P. 147 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14601-14700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gsumfzfsum 14601*	Relate a group sum on ℂfld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> B e. CC )   =>    |- ( ph -> (ℂfld gsumg ( k e. ( M ... N ) |-> B ) ) = sum_ k e. ( M ... N ) B )
Theorem	cnfldui 14602	The invertible complex numbers are exactly those apart from zero. This is recapb 8850 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
|- { z e. CC | z # 0 } = (Unit ` ℂfld )
7.7.2  Ring of integers According to Wikipedia ("Integer", 25-May-2019, https://en.wikipedia.org/wiki/Integer) "The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of [unital] rings, characterizes the ring Z." In set.mm, there was no explicit definition for the ring of integers until June 2019, but it was denoted by (ℂfld ↾s ZZ ), the field of complex numbers restricted to the integers. In zringring 14606 it is shown that this restriction is a ring, and zringbas 14609 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 14604 of the ring of integers. Remark: Instead of using the symbol "ZZrng" analogous to ℂfld used for the field of complex numbers, we have chosen the version with an "i" to indicate that the ring of integers is a unital ring, see also Wikipedia ("Rng (algebra)", 9-Jun-2019, https://en.wikipedia.org/wiki/Rng_(algebra) 14604).
Syntax	czring 14603	Extend class notation with the (unital) ring of integers.
class ℤring
Definition	df-zring 14604	The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
|- ℤring = (ℂfld ↾s ZZ )
Theorem	zringcrng 14605	The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
|- ℤring e. CRing
Theorem	zringring 14606	The ring of integers is a ring. (Contributed by AV, 20-May-2019.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 13-Jun-2019.)
|- ℤring e. Ring
Theorem	zringabl 14607	The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
|- ℤring e. Abel
Theorem	zringgrp 14608	The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
|- ℤring e. Grp
Theorem	zringbas 14609	The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
|- ZZ = ( Base ` ℤring )
Theorem	zringplusg 14610	The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
|- + = ( +g ` ℤring )
Theorem	zringmulg 14611	The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A (.g ` ℤring ) B ) = ( A x. B ) )
Theorem	zringmulr 14612	The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
|- x. = ( .r ` ℤring )
Theorem	zring0 14613	The zero element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
|- 0 = ( 0g ` ℤring )
Theorem	zring1 14614	The unity element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
|- 1 = ( 1r ` ℤring )
Theorem	zringnzr 14615	The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
|- ℤring e. NzRing
Theorem	dvdsrzring 14616	Ring divisibility in the ring of integers corresponds to ordinary divisibility in ZZ. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
|- || = ( ||r ` ℤring )
Theorem	zringinvg 14617	The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
|- ( A e. ZZ -> -u A = ( ( invg ` ℤring ) ` A ) )
Theorem	zringsubgval 14618	Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
|- .- = ( -g ` ℤring )   =>    |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( X - Y ) = ( X .- Y ) )
Theorem	zringmpg 14619	The multiplicative group of the ring of integers is the restriction of the multiplicative group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019.)
|- ( (mulGrp ` ℂfld ) ↾s ZZ ) = (mulGrp ` ℤring )
Theorem	expghmap 14620*	Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)
|- M = (mulGrp ` ℂfld )   &    |- U = ( M ↾s { z e. CC | z # 0 } )   =>    |- ( ( A e. CC /\ A # 0 ) -> ( x e. ZZ |-> ( A ^ x ) ) e. (ℤring GrpHom U ) )
Theorem	mulgghm2 14621*	The powers of a group element give a homomorphism from ZZ to a group. The name .1. should not be taken as a constraint as it may be any group element. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- .x. = (.g ` R )   &    |- F = ( n e. ZZ |-> ( n .x. .1. ) )   &    |- B = ( Base ` R )   =>    |- ( ( R e. Grp /\ .1. e. B ) -> F e. (ℤring GrpHom R ) )
Theorem	mulgrhm 14622*	The powers of the element 1 give a ring homomorphism from ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- .x. = (.g ` R )   &    |- F = ( n e. ZZ |-> ( n .x. .1. ) )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. Ring -> F e. (ℤring RingHom R ) )
Theorem	mulgrhm2 14623*	The powers of the element 1 give the unique ring homomorphism from ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- .x. = (.g ` R )   &    |- F = ( n e. ZZ |-> ( n .x. .1. ) )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. Ring -> (ℤring RingHom R ) = { F } )
7.7.3  Algebraic constructions based on the complex numbers
Syntax	czrh 14624	Map the rationals into a field, or the integers into a ring.
