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Theorem List for Metamath Proof Explorer - 23301-23400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmndpluscn 23301* 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 23302 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 23303* 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 23304* 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 23305* 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 ) )
 
18.3.23  Extended reals Structure - misc additions
 
Theoremxaddeq0 23306 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A + e B )  =  0  <->  A  =  - e B ) )
 
Theoremxrs0 23307 The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 10571 and df-xrs 13405), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  0  =  ( 0g `  RR* s )
 
Theoremxrsinvgval 23308 The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 10571 and df-xrs 13405), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  ( B  e.  RR*  ->  (
 ( inv g `  RR* s ) `
  B )  =  - e B )
 
Theoremxrsmulgzz 23309 The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  RR* )  ->  ( A (.g `  RR* s ) B )  =  ( A x e B ) )
 
Theoremressmulgnn 23310 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 12-Jun-2017.)
 |-  H  =  ( Gs  A )   &    |-  A  C_  ( Base `  G )   &    |-  .*  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  A )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
 
Theoremressmulgnn0 23311 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  H  =  ( Gs  A )   &    |-  A  C_  ( Base `  G )   &    |-  .*  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   &    |-  ( 0g `  G )  =  ( 0g `  H )   =>    |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
 
18.3.24  The extended non-negative real numbers monoid
 
Theoremxrge0base 23312 The base of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  (
 0 [,]  +oo )  =  ( Base `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge00 23313 The zero of the extended non-negative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  0  =  ( 0g `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge0plusg 23314 The additive law of the extended non-negative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  + e  =  ( +g  `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge0mulgnn0 23315 The group multiple function in the extended non-negative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  (
 ( A  e.  NN0  /\  B  e.  ( 0 [,]  +oo ) )  ->  ( A (.g `  ( RR* ss  ( 0 [,]  +oo ) ) ) B )  =  ( A x e B ) )
 
Theoremxrge0hmph 23316 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 23317* 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 23318* 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 23319* 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 23320* 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 23321* 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 23322* 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 23323* 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 23324* 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 23325* 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 23326 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 23327 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 23328 The topology of the extended non-negative real numbers is Hausdorf. (Contributed by Thierry Arnoux, 26-Jul-2017.)
 |-  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )  e. 
 Haus
 
Theoremxrge0tmdALT 23329 The extended non-negative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.)
 |-  ( RR* ss  ( 0 [,]  +oo ) )  e. TopMnd
 
Theoremxrge0tmd 23330 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
 
Theoremxrge0addass 23331 Associativity of extended non-negative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo ) )  ->  ( ( A + e B ) + e C )  =  ( A + e ( B + e C ) ) )
 
Theoremxrge0neqmnf 23332 An extended non-negative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.)
 |-  ( A  e.  ( 0 [,]  +oo )  ->  A  =/=  -oo )
 
Theoremxrge0addgt0 23333 The sum of nonnegative and positive numbers is positive. See addgtge0 9264 (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  (
 ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
 0 [,]  +oo ) ) 
 /\  0  <  A )  ->  0  <  ( A + e B ) )
 
Theoremxrge0adddir 23334 Distributivity of extended non-negative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo ) )  ->  ( ( A + e B ) x e C )  =  ( ( A x e C ) + e ( B x e C ) ) )
 
Theoremxrge0npcan 23335 Extended non-negative real version of npcan 9062. (Contributed by Thierry Arnoux, 9-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  B  <_  A )  ->  (
 ( A + e  - e B ) + e B )  =  A )
 
Theoremfsumrp0cl 23336* Closure of a finite sum of positive integers. (Contributed by Thierry Arnoux, 25-Jun-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  (
 0 [,)  +oo ) )
 
18.3.25  Countable Sets
 
Theoremnnct 23337  NN is countable (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  NN  ~<_  om
 
Theoremctex 23338 A countable set is a set (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  A  e.  _V )
 
Theoremssct 23339 The subset of a countable set is countable (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
 
Theoremxpct 23340 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( A  X.  B )  ~<_  om )
 
Theoremsnct 23341 A singleton is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  ( A  e.  V  ->  { A }  ~<_  om )
 
Theoremprct 23342 An unordered pair is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  { A ,  B } 
 ~<_  om )
 
Theoremfnct 23343 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  (
 ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  om )
 
Theoremdmct 23344 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  dom  A  ~<_  om )
 
Theoremcnvct 23345 If a set is countable, its converse is as well. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  `' A  ~<_  om )
 
Theoremrnct 23346 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ran  A  ~<_  om )
 
