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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnre2csqima 24301* Image of a centered square by the canonical bijection from  ( RR  X.  RR ) to  CC. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y ) ) )   =>    |-  ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR 
 X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( (
 ( ( 1st `  X )  -  D ) (,) ( ( 1st `  X )  +  D )
 )  X.  ( (
 ( 2nd `  X )  -  D ) (,) (
 ( 2nd `  X )  +  D ) ) ) 
 ->  ( ( abs `  ( Re `  ( ( F `
  Y )  -  ( F `  X ) ) ) )  <  D  /\  ( abs `  ( Im `  ( ( F `
  Y )  -  ( F `  X ) ) ) )  <  D ) ) )
 
Theoremtpr2rico 24302* For any point of an open set of the usual topology on  ( RR  X.  RR ) there is an opened square which contains that point and is entirely in the open set. This is square is actually a ball by the  ( l ^  +oo ) norm  X. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  G  =  ( u  e.  RR ,  v  e.  RR  |->  ( u  +  ( _i  x.  v ) ) )   &    |-  B  =  ran  ( x  e.  ran  (,)
 ,  y  e.  ran  (,)  |->  ( x  X.  y
 ) )   =>    |-  ( ( A  e.  ( J  tX  J ) 
 /\  X  e.  A )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
 
19.3.8.5  Order topology - misc. additions
 
Theoremcnvordtrestixx 24303* The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  A  C_  RR*   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x [,] y )  C_  A )   =>    |-  ( (ordTop `  <_  )t  A )  =  (ordTop `  ( `'  <_  i^i  ( A  X.  A ) ) )
 
19.3.8.6  Continuity in topological spaces - misc. additions
 
Theoremmndpluscn 24304* A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
 |-  F  e.  ( J  Homeo  K )   &    |-  .+ 
 : ( B  X.  B ) --> B   &    |-  .*  : ( C  X.  C ) --> C   &    |-  J  e.  (TopOn `  B )   &    |-  K  e.  (TopOn `  C )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .*  ( F `  y ) ) )   &    |-  .+  e.  ( ( J 
 tX  J )  Cn  J )   =>    |- 
 .*  e.  ( ( K  tX  K )  Cn  K )
 
Theoremmhmhmeotmd 24305 Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
 |-  F  e.  ( S MndHom  T )   &    |-  F  e.  ( ( TopOpen `  S )  Homeo  ( TopOpen `  T ) )   &    |-  S  e. TopMnd   &    |-  T  e.  TopSp   =>    |-  T  e. TopMnd
 
Theoremrmulccn 24306* Multiplication by a real constant is a continuous function (Contributed by Thierry Arnoux, 23-May-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( x  e.  RR  |->  ( x  x.  C ) )  e.  ( J  Cn  J ) )
 
Theoremraddcn 24307* Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y ) )  e.  ( ( J  tX  J )  Cn  J )
 
Theoremxrmulc1cn 24308* The operation multiplying an extended real number by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  J  =  (ordTop `  <_  )   &    |-  F  =  ( x  e.  RR*  |->  ( x x e C ) )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  F  e.  ( J  Cn  J ) )
 
Theoremfmcncfil 24309 The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  ( MetOpen `  E )   =>    |-  (
 ( ( D  e.  ( CMet `  X )  /\  E  e.  ( * Met `  Y )  /\  F  e.  ( J  Cn  K ) )  /\  B  e.  (CauFil `  D ) )  ->  ( ( Y  FilMap  F ) `  B )  e.  (CauFil `  E ) )
 
19.3.8.7  Topology of the extended non-negative real numbers monoid
 
Theoremxrge0hmph 24310 The extended non-negative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  II  ~=  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
 )
 
Theoremxrge0iifcnv 24311* Define a bijection from  [ 0 ,  1 ] to  [ 0 , 
+oo ]. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   =>    |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  (
 y  e.  ( 0 [,]  +oo )  |->  if (
 y  =  +oo , 
 0 ,  ( exp `  -u y ) ) ) )
 
