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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremissibf 24601* The predicate " F is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   =>    |-  ( ph  ->  ( F  e.  dom  ( Wsitg M )  <->  ( F  e.  ( dom  MMblFnM S )  /\  ran 
 F  e.  Fin  /\  A. x  e.  ( ran 
 F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) ) ) )
 
Theoremsibf0 24602 The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  W  e.  TopSp )   &    |-  ( ph  ->  W  e.  Grp )   =>    |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
 
Theoremsibfmbl 24603 A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  F  e.  ( dom  MMblFnM S ) )
 
Theoremsibff 24604 A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  F : U. dom  M --> U. J )
 
Theoremsibfrn 24605 A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ran  F  e.  Fin )
 
Theoremsibfima 24606 Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ( ph  /\  A  e.  ( ran  F  \  {  .0.  } ) ) 
 ->  ( M `  ( `' F " { A } ) )  e.  ( 0 [,)  +oo ) )
 
Theoremsibfof 24607 Applying function operations on simple functions results in simple functions with regard to the the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  C  =  ( Base `  K )   &    |-  ( ph  ->  W  e.  TopSp )   &    |-  ( ph  ->  .+ 
 : ( B  X.  B ) --> C )   &    |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  K  e.  TopSp )   &    |-  ( ph  ->  J  e.  Fre )   &    |-  ( ph  ->  (  .0.  .+  .0.  )  =  ( 0g
 `  K ) )   =>    |-  ( ph  ->  ( F  o F  .+  G )  e.  dom  ( Ksitg M ) )
 
Theoremsitgfval 24608* Value of the Bochner integral for a simple function  F. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ( ( Wsitg M ) `  F )  =  ( W  gsumg  ( x  e.  ( ran  F 
 \  {  .0.  }
 )  |->  ( ( H `
  ( M `  ( `' F " { x } ) ) ) 
 .x.  x ) ) ) )
 
Theoremsitgclg 24609* Closure of the Bochner integral on a simple functions. This version is very generic, thus the many hypothesis. See sitgclbn 24610 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  G  =  (Scalar `  W )   &    |-  D  =  ( (
 dist `  G )  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )   &    |-  ( ph  ->  W  e.  TopSp )   &    |-  ( ph  ->  W  e. CMnd )   &    |-  ( ph  ->  G  e.  DivRing )   &    |-  ( ph  ->  G  e. NrmRing )   &    |-  ( ph  ->  ( ZMod `  G )  e. NrmMod )   &    |-  ( ph  ->  (chr `  G )  =  0 )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  G  e. CUnifSp )   &    |-  ( ph  ->  (
 TopOpen `  G )  e. 
 Haus )   &    |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D ) )   &    |-  (
 ( ph  /\  m  e.  ( H " (
 0 [,)  +oo ) ) 
 /\  x  e.  B )  ->  ( m  .x.  x )  e.  B )   =>    |-  ( ph  ->  (
 ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgclbn 24610 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  G  =  (Scalar `  W )   &    |-  D  =  ( (
 dist `  G )  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )   &    |-  ( ph  ->  W  e. Ban )   &    |-  ( ph  ->  G  e. CUnifSp )   &    |-  ( ph  ->  ( TopOpen `  G )  e.  Haus )   &    |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
 )   &    |-  ( ph  ->  ( ZMod `  G )  e. NrmMod )   &    |-  ( ph  ->  (chr `  G )  =  0 )   =>    |-  ( ph  ->  (
 ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgclcn 24611 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  W  e. Ban )   &    |-  ( ph  ->  (Scalar `  W )  =fld )   =>    |-  ( ph  ->  (
 ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgclre 24612 Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  W  e. Ban )   &    |-  ( ph  ->  (Scalar `  W )  =  (flds  RR ) )   =>    |-  ( ph  ->  ( ( Wsitg M ) `
  F )  e.  B )
 
