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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnfldexp 16401 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 ` fld ) ) A )  =  ( A ^ B ) )
 
Theoremcnsrng 16402 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-fld  e.  *Ring
 
Theoremxrs1mnd 16403 The extended real numbers, restricted to  RR*  \  {  -oo }, form a monoid. The full structure is not a monoid or even a semigroup because associativity fails for  1  +  ( 
-oo  +  +oo )  =  1  =/=  ( 1  +  -oo )  + 
+oo  =  0. (Contributed by Mario Carneiro, 27-Nov-2014.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  R  e.  Mnd
 
Theoremxrs10 16404 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  0  =  ( 0g
 `  R )
 
Theoremxrs1cmn 16405 The extended real numbers restricted to  RR*  \  {  -oo } form a commutative monoid. They are not a group because  1  +  +oo  =  2  + 
+oo even though  1  =/=  2. (Contributed by Mario Carneiro, 27-Nov-2014.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  R  e. CMnd
 
Theoremxrge0subm 16406 The nonnegative extended real numbers are a submonoid of the non-negative-infinite extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  ( 0 [,]  +oo )  e.  (SubMnd `  R )
 
Theoremxrge0cmn 16407 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
 
Theoremxrsds 16408* The metric of the extended real number structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  D  =  ( x  e.  RR* ,  y  e.  RR*  |->  if ( x  <_  y ,  ( y + e  - e x ) ,  ( x + e  - e
 y ) ) )
 
Theoremxrsdsval 16409 The metric of the extended real number structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A D B )  =  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) ) )
 
Theoremxrsdsreval 16410 The metric of the extended real number structure coincides with the real number metric on the reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B ) ) )
 
Theoremxrsdsreclblem 16411 Lemma for xrsdsreclb 16412. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  A  <_  B )  ->  ( ( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR )
 ) )
 
Theoremxrsdsreclb 16412 The metric of the extended real number structure is only real when both arguments are real. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  D  =  ( dist ` 
 RR* s )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B ) 
 ->  ( ( A D B )  e.  RR  <->  ( A  e.  RR  /\  B  e.  RR ) ) )
 
Theoremcnsubmlem 16413* Lemma for nn0subm 16421 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 ` fld )
 
Theoremcnsubglem 16414* Lemma for resubdrg 16417 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 ` fld )
 
Theoremcnsubrglem 16415* Lemma for resubdrg 16417 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 ` fld )
 
Theoremcnsubdrglem 16416* Lemma for resubdrg 16417 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 )   &    |-  ( ( x  e.  A  /\  x  =/=  0 )  ->  (
 1  /  x )  e.  A )   =>    |-  ( A  e.  (SubRing ` fld ) 
 /\  (flds  A )  e.  DivRing )
 
Theoremresubdrg 16417 The real numbers form a division subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  ( RR  e.  (SubRing ` fld ) 
 /\  (flds  RR )  e.  DivRing )
 
Theoremqsubdrg 16418 The rational numbers form a division subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  ( QQ  e.  (SubRing ` fld ) 
 /\  (flds  QQ )  e.  DivRing )
 
Theoremzsubrg 16419 The integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ  e.  (SubRing ` fld )
 
Theoremgzsubrg 16420 The gaussian integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ [ _i ]  e.  (SubRing ` fld )
 
Theoremnn0subm 16421 The nonnegative integers form a submonoid of the complexes. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |- 
 NN0  e.  (SubMnd ` fld )
 
Theoremrege0subm 16422 The nonnegative reals form a submonoid of the complexes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
 
Theoremabsabv 16423 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 16424 The integers are a subset of any subring of the complexes. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( R  e.  (SubRing ` fld ) 
 ->  ZZ  C_  R )
 
Theoremqsssubdrg 16425 The rational numbers are a subset of any subfield of the complexes. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( R  e.  (SubRing ` fld )  /\  (flds  R )  e.  DivRing ) 
 ->  QQ  C_  R )
 
Theoremcnsubrg 16426 There are no subrings of the complexes strictly between  RR and  CC. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  R  e.  { RR ,  CC }
 )
 
Theoremcnmgpabl 16427 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 16428* Lemma for rpmsubg 16429 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 16429 The positive reals form a multiplicative subgroup of the complexes. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )   =>    |-  RR+  e.  (SubGrp `  M )
 