Theoremxrge0iifcv 24312* The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   =>    |-  ( X  e.  (
 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X ) )
 
Theoremxrge0iifiso 24313* The defined bijection from the closed unit interval and the extended non-negative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   =>    |-  F  Isom  <  ,  `'  <  ( ( 0 [,] 1 ) ,  (
 0 [,]  +oo ) )
 
Theoremxrge0iifhmeo 24314* Expose a homeomorphism from the closed unit interval and the extended non-negative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )   =>    |-  F  e.  ( II  Homeo  J )
 
Theoremxrge0iifhom 24315* The defined function from the closed unit interval and the extended non-negative reals is also a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )   =>    |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1
 ) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) + e
 ( F `  Y ) ) )
 
Theoremxrge0iif1 24316* Condition for the defined function,  -u ( log `  x
) to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )   =>    |-  ( F `  1 )  =  0
 
Theoremxrge0iifmhm 24317* The defined function from the closed unit interval and the extended non-negative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )   =>    |-  F  e.  (
 ( (mulGrp ` fld )s  ( 0 [,] 1
 ) ) MndHom  ( RR* ss  ( 0 [,]  +oo )
 ) )
 
Theoremxrge0pluscn 24318* The addition operation of the extended non-negative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) ) )   &    |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )   &    |-  .+  =  ( + e  |`  ( ( 0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) )   =>    |-  .+  e.  (
 ( J  tX  J )  Cn  J )
 
Theoremxrge0mulc1cn 24319* The operation multiplying a non-negative real numbers by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
 )   &    |-  F  =  ( x  e.  ( 0 [,]  +oo )  |->  ( x x e C ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  J ) )
 
Theoremxrge0tps 24320 The extended non-negative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e.  TopSp
 
Theoremxrge0topn 24321 The topology of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
 |-  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
 )
 
Theoremxrge0haus 24322 The topology of the extended non-negative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
 |-  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )  e. 
 Haus
 
Theoremxrge0tmdOLD 24323 The extended non-negative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. TopMnd
 
Theoremxrge0tmd 24324 The extended non-negative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof Shortened by Thierry Arnoux, 21-Jun-2017.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. TopMnd
 
19.3.8.8  Limits - misc additions
 
Theoremlmlim 24325 Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on  CC on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.)
 |-  J  e.  (TopOn `  Y )   &    |-  ( ph  ->  F : NN --> X )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( Jt  X )  =  (
 ( TopOpen ` fld )t  X )   &    |-  X  C_  CC   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  F  ~~>  P ) )
 
Theoremlmlimxrge0 24326 Relate a limit in the non-negative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.)
 |-  J  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  ( ph  ->  P  e.  X )   &    |-  X  C_  ( 0 [,)  +oo )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  F  ~~>  P ) )
 
Theoremrge0scvg 24327 Implication of convergence for a non-negative series. This could be used to shorten prmreclem6 13281 (Contributed by Thierry Arnoux, 28-Jul-2017.)
 |-  (
 ( F : NN --> ( 0 [,)  +oo )  /\  seq  1 (  +  ,  F )  e.  dom  ~~>  )  ->  sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )
 
Theorempnfneige0 24328* A neighborhood of  +oo contains an unbounded interval based at a real number. See pnfnei 17276 (Contributed by Thierry Arnoux, 31-Jul-2017.)
 |-  J  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )   =>    |-  ( ( A  e.  J  /\  +oo  e.  A )  ->  E. x  e.  RR  ( x (,]  +oo )  C_  A )
 
Theoremlmxrge0 24329* Express "sequence  F converges to plus infinity" (i.e. diverges), for a sequence of non-negative extended real numbers. (Contributed by Thierry Arnoux, 2-Aug-2017.)
 |-  J  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )   &    |-  ( ph  ->  F : NN
 --> ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k
 )  =  A )   =>    |-  ( ph  ->  ( F (
 ~~> t `  J ) 
 +oo 
 <-> 
 A. x  e.  RR  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) x  <  A ) )
 