Theoremsitgf 24613* The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  S  =  (sigaGen `  J )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  H  =  (RRHom `  (Scalar `  W )
 )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  (
 ( ph  /\  f  e. 
 dom  ( Wsitg M ) )  ->  ( ( Wsitg M ) `  f )  e.  B )   =>    |-  ( ph  ->  ( Wsitg M ) : dom  ( Wsitg M ) --> B )
 
Theoremsitmval 24614* Value of the simple function integral metric for a given space  W and measure  M. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  D  =  ( dist `  W )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   =>    |-  ( ph  ->  ( Wsitm M )  =  ( f  e.  dom  ( Wsitg M ) ,  g  e.  dom  ( Wsitg M )  |->  ( ( (
 RR* ss  ( 0 [,]  +oo ) )sitg M ) `  ( f  o F D g ) ) ) )
 
Theoremsitmfval 24615 Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  D  =  ( dist `  W )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ( F ( Wsitm M ) G )  =  ( ( ( RR* ss  ( 0 [,]  +oo ) )sitg M ) `  ( F  o F D G ) ) )
 
Theoremsitmcl 24616 Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
 |-  ( ph  ->  W  e.  Mnd )   &    |-  ( ph  ->  W  e.  * MetSp )   &    |-  ( ph  ->  M  e.  U. ran measures )   &    |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )   &    |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )   =>    |-  ( ph  ->  ( F ( Wsitm M ) G )  e.  (
 0 [,]  +oo ) )
 
Definitiondf-itgm 24617* Define the Bochner integral as the extension by continuity of the Bochnel integral for simple functions.

Bogachev first defines 'fundamental in the mean' sequences, in definition 2.3.1 of [Bogachev] p. 116, and notes that those are actually Cauchy sequences for the pseudometric  ( wsitm m ).

He then defines the Bochner integral in chapter 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 19469.

(Contributed by Thierry Arnoux, 13-Feb-2018.)

 |- itgm  =  ( w  e.  _V ,  m  e.  U. ran measures  |->  ( ( (metUnif `  ( wsitm m ) )CnExt (UnifSt `  w ) ) `  ( wsitg m ) ) )
 
19.3.15  Probability
 
19.3.15.1  Probability Theory
 
Syntaxcprb 24618 Extend class notation to include the class of probability measures.
 class Prob
 
Definitiondf-prob 24619 Define the class of probability measures as the set of measures with total measure 1. (Contributed by Thierry Arnoux, 14-Sep-2016.)
 |- Prob  =  { p  e.  U. ran measures  |  ( p `  U. dom  p )  =  1 }
 
Theoremelprob 24620 The property of being a probability measure (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( P  e. Prob  <->  ( P  e.  U.
 ran measures  /\  ( P `  U.
 dom  P )  =  1 ) )
 
Theoremdomprobmeas 24621 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( P  e. Prob  ->  P  e.  (measures `  dom  P ) )
 
Theoremdomprobsiga 24622 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( P  e. Prob  ->  dom  P  e.  U. ran sigAlgebra )
 
Theoremprobtot 24623 The Probbiliy of the universe set is 1 (Second axiom of Kolmogorov) (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( P  e. Prob  ->  ( P `
  U. dom  P )  =  1 )
 
Theoremprob01 24624 A Probbiliy is bounded in [ 0 , 1 ] (First axiom of Kolmogorov) (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P )  ->  ( P `  A )  e.  ( 0 [,] 1 ) )
 
Theoremprobnul 24625 The Probbiliy of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( P  e. Prob  ->  ( P `
  (/) )  =  0 )
 
Theoremunveldomd 24626 The universe is an element of the domain of the probability, the universe (entire probability space) being  U.
dom  P in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  U. dom  P  e.  dom 
 P )
 
Theoremunveldom 24627 The universe is an element of the domain of the probability, the universe (entire probability space) being  U.
dom  P in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( P  e. Prob  ->  U. dom  P  e.  dom  P )
 
