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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
10.11  The complex numbers as an extensible structure
 
10.11.1  Definition and basic properties
 
Syntaxcxmt 16201 Extend class notation with the class of all extended metric spaces.
 class  * Met
 
Syntaxcme 16202 Extend class notation with the class of all metrics.
 class  Met
 
Syntaxcbl 16203 Extend class notation with the metric space ball function.
 class  ball
 
Syntaxcmopn 16204 Extend class notation with a function mapping each metric space to the family of its open sets.
 class  MetOpen
 
Definitiondf-xmet 16205* Define the set of all extended metrics on a given base set. The definition is similar to df-met 16206, but we also allow the metric to take on the value  +oo. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |- 
 * Met  =  ( x  e.  _V  |->  { d  e.  ( RR*  ^m  ( x  X.  x ) )  |  A. y  e.  x  A. z  e.  x  ( ( ( y d z )  =  0  <->  y  =  z
 )  /\  A. w  e.  x  ( y d z )  <_  (
 ( w d y ) + e ( w d z ) ) ) } )
 
Definitiondf-met 16206* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 17718. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 17740, metgt0 17755, metsym 17746, and mettri 17748. (Contributed by NM, 25-Aug-2006.)
 |- 
 Met  =  ( x  e.  _V  |->  { d  e.  ( RR  ^m  ( x  X.  x ) )  | 
 A. y  e.  x  A. z  e.  x  ( ( ( y d z )  =  0  <-> 
 y  =  z ) 
 /\  A. w  e.  x  ( y d z )  <_  ( ( w d y )  +  ( w d z ) ) ) } )
 
Definitiondf-bl 16207* Define the metric space ball function. See blval 17780 for its value. (Contributed by NM, 30-Aug-2006.)
 |- 
 ball  =  ( d  e.  U. ran  * Met  |->  ( x  e.  dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  < 
 z } ) )
 
Definitiondf-mopn 16208 Define a function whose value is the family of open sets of a metric space. See elmopn 17820 for its main property. (Contributed by NM, 1-Sep-2006.)
 |-  MetOpen  =  ( d  e. 
 U. ran  * Met  |->  ( topGen `  ran  ( ball `  d ) ) )
 
Syntaxccnfld 16209 Extend class notation with the field of complex numbers.
 classfld
 
Definitiondf-cnfld 16210 The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator, for instance  (fld  |`  AA ) is the field of algebraic numbers.

The contract of this set is defined entirely by cnfldex 16212, cnfldadd 16216, cnfldmul 16217, cnfldcj 16218, cnfldtset 16219, cnfldle 16220, cnfldds 16221, and cnfldbas 16215. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (New usage is discouraged.)

 |-fld  =  ( ( { <. (
 Base `  ndx ) ,  CC >. ,  <. ( +g  ` 
 ndx ) ,  +  >. ,  <. ( .r `  ndx ) ,  x.  >. }  u.  { <. ( * r `  ndx ) ,  * >. } )  u. 
 { <. (TopSet `  ndx ) ,  ( MetOpen `  ( abs  o. 
 -  ) ) >. , 
 <. ( le `  ndx ) ,  <_  >. ,  <. (
 dist `  ndx ) ,  ( abs  o.  -  ) >. } )
 
Theoremcnfldstr 16211 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-fld Struct  <. 1 , ; 1 2 >.
 
Theoremcnfldex 16212 The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-fld  e.  _V
 
Theoremxrsstr 16213 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s Struct  <. 1 , ; 1 2 >.
 
Theoremxrsex 16214 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s  e.  _V
 
Theoremcnfldbas 16215 The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |- 
 CC  =  ( Base ` fld )
 
Theoremcnfldadd 16216 The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |- 
 +  =  ( +g  ` fld )
 
Theoremcnfldmul 16217 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |- 
 x.  =  ( .r
 ` fld
 )
 
Theoremcnfldcj 16218 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  *  =  ( * r ` fld )
 
Theoremcnfldtset 16219 The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |-  ( MetOpen `  ( abs  o. 
 -  ) )  =  (TopSet ` fld )
 
Theoremcnfldle 16220 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  (flds  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.)
 |- 
 <_  =  ( le ` fld )
 
Theoremcnfldds 16221 The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |-  ( abs  o.  -  )  =  ( dist ` fld )
 
Theoremxrsbas 16222 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR*  =  ( Base `  RR* s )
 