Theoremlmdvg 24330* If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.)
 |-  ( ph  ->  F : NN --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   &    |-  ( ph  ->  -.  F  e.  dom  ~~>  )   =>    |-  ( ph  ->  A. x  e.  RR  E. j  e. 
 NN  A. k  e.  ( ZZ>=
 `  j ) x  <  ( F `  k ) )
 
Theoremlmdvglim 24331* If a monotonic real number sequence 
F diverges, it converges in the extended real numbers and its limit is plus infinity. (Contributed by Thierry Arnoux, 3-Aug-2017.)
 |-  J  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )   &    |-  ( ph  ->  F : NN
 --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   &    |-  ( ph  ->  -.  F  e.  dom  ~~>  )   =>    |-  ( ph  ->  F (
 ~~> t `  J ) 
 +oo )
 
19.3.9  Uniform Stuctures and Spaces
 
19.3.9.1  Hausdorff Completion
 
Syntaxchcmp 24332 Extend class notation with the Hausdorff completion relation.
 class HCmp
 
Definitiondf-hcmp 24333* Definition of the Hausdorff completion. In this definition, a structure  w is a Hausdorff completion of a uniform structure  u if  w is a complete uniform space, in which  u is dense, and which admits the same uniform structure. Theorem 3 of [BourbakiTop1] p. II.21. states the existence and unicity of such a completion. (Contributed by Thierry Arnoux, 5-Mar-2018.)
 |- HCmp  =  { <. u ,  w >.  |  ( ( u  e. 
 U. ran UnifOn  /\  w  e. CUnifSp )  /\  (UnifSt `  w )  =  u  /\  ( ( cls `  ( TopOpen `  w ) ) `  dom  U. u )  =  ( Base `  w )
 ) }
 
19.3.10  Topology and algebraic structures
 
19.3.10.1  The norm on the ring of the integer numbers
 
Theoremzzsnm 24334 The norm of the ring of the integers (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  Z  =  (flds  ZZ )   =>    |-  ( M  e.  ZZ  ->  ( abs `  M )  =  ( ( norm `  Z ) `  M ) )
 
19.3.10.2  The complete ordered field of the real numbers
 
Theoremrecms 24335 The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |-  R  e. CMetSp
 
Theoremreust 24336 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
 |-  R  =  (flds  RR )   =>    |-  (UnifSt `  R )  =  (metUnif `  ( ( dist `  R )  |`  ( RR  X.  RR )
 ) )
 
Theoremrecusp 24337 The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |-  R  =  (flds  RR )   =>    |-  R  e. CUnifSp
 
19.3.10.3  Topological ` ZZ ` -modules
 
Theoremzlm0 24338 Zero of a  ZZ-module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |- 
 .0.  =  ( 0g `  G )   =>    |- 
 .0.  =  ( 0g `  W )
 
Theoremzlm1 24339 Unit of a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |- 
 .1.  =  ( 1r `  G )   =>    |- 
 .1.  =  ( 1r `  W )
 
Theoremzlmds 24340 Distance in a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( G  e.  V  ->  D  =  ( dist `  W ) )
 
Theoremzlmtset 24341 Topology in a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |-  J  =  (TopSet `  G )   =>    |-  ( G  e.  V  ->  J  =  (TopSet `  W ) )
 
Theoremzlmnm 24342 Norm of a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   &    |-  N  =  ( norm `  G )   =>    |-  ( G  e.  V  ->  N  =  ( norm `  W ) )
 
Theoremzhmnrg 24343 The  ZZ-module built from a normed ring is also a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e. NrmRing  ->  W  e. NrmRing )
 