Theoremmptct 23347* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremmpt2cti 23348* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  A. x  e.  A  A. y  e.  B  C  e.  V   =>    |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( x  e.  A ,  y  e.  B  |->  C )  ~<_ 
 om )
 
Theoremabrexct 23349* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  { y  |  E. x  e.  A  y  =  B }  ~<_  om )
 
Theoremmptctf 23350 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremabrexctf 23351* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  { y  |  E. x  e.  A  y  =  B } 
 ~<_  om )
 
18.3.26  Disjointness - misc additions
 
Theoremcbvdisjf 23352* Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y  ->  B  =  C )   =>    |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
 
Theoremdisjss1f 23353 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )
 
Theoremdisjdifprg 23354* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 -> Disj 
 x  e.  { ( B  \  A ) ,  A } x )
 
Theoremdisjdifprg2 23355* A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( A  e.  V  -> Disj  x  e.  { ( A  \  B ) ,  ( A  i^i  B ) } x )
 
Theoremdisji2f 23356* Property of a disjoint collection: if  B ( x )  =  C and  B ( Y )  =  D, and  x  =/=  Y, then  B and  C are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A B  /\  ( x  e.  A  /\  Y  e.  A )  /\  x  =/= 
 Y )  ->  ( B  i^i  C )  =  (/) )
 
Theoremdisjif 23357* Property of a disjoint collection: if  B ( x ) and 
B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A B  /\  ( x  e.  A  /\  Y  e.  A )  /\  ( Z  e.  B  /\  Z  e.  C ) )  ->  x  =  Y )
 
Theoremdisjorf 23358* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ i A   &    |-  F/_ j A   &    |-  ( i  =  j  ->  B  =  C )   =>    |-  (Disj  i  e.  A B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
Theoremdisjorsf 23359* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  (Disj  x  e.  A B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
 
Theoremdisjif2 23360* Property of a disjoint collection: if  B ( x ) and 
B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  Y  ->  B  =  C )   =>    |-  ( (Disj  x  e.  A B  /\  ( x  e.  A  /\  Y  e.  A )  /\  ( Z  e.  B  /\  Z  e.  C )
 )  ->  x  =  Y )
 
Theoremdisjabrex 23361* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  (Disj  x  e.  A B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B } y )
 
Theoremdisjabrexf 23362* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   =>    |-  (Disj  x  e.  A B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B }
 y )
 
Theoremdisjpreima 23363* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
 |-  (
 ( Fun  F  /\ Disj  x  e.  A B )  -> Disj  x  e.  A ( `' F " B ) )
 
Theoremdisjin 23364 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  (Disj  x  e.  B C  -> Disj  x  e.  B ( C  i^i  A ) )
 
Theoremiundisjfi 23365* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 18907 (Contributed by Thierry Arnoux, 15-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |-  U_ n  e.  ( 1..^ N ) A  =  U_ n  e.  ( 1..^ N ) ( A 
 \  U_ k  e.  (
 1..^ n ) B )
 
Theoremiundisj2fi 23366* A disjoint union is disjoint, finite version. Cf. iundisj2 18908 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |- Disj  n  e.  ( 1..^ N ) ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisjf 23367* Rewrite a countable union as a disjoint union. Cf. iundisj 18907 (Contributed by Thierry Arnoux, 31-Dec-2016.)
 |-  F/_ k A   &    |-  F/_ n B   &    |-  ( n  =  k  ->  A  =  B )   =>    |-  U_ n  e.  NN  A  =  U_ n  e. 
 NN  ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisj2f 23368* A disjoint union is disjoint. Cf. iundisj2 18908 (Contributed by Thierry Arnoux, 30-Dec-2016.)
 |-  F/_ k A   &    |-  F/_ n B   &    |-  ( n  =  k  ->  A  =  B )   =>    |- Disj  n  e.  NN ( A 
 \  U_ k  e.  (
 1..^ n ) B )
 
Theoremiundisjcnt 23369* Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  ->  U_ n  e.  N  A  =  U_ n  e.  N  ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
Theoremiundisj2cnt 23370* A countable disjoint union is disjoint. Cf. iundisj2 18908 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  -> Disj  n  e.  N ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
Theoremdisjdsct 23371* A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 5312) (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  ( V  \  { (/)
 } ) )   &    |-  ( ph  -> Disj  x  e.  A B )   =>    |-  ( ph  ->  Fun  `' ( x  e.  A  |->  B ) )
 
Theoremdisjrdx 23372* Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
 |-  ( ph  ->  F : A -1-1-onto-> C )   &    |-  ( ( ph  /\  y  =  ( F `  x ) )  ->  D  =  B )   =>    |-  ( ph  ->  (Disj  x  e.  A B  <-> Disj  y  e.  C D ) )
 
