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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrege0subm 18601 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( 0 [,) +oo )  e.  (SubMnd ` fld )
 
Theoremabsabv 18602 The regular absolute value function on the complex numbers is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 abs  e.  (AbsVal ` fld )
 
Theoremzsssubrg 18603 The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( R  e.  (SubRing ` fld ) 
 ->  ZZ  C_  R )
 
Theoremqsssubdrg 18604 The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing ) 
 ->  QQ  C_  R )
 
Theoremcnsubrg 18605 There are no subrings of the complex numbers strictly between  RR and  CC. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  R  e.  { RR ,  CC }
 )
 
Theoremcnmgpabl 18606 The unit group of the complex numbers is an abelian group. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )   =>    |-  M  e.  Abel
 
Theoremcnmsubglem 18607* Lemma for rpmsubg 18608 and friends. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )   &    |-  ( x  e.  A  ->  x  e.  CC )   &    |-  ( x  e.  A  ->  x  =/=  0 )   &    |-  (
 ( x  e.  A  /\  y  e.  A )  ->  ( x  x.  y )  e.  A )   &    |-  1  e.  A   &    |-  ( x  e.  A  ->  ( 1  /  x )  e.  A )   =>    |-  A  e.  (SubGrp `  M )
 
Theoremrpmsubg 18608 The positive reals form a multiplicative subgroup of the complex numbers. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )   =>    |-  RR+  e.  (SubGrp `  M )
 
Theoremgzrngunitlem 18609 Lemma for gzrngunit 18610. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ[_i]
 )   =>    |-  ( A  e.  (Unit `  Z )  ->  1  <_  ( abs `  A ) )
 
Theoremgzrngunit 18610 The units on  ZZ [ _i ] are the gaussian integers with norm  1. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ[_i]
 )   =>    |-  ( A  e.  (Unit `  Z )  <->  ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 ) )
 
Theoremgsumfsum 18611* 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  ->  (fld 
 gsumg  ( k  e.  A  |->  B ) )  = 
 sum_ k  e.  A  B )
 
Theoremexpmhm 18612* Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  N  =  (flds  NN0 )   &    |-  M  =  (mulGrp ` fld )   =>    |-  ( A  e.  CC  ->  ( x  e.  NN0  |->  ( A ^ x ) )  e.  ( N MndHom  M ) )
 
Theoremnn0srg 18613 The nonnegative integers form a semiring (commutative by subcmn 16972). (Contributed by Thierry Arnoux, 1-May-2018.)
 |-  (flds  NN0 )  e. SRing
 
Theoremrge0srg 18614 The nonnegative real numbers form a semiring (commutative by subcmn 16972). (Contributed by Thierry Arnoux, 6-Sep-2018.)
 |-  (flds  ( 0 [,) +oo )
 )  e. SRing
 
10.11.2  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  (flds  ZZ ), the field of complex numbers restricted to the integers. In zringring 18618 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 18639), and zringbas 18621 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition df-zring 18616 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)).

 
Syntaxzring 18615 Extend class notation with the (unital) ring of integers.
 classring
 
Definitiondf-zring 18616 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
 |-ring  =  (flds  ZZ )
 
Theoremzringcrng 18617 The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
 |-ring  e.  CRing
 
Theoremzringring 18618 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.)
 |-ring  e.  Ring
 
Theoremzringabl 18619 The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
 |-ring  e.  Abel
 
Theoremzringgrp 18620 The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
 |-ring  e.  Grp
 
Theoremzringbas 18621 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 ` ring )
 
Theoremzringplusg 18622 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
 |- 
 +  =  ( +g  ` ring )
 
Theoremzringmulg 18623 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 ` ring ) B )  =  ( A  x.  B ) )
 
Theoremzringmulr 18624 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
 |- 
 x.  =  ( .r
 ` ring
 )
 
Theoremzring0 18625 The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
 |-  0  =  ( 0g
 ` ring
 )
 
Theoremzring1 18626 The multiplicative neutral element of the ring of integers (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
 |-  1  =  ( 1r
 ` ring
 )
 
Theoremzringnzr 18627 The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
 |-ring  e. NzRing
 
Theoremzrngbas 18628 The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) Obsolete version of zringbas 18621 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   =>    |- 
 ZZ  =  ( Base `  Z )
 
Theoremzrngplusg 18629 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (New usage is discouraged.) Use zringplusg 18622 instead.
 |-  Z  =  (flds  ZZ )   =>    |- 
 +  =  ( +g  `  Z )
 
