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Theorem List for Metamath Proof Explorer - 32701-32800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsibfof 32701 Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   πΆ = (Baseβ€˜πΎ)    &   (πœ‘ β†’ π‘Š ∈ TopSp)    &   (πœ‘ β†’ + :(𝐡 Γ— 𝐡)⟢𝐢)    &   (πœ‘ β†’ 𝐺 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ 𝐾 ∈ TopSp)    &   (πœ‘ β†’ 𝐽 ∈ Fre)    &   (πœ‘ β†’ ( 0 + 0 ) = (0gβ€˜πΎ))    β‡’   (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀))
 
Theoremsitgfval 32702* Value of the Bochner integral for a simple function 𝐹. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    β‡’   (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜πΉ) = (π‘Š Ξ£g (π‘₯ ∈ (ran 𝐹 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))) Β· π‘₯))))
 
Theoremsitgclg 32703* Closure of the Bochner integral on simple functions, generic version. See sitgclbn 32704 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   πΊ = (Scalarβ€˜π‘Š)    &   π· = ((distβ€˜πΊ) β†Ύ ((Baseβ€˜πΊ) Γ— (Baseβ€˜πΊ)))    &   (πœ‘ β†’ π‘Š ∈ TopSp)    &   (πœ‘ β†’ π‘Š ∈ CMnd)    &   (πœ‘ β†’ (Scalarβ€˜π‘Š) ∈ ℝExt )    &   ((πœ‘ ∧ π‘š ∈ (𝐻 β€œ (0[,)+∞)) ∧ π‘₯ ∈ 𝐡) β†’ (π‘š Β· π‘₯) ∈ 𝐡)    β‡’   (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜πΉ) ∈ 𝐡)
 
Theoremsitgclbn 32704 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.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ π‘Š ∈ Ban)    &   (πœ‘ β†’ (Scalarβ€˜π‘Š) ∈ ℝExt )    β‡’   (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜πΉ) ∈ 𝐡)
 
Theoremsitgclcn 32705 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.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ π‘Š ∈ Ban)    &   (πœ‘ β†’ (Scalarβ€˜π‘Š) = β„‚fld)    β‡’   (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜πΉ) ∈ 𝐡)
 
Theoremsitgclre 32706 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.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ π‘Š ∈ Ban)    &   (πœ‘ β†’ (Scalarβ€˜π‘Š) = ℝfld)    β‡’   (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜πΉ) ∈ 𝐡)
 
Theoremsitg0 32707 The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-Mar-2018.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ π‘Š ∈ TopSp)    &   (πœ‘ β†’ π‘Š ∈ Mnd)    β‡’   (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜(βˆͺ dom 𝑀 Γ— { 0 })) = 0 )
 
Theoremsitgf 32708* The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   ((πœ‘ ∧ 𝑓 ∈ dom (π‘Šsitg𝑀)) β†’ ((π‘Šsitg𝑀)β€˜π‘“) ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘Šsitg𝑀):dom (π‘Šsitg𝑀)⟢𝐡)
 
Theoremsitgaddlemb 32709 Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019.)
𝐡 = (Baseβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   π‘† = (sigaGenβ€˜π½)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π» = (ℝHomβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ π‘Š ∈ TopSp)    &   (πœ‘ β†’ (π‘Š β†Ύv (𝐻 β€œ (0[,)+∞))) ∈ SLMod)    &   (πœ‘ β†’ 𝐽 ∈ Fre)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ 𝐺 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ (Scalarβ€˜π‘Š) ∈ ℝExt )    &    + = (+gβ€˜π‘Š)    β‡’   ((πœ‘ ∧ 𝑝 ∈ ((ran 𝐹 Γ— ran 𝐺) βˆ– {⟨ 0 , 0 ⟩})) β†’ ((π»β€˜(π‘€β€˜((◑𝐹 β€œ {(1st β€˜π‘)}) ∩ (◑𝐺 β€œ {(2nd β€˜π‘)})))) Β· (2nd β€˜π‘)) ∈ 𝐡)
 
