P. 356 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elrfirn2 35501*	Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( ( B e. V /\ A. y e. I C C_ B ) -> ( A e. ( fi ` ( { B } u. ran ( y e. I |-> C ) ) ) <-> E. v e. ( ~P I i^i Fin ) A = ( B i^i |^|_ y e. v C ) ) )
Theorem	cmpfiiin 35502*	In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- X = U. J   &    |- ( ph -> J e. Comp )   &    |- ( ( ph /\ k e. I ) -> S e. ( Clsd ` J ) )   &    |- ( ( ph /\ ( l C_ I /\ l e. Fin ) ) -> ( X i^i |^|_ k e. l S ) =/= (/) )   =>    |- ( ph -> ( X i^i |^|_ k e. I S ) =/= (/) )
21.23.4  Characterization of closure operators. Kuratowski closure axioms
Theorem	ismrcd1 35503*	Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 15516), isotone (satisfies mrcss 15515), and idempotent (satisfies mrcidm 15518) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 35504 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( ph -> B e. V )   &    |- ( ph -> F : ~P B --> ~P B )   &    |- ( ( ph /\ x C_ B ) -> x C_ ( F ` x ) )   &    |- ( ( ph /\ x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) )   &    |- ( ( ph /\ x C_ B ) -> ( F ` ( F ` x ) ) = ( F ` x ) )   =>    |- ( ph -> dom ( F i^i _I ) e. (Moore ` B ) )
Theorem	ismrcd2 35504*	Second half of ismrcd1 35503. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( ph -> B e. V )   &    |- ( ph -> F : ~P B --> ~P B )   &    |- ( ( ph /\ x C_ B ) -> x C_ ( F ` x ) )   &    |- ( ( ph /\ x C_ B /\ y C_ x ) -> ( F ` y ) C_ ( F ` x ) )   &    |- ( ( ph /\ x C_ B ) -> ( F ` ( F ` x ) ) = ( F ` x ) )   =>    |- ( ph -> F = (mrCls ` dom ( F i^i _I ) ) )
Theorem	istopclsd 35505*	A closure function which satisfies sscls 20063, clsidm 20075, cls0 20088, and clsun 30983 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( ph -> B e. V )   &    |- ( ph -> F : ~P B --> ~P B )   &    |- ( ( ph /\ x C_ B ) -> x C_ ( F ` x ) )   &    |- ( ( ph /\ x C_ B ) -> ( F ` ( F ` x ) ) = ( F ` x ) )   &    |- ( ph -> ( F ` (/) ) = (/) )   &    |- ( ( ph /\ x C_ B /\ y C_ B ) -> ( F ` ( x u. y ) ) = ( ( F ` x ) u. ( F ` y ) ) )   &    |- J = { z e. ~P B | ( F ` ( B \ z ) ) = ( B \ z ) }   =>    |- ( ph -> ( J e. (TopOn ` B ) /\ ( cls ` J ) = F ) )
Theorem	ismrc 35506*	A function is a Moore closure operator iff it satisfies mrcssid 15516, mrcss 15515, and mrcidm 15518. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( F e. (mrCls " (Moore ` B ) ) <-> ( B e. _V /\ F : ~P B --> ~P B /\ A. x A. y ( ( x C_ B /\ y C_ x ) -> ( x C_ ( F ` x ) /\ ( F ` y ) C_ ( F ` x ) /\ ( F ` ( F ` x ) ) = ( F ` x ) ) ) ) )
21.23.5  Algebraic closure systems
Syntax	cnacs 35507	Class of Noetherian closure systems.