Theoremzrngmulg 18630 The multiplication (group power) opereation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) Obsolete version of zringmulg 18623 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A (.g `  Z ) B )  =  ( A  x.  B ) )
 
Theoremzrngmulr 18631 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (New usage is discouraged.) Use zringmulr 18624 instead.
 |-  Z  =  (flds  ZZ )   =>    |- 
 x.  =  ( .r
 `  Z )
 
Theoremzrng0 18632 The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (New usage is discouraged.) Use zring 18615 instead.
 |-  Z  =  (flds  ZZ )   =>    |-  0  =  ( 0g
 `  Z )
 
Theoremzrng1 18633 The multiplicative neutral element of the ring of integers (Contributed by Thierry Arnoux, 1-Nov-2017.) (New usage is discouraged.) Use zring1 18626 instead.
 |-  Z  =  (flds  ZZ )   =>    |-  1  =  ( 1r
 `  Z )
 
Theoremdvdsrzring 18634 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 ` ring )
 
Theoremdvdsrz 18635 Ring divisibility in  ZZ corresponds to ordinary divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) Obsolete version of dvdsrzring 18634 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   =>    |-  ||  =  ( ||r `  Z )
 
Theoremzringlpirlem1 18636 Lemma for zringlpir 18639. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
 |-  ( ph  ->  I  e.  (LIdeal ` ring ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   =>    |-  ( ph  ->  ( I  i^i  NN )  =/=  (/) )
 
Theoremzringlpirlem2 18637 Lemma for zringlpir 18639. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
 |-  ( ph  ->  I  e.  (LIdeal ` ring ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  =  sup ( ( I  i^i  NN ) ,  RR ,  `'  <  )   =>    |-  ( ph  ->  G  e.  I )
 
Theoremzringlpirlem3 18638 Lemma for zringlpir 18639. All elements of a nonzero ideal of integers are divided by the least one. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
 |-  ( ph  ->  I  e.  (LIdeal ` ring ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  =  sup ( ( I  i^i  NN ) ,  RR ,  `'  <  )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  G 
 ||  X )
 
Theoremzringlpir 18639 The integers are a principal ideal ring but not a field. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
 |-ring  e. LPIR
 
Theoremzringcyg 18640 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.)
 |-ring  e. CycGrp
 
Theoremzlpirlem1 18641 Lemma for zlpir 18644. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) Obsolete version of zringlpirlem1 18636 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  ( ph  ->  I  e.  (LIdeal `  Z ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   =>    |-  ( ph  ->  ( I  i^i  NN )  =/=  (/) )
 
Theoremzlpirlem2 18642 Lemma for zlpir 18644. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) Obsolete version of zringlpirlem2 18637 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  ( ph  ->  I  e.  (LIdeal `  Z ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  =  sup ( ( I  i^i  NN ) ,  RR ,  `'  <  )   =>    |-  ( ph  ->  G  e.  I )
 
Theoremzlpirlem3 18643 Lemma for zlpir 18644. All elements of a nonzero ideal of integers are divided by the least one. (Contributed by Stefan O'Rear, 3-Jan-2015.) Obsolete version of zringlpirlem3 18638 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  ( ph  ->  I  e.  (LIdeal `  Z ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  =  sup ( ( I  i^i  NN ) ,  RR ,  `'  <  )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  G 
 ||  X )
 
Theoremzlpir 18644 The integers are a principal ideal ring but not a field. (Contributed by Stefan O'Rear, 3-Jan-2015.) Obsolete version of zringlpir 18639 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   =>    |-  Z  e. LPIR
 
Theoremzcyg 18645 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) Obsolete version of zringcyg 18640 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   =>    |-  Z  e. CycGrp
 
Theoremzringinvg 18646 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 ` ring ) `  A ) )
 
Theoremzringunit 18647 The units of  ZZ are the integers with norm  1, i.e.  1 and  -u 1. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
 |-  ( A  e.  (Unit ` ring )  <-> 
 ( A  e.  ZZ  /\  ( abs `  A )  =  1 )
 )
 
Theoremzrngunit 18648 The units of  ZZ are the integers with norm  1, i.e.  1 and  -u 1. (Contributed by Mario Carneiro, 5-Dec-2014.) Obsolete version of zringunit 18647 as of 9-Jun-2019. (New usage is discouraged.)
 |-  Z  =  (flds  ZZ )   =>    |-  ( A  e.  (Unit `  Z )  <->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 )
 )
 