Theoremgzrngunitlem 16430 Lemma for gzrngunit 16431. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ [ _i ]
 )   =>    |-  ( A  e.  (Unit `  Z )  ->  1  <_  ( abs `  A ) )
 
Theoremgzrngunit 16431 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 ) )
 
Theoremzrngunit 16432 The units of  ZZ are the integers with norm  1, i.e.  1 and  -u 1. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  Z  =  (flds  ZZ )   =>    |-  ( A  e.  (Unit `  Z )  <->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 )
 )
 
Theoremgsumfsum 16433* Relate a group sum on ℂfld to a finite sum on the complexes. (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 )
 
Theoremdvdsrz 16434 Ring divisibility in  ZZ corresponds to ordinary divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  Z  =  (flds  ZZ )   =>    |-  ||  =  ( ||r `  Z )
 
Theoremzlpirlem1 16435 Lemma for zlpir 16438. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  ( ph  ->  I  e.  (LIdeal `  Z ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   =>    |-  ( ph  ->  ( I  i^i  NN )  =/=  (/) )
 
Theoremzlpirlem2 16436 Lemma for zlpir 16438. A nonzero ideal of integers contains a least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  ( ph  ->  I  e.  (LIdeal `  Z ) )   &    |-  ( ph  ->  I  =/=  { 0 } )   &    |-  G  =  sup ( ( I  i^i  NN ) ,  RR ,  `'  <  )   =>    |-  ( ph  ->  G  e.  I )
 
Theoremzlpirlem3 16437 Lemma for zlpir 16438. All elements of a nonzero ideal of integers are divided by the least one. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  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 16438 The integers are a principal ideal ring but not a field. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  Z  =  (flds  ZZ )   =>    |-  Z  e. LPIR
 
Theoremzcyg 16439 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Z  =  (flds  ZZ )   =>    |-  Z  e. CycGrp
 
Theoremprmirredlem 16440 A natural number is irreducible over  ZZ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  Z  =  (flds  ZZ )   &    |-  I  =  (Irred `  Z )   =>    |-  ( A  e.  NN  ->  ( A  e.  I  <->  A  e.  Prime ) )
 
Theoremdfprm2 16441 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.)
 |-  Z  =  (flds  ZZ )   &    |-  I  =  (Irred `  Z )   =>    |- 
 Prime  =  ( NN  i^i  I )
 
Theoremprmirred 16442 The irreducible elements of  ZZ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  Z  =  (flds  ZZ )   &    |-  I  =  (Irred `  Z )   =>    |-  ( A  e.  I  <->  ( A  e.  ZZ  /\  ( abs `  A )  e.  Prime ) )
 
Theoremexpmhm 16443* 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 ) )
 
Theoremexpghm 16444* Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  M  =  (mulGrp ` fld )   &    |-  U  =  ( Ms  ( CC  \  { 0 } ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  ( x  e.  ZZ  |->  ( A ^ x ) )  e.  ( Z  GrpHom  U ) )
 
10.11.2  Algebraic constructions based on the complexes
 
Syntaxczrh 16445 Map the rationals into a field, or the integers into a ring.
 class  ZRHom
 
Syntaxczlm 16446 Augment an abelian group with vector space operations to turn it into a  ZZ-module.
 class  ZMod
 
Syntaxcchr 16447 Syntax for ring characteristic.
 class chr
 
Syntaxczn 16448 The ring of integers modulo  n.
 class ℤ/n
 
Definitiondf-zrh 16449 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 14486). (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  ZRHom  =  ( r  e.  _V  |->  U. (
 (flds  ZZ ) RingHom  r ) )
 
Definitiondf-zlm 16450 Augment an abelian group with vector space operations to turn it into a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ZMod  =  ( g  e.  _V  |->  ( ( g sSet  <. (Scalar `  ndx ) ,  (flds  ZZ ) >. ) sSet  <. ( .s
 `  ndx ) ,  (.g `  g ) >. ) )
 
Definitiondf-chr 16451 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 16452* 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.)
 |- ℤ/n =  ( n  e.  NN0  |->  [_ (flds  ZZ )  /  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
 ) >. ) )
 
Theoremmulgghm2 16453* The powers of a group element give a homomorphism from  ZZ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  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 ) )
 
Theoremmulgrhm 16454* The powers of the element  1 give a ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R ) )
 