Theoremsitmval 32710* Value of the simple function integral metric for a given space π‘Š and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐷 = (distβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    β‡’   (πœ‘ β†’ (π‘Šsitm𝑀) = (𝑓 ∈ dom (π‘Šsitg𝑀), 𝑔 ∈ dom (π‘Šsitg𝑀) ↦ (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝑓 ∘f 𝐷𝑔))))
 
Theoremsitmfval 32711 Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐷 = (distβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ 𝐺 ∈ dom (π‘Šsitg𝑀))    β‡’   (πœ‘ β†’ (𝐹(π‘Šsitm𝑀)𝐺) = (((ℝ*𝑠 β†Ύs (0[,]+∞))sitg𝑀)β€˜(𝐹 ∘f 𝐷𝐺)))
 
Theoremsitmcl 32712 Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
(πœ‘ β†’ π‘Š ∈ Mnd)    &   (πœ‘ β†’ π‘Š ∈ ∞MetSp)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    &   (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))    &   (πœ‘ β†’ 𝐺 ∈ dom (π‘Šsitg𝑀))    β‡’   (πœ‘ β†’ (𝐹(π‘Šsitm𝑀)𝐺) ∈ (0[,]+∞))
 
Theoremsitmf 32713 The integral metric as a function. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(πœ‘ β†’ π‘Š ∈ Mnd)    &   (πœ‘ β†’ π‘Š ∈ ∞MetSp)    &   (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)    β‡’   (πœ‘ β†’ (π‘Šsitm𝑀):(dom (π‘Šsitg𝑀) Γ— dom (π‘Šsitg𝑀))⟢(0[,]+∞))
 
Definitiondf-itgm 32714* 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 (𝑀sitmπ‘š).

He then defines the Bochner integral in chapter 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 24909.

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

itgm = (𝑀 ∈ V, π‘š ∈ βˆͺ ran measures ↦ (((metUnifβ€˜(𝑀sitmπ‘š))CnExt(UnifStβ€˜π‘€))β€˜(𝑀sitgπ‘š)))
 
21.3.19  Euler's partition theorem
 
Theoremoddpwdc 32715* Lemma for eulerpart 32743. The function 𝐹 that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.)
𝐽 = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    β‡’   πΉ:(𝐽 Γ— β„•0)–1-1-ontoβ†’β„•
 
Theoremoddpwdcv 32716* Lemma for eulerpart 32743: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.)
𝐽 = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    β‡’   (π‘Š ∈ (𝐽 Γ— β„•0) β†’ (πΉβ€˜π‘Š) = ((2↑(2nd β€˜π‘Š)) Β· (1st β€˜π‘Š)))
 
Theoremeulerpartlemsv1 32717* Lemma for eulerpart 32743. Value of the sum of a partition 𝐴. (Contributed by Thierry Arnoux, 26-Aug-2018.)
𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (π‘†β€˜π΄) = Ξ£π‘˜ ∈ β„• ((π΄β€˜π‘˜) Β· π‘˜))
 
Theoremeulerpartlemelr 32718* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 8-Aug-2018.)
𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (𝐴:β„•βŸΆβ„•0 ∧ (◑𝐴 β€œ β„•) ∈ Fin))
 
Theoremeulerpartlemsv2 32719* Lemma for eulerpart 32743. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.)
𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (π‘†β€˜π΄) = Ξ£π‘˜ ∈ (◑𝐴 β€œ β„•)((π΄β€˜π‘˜) Β· π‘˜))
 
Theoremeulerpartlemsf 32720* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 8-Aug-2018.)
𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   π‘†:((β„•0 ↑m β„•) ∩ 𝑅)βŸΆβ„•0
 
Theoremeulerpartlems 32721* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 6-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„€β‰₯β€˜((π‘†β€˜π΄) + 1))) β†’ (π΄β€˜π‘‘) = 0)
 
Theoremeulerpartlemsv3 32722* Lemma for eulerpart 32743. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.)
𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (π‘†β€˜π΄) = Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
 
Theoremeulerpartlemgc 32723* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 9-Aug-2018.)
𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((2↑𝑛) Β· 𝑑) ≀ (π‘†β€˜π΄))
 