class ZRHom
Syntax	czlm 14625	Augment an abelian group with vector space operations to turn it into a ZZ-module.
class ZMod
Syntax	czn 14626	The ring of integers modulo n.
class ℤ/nℤ
Definition	df-zrh 14627	Define the unique homomorphism from the integers into a ring. This encodes the usual notation of n = 1r + 1r + ... + 1r for integers (see also df-mulg 13706). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- ZRHom = ( r e. _V |-> U. (ℤring RingHom r ) )
Definition	df-zlm 14628	Augment an abelian group with vector space operations to turn it into a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
|- ZMod = ( g e. _V |-> ( ( g sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , (.g ` g ) >. ) )
Definition	df-zn 14629*	Define the ring of integers mod n. This is literally the quotient ring of ZZ by the ideal n ZZ, but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- ℤ/nℤ = ( n e. NN0 |-> [_ℤring / z ]_ [_ ( z /.s ( z ~QG ( (RSpan ` z ) ` { n } ) ) ) / s ]_ ( s sSet <. ( le ` ndx ) , [_ ( ( ZRHom ` s ) |` if ( n = 0 , ZZ , ( 0..^ n ) ) ) / f ]_ ( ( f o. <_ ) o. `' f ) >. ) )
Theorem	zrhval 14630	Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- L = ( ZRHom ` R )   =>    |- L = U. (ℤring RingHom R )
Theorem	zrhvalg 14631	Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- L = ( ZRHom ` R )   =>    |- ( R e. V -> L = U. (ℤring RingHom R ) )
Theorem	zrhval2 14632*	Alternate value of the ZRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- L = ( ZRHom ` R )   &    |- .x. = (.g ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. Ring -> L = ( n e. ZZ |-> ( n .x. .1. ) ) )
Theorem	zrhmulg 14633	Value of the ZRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- L = ( ZRHom ` R )   &    |- .x. = (.g ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( ( R e. Ring /\ N e. ZZ ) -> ( L ` N ) = ( N .x. .1. ) )
Theorem	zrhex 14634	Set existence for ZRHom. (Contributed by Jim Kingdon, 19-May-2025.)
|- L = ( ZRHom ` R )   =>    |- ( R e. V -> L e. _V )
Theorem	zrhrhmb 14635	The ZRHom homomorphism is the unique ring homomorphism from ZZ. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- L = ( ZRHom ` R )   =>    |- ( R e. Ring -> ( F e. (ℤring RingHom R ) <-> F = L ) )
Theorem	zrhrhm 14636	The ZRHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- L = ( ZRHom ` R )   =>    |- ( R e. Ring -> L e. (ℤring RingHom R ) )
Theorem	zrh1 14637	Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- L = ( ZRHom ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. Ring -> ( L ` 1 ) = .1. )
Theorem	zrh0 14638	Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- L = ( ZRHom ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> ( L ` 0 ) = .0. )
Theorem	zrhpropd 14639*	The ZZ ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )   =>    |- ( ph -> ( ZRHom ` K ) = ( ZRHom ` L ) )
Theorem	zlmval 14640	Augment an abelian group with vector space operations to turn it into a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
|- W = ( ZMod ` G )   &    |- .x. = (.g ` G )   =>    |- ( G e. V -> W = ( ( G sSet <. (Scalar ` ndx ) , ℤring >. ) sSet <. ( .s ` ndx ) , .x. >. ) )
Theorem	zlmlemg 14641	Lemma for zlmbasg 14642 and zlmplusgg 14643. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- E = Slot ( E ` ndx )   &    |- ( E ` ndx ) e. NN   &    |- ( E ` ndx ) =/= (Scalar ` ndx )   &    |- ( E ` ndx ) =/= ( .s ` ndx )   =>    |- ( G e. V -> ( E ` G ) = ( E ` W ) )
Theorem	zlmbasg 14642	Base set of a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- B = ( Base ` G )   =>    |- ( G e. V -> B = ( Base ` W ) )
Theorem	zlmplusgg 14643	Group operation of a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. V -> .+ = ( +g ` W ) )
Theorem	zlmmulrg 14644	Ring operation of a ZZ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- .x. = ( .r ` G )   =>    |- ( G e. V -> .x. = ( .r ` W ) )