Theoremnmmulg 24344 The norm of a group product, provided the  ZZ-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  N  =  ( norm `  R )   &    |-  Z  =  ( ZMod `  R )   &    |- 
 .x.  =  (.g `  R )   =>    |-  ( ( Z  e. NrmMod  /\  M  e.  ZZ  /\  X  e.  B )  ->  ( N `  ( M  .x.  X ) )  =  ( ( abs `  M )  x.  ( N `  X ) ) )
 
Theoremzrhnm 24345 The norm of the image by  ZRHom of an integer in a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  N  =  ( norm `  R )   &    |-  Z  =  ( ZMod `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( ( ( Z  e. NrmMod  /\  Z  e. NrmRing  /\  R  e. NzRing )  /\  M  e.  ZZ )  ->  ( N `
  ( L `  M ) )  =  ( abs `  M ) )
 
Theoremcnzh 24346 The  ZZ-module of  CC is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
 |-  ( ZMod ` fld )  e. NrmMod
 
Theoremrezh 24347 The  ZZ-module of  RR is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)
 |-  R  =  (flds  RR )   =>    |-  ( ZMod `  R )  e. NrmMod
 
19.3.10.4  The canonical embedding of the rational numbers into a division ring
 
Syntaxcqqh 24348 Map the rationals into a field.
 class QQHom
 
Definitiondf-qqh 24349* Define the canonical homomorphism from the rationals into any field. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
 |- QQHom  =  ( r  e.  _V  |->  ran  ( x  e.  ZZ ,  y  e.  ( `' ( ZRHom `  r
 ) " (Unit `  r
 ) )  |->  <. ( x 
 /  y ) ,  ( ( ( ZRHom `  r ) `  x ) (/r `  r ) ( ( ZRHom `  r
 ) `  y )
 ) >. ) )
 
Theoremqqhval 24350* Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( R  e.  _V  ->  (QQHom `  R )  =  ran  ( x  e. 
 ZZ ,  y  e.  ( `' L "
 (Unit `  R )
 )  |->  <. ( x  /  y ) ,  (
 ( L `  x )  ./  ( L `  y ) ) >. ) )
 
Theoremzrhf1ker 24351 The kernel of the homomorphism from the integers to a ring, if it is injective. (Contributed by Thierry Arnoux, 26-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( L : ZZ -1-1-> B  <->  ( `' L " {  .0.  } )  =  { 0 } ) )
 
Theoremzrhchr 24352 The kernel of the homomorphism from the integers to a ring is injective if and only if the ring has characteristic 0 . (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( (chr `  R )  =  0  <->  L : ZZ -1-1-> B ) )
 
Theoremzrhker 24353 The kernel of the homomorphism from the integers to a ring with characteristic 0. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( (chr `  R )  =  0  <->  ( `' L " {  .0.  } )  =  { 0 } )
 )
 
Theoremzrhunitpreima 24354 The preimage by  ZRHom of the unit of a division ring is  ( ZZ  \  { 0 } ). (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " (Unit `  R ) )  =  ( ZZ  \  {
 0 } ) )
 
Theoremelzrhunit 24355 Condition for the image by  ZRHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  L  =  ( ZRHom `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  ( M  e.  ZZ  /\  M  =/=  0 ) )  ->  ( L `  M )  e.  (Unit `  R ) )
 
Theoremelzdif0 24356 Lemma for qqhval2 24358 (Contributed by Thierry Arnoux, 29-Oct-2017.)
 |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  e.  NN  \/  -u M  e.  NN ) )
 
Theoremqqhval2lem 24357 Lemma for qqhval2 24358 (Contributed by Thierry Arnoux, 29-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ  /\  Y  =/=  0 ) ) 
 ->  ( ( L `  (numer `  ( X  /  Y ) ) ) 
 ./  ( L `  (denom `  ( X  /  Y ) ) ) )  =  ( ( L `  X ) 
 ./  ( L `  Y ) ) )
 
Theoremqqhval2 24358* Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  =  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )
 
Theoremqqhvval 24359 Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( (QQHom `  R ) `  Q )  =  ( ( L `  (numer `  Q ) ) 
 ./  ( L `  (denom `  Q ) ) ) )
 
