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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzlmds 24301 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 24302 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 24303 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 24304 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 24305 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 24306 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 24307 The  ZZ-module of  CC is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
 |-  ( ZMod ` fld )  e. NrmMod
 
Theoremrezh 24308 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 24309 Map the rationals into a field.
 class QQHom
 
Definitiondf-qqh 24310* 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 24311* 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 24312 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 24313 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 24314 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 24315 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 24316 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 24317 Lemma for qqhval2 24319 (Contributed by Thierry Arnoux, 29-Oct-2017.)
 |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  e.  NN  \/  -u M  e.  NN ) )
 
Theoremqqhval2lem 24318 Lemma for qqhval2 24319 (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 24319* 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 24320 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 24321 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 24322 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 24323 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 24324 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 24325 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 24326 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 24327 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 24328 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 24329 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 24330 Map the real numbers into a complete field.
 class RRHom
 
Definitiondf-rrh 24331 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 24332 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 24333 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 24334 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 24335 Map the extended real numbers into a complete lattice.
 class RR*Hom
 
Definitiondf-xrh 24336* 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 24337* 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 24338 The  ZRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
 |-  ( ZRHom `  (flds  RR ) )  =  (  _I  |`  ZZ )
 
Theoremqqhre 24339 The QQHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
 |-  (QQHom `  (flds  RR ) )  =  (  _I  |`  QQ )
 
Theoremrrhre 24340 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 24341 Extend class notation to include the logarithm generalized to an arbitrary base.
 class logb
 
Definitiondf-logb 24342* 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 24343 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 11207. (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 24344 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 24345 Membership in a set with two elements removed. Similar to eldifsn 3887 and eldiftp 24346. (Contributed by Mario Carneiro, 18-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D } )  <->  ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D ) )
 
Theoremeldiftp 24346 Membership in a set with three elements removed. Similar to eldifsn 3887 and eldifpr 24345. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  ( A  =/=  C 
 /\  A  =/=  D  /\  A  =/=  E ) ) )
 
Theoremlogeq0im1 24347 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 24348 log isn't 0 if argument isn't 0 or 1. Unlike logne0 20450, 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 24349 log isn't 0 if argument isn't 0 or 1. Unlike logne0 20450, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  =/=  0 )
 
Theoremlogbcl 24350 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 24351 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 24352 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 24353 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 24354 Closure of the general logarithm with integer base on positive reals. See logbcl 24350. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+ )  ->  ( Blogb X )  e. 
 RR )
 
Theoremrelogbcl 24355 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 24356 The natural logarithm of  1 in base  B. See log1 20433 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  ( Blogb 1 )  =  0 )
 
Theoremnnlogbexp 24357 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 24358 Logarithm of a reciprocal changes sign. See logrec 20614 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  A  e.  RR+ )  ->  ( Blogb ( 1  /  A ) )  =  -u ( Blogb A ) )
 
Theoremlogblt 24359 The general logarithm function is monotone. See logltb 20447 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+  /\  Y  e.  RR+ )  ->  ( X  <  Y  <->  ( Blogb X )  <  ( Blogb Y ) ) )
 
Theoremlog2le1 24360  log 2 is less than  1. This is just a weaker form of log2ub 20742 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( log `  2 )  < 
 1
 
19.3.11.2  Indicator Functions
 
Syntaxcind 24361 Extend class notation with the indicator function generator.
 class 𝟭
 
Definitiondf-ind 24362* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
 |- 𝟭  =  ( o  e.  _V  |->  ( a  e.  ~P o  |->  ( x  e.  o  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
 
Theoremindv 24363* Value of the indicator function generator with domain  O. (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |-  ( O  e.  V  ->  (𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
 
Theoremindval 24364* Value of the indicator function generator for a set  A and a domain  O. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (𝟭 `  O ) `  A )  =  ( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) ) )
 
Theoremindval2 24365 Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (𝟭 `  O ) `  A )  =  ( ( A  X.  { 1 } )  u.  ( ( O  \  A )  X.  { 0 } ) ) )
 
Theoremindf 24366 An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 } )
 
Theoremindfval 24367 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  ( ( (𝟭 `  O ) `  A ) `  X )  =  if ( X  e.  A ,  1 ,  0 ) )
 
Theorempr01ssre 24368 The range of the indicator function is a subset of  RR. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  { 0 ,  1 }  C_  RR
 
Theoremind1 24369 Value of the indicator function where it is  1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  A )  ->  ( ( (𝟭 `  O ) `  A ) `  X )  =  1
 )
 
Theoremind0 24370 Value of the indicator function where it is  0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  ( O  \  A ) )  ->  ( ( (𝟭 `  O ) `  A ) `  X )  =  0
 )
 
Theoremind1a 24371 Value of the indicator function where it is  1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  ( ( ( (𝟭 `  O ) `  A ) `  X )  =  1  <->  X  e.  A ) )
 
Theoremindpi1 24372 Preimage of the singleton  { 1 } by the indicator function. See i1f1lem 19534. (Contributed by Thierry Arnoux, 21-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( `' ( (𝟭 `  O ) `  A ) " { 1 } )  =  A )
 
Theoremindsum 24373* Finite sum of a product with the indicator function / cross-product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  ( ph  ->  O  e.  Fin )   &    |-  ( ph  ->  A  C_  O )   &    |-  ( ( ph  /\  x  e.  O ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ x  e.  O  ( ( ( (𝟭 `  O ) `  A ) `  x )  x.  B )  =  sum_ x  e.  A  B )
 
Theoremindf1o 24374 The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O
 -1-1-onto-> ( { 0 ,  1 }  ^m  O ) )
 
