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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrege0subm 19101 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( 0 [,) +oo )  e.  (SubMnd ` fld )
 
Theoremabsabv 19102 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 19103 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 19104 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 19105 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 19106 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 19107* Lemma for rpmsubg 19108 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 19108 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 19109 Lemma for gzrngunit 19110. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ[_i]
 )   =>    |-  ( A  e.  (Unit `  Z )  ->  1  <_  ( abs `  A ) )
 
Theoremgzrngunit 19110 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 19111* 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 )
 
Theoremregsumfsum 19112* Relate a group sum on  (flds  RR ) to a finite sum on the reals. Cf. gsumfsum 19111. (Contributed by Thierry Arnoux, 7-Sep-2018.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  RR )   =>    |-  ( ph  ->  ( (flds  RR )  gsumg  ( k  e.  A  |->  B ) )  = 
 sum_ k  e.  A  B )
 
Theoremexpmhm 19113* 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 19114 The nonnegative integers form a semiring (commutative by subcmn 17555). (Contributed by Thierry Arnoux, 1-May-2018.)
 |-  (flds  NN0 )  e. SRing
 
Theoremrge0srg 19115 The nonnegative real numbers form a semiring (commutative by subcmn 17555). (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 19119 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 19135), and zringbas 19122 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition df-zring 19117 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 19116 Extend class notation with the (unital) ring of integers.
 classring
 
Definitiondf-zring 19117 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
 |-ring  =  (flds  ZZ )
 
Theoremzringcrng 19118 The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
 |-ring  e.  CRing
 
Theoremzringring 19119 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 19120 The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
 |-ring  e.  Abel
 
Theoremzringgrp 19121 The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
 |-ring  e.  Grp
 
Theoremzringbas 19122 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 19123 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
 |- 
 +  =  ( +g  ` ring )
 
Theoremzringmulg 19124 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 19125 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 19126 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 19127 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 19128 The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
 |-ring  e. NzRing
 
Theoremdvdsrzring 19129 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 )
 
Theoremzringlpirlem1 19130 Lemma for zringlpir 19135. 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 )  =/=  (/) )
 
Theoremzringlpirlem2OLD 19131 Lemma for zringlpir 19135. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) Obsolete version of zringlpirlem2 19133 as of 27-Sep-2020. (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  I  e.  (LIdeal ` ring ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  =  sup ( ( I  i^i  NN ) ,  RR ,  `'  <  )   =>    |-  ( ph  ->  G  e.  I )
 
Theoremzringlpirlem3OLD 19132 Lemma for zringlpir 19135. 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.) Obsolete version of zringlpirlem3 19134 as of 27-Sep-2020. (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  I  e.  (LIdeal ` ring ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  =  sup ( ( I  i^i  NN ) ,  RR ,  `'  <  )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  G 
 ||  X )
 
Theoremzringlpirlem2 19133 Lemma for zringlpir 19135. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Revised by AV, 27-Sep-2020.)
 |-  ( ph  ->  I  e.  (LIdeal ` ring ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  = inf (
 ( I  i^i  NN ) ,  RR ,  <  )   =>    |-  ( ph  ->  G  e.  I )
 
Theoremzringlpirlem3 19134 Lemma for zringlpir 19135. 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.) (Proof shortened by AV, 27-Sep-2020.)
 |-  ( ph  ->  I  e.  (LIdeal ` ring ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  = inf (
 ( I  i^i  NN ) ,  RR ,  <  )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  G  ||  X )
 
Theoremzringlpir 19135 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.) (Proof shortened by AV, 27-Sep-2020.)
 |-ring  e. LPIR
 
TheoremzringlpirOLD 19136 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.) Obsolete version of zringlpir 19135 as of 27-Sep-2020. (Proof modification is discouraged.) (New usage is discouraged.)
 |-ring  e. LPIR
 
Theoremzringcyg 19137 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.)
 |-ring  e. CycGrp
 
Theoremzringinvg 19138 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 19139 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 )
 )
 
Theoremzringmpg 19140 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 19141 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 19142 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 19143 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 ) )
 
Theoremexpghm 19144* 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 ) )
 
Theoremmulgghm2 19145* 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 19146* 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 19147* 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 } )
 
10.11.3  Algebraic constructions based on the complex numbers
 
Syntaxczrh 19148 Map the rationals into a field, or the integers into a ring.
 class  ZRHom
 