Theorem	zlmsca 14645	Scalar ring of a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
|- W = ( ZMod ` G )   =>    |- ( G e. V -> ℤring = (Scalar ` W ) )
Theorem	zlmvscag 14646	Scalar multiplication operation of a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- W = ( ZMod ` G )   &    |- .x. = (.g ` G )   =>    |- ( G e. V -> .x. = ( .s ` W ) )
Theorem	znlidl 14647	The set n ZZ is an ideal in ZZ. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   =>    |- ( N e. ZZ -> ( S ` { N } ) e. (LIdeal ` ℤring ) )
Theorem	zncrng2 14648	Making a commutative ring as a quotient of ZZ and n ZZ. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   =>    |- ( N e. ZZ -> U e. CRing )
Theorem	znval 14649	The value of the ℤ/nℤ structure. It is defined as the quotient ring ZZ / n ZZ, with an "artificial" ordering added. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   &    |- F = ( ( ZRHom ` U ) |` W )   &    |- W = if ( N = 0 , ZZ , ( 0..^ N ) )   &    |- .<_ = ( ( F o. <_ ) o. `' F )   =>    |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
Theorem	znle 14650	The value of the ℤ/nℤ structure. It is defined as the quotient ring ZZ / n ZZ, with an "artificial" ordering added. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   &    |- F = ( ( ZRHom ` U ) |` W )   &    |- W = if ( N = 0 , ZZ , ( 0..^ N ) )   &    |- .<_ = ( le ` Y )   =>    |- ( N e. NN0 -> .<_ = ( ( F o. <_ ) o. `' F ) )
Theorem	znval2 14651	Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   &    |- .<_ = ( le ` Y )   =>    |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) )
Theorem	znbaslemnn 14652	Lemma for znbas 14657. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   &    |- E = Slot ( E ` ndx )   &    |- ( E ` ndx ) e. NN   &    |- ( E ` ndx ) =/= ( le ` ndx )   =>    |- ( N e. NN0 -> ( E ` U ) = ( E ` Y ) )
Theorem	znbas2 14653	The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> ( Base ` U ) = ( Base ` Y ) )
Theorem	znadd 14654	The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> ( +g ` U ) = ( +g ` Y ) )
Theorem	znmul 14655	The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> ( .r ` U ) = ( .r ` Y ) )
Theorem	znzrh 14656	The ZZ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> ( ZRHom ` U ) = ( ZRHom ` Y ) )
Theorem	znbas 14657	The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- Y = (ℤ/nℤ ` N )   &    |- R = (ℤring ~QG ( S ` { N } ) )   =>    |- ( N e. NN0 -> ( ZZ /. R ) = ( Base ` Y ) )
Theorem	zncrng 14658	ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> Y e. CRing )
Theorem	znzrh2 14659*	The ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- .~ = (ℤring ~QG ( S ` { N } ) )   &    |- Y = (ℤ/nℤ ` N )   &    |- L = ( ZRHom ` Y )   =>    |- ( N e. NN0 -> L = ( x e. ZZ |-> [ x ] .~ ) )
Theorem	znzrhval 14660	The ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- S = (RSpan ` ℤring )   &    |- .~ = (ℤring ~QG ( S ` { N } ) )   &    |- Y = (ℤ/nℤ ` N )   &    |- L = ( ZRHom ` Y )   =>    |- ( ( N e. NN0 /\ A e. ZZ ) -> ( L ` A ) = [ A ] .~ )
Theorem	znzrhfo 14661	The ZZ ring homomorphism is a surjection onto ℤ/nℤ. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   &    |- L = ( ZRHom ` Y )   =>    |- ( N e. NN0 -> L : ZZ -onto-> B )
Theorem	zndvds 14662	Express equality of equivalence classes in ZZ / n ZZ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- Y = (ℤ/nℤ ` N )   &    |- L = ( ZRHom ` Y )   =>    |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( L ` A ) = ( L ` B ) <-> N || ( A - B ) ) )
Theorem	zndvds0 14663	Special case of zndvds 14662 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- Y = (ℤ/nℤ ` N )   &    |- L = ( ZRHom ` Y )   &    |- .0. = ( 0g ` Y )   =>    |- ( ( N e. NN0 /\ A e. ZZ ) -> ( ( L ` A ) = .0. <-> N || A ) )