Theoremmulgrhm2 16455* The powers of the element  1 give the unique ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
 
Theoremzrhval 16456 Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   =>    |-  L  =  U. ( Z RingHom  R )
 
Theoremzrhval2 16457* Alternate value of the  ZRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  L  =  ( n  e. 
 ZZ  |->  ( n  .x.  .1.  ) ) )
 
Theoremzrhmulg 16458 Value of the  ZRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  N  e.  ZZ )  ->  ( L `  N )  =  ( N  .x.  .1.  ) )
 
Theoremzrhrhmb 16459 The  ZRHom homomorphism is the unique ring homomorphism from  Z. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( R  e.  Ring  ->  ( F  e.  ( Z RingHom  R )  <->  F  =  L ) )
 
Theoremzrhrhm 16460 The  ZRHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( R  e.  Ring  ->  L  e.  ( Z RingHom  R )
 )
 
Theoremzrh1 16461 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 16462 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 16463* 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 16464 Augment an abelian group with vector space operations to turn it into a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  G )   =>    |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) sSet  <. ( .s
 `  ndx ) ,  .x.  >.
 ) )
 
Theoremzlmlem 16465 Lemma for zlmbas 16466 and zlmplusg 16467. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  5   =>    |-  ( E `  G )  =  ( E `  W )
 
Theoremzlmbas 16466 Base set of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  B  =  (
 Base `  W )
 
Theoremzlmplusg 16467 Group operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  .+  =  ( +g  `  W )
 
Theoremzlmmulr 16468 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 16469 Scalar ring of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  Z  =  (flds  ZZ )   =>    |-  ( G  e.  V  ->  Z  =  (Scalar `  W ) )
 
Theoremzlmvsca 16470 Scalar multiplication operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .x.  =  (.g `  G )   =>    |- 
 .x.  =  ( .s `  W )
 
Theoremzlmlmod 16471 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 16472 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 16473 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 16474 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   =>    |-  ( R  e.  Ring  ->  C  e.  NN0 )
 
Theoremchrid 16475 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 16476 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 16477 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 16478 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 16479 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  (chr `  S )  ||  (chr `  R ) )
 
Theoremdomnchr 16480 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 16481 The set  n ZZ is an ideal in  ZZ. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   =>    |-  ( N  e.  ZZ  ->  ( S `  { N } )  e.  (LIdeal `  Z ) )
 
Theoremzncrng2 16482 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.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   =>    |-  ( N  e.  ZZ  ->  U  e.  CRing )
 
Theoremznval 16483 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.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~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 16484 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.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~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 16485 Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <.
 ( le `  ndx ) ,  .<_  >. ) )
 
Theoremznbaslem 16486 Lemma for znbas 16491. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  E  = Slot  K   &    |-  K  e.  NN   &    |-  K  <  10   =>    |-  ( N  e.  NN0  ->  ( E `  U )  =  ( E `  Y ) )
 
Theoremznbas2 16487 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( Base `  U )  =  ( Base `  Y )
 )
 
Theoremznadd 16488 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( +g  `  U )  =  ( +g  `  Y ) )
 
Theoremznmul 16489 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( .r `  U )  =  ( .r `  Y ) )
 
Theoremznzrh 16490 The  ZZ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( ZRHom `  U )  =  ( ZRHom `  Y ) )
 
Theoremznbas 16491 The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  Y  =  (ℤ/n `  N )   &    |-  R  =  ( Z ~QG  ( S `  { N } ) )   =>    |-  ( N  e.  NN0 
 ->  ( ZZ /. R )  =  ( Base `  Y ) )
 
Theoremzncrng 16492 ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e.  CRing )
 
Theoremznzrh2 16493* The  ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  .~  =  ( Z ~QG  ( S `  { N } ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L  =  ( x  e.  ZZ  |->  [ x ]  .~  )
 )
 
Theoremznzrhval 16494 The  ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  .~  =  ( Z ~QG  ( S `  { N } ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  (
 ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A )  =  [ A ]  .~  )
 
Theoremznzrhfo 16495 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 16496 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 16497 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 16498 Special case of zndvds 16497 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 16499 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.)
 |-  Z  =  (flds  ZZ )   &    |-  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 16500 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.)
 |-  Y  =  (ℤ/n `  0
 )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  L  e.  ( (flds  ZZ ) GrpIso  Y )
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