18.3.27  Limits - misc additions
 
Theoremlmlim 23373 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 23374 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 23375 Implication of convergence for a non-negative series. This could be used to shorten prmreclem6 12970 (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 23376* A neighborhood of  +oo contains an unbounded interval based at a real number. See pnfnei 16952 (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 23377* 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 23378* 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 23379* 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 )
 
18.3.28  Finitely supported group sums - misc additions
 
Theoremgsumsn2 23380* Group sum of a singleton. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  B  =  ( Base `  G )   &    |-  G  e.  Mnd   &    |-  ( ( ph  /\  k  =  M )  ->  A  =  C )   &    |-  ( ph  ->  M  e.  V )   &    |-  ( ph  ->  C  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  { M }  |->  A ) )  =  C )
 
Theoremgsumpropd2lem 23381* Lemma for gsumpropd2 23382 (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  (
 ( ph  /\  ( s  e.  ( Base `  G )  /\  t  e.  ( Base `  G ) ) )  ->  ( s
 ( +g  `  G ) t )  e.  ( Base `  G ) )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   &    |-  A  =  ( `' F "
 ( _V  \  {
 s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G ) t )  =  t  /\  ( t ( +g  `  G ) s )  =  t ) } )
 )   &    |-  B  =  ( `' F " ( _V  \  { s  e.  ( Base `  H )  | 
 A. t  e.  ( Base `  H ) ( ( s ( +g  `  H ) t )  =  t  /\  (
 t ( +g  `  H ) s )  =  t ) } )
 )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumpropd2 23382* A stronger version of gsumpropd 14455, working for magma, where only the closure of the addition operation on a common base is required. (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  (
 ( ph  /\  ( s  e.  ( Base `  G )  /\  t  e.  ( Base `  G ) ) )  ->  ( s
 ( +g  `  G ) t )  e.  ( Base `  G ) )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumconstf 23383* Sum of a constant series (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  F/_ k X   &    |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  Fin  /\  X  e.  B )  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( # `  A )  .x.  X ) )
 
Theoremxrge0tsmsd 23384* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  S  =  sup ( ran  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) ,  RR* ,  <  ) )   =>    |-  ( ph  ->  ( G tsums  F )  =  { S } )
 
Theoremxrge0tsmsbi 23385 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   =>    |-  ( ph  ->  ( C  e.  ( G tsums  F )  <->  C  =  U. ( G tsums  F ) ) )
 
Theoremxrge0tsmseq 23386 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  C  =  U. ( G tsums  F ) )
 
Theoremhashunif 23387* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 12285 (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Fin )   &    |-  ( ph  -> Disj  x  e.  A x )   =>    |-  ( ph  ->  ( # `
  U. A )  = 
 sum_ x  e.  A  ( # `  x ) )
 
Theoremhashge0 23388 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  ( A  e.  V  ->  0 
 <_  ( # `  A ) )
 
Theoremhashgt0 23389 The cardinality of a non-empty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  (
 ( A  e.  V  /\  A  =/=  (/) )  -> 
 0  <  ( # `  A ) )
 
Theoremhashge1 23390 The cardinality of a non-empty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
 |-  (
 ( A  e.  V  /\  A  =/=  (/) )  -> 
 1  <_  ( # `  A ) )
 
Theoremishashinf 23391* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 7078 (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  ( -.  A  e.  Fin  ->  A. n  e.  NN  E. x  e.  ~P  A ( # `  x )  =  n )
 
18.3.29  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 from 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 23392 Extend class notation to include the logarithm generalized to an arbitrary base.
 class logb
 
Definitiondf-logb 23393* 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 23394 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 10977. (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 23395 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 23396 Membership in a set with two elements removed. Similar to eldifsn 3751 and eldiftp 23397. (Contributed by Mario Carneiro, 18-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D } )  <->  ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D ) )
 
Theoremeldiftp 23397 Membership in a set with three elements removed. Similar to eldifsn 3751 and eldifpr 23396. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  ( A  =/=  C 
 /\  A  =/=  D  /\  A  =/=  E ) ) )
 
Theoremlogeq0im1 23398 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 )
 
Theoremlogccne0 23399 log isn't 0 if argument isn't 0 or 1. Unlike logne0 19958, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  =/=  0 )
 
Theoremlogccne0ALT 23400 log isn't 0 if argument isn't 0 or 1. Unlike logne0 19958, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  =/=  0 )
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