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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubrgvr1 16301 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   =>    |-  ( ph  ->  X  =  (var1 `  H ) )
 
Theoremsubrgvr1cl 16302 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremcoe1z 16303 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  Y  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  (coe1 ` 
 .0.  )  =  (
 NN0  X.  { Y } ) )
 
Theoremcoe1add 16304 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .+b  G ) )  =  ( (coe1 `  F )  o F  .+  (coe1 `  G ) ) )
 
Theoremcoe1addfv 16305 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .+b  G ) ) `  X )  =  ( (
 (coe1 `
  F ) `  X )  .+  ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1subfv 16306 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  N  =  ( -g `  R )   =>    |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .-  G ) ) `
  X )  =  ( ( (coe1 `  F ) `  X ) N ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1mul2lem1 16307 An equivalence for coe1mul2 16309. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  ( ( A  e.  NN0  /\  X  e.  ( NN0  ^m 
 1o ) )  ->  ( X  o R  <_  ( 1o  X.  { A } )  <->  ( X `  (/) )  e.  ( 0
 ... A ) ) )
 
Theoremcoe1mul2lem2 16308* An equivalence for coe1mul2 16309. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  H  =  { d  e.  ( NN0  ^m  1o )  |  d  o R  <_  ( 1o  X.  { k } ) }   =>    |-  (
 k  e.  NN0  ->  ( c  e.  H  |->  ( c `  (/) ) ) : H -1-1-onto-> ( 0 ... k
 ) )
 
Theoremcoe1mul2 16309* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  .xb  =  ( .r `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  S )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremcoe1mul 16310* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  .xb  =  ( .r `  Y )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremply1tmcl 16311 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  B )
 
Theoremcoe1tm 16312* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  ( x  e.  NN0  |->  if ( x  =  D ,  C ,  .0.  )
 ) )
 
Theoremcoe1tmfv1 16313 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D )  =  C )
 
Theoremcoe1tmfv2 16314 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  F  e.  NN0 )   &    |-  ( ph  ->  D  =/=  F )   =>    |-  ( ph  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  F )  =  .0.  )
 
Theoremcoe1tmmul2 16315* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( A  .xb  ( C 
 .x.  ( D  .^  X ) ) ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 ( (coe1 `  A ) `  ( x  -  D ) )  .X.  C ) ,  .0.  ) ) )
 
Theoremcoe1tmmul 16316* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( C  .x.  ( D  .^  X ) )  .xb  A )
 )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  ( x  -  D ) ) ) ,  .0.  ) ) )
 
Theoremcoe1tmmul2fv 16317 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( A  .xb  ( C  .x.  ( D 
 .^  X ) ) ) ) `  ( D  +  Y )
 )  =  ( ( (coe1 `  A ) `  Y )  .X.  C ) )
 
Theoremcoe1pwmul 16318* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( D  .^  X )  .x.  A ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 (coe1 `
  A ) `  ( x  -  D ) ) ,  .0.  ) ) )
 
Theoremcoe1pwmulfv 16319 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( ( D 
 .^  X )  .x.  A ) ) `  ( D  +  Y )
 )  =  ( (coe1 `  A ) `  Y ) )
 
Theoremply1scltm 16320 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K ) 
 ->  ( A `  F )  =  ( F  .x.  ( 0  .^  X ) ) )
 
Theoremcoe1sclmul 16321 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( ( A `
  X )  .xb  Y ) )  =  ( ( NN0  X.  { X } )  o F  .x.  (coe1 `  Y ) ) )
 
Theoremcoe1sclmulfv 16322 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  K  /\  Y  e.  B ) 
 /\  .0.  e.  NN0 )  ->  ( (coe1 `  ( ( A `
  X )  .xb  Y ) ) `  .0.  )  =  ( X  .x.  ( (coe1 `  Y ) `  .0.  ) ) )
 
Theoremcoe1sclmul2 16323 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( Y  .xb  ( A `  X ) ) )  =  ( (coe1 `  Y )  o F  .x.  ( NN0  X. 
 { X } )
 ) )
 
Theoremply1sclf 16324 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K --> B )
 
Theoremcoe1scl 16325* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K ) 
 ->  (coe1 `  ( A `  X ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  X ,  .0.  ) ) )
 
Theoremply1sclid 16326 Recover the base scalar from a scalar polynomial.. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K )  ->  X  =  ( (coe1 `  ( A `  X ) ) `  0 ) )
 
Theoremply1sclf1 16327 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K -1-1-> B )
 
Theoremply1scl0 16328 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .0.  )  =  Y )
 
Theoremply1scln0 16329 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  X  =/=  .0.  )  ->  ( A `  X )  =/=  Y )
 
Theoremply1scl1 16330 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( 1r `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .1.  )  =  N )
 