Theorem	znf1o 14664	The function F enumerates all equivalence classes in ℤ/nℤ for each n. When n = 0, ZZ / 0 ZZ = ZZ / { 0 } ~~ ZZ so we let W = ZZ; otherwise W = { 0 , ... , n - 1 } enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   &    |- F = ( ( ZRHom ` Y ) |` W )   &    |- W = if ( N = 0 , ZZ , ( 0..^ N ) )   =>    |- ( N e. NN0 -> F : W -1-1-onto-> B )
Theorem	znle2 14665	The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- Y = (ℤ/nℤ ` N )   &    |- F = ( ( ZRHom ` Y ) |` W )   &    |- W = if ( N = 0 , ZZ , ( 0..^ N ) )   &    |- .<_ = ( le ` Y )   =>    |- ( N e. NN0 -> .<_ = ( ( F o. <_ ) o. `' F ) )
Theorem	znleval 14666	The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- Y = (ℤ/nℤ ` N )   &    |- F = ( ( ZRHom ` Y ) |` W )   &    |- W = if ( N = 0 , ZZ , ( 0..^ N ) )   &    |- .<_ = ( le ` Y )   &    |- X = ( Base ` Y )   =>    |- ( N e. NN0 -> ( A .<_ B <-> ( A e. X /\ B e. X /\ ( `' F ` A ) <_ ( `' F ` B ) ) ) )
Theorem	znleval2 14667	The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
|- Y = (ℤ/nℤ ` N )   &    |- F = ( ( ZRHom ` Y ) |` W )   &    |- W = if ( N = 0 , ZZ , ( 0..^ N ) )   &    |- .<_ = ( le ` Y )   &    |- X = ( Base ` Y )   =>    |- ( ( N e. NN0 /\ A e. X /\ B e. X ) -> ( A .<_ B <-> ( `' F ` A ) <_ ( `' F ` B ) ) )
Theorem	znfi 14668	The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   =>    |- ( N e. NN -> B e. Fin )
Theorem	znhash 14669	The ℤ/nℤ structure has n elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   =>    |- ( N e. NN -> (♯ ` B ) = N )
Theorem	znidom 14670	The ℤ/nℤ structure is an integral domain when n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Jim Kingdon, 13-Aug-2025.)
|- Y = (ℤ/nℤ ` N )   =>    |- ( N e. Prime -> Y e. IDomn )
Theorem	znidomb 14671	The ℤ/nℤ structure is a domain precisely when n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN -> ( Y e. IDomn <-> N e. Prime ) )
Theorem	znunit 14672	The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.)
|- Y = (ℤ/nℤ ` N )   &    |- U = (Unit ` Y )   &    |- L = ( ZRHom ` Y )   =>    |- ( ( N e. NN0 /\ A e. ZZ ) -> ( ( L ` A ) e. U <-> ( A gcd N ) = 1 ) )
Theorem	znrrg 14673	The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for N = 0, where all nonzero integers are regular, but only pm 1 are units.) (Contributed by Mario Carneiro, 18-Apr-2016.)
|- Y = (ℤ/nℤ ` N )   &    |- U = (Unit ` Y )   &    |- E = (RLReg ` Y )   =>    |- ( N e. NN -> E = U )
PART 8  BASIC LINEAR ALGEBRA According to Wikipedia ("Linear algebra", 03-Mar-2019, https://en.wikipedia.org/wiki/Linear_algebra) "Linear algebra is the branch of mathematics concerning linear equations [...], linear functions [...] and their representations through matrices and vector spaces." Or according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019, https://www.merriam-webster.com/dictionary/linear%20algebra) "Definition of linear algebra: a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations." Dealing with modules (over rings) instead of vector spaces (over fields) allows for a more unified approach. Therefore, linear equations, matrices, determinants, are usually regarded as "over a ring" in this part. Unless otherwise stated, the rings of scalars need not be commutative (see df-cring 14011), but the existence of a unity element is always assumed (our rings are unital, see df-ring 14010). For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under addition, with a zero vector as neutral element, like in a vector space. Like in a vector space, vectors can be multiplied by scalars, with the usual rules, the only difference being that the scalars are only required to form a ring, and not necessarily a field or a division ring. Note that any vector space is a (special kind of) module, so any theorem proved below for modules applies to any vector space.
