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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhashge0 23401 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 23402 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 23403 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 23404* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 7092 (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 23405 Extend class notation to include the logarithm generalized to an arbitrary base.
 class logb
 
Definitiondf-logb 23406* 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 23407 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 10991. (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 23408 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 23409 Membership in a set with two elements removed. Similar to eldifsn 3762 and eldiftp 23410. (Contributed by Mario Carneiro, 18-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D } )  <->  ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D ) )
 
Theoremeldiftp 23410 Membership in a set with three elements removed. Similar to eldifsn 3762 and eldifpr 23409. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D ,  E } )  <->  ( A  e.  B  /\  ( A  =/=  C 
 /\  A  =/=  D  /\  A  =/=  E ) ) )
 
Theoremlogeq0im1 23411 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 23412 log isn't 0 if argument isn't 0 or 1. Unlike logne0 19972, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  =/=  0 )
 
Theoremlogccne0ALT 23413 log isn't 0 if argument isn't 0 or 1. Unlike logne0 19972, this handles complex numbers. (Contributed by David A. Wheeler, 17-Jul-2017.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  =/=  0 )
 
Theoremlogbcl 23414 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 23415 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 23416 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 23417 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 23418 Closure of the general logarithm with integer base on positive reals. See logbcl 23414. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+ )  ->  ( Blogb X )  e. 
 RR )
 
Theoremrelogbcl 23419 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 23420 The natural logarithm of  1 in base  B. See log1 19955 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  ( Blogb 1 )  =  0 )
 
Theoremnnlogbexp 23421 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 23422 Logarithm of a reciprocal changes sign. See logrec 20133 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  A  e.  RR+ )  ->  ( Blogb ( 1  /  A ) )  =  -u ( Blogb A ) )
 
Theoremlogblt 23423 The general logarithm function is monotone. See logltb 19969 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  X  e.  RR+  /\  Y  e.  RR+ )  ->  ( X  <  Y  <->  ( Blogb X )  <  ( Blogb Y ) ) )
 
Theoremlog2le1 23424  log 2 is less than  1. This is just a weaker form of log2ub 20261 when no tight upper bound is required. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( log `  2 )  < 
 1
 
18.3.30  Extended sum
 
Syntaxcesum 23425 Extend class notation to include infinite summations.
 class Σ* k  e.  A B
 
Definitiondf-esum 23426 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 23427 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
 |- Σ* k  e.  A B  e.  _V
 
Theoremesumcl 23428* 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 23429 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 23430* 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 23431* 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 23432* 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 23433* 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 23434 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 23435* 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 23436* 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 23437* 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 23438* 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 23439 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 23440* 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 23441* 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 23442 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
 |- Σ* x  e.  (/) A  =  0
 
Theoremesum0 23443* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  F/_ k A   =>    |-  ( A  e.  V  -> Σ* k  e.  A 0  =  0 )
 
Theoremesumf1o 23444* 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 23445* 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 23446 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 23447* 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 23448* 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 23449* 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 23450* 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 23451* 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 23452* 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 23453* 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 23454* Extended sum over a pair, with a relaxed condition compared to esumpr 23453. (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 23455 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 23456* 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 23457 Lemma for esumpfinval 23458 (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  (
 ( A  e.  V  /\  F : A --> ( 0 [,)  +oo ) )  ->  (fld  gsumg  F )  =  ( ( RR* ss  ( 0 [,]  +oo ) )  gsumg 
 F ) )
 
Theoremesumpfinval 23458* 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 23459 Same as esumpfinval 23458, 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 23460* 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 23461* 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 23462* 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 23463* Lemma for esummulc2 23465 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 23464* 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 23465* 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 23466* 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 23467 Lemma for hasheuni 23468 (Contributed by Thierry Arnoux, 19-Nov-2016.)
 |-  # : _V --> ( 0 [,]  +oo )
 
Theoremhasheuni 23468* The cardinality of a disjoint union, not necessarily finite. cf. hashuni 12299. (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 23469* The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 12216. (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 23470* Simpler version of esumcvg 23469. (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 23471 Extend class notation to include mapping of an operation to an operation for a function and a constant.
 class𝑓/𝑐 R
 
Definitiondf-ofc 23472* 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 23473 Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  ( R  =  S  ->𝑓/𝑐 R  =𝑓/𝑐 S )
 
Theoremofcfval 23474* 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 23475 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 23476 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 23477* 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 23478* 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 23479* 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 23480* 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 23481* 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 23482 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 23483 Extend class notation to include the function giving the sigma-algebras on a given base set.
 class sigAlgebra
 
Definitiondf-siga 23484* 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 23485* Lemma for issiga 23487 and isrnsiga 23489 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 23486. (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 23486* 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 23487* 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 23488* 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 23489* 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 23490 A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
 |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S )
 
Theorembaselsiga 23491 A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
 |-  ( S  e.  (sigAlgebra `  A )  ->  A  e.  S )
 
Theoremsigasspw 23492 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 23493 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 23494* 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 23495 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 23496* 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 23497* 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 23498* 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 23499 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 23500 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 ) )
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