Theoremnuleldmp 24628 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( P  e. Prob  ->  (/)  e.  dom  P )
 
Theoremprobcun 24629* The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the  sum_ construct cannot be used as it can handle infinite indexing set only if they are subsets of 
ZZ, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  ~P dom  P  /\  ( A  ~<_  om  /\ Disj  x  e.  A x ) ) 
 ->  ( P `  U. A )  = Σ* x  e.  A ( P `  x ) )
 
Theoremprobun 24630 The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( ( A  i^i  B )  =  (/)  ->  ( P `  ( A  u.  B ) )  =  ( ( P `  A )  +  ( P `  B ) ) ) )
 
Theoremprobdif 24631 The probabiliy of the difference of two event sets (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( P `  ( A 
 \  B ) )  =  ( ( P `
  A )  -  ( P `  ( A  i^i  B ) ) ) )
 
Theoremprobinc 24632 A probabiliy law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  C_  B )  ->  ( P `  A )  <_  ( P `
  B ) )
 
Theoremprobdsb 24633 The probability of the complement of a set. That is, the probability that the event  A does not occur. (Contributed by Thierry Arnoux, 15-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P )  ->  ( P `  ( U. dom  P  \  A ) )  =  ( 1  -  ( P `  A ) ) )
 
Theoremprobmeasd 24634 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  P  e.  U. ran measures )
 
Theoremprobvalrnd 24635 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  A  e.  dom  P )   =>    |-  ( ph  ->  ( P `  A )  e. 
 RR )
 
Theoremprobtotrnd 24636 The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( P `  U.
 dom  P )  e.  RR )
 
Theoremtotprobd 24637* Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  A  e.  dom  P )   &    |-  ( ph  ->  B  e.  ~P dom  P )   &    |-  ( ph  ->  U. B  =  U. dom  P )   &    |-  ( ph  ->  B  ~<_  om )   &    |-  ( ph  -> Disj  b  e.  B b )   =>    |-  ( ph  ->  ( P `  A )  = Σ* b  e.  B ( P `
  ( b  i^i 
 A ) ) )
 
Theoremtotprob 24638* Law of total probability (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( U. B  =  U. dom  P  /\  B  e.  ~P
 dom  P  /\  ( B  ~<_ 
 om  /\ Disj  b  e.  B b ) ) ) 
 ->  ( P `  A )  = Σ* b  e.  B ( P `  ( b  i^i  A ) ) )
 
TheoremprobfinmeasbOLD 24639* Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( M `  U. S )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  ( M `  U. S ) ) )  e. Prob
 )
 
Theoremprobfinmeasb 24640 Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( M `  U. S )  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  ( M ` 
 U. S ) )  e. Prob )
 
Theoremprobmeasb 24641* Build a probability from a measure and a set with finite measure (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `
  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) ) 
 /  ( M `  A ) ) )  e. Prob )
 
19.3.15.2  Conditional Probabilities
 
Syntaxccprob 24642 Extends class notation with the conditional probability builder.
 class cprob
 
Definitiondf-cndprob 24643* Define the conditional probability. (Contributed by Thierry Arnoux, 14-Sep-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |- cprob  =  ( p  e. Prob  |->  ( a  e.  dom  p ,  b  e.  dom  p  |->  ( ( p `  (
 a  i^i  b )
 )  /  ( p `  b ) ) ) )
 
Theoremcndprobval 24644 The value of the conditional probability , i.e. the probability for the event  A, given  B, under the probability law  P. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( (cprob `  P ) `  <. A ,  B >. )  =  ( ( P `  ( A  i^i  B ) ) 
 /  ( P `  B ) ) )
 
Theoremcndprobin 24645 An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  B )  =/=  0
 )  ->  ( (
 (cprob `  P ) `  <. A ,  B >. )  x.  ( P `
  B ) )  =  ( P `  ( A  i^i  B ) ) )
 