Theoremeulerpartleme 32724* Lemma for eulerpart 32743. (Contributed by Mario Carneiro, 26-Jan-2015.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    β‡’   (𝐴 ∈ 𝑃 ↔ (𝐴:β„•βŸΆβ„•0 ∧ (◑𝐴 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π΄β€˜π‘˜) Β· π‘˜) = 𝑁))
 
Theoremeulerpartlemv 32725* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 19-Aug-2018.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    β‡’   (𝐴 ∈ 𝑃 ↔ (𝐴:β„•βŸΆβ„•0 ∧ (◑𝐴 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ (◑𝐴 β€œ β„•)((π΄β€˜π‘˜) Β· π‘˜) = 𝑁))
 
Theoremeulerpartlemo 32726* Lemma for eulerpart 32743: 𝑂 is the set of odd partitions of 𝑁. (Contributed by Thierry Arnoux, 10-Aug-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    β‡’   (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ βˆ€π‘› ∈ (◑𝐴 β€œ β„•) Β¬ 2 βˆ₯ 𝑛))
 
Theoremeulerpartlemd 32727* Lemma for eulerpart 32743: 𝐷 is the set of distinct part. of 𝑁. (Contributed by Thierry Arnoux, 11-Aug-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    β‡’   (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 β€œ β„•) βŠ† {0, 1}))
 
Theoremeulerpartlem1 32728* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    β‡’   π‘€:𝐻–1-1-ontoβ†’(𝒫 (𝐽 Γ— β„•0) ∩ Fin)
 
Theoremeulerpartlemb 32729* Lemma for eulerpart 32743. The set of all partitions of 𝑁 is finite. (Contributed by Mario Carneiro, 26-Jan-2015.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    β‡’   π‘ƒ ∈ Fin
 
Theoremeulerpartlemt0 32730* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    β‡’   (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (β„•0 ↑m β„•) ∧ (◑𝐴 β€œ β„•) ∈ Fin ∧ (◑𝐴 β€œ β„•) βŠ† 𝐽))
 
Theoremeulerpartlemf 32731* Lemma for eulerpart 32743: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    β‡’   ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– 𝐽)) β†’ (π΄β€˜π‘‘) = 0)
 
Theoremeulerpartlemt 32732* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    β‡’   ((β„•0 ↑m 𝐽) ∩ 𝑅) = ran (π‘š ∈ (𝑇 ∩ 𝑅) ↦ (π‘š β†Ύ 𝐽))
 
Theoremeulerpartgbij 32733* Lemma for eulerpart 32743: The 𝐺 function is a bijection. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    β‡’   πΊ:(𝑇 ∩ 𝑅)–1-1-ontoβ†’(({0, 1} ↑m β„•) ∩ 𝑅)
 
Theoremeulerpartlemgv 32734* Lemma for eulerpart 32743: value of the function 𝐺. (Contributed by Thierry Arnoux, 13-Nov-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    β‡’   (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (πΊβ€˜π΄) = ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (𝐴 β†Ύ 𝐽))))))
 
Theoremeulerpartlemr 32735* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 13-Nov-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    β‡’   π‘‚ = ((𝑇 ∩ 𝑅) ∩ 𝑃)
 
Theoremeulerpartlemmf 32736* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 30-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    β‡’   (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (bits ∘ (𝐴 β†Ύ 𝐽)) ∈ 𝐻)
 
Theoremeulerpartlemgvv 32737* Lemma for eulerpart 32743: value of the function 𝐺 evaluated. (Contributed by Thierry Arnoux, 10-Aug-2018.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    β‡’   ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝐡 ∈ β„•) β†’ ((πΊβ€˜π΄)β€˜π΅) = if(βˆƒπ‘‘ ∈ β„• βˆƒπ‘› ∈ (bitsβ€˜(π΄β€˜π‘‘))((2↑𝑛) Β· 𝑑) = 𝐡, 1, 0))
 
Theoremeulerpartlemgu 32738* Lemma for eulerpart 32743: Rewriting the π‘ˆ set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9306. (Contributed by Thierry Arnoux, 10-Aug-2018.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    &   π‘ˆ = βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘)))    β‡’   (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ π‘ˆ = {βŸ¨π‘‘, π‘›βŸ© ∣ (𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)β€˜π‘‘))})
 
