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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlogbcl 23401 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 23402 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 23403 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 23404 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 23405 Closure of the general logarithm with integer base on positive reals. See logbcl 23401. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+ )  ->  ( Blogb X )  e. 
 RR )
 
Theoremrelogbcl 23406 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 23407 The natural logarithm of  1 in base  B. See log1 19941 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  ( Blogb 1 )  =  0 )
 
Theoremnnlogbexp 23408 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 23409 Logarithm of a reciprocal changes sign. See logrec 20119 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  A  e.  RR+ )  ->  ( Blogb ( 1  /  A ) )  =  -u ( Blogb A ) )
 
Theoremlogblt 23410 The general logarithm function is monotone. See logltb 19955 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+  /\  Y  e.  RR+ )  ->  ( X  <  Y  <->  ( Blogb X )  <  ( Blogb Y ) ) )
 
Theoremlog2le1 23411  log 2 is less than  1. This is just a weaker form of log2ub 20247 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( log `  2 )  < 
 1
 
18.3.30  Extended sum
 
Syntaxcesum 23412 Extend class notation to include infinite summations.
 class Σ* k  e.  A B
 
Definitiondf-esum 23413 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 23414 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
 |- Σ* k  e.  A B  e.  _V
 
Theoremesumcl 23415* 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 23416 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 23417* 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 23418* 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 23419* 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 )
 
Theoremesumeq2 23420* 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 23421 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 23422* 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 23423* 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 )
 
Theoremcbvesum 23424* 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 23425* 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 23426 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 23427* 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 23428* 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 23429 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
 |- Σ* x  e.  (/) A  =  0
 
Theoremesum0 23430* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  F/_ k A   =>    |-  ( A  e.  V  -> Σ* k  e.  A 0  =  0 )
 
Theoremesumf1o 23431* 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 23432* 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 23433 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 ) )
 
Theoremesumadd 23434* Addition of infinite sums. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,]  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  A ( B + e C )  =  (Σ* k  e.  A B + eΣ* k  e.  A C ) )
 
Theoremesumle 23435* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,]  +oo ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  <_  C )   =>    |-  ( ph  -> Σ* k  e.  A B  <_ Σ* k  e.  A C )
 
Theoremesumaddf 23436* Addition of infinite sums. (Contributed by Thierry Arnoux, 22-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,]  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  A ( B + e C )  =  (Σ* k  e.  A B + eΣ* k  e.  A C ) )
 
Theoremesumlef 23437* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  ( 0 [,]  +oo ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  <_  C )   =>    |-  ( ph  -> Σ* k  e.  A B  <_ Σ* k  e.  A C )
 
Theoremesumcst 23438* The extended sum of a constant. (Contributed by Thierry Arnoux, 3-Mar-2017.) (Revised by Thierry Arnoux, 5-Jul-2017.)
 |-  F/_ k A   &    |-  F/_ k B   =>    |-  ( ( A  e.  V  /\  B  e.  (
 0 [,]  +oo ) ) 
 -> Σ* k  e.  A B  =  ( ( # `  A ) x e B ) )
 
Theoremesumsn 23439* The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.)
 |-  (
 ( ph  /\  k  =  M )  ->  A  =  B )   &    |-  ( ph  ->  M  e.  V )   &    |-  ( ph  ->  B  e.  (
 0 [,]  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  { M } A  =  B )
 
Theoremesumpr 23440* Extended sum over a pair. (Contributed by Thierry Arnoux, 2-Jan-2017.)
 |-  (
 ( ph  /\  k  =  A )  ->  C  =  D )   &    |-  ( ( ph  /\  k  =  B ) 
 ->  C  =  E )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  (
 0 [,]  +oo ) )   &    |-  ( ph  ->  E  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  -> Σ* k  e. 
 { A ,  B } C  =  ( D + e E ) )
 
Theoremesumpr2 23441* Extended sum over a pair, with a relaxed condition compared to esumpr 23440. (Contributed by Thierry Arnoux, 2-Jan-2017.)
 |-  (
 ( ph  /\  k  =  A )  ->  C  =  D )   &    |-  ( ( ph  /\  k  =  B ) 
 ->  C  =  E )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  (
 0 [,]  +oo ) )   &    |-  ( ph  ->  E  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  ( A  =  B  ->  ( D  =  0  \/  D  =  +oo )
 ) )   =>    |-  ( ph  -> Σ* k  e.  { A ,  B } C  =  ( D + e E ) )
 
Theoremesumss 23442 Change the index set to a subset by adding zeroes. (Contributed by Thierry Arnoux, 19-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  F/_ k B   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  e.  V )   &    |-  (
 ( ph  /\  k  e.  B )  ->  C  e.  ( 0 [,]  +oo ) )   &    |-  ( ( ph  /\  k  e.  ( B 
 \  A ) ) 
 ->  C  =  0 )   =>    |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B C )
 
