P. 151 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	metuex 15001	Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
|- ( A e. V -> (metUnif ` A ) e. _V )
Syntax	ccnfld 15002	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-cnfld 15003*	The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator. The contract of this set is defined entirely by cnfldex 15005, cnfldadd 15008, cnfldmul 15010, cnfldcj 15011, cnfldtset 15012, cnfldle 15013, cnfldds 15014, and cnfldbas 15006. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)
|- ℂfld = ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
Theorem	cnfldstr 15004	The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- ℂfld Struct <. 1 , ; 1 3 >.
Theorem	cnfldex 15005	The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- ℂfld e. _V
Theorem	cnfldbas 15006	The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- CC = ( Base ` ℂfld )
Theorem	mpocnfldadd 15007*	The addition operation of the field of complex numbers. Version of cnfldadd 15008 using maps-to notation, which does not require ax-addf 8307. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x + y ) ) = ( +g ` ℂfld )
Theorem	cnfldadd 15008	The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
|- + = ( +g ` ℂfld )
Theorem	mpocnfldmul 15009*	The multiplication operation of the field of complex numbers. Version of cnfldmul 15010 using maps-to notation, which does not require ax-mulf 8308. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` ℂfld )
Theorem	cnfldmul 15010	The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
|- x. = ( .r ` ℂfld )
Theorem	cnfldcj 15011	The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- * = ( *r ` ℂfld )
Theorem	cnfldtset 15012	The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 31-Mar-2025.)
|- ( MetOpen ` ( abs o. - ) ) = (TopSet ` ℂfld )
Theorem	cnfldle 15013	The ordering of the field of complex numbers. Note that this is not actually an ordering on CC, but we put it in the structure anyway because restricting to RR does not affect this component, so that (ℂfld ↾s RR ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 15003. (Revised by GG, 31-Mar-2025.)
|- <_ = ( le ` ℂfld )
Theorem	cnfldds 15014	The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 15003. (Revised by GG, 31-Mar-2025.)
|- ( abs o. - ) = ( dist ` ℂfld )
Theorem	cncrng 15015	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
|- ℂfld e. CRing
Theorem	cnring 15016	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- ℂfld e. Ring
Theorem	cnfld0 15017	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- 0 = ( 0g ` ℂfld )
Theorem	cnfld1 15018	One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- 1 = ( 1r ` ℂfld )
Theorem	cnfldneg 15019	The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- ( X e. CC -> ( ( invg ` ℂfld ) ` X ) = -u X )
Theorem	cnfldplusf 15020	The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- + = ( +f ` ℂfld )
Theorem	cnfldsub 15021	The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- - = ( -g ` ℂfld )
Theorem	cnfldmulg 15022	The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- ( ( A e. ZZ /\ B e. CC ) -> ( A (.g ` ℂfld ) B ) = ( A x. B ) )
Theorem	cnfldexp 15023	The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
|- ( ( A e. CC /\ B e. NN0 ) -> ( B (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ B ) )
Theorem	cnsubmlem 15024*	Lemma for nn0subm 15029 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( x e. A -> x e. CC )   &    |- ( ( x e. A /\ y e. A ) -> ( x + y ) e. A )   &    |- 0 e. A   =>    |- A e. (SubMnd ` ℂfld )
Theorem	cnsubglem 15025*	Lemma for cnsubrglem 15026 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- ( x e. A -> x e. CC )   &    |- ( ( x e. A /\ y e. A ) -> ( x + y ) e. A )   &    |- ( x e. A -> -u x e. A )   &    |- B e. A   =>    |- A e. (SubGrp ` ℂfld )
Theorem	cnsubrglem 15026*	Lemma for zsubrg 15027 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- ( x e. A -> x e. CC )   &    |- ( ( x e. A /\ y e. A ) -> ( x + y ) e. A )   &    |- ( x e. A -> -u x e. A )   &    |- 1 e. A   &    |- ( ( x e. A /\ y e. A ) -> ( x x. y ) e. A )   =>    |- A e. (SubRing ` ℂfld )
Theorem	zsubrg 15027	The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- ZZ e. (SubRing ` ℂfld )
Theorem	gzsubrg 15028	The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- ZZ[_i] e. (SubRing ` ℂfld )
Theorem	nn0subm 15029	The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- NN0 e. (SubMnd ` ℂfld )
Theorem	rege0subm 15030	The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- ( 0 [,) +oo ) e. (SubMnd ` ℂfld )
Theorem	zsssubrg 15031	The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- ( R e. (SubRing ` ℂfld ) -> ZZ C_ R )
Theorem	gsumfsum 15032*	Relate a group sum on ℂfld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )
Theorem	cnfldui 15033	The invertible complex numbers are exactly those apart from zero. This is recapb 9009 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 15037 it is shown that this restriction is a ring, and zringbas 15040 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 15035 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) 15035).
Syntax	czring 15034	Extend class notation with the (unital) ring of integers.
class ℤring
Definition	df-zring 15035	The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
|- ℤring = (ℂfld ↾s ZZ )
Theorem	zringcrng 15036	The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
|- ℤring e. CRing
Theorem	zringring 15037	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 15038	The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
|- ℤring e. Abel
Theorem	zringgrp 15039	The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
|- ℤring e. Grp
Theorem	zringbas 15040	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 15041	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 15042	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 15043	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 15044	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 15045	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 15046	The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
|- ℤring e. NzRing
Theorem	dvdsrzring 15047	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 15048	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 15049	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 15050	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 15051*	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 15052*	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 15053*	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 15054*	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 15055	Map the rationals into a field, or the integers into a ring.
class ZRHom
Syntax	czlm 15056	Augment an abelian group with vector space operations to turn it into a ZZ-module.
class ZMod
Syntax	czn 15057	The ring of integers modulo n.
class ℤ/nℤ
Definition	df-zrh 15058	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 13982). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
|- ZRHom = ( r e. _V |-> U. (ℤring RingHom r ) )
Definition	df-zlm 15059	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 15060*	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 15061	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 15062	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 15063*	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 15064	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 15065	Set existence for ZRHom. (Contributed by Jim Kingdon, 19-May-2025.)
|- L = ( ZRHom ` R )   =>    |- ( R e. V -> L e. _V )
Theorem	zrhrhmb 15066	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 15067	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 15068	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 15069	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 15070*	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 15071	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 15072	Lemma for zlmbasg 15073 and zlmplusgg 15074. (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 15073	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 15074	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 15075	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 15076	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 15077	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 15078	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 15079	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 15080	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 15081	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 15082	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 15083	Lemma for znbas 15088. (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 15084	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 15085	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 15086	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 15087	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 15088	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 15089	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 15090*	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 15091	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 15092	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 15093	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 15094	Special case of zndvds 15093 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 15095	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 15096	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 15097	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 15098	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 15099	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 15100	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 )