Theoremeulerpartlemgh 32739* Lemma for eulerpart 32743: The 𝐹 function is a bijection on the π‘ˆ subsets. (Contributed by Thierry Arnoux, 15-Aug-2018.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    &   π‘ˆ = βˆͺ 𝑑 ∈ ((◑𝐴 β€œ β„•) ∩ 𝐽)({𝑑} Γ— (bitsβ€˜(π΄β€˜π‘‘)))    β‡’   (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (𝐹 β†Ύ π‘ˆ):π‘ˆβ€“1-1-ontoβ†’{π‘š ∈ β„• ∣ βˆƒπ‘‘ ∈ β„• βˆƒπ‘› ∈ (bitsβ€˜(π΄β€˜π‘‘))((2↑𝑛) Β· 𝑑) = π‘š})
 
Theoremeulerpartlemgf 32740* Lemma for eulerpart 32743: Images under 𝐺 have finite support. (Contributed by Thierry Arnoux, 29-Aug-2018.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    β‡’   (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (β—‘(πΊβ€˜π΄) β€œ β„•) ∈ Fin)
 
Theoremeulerpartlemgs2 32741* Lemma for eulerpart 32743: The 𝐺 function also preserves partition sums. (Contributed by Thierry Arnoux, 10-Sep-2017.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   (𝐴 ∈ (𝑇 ∩ 𝑅) β†’ (π‘†β€˜(πΊβ€˜π΄)) = (π‘†β€˜π΄))
 
Theoremeulerpartlemn 32742* Lemma for eulerpart 32743. (Contributed by Thierry Arnoux, 30-Aug-2018.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    &   π½ = {𝑧 ∈ β„• ∣ Β¬ 2 βˆ₯ 𝑧}    &   πΉ = (π‘₯ ∈ 𝐽, 𝑦 ∈ β„•0 ↦ ((2↑𝑦) Β· π‘₯))    &   π» = {π‘Ÿ ∈ ((𝒫 β„•0 ∩ Fin) ↑m 𝐽) ∣ (π‘Ÿ supp βˆ…) ∈ Fin}    &   π‘€ = (π‘Ÿ ∈ 𝐻 ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ (π‘Ÿβ€˜π‘₯))})    &   π‘… = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π‘‡ = {𝑓 ∈ (β„•0 ↑m β„•) ∣ (◑𝑓 β€œ β„•) βŠ† 𝐽}    &   πΊ = (π‘œ ∈ (𝑇 ∩ 𝑅) ↦ ((πŸ­β€˜β„•)β€˜(𝐹 β€œ (π‘€β€˜(bits ∘ (π‘œ β†Ύ 𝐽))))))    &   π‘† = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))    β‡’   (𝐺 β†Ύ 𝑂):𝑂–1-1-onto→𝐷
 
Theoremeulerpart 32743* Euler's theorem on partitions, also known as a special case of Glaisher's theorem. Let 𝑃 be the set of all partitions of 𝑁, represented as multisets of positive integers, which is to say functions from β„• to β„•0 where the value of the function represents the number of repetitions of an individual element, and the sum of all the elements with repetition equals 𝑁. Then the set 𝑂 of all partitions that only consist of odd numbers and the set 𝐷 of all partitions which have no repeated elements have the same cardinality. This is Metamath 100 proof #45. (Contributed by Thierry Arnoux, 14-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑃 = {𝑓 ∈ (β„•0 ↑m β„•) ∣ ((◑𝑓 β€œ β„•) ∈ Fin ∧ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜) = 𝑁)}    &   π‘‚ = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ (◑𝑔 β€œ β„•) Β¬ 2 βˆ₯ 𝑛}    &   π· = {𝑔 ∈ 𝑃 ∣ βˆ€π‘› ∈ β„• (π‘”β€˜π‘›) ≀ 1}    β‡’   (β™―β€˜π‘‚) = (β™―β€˜π·)
 