Theoremzringmpg 18649 The multiplication group of the ring of integers is the restriction of the multiplication group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019.)
 |-  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp ` ring )
 
Theoremprmirredlem 18650 A positive integer is irreducible over  ZZ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
 |-  I  =  (Irred ` ring )   =>    |-  ( A  e.  NN  ->  ( A  e.  I  <->  A  e.  Prime ) )
 
Theoremdfprm2 18651 The positive irreducible elements of  ZZ are the prime numbers. This is an alternative way to define  Prime. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
 |-  I  =  (Irred ` ring )   =>    |-  Prime  =  ( NN  i^i  I
 )
 
Theoremprmirred 18652 The irreducible elements of  ZZ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
 |-  I  =  (Irred ` ring )   =>    |-  ( A  e.  I  <->  ( A  e.  ZZ  /\  ( abs `  A )  e.  Prime ) )
 
TheoremprmirredlemOLD 18653 A positive integer is irreducible over  ZZ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) Obsolete version of prmirredlem 18650 as of 10-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  I  =  (Irred `  Z )   =>    |-  ( A  e.  NN  ->  ( A  e.  I  <->  A  e.  Prime ) )
 
Theoremdfprm2OLD 18654 The positive irreducible elements of  ZZ are the prime numbers. This is an alternative way to define  Prime. (Contributed by Mario Carneiro, 5-Dec-2014.) Obsolete version of dfprm2 18651 as of 10-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  I  =  (Irred `  Z )   =>    |- 
 Prime  =  ( NN  i^i  I )
 
TheoremprmirredOLD 18655 The irreducible elements of  ZZ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) Obsolete version of prmirred 18652 as of 10-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  I  =  (Irred `  Z )   =>    |-  ( A  e.  I  <->  ( A  e.  ZZ  /\  ( abs `  A )  e.  Prime ) )
 
Theoremexpghm 18656* Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.)
 |-  M  =  (mulGrp ` fld )   &    |-  U  =  ( Ms  ( CC  \  {
 0 } ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( x  e.  ZZ  |->  ( A ^ x ) )  e.  (ring  GrpHom  U ) )
 
TheoremexpghmOLD 18657* Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) Obsolete version of expghm 18656 as of 10-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  M  =  (mulGrp ` fld )   &    |-  U  =  ( Ms  ( CC  \  { 0 } ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  ( x  e.  ZZ  |->  ( A ^ x ) )  e.  ( Z  GrpHom  U ) )
 
Theoremmulgghm2 18658* The powers of a group element give a homomorphism from  ZZ to a group. (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.  (ring  GrpHom  R ) )
 
Theoremmulgrhm 18659* 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.  (ring RingHom  R ) )
 
Theoremmulgrhm2 18660* 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  ->  (ring RingHom  R )  =  { F } )
 
Theoremmulgghm2OLD 18661* The powers of a group element give a homomorphism from  ZZ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.) Obsolete version of mulgghm2 18658 as of 12-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( R  e.  Grp  /\  .1.  e.  B ) 
 ->  F  e.  ( Z 
 GrpHom  R ) )
 
TheoremmulgrhmOLD 18662* The powers of the element  1 give a ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) Obsolete version of mulgrhm 18659 as of 12-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R ) )
 
Theoremmulgrhm2OLD 18663* The powers of the element  1 give the unique ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) Obsolete version of mulgrhm2 18660 as of 12-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
 
10.11.3  Algebraic constructions based on the complex numbers
 
Syntaxczrh 18664 Map the rationals into a field, or the integers into a ring.
 class  ZRHom
 
Syntaxczlm 18665 Augment an abelian group with vector space operations to turn it into a  ZZ-module.
 class  ZMod
 
Syntaxcchr 18666 Syntax for ring characteristic.
 class chr
 
Syntaxczn 18667 The ring of integers modulo  n.
 class ℤ/n
 
Definitiondf-zrh 18668 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 16187). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
 |-  ZRHom  =  ( r  e.  _V  |->  U. (ring RingHom  r ) )
 
Definitiondf-zlm 18669 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 ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  g
 ) >. ) )
 
Definitiondf-chr 18670 The characteristic of a ring is the smallest positive integer which is equal to 0 when interpreted in the ring, or 0 if there is no such positive integer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- chr 
 =  ( g  e. 
 _V  |->  ( ( od
 `  g ) `  ( 1r `  g ) ) )
 