Syntaxczlm 19149 Augment an abelian group with vector space operations to turn it into a  ZZ-module.
 class  ZMod
 
Syntaxcchr 19150 Syntax for ring characteristic.
 class chr
 
Syntaxczn 19151 The ring of integers modulo  n.
 class ℤ/n
 
Definitiondf-zrh 19152 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 16754). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
 |-  ZRHom  =  ( r  e.  _V  |->  U. (ring RingHom  r ) )
 
Definitiondf-zlm 19153 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 19154 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 19155* 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 19156 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 19157* 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 19158 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 19159 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 19160 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 19161 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 19162 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 19163* 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 19164 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 19165 Lemma for zlmbas 19166 and zlmplusg 19167. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  5   =>    |-  ( E `  G )  =  ( E `  W )
 
Theoremzlmbas 19166 Base set of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  B  =  (
 Base `  W )
 
Theoremzlmplusg 19167 Group operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  .+  =  ( +g  `  W )
 
Theoremzlmmulr 19168 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 19169 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 19170 Scalar multiplication operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .x.  =  (.g `  G )   =>    |- 
 .x.  =  ( .s `  W )
 
Theoremzlmlmod 19171 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 19172 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 19173 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 19174 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   =>    |-  ( R  e.  Ring  ->  C  e.  NN0 )
 
Theoremchrid 19175 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 19176 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 19177 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 19178 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 19179 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  (chr `  S )  ||  (chr `  R ) )
 
Theoremdomnchr 19180 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 19181 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 19182 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 19183 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 19184 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 ) )
 
Theoremznval2 19185 Self-referential expression for the ℤ/nℤ structure. (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 )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <.
 ( le `  ndx ) ,  .<_  >. ) )
 
Theoremznbaslem 19186 Lemma for znbas 19191. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  E  = Slot  K   &    |-  K  e.  NN   &    |-  K  <  10   =>    |-  ( N  e.  NN0  ->  ( E `  U )  =  ( E `  Y ) )
 
Theoremznbas2 19187 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.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( Base `  U )  =  ( Base `  Y )
 )
 
Theoremznadd 19188 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.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( +g  `  U )  =  ( +g  `  Y ) )
 
Theoremznmul 19189 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.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( .r `  U )  =  ( .r `  Y ) )
 
Theoremznzrh 19190 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 ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( ZRHom `  U )  =  ( ZRHom `  Y ) )
 
Theoremznbas 19191 The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  Y  =  (ℤ/n `  N )   &    |-  R  =  (ring ~QG  ( S `
  { N }
 ) )   =>    |-  ( N  e.  NN0  ->  ( ZZ /. R )  =  ( Base `  Y ) )
 
Theoremzncrng 19192 ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e.  CRing )
 
Theoremznzrh2 19193* 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 ` ring )   &    |-  .~  =  (ring ~QG  ( S `  { N }
 ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L  =  ( x  e.  ZZ  |->  [ x ]  .~  )
 )
 
Theoremznzrhval 19194 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 ` ring )   &    |-  .~  =  (ring ~QG  ( S `  { N }
 ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  (
 ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A )  =  [ A ]  .~  )
 
Theoremznzrhfo 19195 The  ZZ ring homomorphism is a surjection onto 
ZZ  /  n ZZ. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
 
Theoremzncyg 19196 The group  ZZ  /  n ZZ is cyclic for all  n (including  n  =  0). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e. CycGrp )
 
Theoremzndvds 19197 Express equality of equivalence classes in  ZZ 
/  n ZZ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( L `  A )  =  ( L `  B )  <->  N  ||  ( A  -  B ) ) )
 
Theoremzndvds0 19198 Special case of zndvds 19197 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   &    |-  .0.  =  ( 0g `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( ( L `  A )  =  .0.  <->  N  ||  A ) )
 
Theoremznf1o 19199 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/n `  N )   &    |-  B  =  ( Base `  Y )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   =>    |-  ( N  e.  NN0  ->  F : W -1-1-onto-> B )
 
Theoremzzngim 19200 The  ZZ ring homomorphism is an isomorphism for 
N  =  0. (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 13-Jun-2019.)
 |-  Y  =  (ℤ/n `  0
 )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  L  e.  (ring GrpIso  Y )
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