class NoeACS
Definition	df-nacs 35508*	Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- NoeACS = ( x e. _V |-> { c e. (ACS ` x ) | A. s e. c E. g e. ( ~P x i^i Fin ) s = ( (mrCls ` c ) ` g ) } )
Theorem	isnacs 35509*	Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (NoeACS ` X ) <-> ( C e. (ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) )
Theorem	nacsfg 35510*	In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (NoeACS ` X ) /\ S e. C ) -> E. g e. ( ~P X i^i Fin ) S = ( F ` g ) )
Theorem	isnacs2 35511	Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (NoeACS ` X ) <-> ( C e. (ACS ` X ) /\ ( F " ( ~P X i^i Fin ) ) = C ) )
Theorem	mrefg2 35512*	Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (Moore ` X ) -> ( E. g e. ( ~P X i^i Fin ) S = ( F ` g ) <-> E. g e. ( ~P S i^i Fin ) S = ( F ` g ) ) )
Theorem	mrefg3 35513*	Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ S e. C ) -> ( E. g e. ( ~P X i^i Fin ) S = ( F ` g ) <-> E. g e. ( ~P S i^i Fin ) S C_ ( F ` g ) ) )
Theorem	nacsacs 35514	A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- ( C e. (NoeACS ` X ) -> C e. (ACS ` X ) )
Theorem	isnacs3 35515*	A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- ( C e. (NoeACS ` X ) <-> ( C e. (Moore ` X ) /\ A. s e. ~P C ( (toInc ` s ) e. Dirset -> U. s e. s ) ) )
Theorem	incssnn0 35516*	Transitivity induction of subsets, lemma for nacsfix 35517. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- ( ( A. x e. NN0 ( F ` x ) C_ ( F ` ( x + 1 ) ) /\ A e. NN0 /\ B e. ( ZZ>= ` A ) ) -> ( F ` A ) C_ ( F ` B ) )
Theorem	nacsfix 35517*	An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- ( ( C e. (NoeACS ` X ) /\ F : NN0 --> C /\ A. x e. NN0 ( F ` x ) C_ ( F ` ( x + 1 ) ) ) -> E. y e. NN0 A. z e. ( ZZ>= ` y ) ( F ` z ) = ( F ` y ) )
21.23.6  Miscellanea 1. Map utilities
Theorem	constmap 35518	A constant (represented without dummy variables) is an element of a function set. _Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|- A e. _V   &    |- C e. _V   =>    |- ( B e. C -> ( A X. { B } ) e. ( C ^m A ) )
Theorem	mapco2g 35519	Renaming indexes in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
|- ( ( E e. _V /\ A e. ( B ^m C ) /\ D : E --> C ) -> ( A o. D ) e. ( B ^m E ) )
Theorem	mapco2 35520	Post-composition (renaming indexes) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|- E e. _V   =>    |- ( ( A e. ( B ^m C ) /\ D : E --> C ) -> ( A o. D ) e. ( B ^m E ) )
Theorem	mapfzcons 35521	Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|- M = ( N + 1 )   =>    |- ( ( N e. NN0 /\ A e. ( B ^m ( 1 ... N ) ) /\ C e. B ) -> ( A u. { <. M , C >. } ) e. ( B ^m ( 1 ... M ) ) )
Theorem	mapfzcons1 35522	Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|- M = ( N + 1 )   =>    |- ( A e. ( B ^m ( 1 ... N ) ) -> ( ( A u. { <. M , C >. } ) |` ( 1 ... N ) ) = A )
Theorem	mapfzcons1cl 35523	A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|- M = ( N + 1 )   =>    |- ( A e. ( B ^m ( 1 ... M ) ) -> ( A |` ( 1 ... N ) ) e. ( B ^m ( 1 ... N ) ) )
Theorem	mapfzcons2 35524	Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|- M = ( N + 1 )   =>    |- ( ( A e. ( B ^m ( 1 ... N ) ) /\ C e. B ) -> ( ( A u. { <. M , C >. } ) ` M ) = C )
21.23.7  Miscellanea for polynomials
Theorem	mptfcl 35525*	Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( t e. A |-> B ) : A --> C -> ( t e. A -> B e. C ) )
21.23.8  Multivariate polynomials over the integers
Syntax	cmzpcl 35526	Extend class notation to include pre-polynomial rings.
class mzPolyCld
Syntax	cmzp 35527	Extend class notation to include polynomial rings.