Theoremesumpinfval 23443* The value of the extended sum of non-negative terms, with at least one infinite term. (Contributed by Thierry Arnoux, 19-Jun-2017.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  E. k  e.  A  B  =  +oo )   =>    |-  ( ph  -> Σ* k  e.  A B  =  +oo )
 
Theoremesumpfinvallem 23444 Lemma for esumpfinval 23445 (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  (
 ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  (fld  gsumg  F )  =  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg 
 F ) )
 
Theoremesumpfinval 23445* The value of the extended sum of a finite set of non-negative finite terms (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  A B  =  sum_ k  e.  A  B )
 
Theoremesumpfinvalf 23446 Same as esumpfinval 23445, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.)
 |-  F/_ k A   &    |- 
 F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  A B  =  sum_ k  e.  A  B )
 
Theoremesumpinfsum 23447* The value of the extended sum of infinitely many terms greater than one. (Contributed by Thierry Arnoux, 29-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  -.  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ( 0 [,]  +oo ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  M  <_  B )   &    |-  ( ph  ->  M  e.  RR* )   &    |-  ( ph  ->  0  <  M )   =>    |-  ( ph  -> Σ* k  e.  A B  =  +oo )
 
Theoremesumpcvgval 23448* The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)
 |-  (
 ( ph  /\  k  e. 
 NN )  ->  A  e.  ( 0 [,)  +oo ) )   &    |-  ( k  =  l  ->  A  =  B )   &    |-  ( ph  ->  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n ) A )  e.  dom  ~~>  )   =>    |-  ( ph  -> Σ* k  e.  NN A  =  sum_ k  e. 
 NN  A )
 
Theoremesumpmono 23449* The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  NN )  ->  A  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  -> Σ* k  e.  (
 1 ... M ) A 
 <_ Σ* k  e.  ( 1 ... N ) A )
 
Theoremesumcocn 23450* Lemma for esummulc2 23452 and co. Composing with a continuous function preserves extended sums (Contributed by Thierry Arnoux, 29-Jun-2017.)
 |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  ( J  Cn  J ) )   &    |-  ( ph  ->  ( C `  0 )  =  0 )   &    |-  ( ( ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) )  ->  ( C `
  ( x + e y ) )  =  ( ( C `
  x ) + e ( C `  y ) ) )   =>    |-  ( ph  ->  ( C ` Σ* k  e.  A B )  = Σ* k  e.  A ( C `  B ) )
 
Theoremesummulc1 23451* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  (Σ* k  e.  A B x e C )  = Σ* k  e.  A ( B x e C ) )
 
Theoremesummulc2 23452* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  ( C x eΣ* k  e.  A B )  = Σ* k  e.  A ( C x e B ) )
 
Theoremesumdivc 23453* An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (Σ* k  e.  A B /𝑒  C )  = Σ* k  e.  A ( B /𝑒  C )
 )
 
Theoremhashf2 23454 Lemma for hasheuni 23455 (Contributed by Thierry Arnoux, 19-Nov-2016.)
 |-  # : _V --> ( 0 [,]  +oo )
 
Theoremhasheuni 23455* The cardinality of a disjoint union, not necessarily finite. cf. hashuni 12285. (Contributed by Thierry Arnoux, 19-Nov-2016.) (Revised by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 20-Jun-2017.)
 |-  (
 ( A  e.  V  /\ Disj 
 x  e.  A x )  ->  ( # `  U. A )  = Σ* x  e.  A ( # `  x ) )
 
Theoremesumcvg 23456* The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 12202. (Contributed by Thierry Arnoux, 5-Sep-2017.)
 |-  J  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )   &    |-  F  =  ( n  e.  NN  |-> Σ*
 k  e.  ( 1
 ... n ) A )   &    |-  ( ( ph  /\  k  e.  NN )  ->  A  e.  ( 0 [,]  +oo ) )   &    |-  (
 k  =  m  ->  A  =  B )   =>    |-  ( ph  ->  F ( ~~> t `  J )Σ* k  e.  NN A )
 
Theoremesumcvg2 23457* Simpler version of esumcvg 23456. (Contributed by Thierry Arnoux, 5-Sep-2017.)
 |-  J  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )   &    |-  ( ( ph  /\  k  e.  NN )  ->  A  e.  ( 0 [,]  +oo ) )   &    |-  ( k  =  l  ->  A  =  B )   &    |-  ( k  =  m  ->  A  =  C )   =>    |-  ( ph  ->  ( n  e.  NN  |-> Σ* k  e.  (
 1 ... n ) A ) ( ~~> t `  J )Σ* k  e.  NN A )
 