Theoremply1coe 16331* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .s `  P )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   &    |-  A  =  (coe1 `  K )   &    |-  R  e.  _V   =>    |-  (
 ( R  e.  CRing  /\  K  e.  B ) 
 ->  K  =  ( P 
 gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
 .x.  ( k  .^  X ) ) ) ) )
 
10.11  The complex numbers as an extensible structure
 
10.11.1  Definition and basic properties
 
Syntaxcxmt 16332 Extend class notation with the class of all extended metric spaces.
 class  * Met
 
Syntaxcme 16333 Extend class notation with the class of all metrics.
 class  Met
 
Syntaxcbl 16334 Extend class notation with the metric space ball function.
 class  ball
 
Syntaxcmopn 16335 Extend class notation with a function mapping each metric space to the family of its open sets.
 class  MetOpen
 
Definitiondf-xmet 16336* Define the set of all extended metrics on a given base set. The definition is similar to df-met 16337, 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 16337* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 17849. 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 17871, metgt0 17886, metsym 17877, and mettri 17879. (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 16338* Define the metric space ball function. See blval 17911 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 16339 Define a function whose value is the family of open sets of a metric space. See elmopn 17951 for its main property. (Contributed by NM, 1-Sep-2006.)
 |-  MetOpen  =  ( d  e. 
 U. ran  * Met  |->  ( topGen `  ran  ( ball `  d ) ) )
 
Syntaxccnfld 16340 Extend class notation with the field of complex numbers.
 classfld
 
Definitiondf-cnfld 16341 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 16343, cnfldadd 16347, cnfldmul 16348, cnfldcj 16349, cnfldtset 16350, cnfldle 16351, cnfldds 16352, and cnfldbas 16346. 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 16342 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-fld Struct  <. 1 , ; 1 2 >.
 
Theoremcnfldex 16343 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 16344 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s Struct  <. 1 , ; 1 2 >.
 
Theoremxrsex 16345 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s  e.  _V
 
Theoremcnfldbas 16346 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 16347 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 16348 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 16349 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  *  =  ( * r ` fld )
 
Theoremcnfldtset 16350 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 16351 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 16352 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 16353 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR*  =  ( Base `  RR* s )
 
Theoremxrsadd 16354 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e  =  (
 +g  `  RR* s )
 
Theoremxrsmul 16355 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  x e  =  ( .r `  RR* s )
 
Theoremxrstset 16356 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (ordTop `  <_  )  =  (TopSet `  RR* s )
 
Theoremxrsle 16357 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 <_  =  ( le ` 
 RR* s )
 
Theoremcncrng 16358 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-fld  e.  CRing
 
Theoremcnrng 16359 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  Ring
 
Theoremxrsmcmn 16360 The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrs1mnd 16372.) (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (mulGrp `  RR* s )  e. CMnd
 
Theoremcnfld0 16361 The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  0  =  ( 0g
 ` fld
 )
 
Theoremcnfld1 16362 The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  1  =  ( 1r
 ` fld
 )
 
Theoremcnfldneg 16363 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 16364 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 +  =  ( + f ` fld )
 
Theoremcnfldsub 16365 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- 
 -  =  ( -g ` fld )
 
Theoremcndrng 16366 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-fld  e.  DivRing
 
Theoremcnflddiv 16367 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 16368 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 16369 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 16370 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 16371 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-fld  e.  *Ring
 
Theoremxrs1mnd 16372 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 16373 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  R  =  ( RR* ss  (
 RR*  \  {  -oo } )
 )   =>    |-  0  =  ( 0g
 `  R )
 
Theoremxrs1cmn 16374 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 16375 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 16376 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
 
Theoremxrsds 16377* 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 16378 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 16379 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 16380 Lemma for xrsdsreclb 16381. (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 16381 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 16382* Lemma for nn0subm 16390 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 16383* Lemma for resubdrg 16386 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 16384* Lemma for resubdrg 16386 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 16385* Lemma for resubdrg 16386 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 16386 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 16387 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 16388 The integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ  e.  (SubRing ` fld )
 
Theoremgzsubrg 16389 The gaussian integers form a subring of the complexes. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- 
 ZZ [ _i ]  e.  (SubRing ` fld )
 
Theoremnn0subm 16390 The nonnegative integers form a submonoid of the complexes. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |- 
 NN0  e.  (SubMnd ` fld )
 
Theoremrege0subm 16391 The nonnegative reals form a submonoid of the complexes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( 0 [,)  +oo )  e.  (SubMnd ` fld )
 
Theoremabsabv 16392 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 16393 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 16394 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 16395 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 16396 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 16397* Lemma for rpmsubg 16398 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 16398 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 16399 Lemma for gzrngunit 16400. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  Z  =  (flds  ZZ [ _i ]
 )   =>    |-  ( A  e.  (Unit `  Z )  ->  1  <_  ( abs `  A ) )
 
Theoremgzrngunit 16400 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 ) )
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