Theoremqqh0 24360 The image of  0 by the QQHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (
 (QQHom `  R ) `  0 )  =  ( 0g `  R ) )
 
Theoremqqh1 24361 The image of  1 by the QQHom homomorphism is the ring's unit. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (
 (QQHom `  R ) `  1 )  =  ( 1r `  R ) )
 
Theoremqqhf 24362 QQHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R ) : QQ --> B )
 
Theoremqqhvq 24363 The image of a quotient by the QQHom homomorphism. (Contributed by Thierry Arnoux, 28-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   =>    |-  (
 ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ  /\  Y  =/=  0 ) ) 
 ->  ( (QQHom `  R ) `  ( X  /  Y ) )  =  ( ( L `  X )  ./  ( L `
  Y ) ) )
 
Theoremqqhghm 24364 The QQHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  Q  =  (flds  QQ )   =>    |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  e.  ( Q  GrpHom  R ) )
 
Theoremqqhrhm 24365 The QQHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  Q  =  (flds  QQ )   =>    |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  e.  ( Q RingHom  R )
 )
 
Theoremqqhnm 24366 The norm of the image by QQHom of a rational number in a topological division ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  N  =  ( norm `  R )   &    |-  Z  =  ( ZMod `  R )   =>    |-  ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0
 )  /\  Q  e.  QQ )  ->  ( N `
  ( (QQHom `  R ) `  Q ) )  =  ( abs `  Q ) )
 
Theoremqqhcn 24367 The QQHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)
 |-  Q  =  (flds  QQ )   &    |-  J  =  ( TopOpen `  Q )   &    |-  Z  =  ( ZMod `  R )   &    |-  K  =  ( TopOpen `  R )   =>    |-  (
 ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R )  e.  ( J  Cn  K ) )
 
Theoremqqhucn 24368 The QQHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  (flds  QQ )   &    |-  U  =  (UnifSt `  Q )   &    |-  V  =  (metUnif `  (
 ( dist `  R )  |`  ( B  X.  B ) ) )   &    |-  Z  =  ( ZMod `  R )   &    |-  ( ph  ->  R  e. NrmRing )   &    |-  ( ph  ->  R  e.  DivRing )   &    |-  ( ph  ->  Z  e. NrmMod )   &    |-  ( ph  ->  (chr `  R )  =  0 )   =>    |-  ( ph  ->  (QQHom `  R )  e.  ( U Cnu
 V ) )
 
19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field
 
Syntaxcrrh 24369 Map the real numbers into a complete field.
 class RRHom
 
Definitiondf-rrh 24370 Define the canonical homomorphism from the real numbers to any complete field, as the extension by continuity of the canonical homomorphism from the rational numbers. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
 |- RRHom  =  ( r  e.  _V  |->  ( ( ( topGen `  ran  (,) )CnExt ( TopOpen `  r
 ) ) `  (QQHom `  r ) ) )
 
Theoremrrhval 24371 Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  K  =  ( TopOpen `  R )   =>    |-  ( R  e.  V  ->  (RRHom `  R )  =  ( ( JCnExt K ) `
  (QQHom `  R ) ) )
 
Theoremrrhcn 24372 If the topology of  R is Hausdorff, and  R is a complete uniform space, then the canonical homomorphism from the real numbers to  R is continuous. (Contributed by Thierry Arnoux, 17-Jan-2018.)
 |-  D  =  ( ( dist `  R )  |`  ( B  X.  B ) )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  B  =  ( Base `  R )   &    |-  K  =  ( TopOpen `  R )   &    |-  ( ph  ->  R  e.  DivRing )   &    |-  ( ph  ->  R  e. NrmRing )   &    |-  ( ph  ->  ( ZMod `  R )  e. NrmMod )   &    |-  ( ph  ->  (chr `  R )  =  0 )   &    |-  ( ph  ->  R  e.  TopSp )   &    |-  ( ph  ->  R  e. CUnifSp )   &    |-  ( ph  ->  K  e.  Haus
 )   &    |-  ( ph  ->  (UnifSt `  R )  =  (metUnif `  D ) )   =>    |-  ( ph  ->  (RRHom `  R )  e.  ( J  Cn  K ) )
 