18.3.31  Mixed Function/Constant operation
 
Syntaxcofc 23458 Extend class notation to include mapping of an operation to an operation for a function and a constant.
 class𝑓/𝑐 R
 
Definitiondf-ofc 23459* Define the function/constant operation map. The definition is designed so that if  R is a binary operation, then ∘𝑓/𝑐 R is the analogous operation on functions and constants. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-𝑓/𝑐 R  =  ( f  e.  _V ,  c  e. 
 _V  |->  ( x  e. 
 dom  f  |->  ( ( f `  x ) R c ) ) )
 
Theoremofceq 23460 Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( R  =  S  ->𝑓/𝑐 R  =𝑓/𝑐 S )
 
Theoremofcfval 23461* Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  B )   =>    |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremofcval 23462 Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  (
 ( ph  /\  X  e.  A )  ->  ( F `
  X )  =  B )   =>    |-  ( ( ph  /\  X  e.  A )  ->  (
 ( F𝑓/𝑐 R C ) `  X )  =  ( B R C ) )
 
Theoremofcfn 23463 The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  ( F𝑓/𝑐 R C )  Fn  A )
 
Theoremofcfeqd2 23464* Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  ( y R C )  =  ( y P C ) )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F𝑓/𝑐 P C ) )
 
Theoremofcfval3 23465* General value of  ( F𝑓/𝑐 R C ) with no assumptions on functionality of  F. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( F  e.  V  /\  C  e.  W ) 
 ->  ( F𝑓/𝑐 R C )  =  ( x  e.  dom  F  |->  ( ( F `  x ) R C ) ) )
 
Theoremofcf 23466* The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  T ) )  ->  ( x R y )  e.  U )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  T )   =>    |-  ( ph  ->  ( F𝑓/𝑐 R C ) : A --> U )
 
Theoremofcfval2 23467* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  W )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   =>    |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremofcfval4 23468* The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  (
 ( x  e.  B  |->  ( x R C ) )  o.  F ) )
 
Theoremofcc 23469 Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  ( ( A  X.  { B } )𝑓/𝑐 R C )  =  ( A  X.  { ( B R C ) }
 ) )
 
18.3.32  Sigma-Algebra
 
Syntaxcsiga 23470 Extend class notation to include the function giving the sigma-algebras on a given base set.
 class sigAlgebra
 
Definitiondf-siga 23471* Define a sigma-algebra, i.e. a set closed under complement and countable union. Litterature usually uses capital greek sigma and omega letters for the algebra set, and the base set respectively. We are using  S and  O as a parallel. (Contributed by Thierry Arnoux, 3-Sep-2016.)
 |- sigAlgebra  =  ( o  e.  _V  |->  { s  |  ( s 
 C_  ~P o  /\  (
 o  e.  s  /\  A. x  e.  s  ( o  \  x )  e.  s  /\  A. x  e.  ~P  s
 ( x  ~<_  om  ->  U. x  e.  s ) ) ) } )
 
Theoremsigaex 23472* Lemma for issiga 23474 and isrnsiga 23476 The set of sigma algebra with base set  o is a set. Note: a more generic version with  ( O  e. 
_V  ->  ... ) could be useful for sigaval 23473. (Contributed by Thierry Arnoux, 24-Oct-2016.)
 |-  { s  |  ( s  C_  ~P o  /\  ( o  e.  s  /\  A. x  e.  s  ( o  \  x )  e.  s  /\  A. x  e.  ~P  s
 ( x  ~<_  om  ->  U. x  e.  s ) ) ) }  e.  _V
 
Theoremsigaval 23473* The set of sigma-algebra with a given base set. (Contributed by Thierry Arnoux, 23-Sep-2016.)
 |-  ( O  e.  _V  ->  (sigAlgebra `  O )  =  {
 s  |  ( s 
 C_  ~P O  /\  ( O  e.  s  /\  A. x  e.  s  ( O  \  x )  e.  s  /\  A. x  e.  ~P  s
 ( x  ~<_  om  ->  U. x  e.  s ) ) ) } )
 
Theoremissiga 23474* An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)
 |-  ( S  e.  _V  ->  ( S  e.  (sigAlgebra `  O ) 
 <->  ( S  C_  ~P O  /\  ( O  e.  S  /\  A. x  e.  S  ( O  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) ) ) ) )
 
TheoremisrnsigaOLD 23475* The property of being a sigma algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.)
 |-  ( S  e.  U. ran sigAlgebra  <->  ( S  e.  _V 
 /\  E. o ( S 
 C_  ~P o  /\  (
 o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) ) ) ) )
 
Theoremisrnsiga 23476* The property of being a sigma algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (Proof shortened by Thierry Arnoux, 23-Oct-2016.)
 |-  ( S  e.  U. ran sigAlgebra  <->  ( S  e.  _V 
 /\  E. o ( S 
 C_  ~P o  /\  (
 o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) ) ) ) )
 
