( W ↾s B ) ) = B )
Theorem	strressid 12692	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 -> ( W ↾s B ) = W )
Theorem	ressval3d 12693	Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
|- R = ( S ↾s 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 >. ) )
Theorem	resseqnbasd 12694	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 = ( W ↾s 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 ) )
Theorem	ressinbasd 12695	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 -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
Theorem	ressressg 12696	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 ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
Theorem	ressabsg 12697	Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- ( ( A e. X /\ B C_ A /\ W e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s B ) )
6.1.2  Slot definitions
Syntax	cplusg 12698	Extend class notation with group (addition) operation.
class +g
Syntax	cmulr 12699	Extend class notation with ring multiplication.
class .r
Syntax	cstv 12700	Extend class notation with involution.
class *r