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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-met 16301* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 17813. 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 17835, metgt0 17850, metsym 17841, and mettri 17843. (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 16302* Define the metric space ball function. See blval 17875 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 16303 Define a function whose value is the family of open sets of a metric space. See elmopn 17915 for its main property. (Contributed by NM, 1-Sep-2006.)
 |-  MetOpen  =  ( d  e. 
 U. ran  * Met  |->  ( topGen `  ran  ( ball `  d ) ) )
 
Syntaxccnfld 16304 Extend class notation with the field of complex numbers.
 classfld
 
Definitiondf-cnfld 16305 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 16307, cnfldadd 16311, cnfldmul 16312, cnfldcj 16313, cnfldtset 16314, cnfldle 16315, cnfldds 16316, and cnfldbas 16310. 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 16306 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-fld Struct  <. 1 , ; 1 2 >.
 
Theoremcnfldex 16307 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 16308 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s Struct  <. 1 , ; 1 2 >.
 
Theoremxrsex 16309 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s  e.  _V
 
Theoremcnfldbas 16310 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 16311 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 16312 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 16313 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  *  =  ( * r ` fld )
 
Theoremcnfldtset 16314 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 16315 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 16316 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 16317 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR*  =  ( Base `  RR* s )
 
Theoremxrsadd 16318 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e  =  (
 +g  `  RR* s )
 
Theoremxrsmul 16319 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  x e  =  ( .r `  RR* s )
 
Theoremxrstset 16320 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (ordTop `  <_  )  =  (TopSet `  RR* s )
 
Theoremxrsle 16321 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 <_  =  ( le ` 
 RR* s )
 
Theoremcncrng 16322 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-fld  e.  CRing
 
Theoremcnrng 16323 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  Ring
 
Theoremxrsmcmn 16324 The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrs1mnd 16336.) (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (mulGrp `  RR* s )  e. CMnd
 
Theoremcnfld0 16325 The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  0  =  ( 0g
 ` fld
 )
 
Theoremcnfld1 16326 The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  1  =  ( 1r
 ` fld
 )
 
Theoremcnfldneg 16327 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 16328 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 +  =  ( + f ` fld )
 
Theoremcnfldsub 16329 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- 
 -  =  ( -g ` fld )
 
Theoremcndrng 16330 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  DivRing
 
Theoremcnflddiv 16331 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 16332 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 16333 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 16334 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 16335 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-fld  e.  *Ring
 
Theoremxrs1mnd 16336 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 16337 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  0  =  ( 0g
 `  R )
 
Theoremxrs1cmn 16338 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 16339 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 16340 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
 
Theoremxrsds 16341* 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 16342 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 16343 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 16344 Lemma for xrsdsreclb 16345. (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 16345 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 16346* Lemma for nn0subm 16354 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 16347* Lemma for resubdrg 16350 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 16348* Lemma for resubdrg 16350 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 16349* Lemma for resubdrg 16350 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 16350 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 16351 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 16352 The integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ  e.  (SubRing ` fld )
 
Theoremgzsubrg 16353 The gaussian integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ [ _i ]  e.  (SubRing ` fld )
 
Theoremnn0subm 16354 The nonnegative integers form a submonoid of the complexes. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |- 
 NN0  e.  (SubMnd ` fld )
 
Theoremrege0subm 16355 The nonnegative reals form a submonoid of the complexes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
 
Theoremabsabv 16356 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 16357 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 16358 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 16359 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 16360 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 16361* Lemma for rpmsubg 16362 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 16362 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 16363 Lemma for gzrngunit 16364. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ [ _i ]
 )   =>    |-  ( A  e.  (Unit `  Z )  ->  1  <_  ( abs `  A ) )
 
Theoremgzrngunit 16364 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 16365 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 16366* 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 16367 Ring divisibility in  ZZ corresponds to ordinary divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  Z  =  (flds  ZZ )   =>    |-  ||  =  ( ||r `  Z )
 
Theoremzlpirlem1 16368 Lemma for zlpir 16371. 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 16369 Lemma for zlpir 16371. 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 16370 Lemma for zlpir 16371. 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 16371 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 16372 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Z  =  (flds  ZZ )   =>    |-  Z  e. CycGrp
 
Theoremprmirredlem 16373 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 16374 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 16375 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 16376* 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 16377* 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 16378 Map the rationals into a field, or the integers into a ring.
 class  ZRHom
 
Syntaxczlm 16379 Augment an abelian group with vector space operations to turn it into a  ZZ-module.
 class  ZMod
 
Syntaxcchr 16380 Syntax for ring characteristic.
 class chr
 
Syntaxczn 16381 The ring of integers modulo  n.
 class ℤ/n
 
Definitiondf-zrh 16382 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 14419). (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  ZRHom  =  ( r  e.  _V  |->  U. (
 (flds  ZZ ) RingHom  r ) )
 
Definitiondf-zlm 16383 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 16384 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 16385* 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 16386* 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 16387* 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 16388* 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 16389 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 16390* 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 16391 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 16392 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 16393 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 16394 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 16395 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 16396* 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 16397 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 16398 Lemma for zlmbas 16399 and zlmplusg 16400. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  5   =>    |-  ( E `  G )  =  ( E `  W )
 
Theoremzlmbas 16399 Base set of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  B  =  (
 Base `  W )
 
Theoremzlmplusg 16400 Group operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  .+  =  ( +g  `  W )
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