Theoremcndprob01 24646 The conditional probability has values in  [ 0 ,  1 ]. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  B )  =/=  0
 )  ->  ( (cprob `  P ) `  <. A ,  B >. )  e.  (
 0 [,] 1 ) )
 
Theoremcndprobtot 24647 The conditional probability given a certain event is one. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( P `  A )  =/=  0 )  ->  (
 (cprob `  P ) `  <. U. dom  P ,  A >. )  =  1 )
 
Theoremcndprobnul 24648 The conditional probability given empty event is zero. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( P `  A )  =/=  0 )  ->  (
 (cprob `  P ) `  <. (/) ,  A >. )  =  0 )
 
Theoremcndprobprob 24649* The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  B  e.  dom  P  /\  ( P `  B )  =/=  0 )  ->  (
 a  e.  dom  P  |->  ( (cprob `  P ) `  <. a ,  B >. ) )  e. Prob )
 
Theorembayesth 24650 Bayes Theorem (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  A )  =/=  0  /\  ( P `  B )  =/=  0 )  ->  ( (cprob `  P ) `  <. A ,  B >. )  =  ( ( ( (cprob `  P ) `  <. B ,  A >. )  x.  ( P `
  A ) ) 
 /  ( P `  B ) ) )
 
19.3.15.3  Real Valued Random Variables
 
Syntaxcrrv 24651 Extend class notation with the class of real valued random variables.
 class rRndVar
 
Definitiondf-rrv 24652 In its generic definition, a random variable is a measurable function from a probability space to a Borel set. Here, we specifically target real-valued random variables, i.e. measurable function from a probability space to the Borel sigma algebra on the set of real numbers. (Contributed by Thierry Arnoux, 20-Sep-2016.) (Revised by Thierry Arnoux, 25-Jan-2017.)
 |- rRndVar  =  ( p  e. Prob  |->  ( dom 
 pMblFnM𝔅 ) )
 
Theoremrrvmbfm 24653 A real-valued random variable is a measurable function from its sample space to the Borel Sigma Algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  X  e.  ( dom  PMblFnM𝔅 ) ) )
 
Theoremisrrvv 24654* Elementhood to the set of real-valued random variables with respect to the probability  P. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  ( X : U. dom  P --> RR  /\  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P ) ) )
 
Theoremrrvvf 24655 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : U. dom  P --> RR )
 
Theoremrrvfn 24656 A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X  Fn  U. dom  P )
 
Theoremrrvdm 24657 The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  dom 
 X  =  U. dom  P )
 
Theoremrrvrnss 24658 The range of a random variable as a subset of  RR. (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  ran 
 X  C_  RR )
 
Theoremrrvf2 24659 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : dom  X --> RR )
 
Theoremrrvdmss 24660 The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  U.
 dom  P  C_  dom  X )
 
Theoremrrvfinvima 24661* For a real-value random variable  X, any open interval in 
RR is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P )
 
Theorem0rrv 24662* The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( x  e. 
 U. dom  P  |->  0 )  e.  (rRndVar `  P ) )
 
Theoremrrvadd 24663 The sum of two random variables is a random variable (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  Y  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  ( X  o F  +  Y )  e.  (rRndVar `  P ) )
 
Theoremrrvmulc 24664 A random variable multiplied by a constant is a random variable. (Contributed by Thierry Arnoux, 17-Jan-2017.) (Revised by Thierry Arnoux, 22-May-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( X𝑓/𝑐  x.  C )  e.  (rRndVar `  P ) )
 
Theoremrrvsum 24665 An indexed sum of random variables is a random variable. (Contributed by Thierry Arnoux, 22-May-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X : NN --> (rRndVar `  P ) )   &    |-  ( ( ph  /\  N  e.  NN )  ->  S  =  (  seq  1 (  o F  +  ,  X ) `  N ) )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  S  e.  (rRndVar `  P )
 )
 
19.3.15.4  Preimage set mapping operator
 
Syntaxcorvc 24666 Extend class notation to include the preimage set mapping operator.
 classRV/𝑐 R
 
Definitiondf-orvc 24667* Define the preimage set mapping operator. In probability theory, the notation  P ( X  =  A ) denotes the probability that a random variable  X takes the value  A. We introduce here an operator which enables to write this in Metamath as  ( P `  ( XRV/𝑐  _I  A ) ), and keep a similar notation. Because with this notation  ( XRV/𝑐  _I  A ) is a set, we can also apply it to conditional probabilities, like in  ( P `  ( XRV/𝑐  _I  A )  |  ( YRV/𝑐  _I  B ) ) ).