class mzPoly
Definition	df-mzpcl 35528*	Define the polynomially closed function rings over an arbitrary index set v. The set (mzPolyCld ` v ) contains all sets of functions from ( ZZ ^m v ) to ZZ which include all constants and projections and are closed under addition and multiplication. This is a "temporary" set used to define the polynomial function ring itself (mzPoly ` v ); see df-mzp 35529. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- mzPolyCld = ( v e. _V |-> { p e. ~P ( ZZ ^m ( ZZ ^m v ) ) | ( ( A. i e. ZZ ( ( ZZ ^m v ) X. { i } ) e. p /\ A. j e. v ( x e. ( ZZ ^m v ) |-> ( x ` j ) ) e. p ) /\ A. f e. p A. g e. p ( ( f oF + g ) e. p /\ ( f oF x. g ) e. p ) ) } )
Definition	df-mzp 35529	Polynomials over ZZ with an arbitrary index set, that is, the smallest ring of functions containing all constant functions and all projections. This is almost the most general reasonable definition; to reach full generality, we would need to be able to replace ZZ with an arbitrary (semi-)ring (and a coordinate subring), but rings have not been defined yet. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- mzPoly = ( v e. _V |-> |^| (mzPolyCld ` v ) )
Theorem	mzpclval 35530*	Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( V e. _V -> (mzPolyCld ` V ) = { p e. ~P ( ZZ ^m ( ZZ ^m V ) ) | ( ( A. i e. ZZ ( ( ZZ ^m V ) X. { i } ) e. p /\ A. j e. V ( x e. ( ZZ ^m V ) |-> ( x ` j ) ) e. p ) /\ A. f e. p A. g e. p ( ( f oF + g ) e. p /\ ( f oF x. g ) e. p ) ) } )
Theorem	elmzpcl 35531*	Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( V e. _V -> ( P e. (mzPolyCld ` V ) <-> ( P C_ ( ZZ ^m ( ZZ ^m V ) ) /\ ( ( A. i e. ZZ ( ( ZZ ^m V ) X. { i } ) e. P /\ A. j e. V ( x e. ( ZZ ^m V ) |-> ( x ` j ) ) e. P ) /\ A. f e. P A. g e. P ( ( f oF + g ) e. P /\ ( f oF x. g ) e. P ) ) ) ) )
Theorem	mzpclall 35532	The set of all functions with the signature of a polynomial is a polynomially closed set. This is a lemma to show that the intersection in df-mzp 35529 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( V e. _V -> ( ZZ ^m ( ZZ ^m V ) ) e. (mzPolyCld ` V ) )
Theorem	mzpcln0 35533	Corrolary of mzpclall 35532: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( V e. _V -> (mzPolyCld ` V ) =/= (/) )
Theorem	mzpcl1 35534	Defining property 1 of a polynomially closed function set P: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( P e. (mzPolyCld ` V ) /\ F e. ZZ ) -> ( ( ZZ ^m V ) X. { F } ) e. P )
Theorem	mzpcl2 35535*	Defining property 2 of a polynomially closed function set P: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( P e. (mzPolyCld ` V ) /\ F e. V ) -> ( g e. ( ZZ ^m V ) |-> ( g ` F ) ) e. P )
Theorem	mzpcl34 35536	Defining properties 3 and 4 of a polynomially closed function set P: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( P e. (mzPolyCld ` V ) /\ F e. P /\ G e. P ) -> ( ( F oF + G ) e. P /\ ( F oF x. G ) e. P ) )
Theorem	mzpval 35537	Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( V e. _V -> (mzPoly ` V ) = |^| (mzPolyCld ` V ) )
Theorem	dmmzp 35538	mzPoly is defined for all index sets which are sets. This is used with elfvdm 5905 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- dom mzPoly = _V
Theorem	mzpincl 35539	Polynomial closedness is a universal first-order property and passes to intersections. This is where the closure properties of the polynomial ring itself are proved. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( V e. _V -> (mzPoly ` V ) e. (mzPolyCld ` V ) )