Theoremrrhf 24373 If the topology of  R is Hausdorff, Cauchy sequences have at most one limit, i.e. the canonical homomorphism of  RR into  R is a function. (Contributed by Thierry Arnoux, 2-Nov-2017.)
 |-  D  =  ( ( dist `  R )  |`  ( B  X.  B ) )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  B  =  ( Base `  R )   &    |-  K  =  ( TopOpen `  R )   &    |-  ( ph  ->  R  e.  DivRing )   &    |-  ( ph  ->  R  e. NrmRing )   &    |-  ( ph  ->  ( ZMod `  R )  e. NrmMod )   &    |-  ( ph  ->  (chr `  R )  =  0 )   &    |-  ( ph  ->  R  e.  TopSp )   &    |-  ( ph  ->  R  e. CUnifSp )   &    |-  ( ph  ->  K  e.  Haus
 )   &    |-  ( ph  ->  (UnifSt `  R )  =  (metUnif `  D ) )   =>    |-  ( ph  ->  (RRHom `  R ) : RR --> B )
 
19.3.10.6  Embedding into ` RR* `
 
Syntaxcxrh 24374 Map the extended real numbers into a complete lattice.
 class RR*Hom
 
Definitiondf-xrh 24375* Define an embedding from the extended real number into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |- RR*Hom  =  (
 r  e.  _V  |->  ( x  e.  RR*  |->  if ( x  e.  RR ,  (
 (RRHom `  r ) `  x ) ,  if ( x  =  +oo ,  ( ( lub `  r
 ) `  ( (RRHom `  r ) " RR ) ) ,  (
 ( glb `  r ) `  ( (RRHom `  r
 ) " RR ) ) ) ) ) )
 
Theoremxrhval 24376* The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( (RRHom `  R ) " RR )   &    |-  L  =  ( glb `  R )   &    |-  U  =  ( lub `  R )   =>    |-  ( R  e.  V  ->  (RR*Hom `  R )  =  ( x  e.  RR*  |->  if ( x  e.  RR ,  ( (RRHom `  R ) `  x ) ,  if ( x  = 
 +oo ,  ( U `  B ) ,  ( L `  B ) ) ) ) )
 
19.3.10.7  Canonical embeddings into ` RR `
 
Theoremzrhre 24377 The  ZRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
 |-  ( ZRHom `  (flds  RR ) )  =  (  _I  |`  ZZ )
 
Theoremqqhre 24378 The QQHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
 |-  (QQHom `  (flds  RR ) )  =  (  _I  |`  QQ )
 
Theoremrrhre 24379 The RRHom homomorphism for the real numbers structure is the identity. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  (RRHom `  (flds  RR ) )  =  (  _I  |`  RR )
 
19.3.11  Real and complex functions
 
19.3.11.1  Logarithm laws generalized to an arbitrary base - logb

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear. Note that logb is generalized to an arbitrary base and arbitrary parameter in  CC, but it doesn't accept infinities as arguments, unlike  log.

Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log".

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions. The way defined here supports two notations,  (logb `  <. B ,  X >. ) and  ( Blogb X ) where  B is the base and  X is the other parameter. An alternative would be to support the notational form  ( (logb `  B ) `  X
); that looks a little more like traditional notation, but is different than other 2-parameter functions. It's not obvious to me which is better, so here we try out one way as an experiment. Feedback and help welcome.