The oRVC operator transforms a relation  R into an operation taking a random variable  X and a constant  C, and returning the preimage through  X of the equivalence class of  C.

The most commonly used relations are: - equality:  { X  =  A } as  ( XRV/𝑐  _I  A ) cf. ideq 4984- elementhood:  { X  e.  A } as  ( XRV/𝑐  _E  A ) cf. epel 4457- less-than:  { X  <_  A } as  ( XRV/𝑐  <_  A )

Even though it is primarily designed to be used within probability theory and with random variables, this operator is defined on generic functions, and could be used in other fields, e.g. for continuous functions. (Contributed by Thierry Arnoux, 15-Jan-2017.)

 |-RV/𝑐 R  =  ( x  e.  { x  |  Fun  x } ,  a  e.  _V  |->  ( `' x " { y  |  y R a } )
 )
 
Theoremorvcval 24668* Value of the preimage mapping operator applied on a given random variable and constant (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A }
 ) )
 
Theoremorvcval2 24669* Another way to express the value of the preimage mapping operator (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  {
 z  e.  dom  X  |  ( X `  z
 ) R A }
 )
 
Theoremelorvc 24670* Elementhood of a preimage (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ( ph  /\  z  e.  dom  X )  ->  ( z  e.  ( XRV/𝑐 R A )  <->  ( X `  z ) R A ) )
 
Theoremorvcval4 24671* The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 24668 (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
 ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  U. J  |  y R A } )
 )
 
Theoremorvcoel 24672* If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
 ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e. 
 U. J  |  y R A }  e.  J )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S )
 
Theoremorvccel 24673* If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
 ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e. 
 U. J  |  y R A }  e.  ( Clsd `  J )
 )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S )
 
Theoremelorrvc 24674* Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ( ph  /\  z  e.  U. dom  P ) 
 ->  ( z  e.  ( XRV/𝑐 R A )  <->  ( X `  z ) R A ) )
 
Theoremorrvcval4 24675* The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  RR  |  y R A } ) )
 
Theoremorrvcoel 24676* If the relation produces open sets, preimage maps of a random variable are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  RR  |  y R A }  e.  ( topGen `
  ran  (,) ) )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  dom  P )
 
Theoremorrvccel 24677* If the relation produces closed sets, preimage maps are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  RR  |  y R A }  e.  ( Clsd `  ( topGen `  ran  (,) ) ) )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  dom  P )
 
Theoremorvcgteel 24678 Preimage maps produced by the "greater than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( XRV/𝑐 `' 
 <_  A )  e.  dom  P )
 
19.3.15.5  Distribution Functions
 
Theoremorvcelval 24679 Preimage maps produced by the "elementhood" relation (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " A ) )
 
Theoremorvcelel 24680 Preimage maps produced by the "elementhood" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( XRV/𝑐  _E  A )  e.  dom  P )
 
Theoremdstrvval 24681* The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  D  =  ( a  e. 𝔅 
 |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( D `  A )  =  ( P `  ( `' X " A ) ) )
 
Theoremdstrvprob 24682* The distribution of a random variable is a probability law (TODO: could be shortened using dstrvval 24681) (Contributed by Thierry Arnoux, 10-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  D  =  ( a  e. 𝔅 
 |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )   =>    |-  ( ph  ->  D  e. Prob )
 