Theorem	mzpconst 35540	Constant functions are polynomial. See also mzpconstmpt 35545. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( V e. _V /\ C e. ZZ ) -> ( ( ZZ ^m V ) X. { C } ) e. (mzPoly ` V ) )
Theorem	mzpf 35541	A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( F e. (mzPoly ` V ) -> F : ( ZZ ^m V ) --> ZZ )
Theorem	mzpproj 35542*	A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( V e. _V /\ X e. V ) -> ( g e. ( ZZ ^m V ) |-> ( g ` X ) ) e. (mzPoly ` V ) )
Theorem	mzpadd 35543	The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 35546. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( A e. (mzPoly ` V ) /\ B e. (mzPoly ` V ) ) -> ( A oF + B ) e. (mzPoly ` V ) )
Theorem	mzpmul 35544	The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 35547. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( A e. (mzPoly ` V ) /\ B e. (mzPoly ` V ) ) -> ( A oF x. B ) e. (mzPoly ` V ) )
Theorem	mzpconstmpt 35545*	A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 35546, mzpmulmpt 35547, mzpnegmpt 35549, mzpsubmpt 35548, mzpexpmpt 35550) can be used to build proofs that functions which are "manifestly polynomial", in the sense of being a maps-to containing constants, projections, and simple arithmetic operations, are actually polynomial functions. There is no mzpprojmpt because mzpproj 35542 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( V e. _V /\ C e. ZZ ) -> ( x e. ( ZZ ^m V ) |-> C ) e. (mzPoly ` V ) )
Theorem	mzpaddmpt 35546*	Sum of polynomial functions is polynomial. Maps-to version of mzpadd 35543. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( ( x e. ( ZZ ^m V ) |-> A ) e. (mzPoly ` V ) /\ ( x e. ( ZZ ^m V ) |-> B ) e. (mzPoly ` V ) ) -> ( x e. ( ZZ ^m V ) |-> ( A + B ) ) e. (mzPoly ` V ) )
Theorem	mzpmulmpt 35547*	Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 35547. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( ( x e. ( ZZ ^m V ) |-> A ) e. (mzPoly ` V ) /\ ( x e. ( ZZ ^m V ) |-> B ) e. (mzPoly ` V ) ) -> ( x e. ( ZZ ^m V ) |-> ( A x. B ) ) e. (mzPoly ` V ) )
Theorem	mzpsubmpt 35548*	The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( ( x e. ( ZZ ^m V ) |-> A ) e. (mzPoly ` V ) /\ ( x e. ( ZZ ^m V ) |-> B ) e. (mzPoly ` V ) ) -> ( x e. ( ZZ ^m V ) |-> ( A - B ) ) e. (mzPoly ` V ) )
Theorem	mzpnegmpt 35549*	Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.)
|- ( ( x e. ( ZZ ^m V ) |-> A ) e. (mzPoly ` V ) -> ( x e. ( ZZ ^m V ) |-> -u A ) e. (mzPoly ` V ) )
Theorem	mzpexpmpt 35550*	Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( ( x e. ( ZZ ^m V ) |-> A ) e. (mzPoly ` V ) /\ D e. NN0 ) -> ( x e. ( ZZ ^m V ) |-> ( A ^ D ) ) e. (mzPoly ` V ) )
Theorem	mzpindd 35551*	"Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
|- ( ( ph /\ f e. ZZ ) -> ch )   &    |- ( ( ph /\ f e. V ) -> th )   &    |- ( ( ph /\ ( f : ( ZZ ^m V ) --> ZZ /\ ta ) /\ ( g : ( ZZ ^m V ) --> ZZ /\ et ) ) -> ze )   &    |- ( ( ph /\ ( f : ( ZZ ^m V ) --> ZZ /\ ta ) /\ ( g : ( ZZ ^m V ) --> ZZ /\ et ) ) -> si )   &    |- ( x = ( ( ZZ ^m V ) X. { f } ) -> ( ps <-> ch ) )   &    |- ( x = ( g e. ( ZZ ^m V ) |-> ( g ` f ) ) -> ( ps <-> th ) )   &    |- ( x = f -> ( ps <-> ta ) )   &    |- ( x = g -> ( ps <-> et ) )   &    |- ( x = ( f oF + g ) -> ( ps <-> ze ) )   &    |- ( x = ( f oF x. g ) -> ( ps <-> si ) )   &    |- ( x = A -> ( ps <-> rh ) )   =>    |- ( ( ph /\ A e. (mzPoly ` V ) ) -> rh )
Theorem	mzpmfp 35552	Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.)