21.3.20  Sequences defined by strong recursion
 
Syntaxcsseq 32744 Sequences defined by strong recursion.
class seqstr
 
Definitiondf-sseq 32745* Define a builder for sequences by strong recursion, i.e., by computing the value of the n-th element of the sequence from all preceding elements and not just the previous one. (Contributed by Thierry Arnoux, 21-Apr-2019.)
seqstr = (π‘š ∈ V, 𝑓 ∈ V ↦ (π‘š βˆͺ (lastS ∘ seq(β™―β€˜π‘š)((π‘₯ ∈ V, 𝑦 ∈ V ↦ (π‘₯ ++ βŸ¨β€œ(π‘“β€˜π‘₯)β€βŸ©)), (β„•0 Γ— {(π‘š ++ βŸ¨β€œ(π‘“β€˜π‘š)β€βŸ©)})))))
 
Theoremsubiwrd 32746 Lemma for sseqp1 32756. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝐹:β„•0βŸΆπ‘†)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝐹 β†Ύ (0..^𝑁)) ∈ Word 𝑆)
 
Theoremsubiwrdlen 32747 Length of a subword of an infinite word. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝐹:β„•0βŸΆπ‘†)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (β™―β€˜(𝐹 β†Ύ (0..^𝑁))) = 𝑁)
 
Theoremiwrdsplit 32748 Lemma for sseqp1 32756. (Contributed by Thierry Arnoux, 25-Apr-2019.) (Proof shortened by AV, 14-Oct-2022.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝐹:β„•0βŸΆπ‘†)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝐹 β†Ύ (0..^(𝑁 + 1))) = ((𝐹 β†Ύ (0..^𝑁)) ++ βŸ¨β€œ(πΉβ€˜π‘)β€βŸ©))
 
Theoremsseqval 32749* Value of the strong sequence builder function. The set π‘Š represents here the words of length greater than or equal to the lenght of the initial sequence 𝑀. (Contributed by Thierry Arnoux, 21-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    β‡’   (πœ‘ β†’ (𝑀seqstr𝐹) = (𝑀 βˆͺ (lastS ∘ seq(β™―β€˜π‘€)((π‘₯ ∈ V, 𝑦 ∈ V ↦ (π‘₯ ++ βŸ¨β€œ(πΉβ€˜π‘₯)β€βŸ©)), (β„•0 Γ— {(𝑀 ++ βŸ¨β€œ(πΉβ€˜π‘€)β€βŸ©)})))))
 
Theoremsseqfv1 32750 Value of the strong sequence builder function at one of its initial values. (Contributed by Thierry Arnoux, 21-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    &   (πœ‘ β†’ 𝑁 ∈ (0..^(β™―β€˜π‘€)))    β‡’   (πœ‘ β†’ ((𝑀seqstr𝐹)β€˜π‘) = (π‘€β€˜π‘))
 
Theoremsseqfn 32751 A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    β‡’   (πœ‘ β†’ (𝑀seqstr𝐹) Fn β„•0)
 
Theoremsseqmw 32752 Lemma for sseqf 32753 amd sseqp1 32756. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    β‡’   (πœ‘ β†’ 𝑀 ∈ π‘Š)
 
Theoremsseqf 32753 A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.) (Proof shortened by AV, 7-Mar-2022.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    β‡’   (πœ‘ β†’ (𝑀seqstr𝐹):β„•0βŸΆπ‘†)
 
Theoremsseqfres 32754 The first elements in the strong recursive sequence are the sequence initializer. (Contributed by Thierry Arnoux, 23-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    β‡’   (πœ‘ β†’ ((𝑀seqstr𝐹) β†Ύ (0..^(β™―β€˜π‘€))) = 𝑀)
 
Theoremsseqfv2 32755* Value of the strong sequence builder function. (Contributed by Thierry Arnoux, 21-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(β™―β€˜π‘€)))    β‡’   (πœ‘ β†’ ((𝑀seqstr𝐹)β€˜π‘) = (lastSβ€˜(seq(β™―β€˜π‘€)((π‘₯ ∈ V, 𝑦 ∈ V ↦ (π‘₯ ++ βŸ¨β€œ(πΉβ€˜π‘₯)β€βŸ©)), (β„•0 Γ— {(𝑀 ++ βŸ¨β€œ(πΉβ€˜π‘€)β€βŸ©)}))β€˜π‘)))
 