Definitiondf-zn 18671* 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/n =  ( n  e.  NN0  |->  [_ring  /  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
 ) >. ) )
 
Theoremzrhval 18672 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. (ring RingHom  R )
 
Theoremzrhval2 18673* 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.  ) ) )
 
Theoremzrhmulg 18674 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.  ) )
 
Theoremzrhrhmb 18675 The  ZRHom homomorphism is the unique ring homomorphism from  Z. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.)
 |-  L  =  ( ZRHom `  R )   =>    |-  ( R  e.  Ring  ->  ( F  e.  (ring RingHom  R )  <->  F  =  L )
 )
 
Theoremzrhrhm 18676 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.  (ring RingHom  R ) )
 
Theoremzrh1 18677 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.  )
 
Theoremzrh0 18678 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.  )
 
Theoremzrhpropd 18679* 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 ) )
 
Theoremzlmval 18680 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 ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  .x.  >. ) )
 
Theoremzlmlem 18681 Lemma for zlmbas 18682 and zlmplusg 18683. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  5   =>    |-  ( E `  G )  =  ( E `  W )
 
Theoremzlmbas 18682 Base set of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  B  =  (
 Base `  W )
 
Theoremzlmplusg 18683 Group operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  .+  =  ( +g  `  W )
 
Theoremzlmmulr 18684 Ring operation of a  ZZ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .x.  =  ( .r `  G )   =>    |-  .x.  =  ( .r `  W )
 
Theoremzlmsca 18685 Scalar ring of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e.  V  ->ring  =  (Scalar `  W )
 )
 
Theoremzlmvsca 18686 Scalar multiplication operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .x.  =  (.g `  G )   =>    |- 
 .x.  =  ( .s `  W )
 
Theoremzlmlmod 18687 The  ZZ-module operation turns an arbitrary abelian group into a left module over  ZZ. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e.  Abel  <->  W  e.  LMod )
 
Theoremzlmassa 18688 The  ZZ-module operation turns a ring into an associative algebra over  ZZ. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e.  Ring  <->  W  e. AssAlg )
 
Theoremchrval 18689 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  O  =  ( od
 `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  C  =  (chr `  R )   =>    |-  ( O `  .1.  )  =  C
 
Theoremchrcl 18690 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   =>    |-  ( R  e.  Ring  ->  C  e.  NN0 )
 
Theoremchrid 18691 The canonical  ZZ ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( L `  C )  =  .0.  )
 
Theoremchrdvds 18692 The  ZZ ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  N  e.  ZZ )  ->  ( C  ||  N  <->  ( L `  N )  =  .0.  ) )
 
Theoremchrcong 18693 If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, 24-Sep-2015.)
 |-  C  =  (chr `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( C  ||  ( M  -  N )  <->  ( L `  M )  =  ( L `  N ) ) )
 
Theoremchrnzr 18694 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( R  e.  Ring  ->  ( R  e. NzRing  <->  (chr `  R )  =/=  1 ) )
 
Theoremchrrhm 18695 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  (chr `  S )  ||  (chr `  R ) )
 
Theoremdomnchr 18696 The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( R  e. Domn  ->  ( (chr `  R )  =  0  \/  (chr `  R )  e.  Prime ) )
 
Theoremznlidl 18697 The set  n ZZ is an ideal in  ZZ. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   =>    |-  ( N  e.  ZZ  ->  ( S `  { N } )  e.  (LIdeal ` ring ) )
 
Theoremzncrng2 18698 The value of the ℤ/nℤ structure. It is defined as the quotient ring  ZZ  /  n ZZ, with an "artificial" ordering added to make it a Toset. (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, 12-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   =>    |-  ( N  e.  ZZ  ->  U  e.  CRing )
 
Theoremznval 18699 The value of the ℤ/nℤ structure. It is defined as the quotient ring  ZZ  /  n ZZ, with an "artificial" ordering added to make it a Toset. (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 ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  U )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( ( F  o.  <_  )  o.  `' F )   =>    |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <.
 ( le `  ndx ) ,  .<_  >. ) )
 
Theoremznle 18700 The value of the ℤ/nℤ structure. It is defined as the quotient ring  ZZ  /  n ZZ, with an "artificial" ordering added to make it a Toset. (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 ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  U )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  .<_  =  ( ( F  o.  <_  )  o.  `' F ) )
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