Theorem0elsiga 23477 A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
 |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S )
 
Theorembaselsiga 23478 A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
 |-  ( S  e.  (sigAlgebra `  A )  ->  A  e.  S )
 
Theoremsigasspw 23479 A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
 |-  ( S  e.  (sigAlgebra `  A )  ->  S  C_  ~P A )
 
Theoremsigaclcu 23480 A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  U. A  e.  S )
 
Theoremsigaclcuni 23481* A sigma-algebra is closed under countable union: indexed union version (Contributed by Thierry Arnoux, 8-Jun-2017.)
 |-  F/_ k A   =>    |-  ( ( S  e.  U.
 ran sigAlgebra  /\  A. k  e.  A  B  e.  S  /\  A  ~<_  om )  ->  U_ k  e.  A  B  e.  S )
 
Theoremsigaclfu 23482 A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  e.  Fin )  ->  U. A  e.  S )
 
Theoremsigaclcu2 23483* A sigma-algebra is closed under countable union - indexing on  NN (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A. k  e.  NN  A  e.  S )  -> 
 U_ k  e.  NN  A  e.  S )
 
Theoremsigaclfu2 23484* A sigma-algebra is closed under finite union - indexing on  ( 1..^ N ) (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A. k  e.  (
 1..^ N ) A  e.  S )  ->  U_ k  e.  ( 1..^ N ) A  e.  S )
 
Theoremsigaclcu3 23485* A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   &    |-  ( ( ph  /\  k  e.  N )  ->  A  e.  S )   =>    |-  ( ph  ->  U_ k  e.  N  A  e.  S )
 
Theoremissgon 23486 Property of being a sigma-algebra with a given base set, noting that the base set of a sigma algebra is actually its union set. (Contributed by Thierry Arnoux, 24-Sep-2016.) (Revised by Thierry Arnoux, 23-Oct-2016.)
 |-  ( S  e.  (sigAlgebra `  O ) 
 <->  ( S  e.  U. ran sigAlgebra  /\  O  =  U. S ) )
 
Theoremsgon 23487 A sigma alebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
 
Theoremelsigass 23488 An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S ) 
 ->  A  C_  U. S )
 
Theoremelrnsiga 23489 Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
 |-  ( S  e.  (sigAlgebra `  O )  ->  S  e.  U. ran sigAlgebra )
 
Theoremisrnsigau 23490* The property of being a sigma algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
 |-  ( S  e.  U. ran sigAlgebra  ->  ( S  C_  ~P U. S  /\  ( U. S  e.  S  /\  A. x  e.  S  ( U. S  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om 
 ->  U. x  e.  S ) ) ) )
 
Theoremunielsiga 23491 A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
 |-  ( S  e.  U. ran sigAlgebra  ->  U. S  e.  S )
 
Theoremdmvlsiga 23492 Lebesgue-measurable subsets of  RR form a sigma-algebra (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
 |-  dom  vol 
 e.  (sigAlgebra `  RR )
 
Theorempwsiga 23493 Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
 |-  ( O  e.  V  ->  ~P O  e.  (sigAlgebra `  O ) )
 
Theoremprsiga 23494 The smallest possible sigma-algebra containing  O (Contributed by Thierry Arnoux, 13-Sep-2016.)
 |-  ( O  e.  V  ->  { (/) ,  O }  e.  (sigAlgebra `
  O ) )
 
Theoremsigaclci 23495 A sigma-algebra is closed under countable intersection. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)
 |-  (
 ( ( S  e.  U.
 ran sigAlgebra  /\  A  e.  ~P S )  /\  ( A  ~<_ 
 om  /\  A  =/=  (/) ) )  ->  |^| A  e.  S )
 
Theoremdifelsiga 23496 A sigma algebra is closed under set difference. (Contributed by Thierry Arnoux, 13-Sep-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  B  e.  S )  ->  ( A  \  B )  e.  S )
 
Theoremunelsiga 23497 A sigma algebra is closed under set union. (Contributed by Thierry Arnoux, 13-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  B  e.  S )  ->  ( A  u.  B )  e.  S )
 
Theoreminelsiga 23498 A sigma algebra is closed under set intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S  /\  B  e.  S )  ->  ( A  i^i  B )  e.  S )
 
Theoremsigainb 23499 Building a sigma algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
 |-  (
 ( S  e.  U. ran sigAlgebra  /\  A  e.  S ) 
 ->  ( S  i^i  ~P A )  e.  (sigAlgebra `  A ) )
 
Theoreminsiga 23500 The intersection of a collection of sigma-algebras of same base is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
 |-  (
 ( A  =/=  (/)  /\  A  e.  ~P (sigAlgebra `  O ) ) 
 ->  |^| A  e.  (sigAlgebra `  O ) )
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