|- (mzPoly ` I ) = ran ( I eval ℤring )
Theorem	mzpsubst 35553*	Substituting polynomials for the variables of a polynomial results in a polynomial. G is expected to depend on y and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( W e. _V /\ F e. (mzPoly ` V ) /\ A. y e. V G e. (mzPoly ` W ) ) -> ( x e. ( ZZ ^m W ) |-> ( F ` ( y e. V |-> ( G ` x ) ) ) ) e. (mzPoly ` W ) )
Theorem	mzprename 35554*	Simplified version of mzpsubst 35553 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( W e. _V /\ F e. (mzPoly ` V ) /\ R : V --> W ) -> ( x e. ( ZZ ^m W ) |-> ( F ` ( x o. R ) ) ) e. (mzPoly ` W ) )
Theorem	mzpresrename 35555*	A polynomial is a polynomial over all larger index sets. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
|- ( ( W e. _V /\ V C_ W /\ F e. (mzPoly ` V ) ) -> ( x e. ( ZZ ^m W ) |-> ( F ` ( x |` V ) ) ) e. (mzPoly ` W ) )
Theorem	mzpcompact2lem 35556*	Lemma for mzpcompact2 35557. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- B e. _V   =>    |- ( A e. (mzPoly ` B ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) )
Theorem	mzpcompact2 35557*	Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( A e. (mzPoly ` B ) -> E. a e. Fin E. b e. (mzPoly ` a ) ( a C_ B /\ A = ( c e. ( ZZ ^m B ) |-> ( b ` ( c |` a ) ) ) ) )
21.23.9  Miscellanea for Diophantine sets 1
Theorem	coeq0i 35558	coeq0 5361 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( A o. B ) = (/) )
Theorem	fzsplit1nn0 35559	Split a finite 1-based set of integers in the middle, allowing either end to be empty ( ( 1 ... 0 )). (Contributed by Stefan O'Rear, 8-Oct-2014.)
|- ( ( A e. NN0 /\ B e. NN0 /\ A <_ B ) -> ( 1 ... B ) = ( ( 1 ... A ) u. ( ( A + 1 ) ... B ) ) )
21.23.10  Diophantine sets 1: definitions
Syntax	cdioph 35560	Extend class notation to include the family of Diophantine sets.
class Dioph
Definition	df-dioph 35561*	A Diophantine set is a set of positive integers which is a projection of the zero set of some polynomial. This definition somewhat awkwardly mixes ZZ (via mzPoly) and NN0 (to define the zero sets); the former could be avoided by considering coincidence sets of NN0 polynomials at the cost of requiring two, and the second is driven by consistency with our mu-recursive functions and the requirements of the Davis-Putnam-Robinson-Matiyasevich proof. Both are avoidable at a complexity cost. In particular, it is a consequence of 4sq 14907 that implicitly restricting variables to NN0 adds no expressive power over allowing them to range over ZZ. While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 35568. (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- Dioph = ( n e. NN0 |-> ran ( k e. ( ZZ>= ` n ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... n ) ) /\ ( p ` u ) = 0 ) } ) )
Theorem	eldiophb 35562*	Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
|- ( D e. (Dioph ` N ) <-> ( N e. NN0 /\ E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) D = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
Theorem	eldioph 35563*	Condition for a set to be Diophantine (unpacking existential quantifier) (Contributed by Stefan O'Rear, 5-Oct-2014.)
|- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) )
Theorem	diophrw 35564*	Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)
|- ( ( S e. _V /\ M : T -1-1-> S /\ ( M |` O ) = ( _I |` O ) ) -> { a | E. b e. ( NN0 ^m S ) ( a = ( b |` O ) /\ ( ( d e. ( ZZ ^m S ) |-> ( P ` ( d o. M ) ) ) ` b ) = 0 ) } = { a | E. c e. ( NN0 ^m T ) ( a = ( c |` O ) /\ ( P ` c ) = 0 ) } )
Theorem	eldioph2lem1 35565*	Lemma for eldioph2 35567. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
|- ( ( N e. NN0 /\ A e. Fin /\ ( 1 ... N ) C_ A ) -> E. d e. ( ZZ>= ` N ) E. e e. _V ( e : ( 1 ... d ) -1-1-onto-> A /\ ( e |` ( 1 ... N ) ) = ( _I |` ( 1 ... N ) ) ) )
Theorem	eldioph2lem2 35566*	Lemma for eldioph2 35567. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