Theoremsseqp1 32756 Value of the strong sequence builder function at a successor. (Contributed by Thierry Arnoux, 24-Apr-2019.)
(πœ‘ β†’ 𝑆 ∈ V)    &   (πœ‘ β†’ 𝑀 ∈ Word 𝑆)    &   π‘Š = (Word 𝑆 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜π‘€))))    &   (πœ‘ β†’ 𝐹:π‘ŠβŸΆπ‘†)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(β™―β€˜π‘€)))    β‡’   (πœ‘ β†’ ((𝑀seqstr𝐹)β€˜π‘) = (πΉβ€˜((𝑀seqstr𝐹) β†Ύ (0..^𝑁))))
 
21.3.21  Fibonacci Numbers
 
Syntaxcfib 32757 The Fibonacci sequence.
class Fibci
 
Definitiondf-fib 32758 Define the Fibonacci sequence, where that each element is the sum of the two preceding ones, starting from 0 and 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)
Fibci = (βŸ¨β€œ01β€βŸ©seqstr(𝑀 ∈ (Word β„•0 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜2))) ↦ ((π‘€β€˜((β™―β€˜π‘€) βˆ’ 2)) + (π‘€β€˜((β™―β€˜π‘€) βˆ’ 1)))))
 
Theoremfiblem 32759 Lemma for fib0 32760, fib1 32761 and fibp1 32762. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝑀 ∈ (Word β„•0 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜2))) ↦ ((π‘€β€˜((β™―β€˜π‘€) βˆ’ 2)) + (π‘€β€˜((β™―β€˜π‘€) βˆ’ 1)))):(Word β„•0 ∩ (β—‘β™― β€œ (β„€β‰₯β€˜(β™―β€˜βŸ¨β€œ01β€βŸ©))))βŸΆβ„•0
 
Theoremfib0 32760 Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibciβ€˜0) = 0
 
Theoremfib1 32761 Value of the Fibonacci sequence at index 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibciβ€˜1) = 1
 
Theoremfibp1 32762 Value of the Fibonacci sequence at higher indices. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝑁 ∈ β„• β†’ (Fibciβ€˜(𝑁 + 1)) = ((Fibciβ€˜(𝑁 βˆ’ 1)) + (Fibciβ€˜π‘)))
 
Theoremfib2 32763 Value of the Fibonacci sequence at index 2. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibciβ€˜2) = 1
 
Theoremfib3 32764 Value of the Fibonacci sequence at index 3. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibciβ€˜3) = 2
 
Theoremfib4 32765 Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibciβ€˜4) = 3
 
Theoremfib5 32766 Value of the Fibonacci sequence at index 5. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibciβ€˜5) = 5
 
Theoremfib6 32767 Value of the Fibonacci sequence at index 6. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibciβ€˜6) = 8
 
21.3.22  Probability
 
21.3.22.1  Probability Theory
 
Syntaxcprb 32768 Extend class notation to include the class of probability measures.
class Prob
 
Definitiondf-prob 32769 Define the class of probability measures as the set of measures with total measure 1. (Contributed by Thierry Arnoux, 14-Sep-2016.)
Prob = {𝑝 ∈ βˆͺ ran measures ∣ (π‘β€˜βˆͺ dom 𝑝) = 1}
 
Theoremelprob 32770 The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝑃 ∈ Prob ↔ (𝑃 ∈ βˆͺ ran measures ∧ (π‘ƒβ€˜βˆͺ dom 𝑃) = 1))
 
Theoremdomprobmeas 32771 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
(𝑃 ∈ Prob β†’ 𝑃 ∈ (measuresβ€˜dom 𝑃))
 
Theoremdomprobsiga 32772 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
(𝑃 ∈ Prob β†’ dom 𝑃 ∈ βˆͺ ran sigAlgebra)
 