Theoremindpreima 24375 A function with range  { 0 ,  1 } as an indicator of the preimage of  { 1 } (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |-  (
 ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  =  ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) )
 
Theoremindf1ofs 24376* The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
 |-  ( O  e.  V  ->  ( (𝟭 `  O )  |`  Fin ) : ( ~P O  i^i  Fin ) -1-1-onto-> {
 f  e.  ( {
 0 ,  1 } 
 ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
 
19.3.11.3  Extended sum
 
Syntaxcesum 24377 Extend class notation to include infinite summations.
 class Σ* k  e.  A B
 
Definitiondf-esum 24378 Define a short-hand for the possibly infinite sum over the extended non-negative reals. Σ* is relying on the properties of the tsums, developped by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)
 |- Σ* k  e.  A B  =  U. ( (
 RR* ss  ( 0 [,]  +oo ) ) tsums  ( k  e.  A  |->  B ) )
 
Theoremesumex 24379 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
 |- Σ* k  e.  A B  e.  _V
 
Theoremesumcl 24380* Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.)
 |-  F/_ k A   =>    |-  ( ( A  e.  V  /\  A. k  e.  A  B  e.  (
 0 [,]  +oo ) ) 
 -> Σ* k  e.  A B  e.  ( 0 [,]  +oo ) )
 
Theoremesumeq12dvaf 24381 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
 |-  F/ k ph   &    |-  ( ph  ->  A  =  B )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  =  D )   =>    |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B D )
 
Theoremesumeq12dva 24382* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  C  =  D )   =>    |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B D )
 
Theoremesumeq12d 24383* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B D )
 
Theoremesumeq1 24384* Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
 |-  ( A  =  B  -> Σ* k  e.  A C  = Σ* k  e.  B C )
 
Theoremesumeq1d 24385 Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
 |-  F/ k ph   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B C )
 
Theoremesumeq2 24386* Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
 |-  ( A. k  e.  A  B  =  C  -> Σ* k  e.  A B  = Σ* k  e.  A C )
 
Theoremesumeq2d 24387 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
 |-  F/ k ph   &    |-  ( ph  ->  A. k  e.  A  B  =  C )   =>    |-  ( ph  -> Σ* k  e.  A B  = Σ* k  e.  A C )
 
Theoremesumeq2dv 24388* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
 |-  (
 ( ph  /\  k  e.  A )  ->  B  =  C )   =>    |-  ( ph  -> Σ* k  e.  A B  = Σ* k  e.  A C )
 
Theoremesumeq2sdv 24389* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  -> Σ* k  e.  A B  = Σ* k  e.  A C )
 
Theoremnfesum1 24390 Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
 |-  F/_ k A   =>    |-  F/_ kΣ* k  e.  A B
 
Theoremcbvesum 24391* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
 |-  (
 j  =  k  ->  B  =  C )   &    |-  F/_ k A   &    |-  F/_ j A   &    |-  F/_ k B   &    |-  F/_ j C   =>    |- Σ* j  e.  A B  = Σ* k  e.  A C
 
Theoremcbvesumv 24392* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
 |-  (
 j  =  k  ->  B  =  C )   =>    |- Σ* j  e.  A B  = Σ* k  e.  A C
 
Theoremesumid 24393 Identify the extended sum as any limit points of the infinite sum. (Contributed by Thierry Arnoux, 9-May-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  ( ( RR* ss  ( 0 [,]  +oo )
 ) tsums  ( k  e.  A  |->  B ) ) )   =>    |-  ( ph  -> Σ* k  e.  A B  =  C )
 
Theoremesumval 24394* Develop the value of the extended sum. (Contributed by Thierry Arnoux, 4-Jan-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  x  e.  ( ~P A  i^i  Fin )
 )  ->  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg  ( k  e.  x  |->  B ) )  =  C )   =>    |-  ( ph  -> Σ* k  e.  A B  =  sup ( ran  ( x  e.  ( ~P A  i^i  Fin )  |->  C ) ,  RR* ,  <  ) )
 
Theoremesumel 24395* The extended sum is a limit point of the corresponding infinite group sum. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  A B  e.  ( ( RR* ss  ( 0 [,]  +oo ) ) tsums  ( k  e.  A  |->  B ) ) )
 
Theoremesumnul 24396 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
 |- Σ* x  e.  (/) A  =  0
 
Theoremesum0 24397* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  F/_ k A   =>    |-  ( A  e.  V  -> Σ* k  e.  A 0  =  0 )
 
Theoremesumf1o 24398* Re-index an extended sum using a bijection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/ n ph   &    |-  F/_ n A   &    |-  F/_ n C   &    |-  F/_ n F   &    |-  ( k  =  G  ->  B  =  D )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ( ph  /\  n  e.  C )  ->  ( F `  n )  =  G )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ( 0 [,]  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  A B  = Σ* n  e.  C D )
 
Theoremesumc 24399* Convert from the collection form to the class-variable form of a sum. (Contributed by Thierry Arnoux, 10-May-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  (
 y  =  C  ->  D  =  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Fun  `' ( k  e.  A  |->  C ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  W )   =>    |-  ( ph  -> Σ* k  e.  A B  = Σ* y  e.  { z  |  E. k  e.  A  z  =  C } D )
 
Theoremesumsplit 24400 Split an extended sum into two parts. (Contributed by Thierry Arnoux, 9-May-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  F/_ k B   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  C  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  B ) 
 ->  C  e.  ( 0 [,]  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  ( A  u.  B ) C  =  (Σ* k  e.  A C + eΣ* k  e.  B C ) )
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