|- ( ( ( N e. NN0 /\ -. S e. Fin ) /\ ( ( 1 ... N ) C_ S /\ A e. ( ZZ>= ` N ) ) ) -> E. c ( c : ( 1 ... A ) -1-1-> S /\ ( c |` ( 1 ... N ) ) = ( _I |` ( 1 ... N ) ) ) )
Theorem	eldioph2 35567*	Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 35557. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
|- ( ( N e. NN0 /\ ( S e. _V /\ ( 1 ... N ) C_ S ) /\ P e. (mzPoly ` S ) ) -> { t | E. u e. ( NN0 ^m S ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) )
Theorem	eldioph2b 35568*	While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set ( S \ ( 1 ... N ) ). For instance, in diophin 35578 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
|- ( ( ( N e. NN0 /\ S e. _V ) /\ ( -. S e. Fin /\ ( 1 ... N ) C_ S ) ) -> ( A e. (Dioph ` N ) <-> E. p e. (mzPoly ` S ) A = { t | E. u e. ( NN0 ^m S ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
Theorem	eldiophelnn0 35569	Remove antecedent on B from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( A e. (Dioph ` B ) -> B e. NN0 )
Theorem	eldioph3b 35570*	Define Diophantine sets in terms of polynomials with variables indexed by NN. This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 35562 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( A e. (Dioph ` N ) <-> ( N e. NN0 /\ E. p e. (mzPoly ` NN ) A = { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
Theorem	eldioph3 35571*	Inference version of eldioph3b 35570 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( N e. NN0 /\ P e. (mzPoly ` NN ) ) -> { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) )
21.23.11  Diophantine sets 2 miscellanea
Theorem	ellz1 35572	Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( B e. ZZ -> ( A e. ( ZZ \ ( ZZ>= ` ( B + 1 ) ) ) <-> ( A e. ZZ /\ A <_ B ) ) )
Theorem	lzunuz 35573	The union of a lower set of integers and an upper set of integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ B <_ ( A + 1 ) ) -> ( ( ZZ \ ( ZZ>= ` ( A + 1 ) ) ) u. ( ZZ>= ` B ) ) = ZZ )
Theorem	fz1eqin 35574	Express a one-based finite range as the intersection of lower integers with NN. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( N e. NN0 -> ( 1 ... N ) = ( ( ZZ \ ( ZZ>= ` ( N + 1 ) ) ) i^i NN ) )
Theorem	lzenom 35575	Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( N e. ZZ -> ( ZZ \ ( ZZ>= ` ( N + 1 ) ) ) ~~ om )
Theorem	elmapresaun 35576	fresaun 5769 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
|- ( ( F e. ( C ^m A ) /\ G e. ( C ^m B ) /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( F u. G ) e. ( C ^m ( A u. B ) ) )
Theorem	elmapresaunres2 35577	fresaunres2 5770 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( ( F e. ( C ^m A ) /\ G e. ( C ^m B ) /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( ( F u. G ) |` B ) = G )
21.23.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra
Theorem	diophin 35578	If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
|- ( ( A e. (Dioph ` N ) /\ B e. (Dioph ` N ) ) -> ( A i^i B ) e. (Dioph ` N ) )
Theorem	diophun 35579	If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
|- ( ( A e. (Dioph ` N ) /\ B e. (Dioph ` N ) ) -> ( A u. B ) e. (Dioph ` N ) )
Theorem	eldiophss 35580	Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
|- ( A e. (Dioph ` B ) -> A C_ ( NN0 ^m ( 1 ... B ) ) )
21.23.13  Diophantine sets 3: construction
Theorem	diophrex 35581*	Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) -> { t | E. u e. S t = ( u |` ( 1 ... N ) ) } e. (Dioph ` N ) )
Theorem	eq0rabdioph 35582*	This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first-order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( N e. NN0 /\ ( t e. ( ZZ ^m ( 1 ... N ) ) |-> A ) e. (mzPoly ` ( 1 ... N ) ) ) -> { t e. ( NN0 ^m ( 1 ... N ) ) | A = 0 } e. (Dioph ` N ) )
Theorem	eqrabdioph 35583*	Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( N e. NN0 /\ ( t e. ( ZZ ^m ( 1 ... N ) ) |-> A ) e. (mzPoly ` ( 1 ... N ) ) /\ ( t e. ( ZZ ^m ( 1 ... N ) ) |-> B ) e. (mzPoly ` ( 1 ... N ) ) ) -> { t e. ( NN0 ^m ( 1 ... N ) ) | A = B } e. (Dioph ` N ) )