Theoremxrsadd 16223 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e  =  (
 +g  `  RR* s )
 
Theoremxrsmul 16224 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  x e  =  ( .r `  RR* s )
 
Theoremxrstset 16225 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (ordTop `  <_  )  =  (TopSet `  RR* s )
 
Theoremxrsle 16226 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 <_  =  ( le ` 
 RR* s )
 
Theoremcncrng 16227 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-fld  e.  CRing
 
Theoremcnrng 16228 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  Ring
 
Theoremxrsmcmn 16229 The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrs1mnd 16241.) (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (mulGrp `  RR* s )  e. CMnd
 
Theoremcnfld0 16230 The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  0  =  ( 0g
 ` fld
 )
 
Theoremcnfld1 16231 The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  1  =  ( 1r
 ` fld
 )
 
Theoremcnfldneg 16232 The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  ( X  e.  CC  ->  ( ( inv g ` fld ) `  X )  =  -u X )
 
Theoremcnfldplusf 16233 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 +  =  ( + f ` fld )
 
Theoremcnfldsub 16234 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- 
 -  =  ( -g ` fld )
 
Theoremcndrng 16235 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  DivRing
 
Theoremcnflddiv 16236 The division operation in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |- 
 /  =  (/r ` fld )
 
Theoremcnfldinv 16237 The multiplicative inverse in the field of complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  ( ( X  e.  CC  /\  X  =/=  0
 )  ->  ( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
 
Theoremcnfldmulg 16238 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 ` fld ) B )  =  ( A  x.  B ) )
 
Theoremcnfldexp 16239 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 16240 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-fld  e.  *Ring
 
Theoremxrs1mnd 16241 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 16242 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  0  =  ( 0g
 `  R )
 
Theoremxrs1cmn 16243 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 16244 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 16245 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
 
Theoremxrsds 16246* 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 16247 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 16248 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 16249 Lemma for xrsdsreclb 16250. (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 16250 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 16251* Lemma for nn0subm 16259 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 16252* Lemma for resubdrg 16255 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 16253* Lemma for resubdrg 16255 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 16254* Lemma for resubdrg 16255 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 16255 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 16256 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 16257 The integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ  e.  (SubRing ` fld )
 
Theoremgzsubrg 16258 The gaussian integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ [ _i ]  e.  (SubRing ` fld )
 
Theoremnn0subm 16259 The nonnegative integers form a submonoid of the complexes. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |- 
 NN0  e.  (SubMnd ` fld )
 
Theoremrege0subm 16260 The nonnegative reals form a submonoid of the complexes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
 
Theoremabsabv 16261 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 16262 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 16263 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 16264 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 16265 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 16266* Lemma for rpmsubg 16267 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 16267 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 16268 Lemma for gzrngunit 16269. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ [ _i ]
 )   =>    |-  ( A  e.  (Unit `  Z )  ->  1  <_  ( abs `  A ) )
 
Theoremgzrngunit 16269 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 16270 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 16271* 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 16272 Ring divisibility in  ZZ corresponds to ordinary divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  Z  =  (flds  ZZ )   =>    |-  ||  =  ( ||r `  Z )
 
Theoremzlpirlem1 16273 Lemma for zlpir 16276. 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 16274 Lemma for zlpir 16276. 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 16275 Lemma for zlpir 16276. 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 16276 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 16277 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Z  =  (flds  ZZ )   =>    |-  Z  e. CycGrp
 
Theoremprmirredlem 16278 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 ) )
 
Theoremdfprime2 16279 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 16280 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 16281* 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 16282* 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 16283 Map the rationals into a field, or the integers into a ring.
 class  ZRHom
 
Syntaxczlm 16284 Augment an abelian group with vector space operations to turn it into a  ZZ-module.
 class  ZMod
 
Syntaxcchr 16285 Syntax for ring characteristic.
 class chr
 
Syntaxczn 16286 The ring of integers modulo  n.
 class ℤ/n
 
Definitiondf-zrh 16287 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 14327). (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  ZRHom  =  ( r  e.  _V  |->  U. (
 (flds  ZZ ) RingHom  r ) )
 
Definitiondf-zlm 16288 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 16289 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 16290* 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 16291* 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 16292* 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 16293* 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 16294 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 16295* 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 16296 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 16297 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 16298 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 16299 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 16300 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.  )
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