19.3.15.6  Cumulative Distribution Functions
 
Theoremorvclteel 24683 Preimage maps produced by the "lower than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( XRV/𝑐  <_  A )  e.  dom  P )
 
Theoremdstfrvel 24684 Elementhood of preimage maps produced by the "lower than or equal" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  U. dom  P )   &    |-  ( ph  ->  ( X `  B )  <_  A )   =>    |-  ( ph  ->  B  e.  ( XRV/𝑐  <_  A ) )
 
Theoremdstfrvunirn 24685* The limit of all preimage maps by the "lower than or equal" relation is the universe. (Contributed by Thierry Arnoux, 12-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  U.
 ran  ( n  e. 
 NN  |->  ( XRV/𝑐  <_  n ) )  = 
 U. dom  P )
 
Theoremorvclteinc 24686 Preimage maps produced by the "lower than or equal" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( XRV/𝑐  <_  A )  C_  ( XRV/𝑐  <_  B ) )
 
Theoremdstfrvinc 24687* A cumulative distribution function is non-decreasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  F  =  ( x  e.  RR  |->  ( P `  ( XRV/𝑐  <_  x ) ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( F `  A )  <_  ( F `  B ) )
 
Theoremdstfrvclim1 24688* The limit of the cumulative distribution function is one. (Contributed by Thierry Arnoux, 12-Feb-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  F  =  ( x  e.  RR  |->  ( P `  ( XRV/𝑐  <_  x ) ) ) )   =>    |-  ( ph  ->  F  ~~>  1 )
 
19.3.15.7  Probabilities - example
 
Theoremcoinfliplem 24689 Division in the extended real numbers can be used for the coin-flip example. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  P  =  ( ( #  |`  ~P { H ,  T } )𝑓/𝑐 /𝑒  2 )
 
Theoremcoinflipprob 24690 The  P we defined for coin-flip is a probability law. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  P  e. Prob
 
Theoremcoinflipspace 24691 The space of our coin-flip probability (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |- 
 dom  P  =  ~P { H ,  T }
 
Theoremcoinflipuniv 24692 The universe of our coin-flip probability is  { H ,  T }. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |- 
 U. dom  P  =  { H ,  T }
 
Theoremcoinfliprv 24693 The  X we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  X  e.  (rRndVar `  P )
 
Theoremcoinflippv 24694 The probability of heads is one-half. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  ( P `  { H } )  =  (
 1  /  2 )
 
Theoremcoinflippvt 24695 The probability of tails is one-half. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  ( P `  { T } )  =  (
 1  /  2 )
 
19.3.15.8  Bertrand's Ballot Problem
 
Theoremballotlemoex 24696*  O is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  O  e.  _V
 
Theoremballotlem1 24697* The size of the universe is a binomial coefficient. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  ( # `  O )  =  ( ( M  +  N )  _C  M )
 
Theoremballotlemelo 24698* Elementhood in  O. (Contributed by Thierry Arnoux, 17-Apr-2017.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   =>    |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N ) )  /\  ( # `  C )  =  M ) )
 
Theoremballotlem2 24699* The probability that the first vote picked in a count is a B (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   =>    |-  ( P `  { c  e.  O  |  -.  1  e.  c } )  =  ( N  /  ( M  +  N ) )
 
Theoremballotlemfval 24700* The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)
 |-  M  e.  NN   &    |-  N  e.  NN   &    |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
 )  |  ( # `  c )  =  M }   &    |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `  O ) ) )   &    |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
 ( 1 ... i
 )  i^i  c )
 )  -  ( # `  ( ( 1 ... i )  \  c
 ) ) ) ) )   &    |-  ( ph  ->  C  e.  O )   &    |-  ( ph  ->  J  e.  ZZ )   =>    |-  ( ph  ->  (
 ( F `  C ) `  J )  =  ( ( # `  (
 ( 1 ... J )  i^i  C ) )  -  ( # `  (
 ( 1 ... J )  \  C ) ) ) )
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