Theorem	0dioph 35584	The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( A e. NN0 -> (/) e. (Dioph ` A ) )
Theorem	vdioph 35585	The "universal" set (as large as possible given eldiophss 35580) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( A e. NN0 -> ( NN0 ^m ( 1 ... A ) ) e. (Dioph ` A ) )
Theorem	anrabdioph 35586*	Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( { t e. ( NN0 ^m ( 1 ... N ) ) | ph } e. (Dioph ` N ) /\ { t e. ( NN0 ^m ( 1 ... N ) ) | ps } e. (Dioph ` N ) ) -> { t e. ( NN0 ^m ( 1 ... N ) ) | ( ph /\ ps ) } e. (Dioph ` N ) )
Theorem	orrabdioph 35587*	Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( { t e. ( NN0 ^m ( 1 ... N ) ) | ph } e. (Dioph ` N ) /\ { t e. ( NN0 ^m ( 1 ... N ) ) | ps } e. (Dioph ` N ) ) -> { t e. ( NN0 ^m ( 1 ... N ) ) | ( ph \/ ps ) } e. (Dioph ` N ) )
Theorem	3anrabdioph 35588*	Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( { t e. ( NN0 ^m ( 1 ... N ) ) | ph } e. (Dioph ` N ) /\ { t e. ( NN0 ^m ( 1 ... N ) ) | ps } e. (Dioph ` N ) /\ { t e. ( NN0 ^m ( 1 ... N ) ) | ch } e. (Dioph ` N ) ) -> { t e. ( NN0 ^m ( 1 ... N ) ) | ( ph /\ ps /\ ch ) } e. (Dioph ` N ) )
Theorem	3orrabdioph 35589*	Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( ( { t e. ( NN0 ^m ( 1 ... N ) ) | ph } e. (Dioph ` N ) /\ { t e. ( NN0 ^m ( 1 ... N ) ) | ps } e. (Dioph ` N ) /\ { t e. ( NN0 ^m ( 1 ... N ) ) | ch } e. (Dioph ` N ) ) -> { t e. ( NN0 ^m ( 1 ... N ) ) | ( ph \/ ps \/ ch ) } e. (Dioph ` N ) )
21.23.14  Diophantine sets 4 miscellanea
Theorem	2sbcrex 35590*	Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
|- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )
Theorem	sbcrexgOLD 35591*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3376 instead. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A e. V -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
Theorem	2sbcrexOLD 35592*	Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6337 instead. (New usage is discouraged.) (Proof modification is discouraged.)
|- A e. _V   &    |- B e. _V   =>    |- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )
Theorem	sbc2rex 35593*	Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
|- ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph )
Theorem	sbc2rexgOLD 35594*	Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6337 instead. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A e. V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )
Theorem	sbc4rex 35595*	Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
|- ( [. A / a ]. E. b e. B E. c e. C E. d e. D E. e e. E ph <-> E. b e. B E. c e. C E. d e. D E. e e. E [. A / a ]. ph )
Theorem	sbc4rexgOLD 35596*	Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6337 instead. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A e. V -> ( [. A / a ]. E. b e. B E. c e. C E. d e. D E. e e. E ph <-> E. b e. B E. c e. C E. d e. D E. e e. E [. A / a ]. ph ) )
Theorem	sbcrot3 35597*	Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- ( [. A / a ]. [. B / b ]. [. C / c ]. ph <-> [. B / b ]. [. C / c ]. [. A / a ]. ph )
Theorem	sbcrot5 35598*	Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- ( [. A / a ]. [. B / b ]. [. C / c ]. [. D / d ]. [. E / e ]. ph <-> [. B / b ]. [. C / c ]. [. D / d ]. [. E / e ]. [. A / a ]. ph )
Theorem	sbccomieg 35599*	Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- ( a = A -> B = C )   =>    |- ( [. A / a ]. [. B / b ]. ph <-> [. C / b ]. [. A / a ]. ph )
21.23.15  Diophantine sets 4: Quantification
Theorem	rexrabdioph 35600*	Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- M = ( N + 1 )   &    |- ( v = ( t ` M ) -> ( ps <-> ch ) )   &    |- ( u = ( t |` ( 1 ... N ) ) -> ( ch <-> ph ) )   =>    |- ( ( N e. NN0 /\ { t e. ( NN0 ^m ( 1 ... M ) ) | ph } e. (Dioph ` M ) ) -> { u e. ( NN0 ^m ( 1 ... N ) ) | E. v e. NN0 ps } e. (Dioph ` N ) )