Theoremprobtot 32773 The probability of the universe set is 1. Second axiom of Kolmogorov. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝑃 ∈ Prob β†’ (π‘ƒβ€˜βˆͺ dom 𝑃) = 1)
 
Theoremprob01 32774 A probability is an element of [ 0 , 1 ]. First axiom of Kolmogorov. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) β†’ (π‘ƒβ€˜π΄) ∈ (0[,]1))
 
Theoremprobnul 32775 The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑃 ∈ Prob β†’ (π‘ƒβ€˜βˆ…) = 0)
 
Theoremunveldomd 32776 The universe is an element of the domain of the probability, the universe (entire probability space) being βˆͺ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(πœ‘ β†’ 𝑃 ∈ Prob)    β‡’   (πœ‘ β†’ βˆͺ dom 𝑃 ∈ dom 𝑃)
 
Theoremunveldom 32777 The universe is an element of the domain of the probability, the universe (entire probability space) being βˆͺ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝑃 ∈ Prob β†’ βˆͺ dom 𝑃 ∈ dom 𝑃)
 
Theoremnuleldmp 32778 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝑃 ∈ Prob β†’ βˆ… ∈ dom 𝑃)
 
Theoremprobcun 32779* The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the Ξ£ construct cannot be used as it can handle infinite indexing set only if they are subsets of β„€, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘ƒβ€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘ƒβ€˜π‘₯))
 
Theoremprobun 32780 The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ ((𝐴 ∩ 𝐡) = βˆ… β†’ (π‘ƒβ€˜(𝐴 βˆͺ 𝐡)) = ((π‘ƒβ€˜π΄) + (π‘ƒβ€˜π΅))))
 
Theoremprobdif 32781 The probability of the difference of two event sets. (Contributed by Thierry Arnoux, 12-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ (π‘ƒβ€˜(𝐴 βˆ– 𝐡)) = ((π‘ƒβ€˜π΄) βˆ’ (π‘ƒβ€˜(𝐴 ∩ 𝐡))))
 
Theoremprobinc 32782 A probability law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
(((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ 𝐴 βŠ† 𝐡) β†’ (π‘ƒβ€˜π΄) ≀ (π‘ƒβ€˜π΅))
 
Theoremprobdsb 32783 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 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) β†’ (π‘ƒβ€˜(βˆͺ dom 𝑃 βˆ– 𝐴)) = (1 βˆ’ (π‘ƒβ€˜π΄)))
 
Theoremprobmeasd 32784 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(πœ‘ β†’ 𝑃 ∈ Prob)    β‡’   (πœ‘ β†’ 𝑃 ∈ βˆͺ ran measures)
 
Theoremprobvalrnd 32785 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(πœ‘ β†’ 𝑃 ∈ Prob)    &   (πœ‘ β†’ 𝐴 ∈ dom 𝑃)    β‡’   (πœ‘ β†’ (π‘ƒβ€˜π΄) ∈ ℝ)
 
Theoremprobtotrnd 32786 The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(πœ‘ β†’ 𝑃 ∈ Prob)    β‡’   (πœ‘ β†’ (π‘ƒβ€˜βˆͺ dom 𝑃) ∈ ℝ)
 
Theoremtotprobd 32787* Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(πœ‘ β†’ 𝑃 ∈ Prob)    &   (πœ‘ β†’ 𝐴 ∈ dom 𝑃)    &   (πœ‘ β†’ 𝐡 ∈ 𝒫 dom 𝑃)    &   (πœ‘ β†’ βˆͺ 𝐡 = βˆͺ dom 𝑃)    &   (πœ‘ β†’ 𝐡 β‰Ό Ο‰)    &   (πœ‘ β†’ Disj 𝑏 ∈ 𝐡 𝑏)    β‡’   (πœ‘ β†’ (π‘ƒβ€˜π΄) = Ξ£*𝑏 ∈ 𝐡(π‘ƒβ€˜(𝑏 ∩ 𝐴)))
 
