P. 145 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14401-14500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zringgrp 14401	The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
|- ℤring e. Grp
Theorem	zringbas 14402	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 14403	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 14404	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 14405	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 14406	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 14407	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 14408	The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
|- ℤring e. NzRing
Theorem	dvdsrzring 14409	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 14410	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 14411	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 14412	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 14413*	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 14414*	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 14415*	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 14416*	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 14417	Map the rationals into a field, or the integers into a ring.
class ZRHom
Syntax	czlm 14418	Augment an abelian group with vector space operations to turn it into a ZZ-module.
class ZMod
Syntax	czn 14419	The ring of integers modulo n.
class ℤ/nℤ
Definition	df-zrh 14420	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 13500). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- ZRHom = ( r e. _V |-> U. (ℤring RingHom r ) )
Definition	df-zlm 14421	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 14422*	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 14423	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 14424	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 14425*	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 14426	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 14427	Set existence for ZRHom. (Contributed by Jim Kingdon, 19-May-2025.)
|- L = ( ZRHom ` R )   =>    |- ( R e. V -> L e. _V )
Theorem	zrhrhmb 14428	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 14429	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 14430	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 14431	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 14432*	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 14433	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 14434	Lemma for zlmbasg 14435 and zlmplusgg 14436. (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 14435	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 14436	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 14437	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 14438	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 14439	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 14440	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 14441	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 14442	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 14443	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 14444	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 14445	Lemma for znbas 14450. (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 14446	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 14447	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 14448	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 14449	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 14450	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 14451	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 14452*	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 14453	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 14454	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 14455	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 14456	Special case of zndvds 14455 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 14457	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 14458	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 14459	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 14460	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 14461	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 14462	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 14463	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 14464	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 14465	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 14466	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 13805), but the existence of a unity element is always assumed (our rings are unital, see df-ring 13804). 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 14467	Multivariate power series.
class mPwSer
Syntax	cmpl 14468	Multivariate polynomials.
class mPoly
Definition	df-psr 14469*	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 14470*	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 14471	The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- Rel dom mPwSer
Theorem	psrval 14472*	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 14473	The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
|- mPwSer Fn ( _V X. _V )
Theorem	psrvalstrd 14474	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 14475*	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 14476*	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 14477*	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 14478*	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 14479*	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 14480*	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 14481*	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 14482	Simpler form of psrelbas 14481 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 14483	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 14484	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 14485	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 14486	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 14487*	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 14488*	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 14489*	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 14490*	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 14491	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 14492*	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 14493*	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 ) )
Theorem	psr1clfi 14494*	The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Ring )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   &    |- U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) )   &    |- B = ( Base ` S )   =>    |- ( ph -> U e. B )
Theorem	reldmmpl 14495	The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- Rel dom mPoly
Theorem	mplvalcoe 14496*	Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
Theorem	mplbascoe 14497*	Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) } )
Theorem	mplelbascoe 14498*	Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> ( X e. U <-> ( X e. B /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = .0. ) ) ) )
Theorem	fnmpl 14499	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
|- mPoly Fn ( _V X. _V )
Theorem	mplrcl 14500	Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   =>    |- ( X e. B -> I e. _V )