 
Syntaxclogb 24380 Extend class notation to include the logarithm generalized to an arbitrary base.
 class logb
 
Definitiondf-logb 24381* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as  (logb `  <. B ,  X >. ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". You could also use  ( Blogb X ), which looks like a less-common notation that some use where the base is a preceding superscript. Note: This definition doesn't prevent bases of 1 or 0; proofs may need to forbid them. (Contributed by David A. Wheeler, 21-Jan-2017.)
 |- logb  =  ( x  e.  ( CC  \  { 0 ,  1 } ) ,  y  e.  ( CC  \  {
 0 } )  |->  ( ( log `  y
 )  /  ( log `  x ) ) )
 
Theoremlogbval 24382 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the other operand here. Proof is similar to modval 11244. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
 |-  (
 ( B  e.  ( CC  \  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )
 )  ->  ( Blogb X )  =  ( ( log `  X )  /  ( log `  B ) ) )
 
Theoremlogb2aval 24383 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb <. B ,  X >. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
 |-  (
 ( B  e.  ( CC  \  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )
 )  ->  (logb `  <. B ,  X >. )  =  ( ( log `  X )  /  ( log `  B ) ) )
 
Theoremeldifpr 24384 Membership in a set with two elements removed. Similar to eldifsn 3919 and eldiftp 24385. (Contributed by Mario Carneiro, 18-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D } )  <->  ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D ) )
 
Theoremeldiftp 24385 Membership in a set with three elements removed. Similar to eldifsn 3919 and eldifpr 24384. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  ( A  =/=  C 
 /\  A  =/=  D  /\  A  =/=  E ) ) )
 
Theoremlogeq0im1 24386 if  ( log `  A )  =  0 then 
A  =  1 (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  ( log `  A )  =  0 )  ->  A  =  1 )
 
Theoremlogccne0OLD 24387 log isn't 0 if argument isn't 0 or 1. Unlike logne0 20489, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  =/=  0 )
 
Theoremlogccne0 24388 log isn't 0 if argument isn't 0 or 1. Unlike logne0 20489, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  =/=  0 )
 
Theoremlogbcl 24389 General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  (
 ( B  e.  ( CC  \  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )
 )  ->  ( Blogb X )  e.  CC )
 
Theoremlogbid1 24390 General logarithm when base and arg match (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( Alogb A )  =  1 )
 
Theoremrnlogblem 24391 Useful lemma for working with integer logarithm bases (with is a common case, e.g. base 2, base 3 or base 10) (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  ( B  e.  ( ZZ>= `  2 )  ->  ( B  e.  RR+  /\  B  =/=  0  /\  B  =/=  1
 ) )
 
Theoremrnlogbval 24392 Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+ )  ->  ( Blogb X )  =  ( ( log `  X )  /  ( log `  B ) ) )
 
Theoremrnlogbcl 24393 Closure of the general logarithm with integer base on positive reals. See logbcl 24389. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+ )  ->  ( Blogb X )  e. 
 RR )
 
Theoremrelogbcl 24394 Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  RR+  /\  X  e.  RR+  /\  B  =/=  1 )  ->  ( Blogb X )  e.  RR )
 
Theoremlogb1 24395 The natural logarithm of  1 in base  B. See log1 20472 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  ( Blogb 1 )  =  0 )
 
Theoremnnlogbexp 24396 Identity law for general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ )  ->  ( Blogb ( B ^ M ) )  =  M )
 
Theoremlogbrec 24397 Logarithm of a reciprocal changes sign. See logrec 20653 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  A  e.  RR+ )  ->  ( Blogb ( 1  /  A ) )  =  -u ( Blogb A ) )
 
Theoremlogblt 24398 The general logarithm function is monotone. See logltb 20486 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+  /\  Y  e.  RR+ )  ->  ( X  <  Y  <->  ( Blogb X )  <  ( Blogb Y ) ) )
 
Theoremlog2le1 24399  log 2 is less than  1. This is just a weaker form of log2ub 20781 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( log `  2 )  < 
 1
 
19.3.11.2  Indicator Functions
 
Syntaxcind 24400 Extend class notation with the indicator function generator.
 class 𝟭
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