Theoremtotprob 32788* Law of total probability. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ (βˆͺ 𝐡 = βˆͺ dom 𝑃 ∧ 𝐡 ∈ 𝒫 dom 𝑃 ∧ (𝐡 β‰Ό Ο‰ ∧ Disj 𝑏 ∈ 𝐡 𝑏))) β†’ (π‘ƒβ€˜π΄) = Ξ£*𝑏 ∈ 𝐡(π‘ƒβ€˜(𝑏 ∩ 𝐴)))
 
Theoremprobfinmeasb 32789 Build a probability measure from a finite measure. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ (π‘€β€˜βˆͺ 𝑆) ∈ ℝ+) β†’ (𝑀 ∘f/c /𝑒 (π‘€β€˜βˆͺ 𝑆)) ∈ Prob)
 
TheoremprobfinmeasbALTV 32790* Alternate version of probfinmeasb 32789. (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ (π‘€β€˜βˆͺ 𝑆) ∈ ℝ+) β†’ (π‘₯ ∈ 𝑆 ↦ ((π‘€β€˜π‘₯) /𝑒 (π‘€β€˜βˆͺ 𝑆))) ∈ Prob)
 
Theoremprobmeasb 32791* Build a probability from a measure and a set with finite measure. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝑆 ∧ (π‘€β€˜π΄) ∈ ℝ+) β†’ (π‘₯ ∈ 𝑆 ↦ ((π‘€β€˜(π‘₯ ∩ 𝐴)) / (π‘€β€˜π΄))) ∈ Prob)
 
21.3.22.2  Conditional Probabilities
 
Syntaxccprob 32792 Extends class notation with the conditional probability builder.
class cprob
 
Definitiondf-cndprob 32793* Define the conditional probability. (Contributed by Thierry Arnoux, 14-Sep-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
cprob = (𝑝 ∈ Prob ↦ (π‘Ž ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((π‘β€˜(π‘Ž ∩ 𝑏)) / (π‘β€˜π‘))))
 
Theoremcndprobval 32794 The value of the conditional probability , i.e. the probability for the event 𝐴, given 𝐡, under the probability law 𝑃. (Contributed by Thierry Arnoux, 21-Jan-2017.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) β†’ ((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩) = ((π‘ƒβ€˜(𝐴 ∩ 𝐡)) / (π‘ƒβ€˜π΅)))
 
Theoremcndprobin 32795 An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
(((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘ƒβ€˜π΅) β‰  0) β†’ (((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩) Β· (π‘ƒβ€˜π΅)) = (π‘ƒβ€˜(𝐴 ∩ 𝐡)))
 
Theoremcndprob01 32796 The conditional probability has values in [0, 1]. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
(((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘ƒβ€˜π΅) β‰  0) β†’ ((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩) ∈ (0[,]1))
 
Theoremcndprobtot 32797 The conditional probability given a certain event is one. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ (π‘ƒβ€˜π΄) β‰  0) β†’ ((cprobβ€˜π‘ƒ)β€˜βŸ¨βˆͺ dom 𝑃, 𝐴⟩) = 1)
 
Theoremcndprobnul 32798 The conditional probability given empty event is zero. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ (π‘ƒβ€˜π΄) β‰  0) β†’ ((cprobβ€˜π‘ƒ)β€˜βŸ¨βˆ…, 𝐴⟩) = 0)
 
Theoremcndprobprob 32799* The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
((𝑃 ∈ Prob ∧ 𝐡 ∈ dom 𝑃 ∧ (π‘ƒβ€˜π΅) β‰  0) β†’ (π‘Ž ∈ dom 𝑃 ↦ ((cprobβ€˜π‘ƒ)β€˜βŸ¨π‘Ž, 𝐡⟩)) ∈ Prob)
 
Theorembayesth 32800 Bayes Theorem. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
(((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐡 ∈ dom 𝑃) ∧ (π‘ƒβ€˜π΄) β‰  0 ∧ (π‘ƒβ€˜π΅) β‰  0) β†’ ((cprobβ€˜π‘ƒ)β€˜βŸ¨π΄, 𝐡⟩) = ((((cprobβ€˜π‘ƒ)β€˜βŸ¨π΅, 𝐴⟩) Β· (π‘ƒβ€˜π΄)) / (π‘ƒβ€˜π΅)))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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