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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-met 16701* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 18356. 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 18378, metgt0 18394, metsym 18385, and mettri 18387. (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 16702* Define the metric space ball function. See blval 18421 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |- 
 ball  =  ( d  e.  _V  |->  ( x  e. 
 dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  < 
 z } ) )
 
Definitiondf-mopn 16703 Define a function whose value is the family of open sets of a metric space. See elmopn 18477 for its main property. (Contributed by NM, 1-Sep-2006.)
 |-  MetOpen  =  ( d  e. 
 U. ran  * Met  |->  ( topGen `  ran  ( ball `  d ) ) )
 
Definitiondf-fbas 16704* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
 |- 
 fBas  =  ( w  e.  _V  |->  { x  e.  ~P ~P w  |  ( x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
 y  i^i  z )
 )  =/=  (/) ) }
 )
 
Definitiondf-fg 16705* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
 |-  filGen  =  ( w  e. 
 _V ,  x  e.  ( fBas `  w )  |->  { y  e.  ~P w  |  ( x  i^i  ~P y )  =/=  (/) } )
 
Definitiondf-metuOLD 16706* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (New usage is discouraged.)
 |- metUnifOLD  =  ( d  e.  U. ran  * Met  |->  ( ( dom 
 dom  d  X.  dom  dom  d ) filGen ran  (
 a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) ) ) )
 
Definitiondf-metu 16707* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |- metUnif  =  ( d  e.  U. ran PsMet 
 |->  ( ( dom  dom  d  X.  dom  dom  d )
 filGen ran  ( a  e.  RR+  |->  ( `' d " ( 0 [,) a
 ) ) ) ) )
 
Syntaxccnfld 16708 Extend class notation with the field of complex numbers.
 classfld
 
Definitiondf-cnfld 16709 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 16711, cnfldadd 16713, cnfldmul 16714, cnfldcj 16715, cnfldtset 16716, cnfldle 16717, cnfldds 16718, and cnfldbas 16712. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (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. 
 -  ) >. }  u.  {
 <. ( UnifSet `  ndx ) ,  (metUnif `  ( abs  o. 
 -  ) ) >. } ) )
 
Theoremcnfldstr 16710 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |-fld Struct  <. 1 , ; 1 3 >.
 
Theoremcnfldex 16711 The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |-fld  e.  _V
 
Theoremcnfldbas 16712 The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |- 
 CC  =  ( Base ` fld )
 
Theoremcnfldadd 16713 The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |- 
 +  =  ( +g  ` fld )
 
Theoremcnfldmul 16714 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |- 
 x.  =  ( .r
 ` fld
 )
 
Theoremcnfldcj 16715 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |-  *  =  ( * r ` fld )
 
Theoremcnfldtset 16716 The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |-  ( MetOpen `  ( abs  o. 
 -  ) )  =  (TopSet ` fld )
 
Theoremcnfldle 16717 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.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |- 
 <_  =  ( le ` fld )
 
Theoremcnfldds 16718 The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
 |-  ( abs  o.  -  )  =  ( dist ` fld )
 
Theoremcnfldunif 16719 The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |-  (metUnif `  ( abs  o. 
 -  ) )  =  ( UnifSet ` fld )
 
Theoremxrsstr 16720 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s Struct  <. 1 , ; 1 2 >.
 
Theoremxrsex 16721 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s  e.  _V
 
Theoremxrsbas 16722 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR*  =  ( Base `  RR* s )
 
Theoremxrsadd 16723 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e  =  (
 +g  `  RR* s )
 
Theoremxrsmul 16724 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  x e  =  ( .r `  RR* s )
 
Theoremxrstset 16725 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (ordTop `  <_  )  =  (TopSet `  RR* s )
 
Theoremxrsle 16726 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 <_  =  ( le ` 
 RR* s )
 
Theoremcncrng 16727 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-fld  e.  CRing
 
Theoremcnrng 16728 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  Ring
 
Theoremxrsmcmn 16729 The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrs1mnd 16741.) (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (mulGrp `  RR* s )  e. CMnd
 
Theoremcnfld0 16730 The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  0  =  ( 0g
 ` fld
 )
 
Theoremcnfld1 16731 The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  1  =  ( 1r
 ` fld
 )
 
Theoremcnfldneg 16732 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 16733 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 +  =  ( + f ` fld )
 
Theoremcnfldsub 16734 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- 
 -  =  ( -g ` fld )
 
Theoremcndrng 16735 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  DivRing
 
Theoremcnflddiv 16736 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 16737 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 16738 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 16739 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 16740 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-fld  e.  *Ring
 
Theoremxrs1mnd 16741 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 16742 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  0  =  ( 0g
 `  R )
 
Theoremxrs1cmn 16743 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 16744 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 16745 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
 
Theoremxrsds 16746* 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 16747 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 16748 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 16749 Lemma for xrsdsreclb 16750. (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 16750 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 16751* Lemma for nn0subm 16759 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 16752* Lemma for resubdrg 16755 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 16753* Lemma for resubdrg 16755 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 16754* Lemma for resubdrg 16755 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 16755 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 16756 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 16757 The integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ  e.  (SubRing ` fld )
 
Theoremgzsubrg 16758 The gaussian integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ [ _i ]  e.  (SubRing ` fld )
 
Theoremnn0subm 16759 The nonnegative integers form a submonoid of the complexes. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |- 
 NN0  e.  (SubMnd ` fld )
 
Theoremrege0subm 16760 The nonnegative reals form a submonoid of the complexes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
 
Theoremabsabv 16761 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 16762 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 16763 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 16764 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 16765 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 16766* Lemma for rpmsubg 16767 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 16767 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 16768 Lemma for gzrngunit 16769. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ [ _i ]
 )   =>    |-  ( A  e.  (Unit `  Z )  ->  1  <_  ( abs `  A ) )
 
Theoremgzrngunit 16769 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 16770 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 16771* 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 16772 Ring divisibility in  ZZ corresponds to ordinary divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  Z  =  (flds  ZZ )   =>    |-  ||  =  ( ||r `  Z )
 
Theoremzlpirlem1 16773 Lemma for zlpir 16776. 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 16774 Lemma for zlpir 16776. A nonzero ideal of integers contains the 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 16775 Lemma for zlpir 16776. 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 16776 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 16777 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Z  =  (flds  ZZ )   =>    |-  Z  e. CycGrp
 
Theoremprmirredlem 16778 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 16779 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 16780 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 16781* 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 16782* 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 16783 Map the rationals into a field, or the integers into a ring.
 class  ZRHom
 
Syntaxczlm 16784 Augment an abelian group with vector space operations to turn it into a  ZZ-module.
 class  ZMod
 
Syntaxcchr 16785 Syntax for ring characteristic.
 class chr
 
Syntaxczn 16786 The ring of integers modulo  n.
 class ℤ/n
 
Definitiondf-zrh 16787 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 14820). (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  ZRHom  =  ( r  e.  _V  |->  U. (
 (flds  ZZ ) RingHom  r ) )
 
Definitiondf-zlm 16788 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 16789 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 16790* 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 16791* 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 16792* 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 16793* 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 16794 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 16795* 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 16796 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 16797 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 16798 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 16799 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 16800 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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