8.1  Abstract multivariate polynomials
8.1.1  Definition and basic properties
Syntax	cmps 14674	Multivariate power series.
class mPwSer
Syntax	cmpl 14675	Multivariate polynomials.
class mPoly
Definition	df-psr 14676*	Define the algebra of power series over the index set i and with coefficients from the ring r. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- mPwSer = ( i e. _V , r e. _V |-> [_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]_ [_ ( ( Base ` r ) ^m d ) / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( oF ( +g ` r ) |` ( b X. b ) ) >. , <. ( .r ` ndx ) , ( f e. b , g e. b |-> ( k e. d |-> ( r gsumg ( x e. { y e. d | y oR <_ k } |-> ( ( f ` x ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } ) )
Definition	df-mplcoe 14677*	Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). The index set (which has an element for each variable) is i, the coefficients are in ring r, and for each variable there is a "degree" such that the coefficient is zero for a term where the powers are all greater than those degrees. (Degree is in quotes because there is no guarantee that coefficients below that degree are nonzero, as we do not assume decidable equality for r). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 7-Oct-2025.)
|- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / w ]_ ( w ↾s { f e. ( Base ` w ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } ) )
Theorem	reldmpsr 14678	The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- Rel dom mPwSer
Theorem	psrval 14679*	Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   &    |- O = ( TopOpen ` R )   &    |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }   &    |- ( ph -> B = ( K ^m D ) )   &    |- .+b = ( oF .+ |` ( B X. B ) )   &    |- .X. = ( f e. B , g e. B |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( f ` x ) .x. ( g ` ( k oF - x ) ) ) ) ) ) )   &    |- .xb = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) )   &    |- ( ph -> J = ( Xt_ ` ( D X. { O } ) ) )   &    |- ( ph -> I e. W )   &    |- ( ph -> R e. X )   =>    |- ( ph -> S = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , .xb >. , <. (TopSet ` ndx ) , J >. } ) )
Theorem	fnpsr 14680	The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
|- mPwSer Fn ( _V X. _V )
Theorem	psrvalstrd 14681	The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
|- ( ph -> B e. X )   &    |- ( ph -> .+ e. Y )   &    |- ( ph -> .X. e. Z )   &    |- ( ph -> R e. W )   &    |- ( ph -> .x. e. P )   &    |- ( ph -> J e. Q )   =>    |- ( ph -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , .x. >. , <. (TopSet ` ndx ) , J >. } ) Struct <. 1 , 9 >. )
Theorem	psrbag 14682*	Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
Theorem	psrbagf 14683*	A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( F e. D -> F : I --> NN0 )
Theorem	fczpsrbag 14684*	The constant function equal to zero is a finite bag. (Contributed by AV, 8-Jul-2019.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. V -> ( x e. I |-> 0 ) e. D )
Theorem	psrbaglesuppg 14685*	The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' G " NN ) C_ ( `' F " NN ) )
Theorem	psrbagfi 14686*	A finite index set gives a simpler expression for finite bags. (Contributed by Jim Kingdon, 23-Nov-2025.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. Fin -> D = ( NN0 ^m I ) )
Theorem	psrbasg 14687*	The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- B = ( Base ` S )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. W )   =>    |- ( ph -> B = ( K ^m D ) )
Theorem	psrelbas 14688*	An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : D --> K )
Theorem	psrelbasfi 14689	Simpler form of psrelbas 14688 when the index set is finite. (Contributed by Jim Kingdon, 27-Nov-2025.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- ( ph -> I e. Fin )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : ( NN0 ^m I ) --> K )
Theorem	psrelbasfun 14690	An element of the set of power series is a function. (Contributed by AV, 17-Jul-2019.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   =>    |- ( X e. B -> Fun X )
Theorem	psrplusgg 14691	The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` S )   =>    |- ( ( I e. V /\ R e. W ) -> .+b = ( oF .+ |` ( B X. B ) ) )
Theorem	psradd 14692	The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` S )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )
Theorem	psraddcl 14693	Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- ( ph -> R e. Mgm )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+ Y ) e. B )
Theorem	psr0cl 14694*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- B = ( Base ` S )   =>    |- ( ph -> ( D X. { .0. } ) e. B )
Theorem	psr0lid 14695*	The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( ( D X. { .0. } ) .+ X ) = X )
Theorem	psrnegcl 14696*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N o. X ) e. B )
Theorem	psrlinv 14697*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   &    |- .0. = ( 0g ` R )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )
Theorem	psrgrp 14698	The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> S e. Grp )
Theorem	psr0 14699*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- O = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( ph -> .0. = ( D X. { O } ) )
Theorem	psrneg 14700*	The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- M = ( invg ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )