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Theorem List for Metamath Proof Explorer - 32301-32400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempmeasadd 32301* A premeasure on a ring of sets is additive on disjoint countable collections. This is called sigma-additivity. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(𝜑𝑃:𝑅⟶(0[,]+∞))    &   (𝜑 → (𝑃‘∅) = 0)    &   ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥𝑅Disj 𝑦𝑥 𝑦)) → (𝑃 𝑥) = Σ*𝑦𝑥(𝑃𝑦))    &   𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}    &   (𝜑𝑅𝑄)    &   (𝜑𝐴 ≼ ω)    &   ((𝜑𝑘𝐴) → 𝐵𝑅)    &   (𝜑Disj 𝑘𝐴 𝐵)       (𝜑 → (𝑃 𝑘𝐴 𝐵) = Σ*𝑘𝐴(𝑃𝐵))
 
20.3.18  Integration
 
20.3.18.1  Lebesgue integral - misc additions
 
Theoremitgeq12dv 32302* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → 𝐶 = 𝐷)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
 
20.3.18.2  Bochner integral
 
Syntaxcitgm 32303 Extend class notation with the (measure) Bochner integral.
class itgm
 
Syntaxcsitm 32304 Extend class notation with the integral metric for simple functions.
class sitm
 
Syntaxcsitg 32305 Extend class notation with the integral of simple functions.
class sitg
 
Definitiondf-sitg 32306* Define the integral of simple functions from a measurable space dom 𝑚 to a generic space 𝑤 equipped with the right scalar product. 𝑤 will later be required to be a Banach space.

These simple functions are required to take finitely many different values: this is expressed by ran 𝑔 ∈ Fin in the definition.

Moreover, for each 𝑥, the pre-image (𝑔 “ {𝑥}) is requested to be measurable, of finite measure.

In this definition, (sigaGen‘(TopOpen‘𝑤)) is the Borel sigma-algebra on 𝑤, and the functions 𝑔 range over the measurable functions over that Borel algebra.

Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.)

sitg = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥)))))
 
Definitiondf-sitm 32307* Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in *, the range structure for this integral is (ℝ*𝑠s (0[,]+∞)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.)
sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓f (dist‘𝑤)𝑔))))
 
Theoremsitgval 32308* Value of the simple function integral builder for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)       (𝜑 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)))))
 
Theoremissibf 32309* 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)       (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
 
Theoremsibf0 32310 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ Mnd)       (𝜑 → ( dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))
 
Theoremsibfmbl 32311 A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
 
Theoremsibff 32312 A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑𝐹: dom 𝑀 𝐽)
 
Theoremsibfrn 32313 A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑 → ran 𝐹 ∈ Fin)
 
Theoremsibfima 32314 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       ((𝜑𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝐴})) ∈ (0[,)+∞))
 
Theoremsibfinima 32315 The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝐽 ∈ Fre)       (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞))
 
Theoremsibfof 32316 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.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   𝐶 = (Base‘𝐾)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐾 ∈ TopSp)    &   (𝜑𝐽 ∈ Fre)    &   (𝜑 → ( 0 + 0 ) = (0g𝐾))       (𝜑 → (𝐹f + 𝐺) ∈ dom (𝐾sitg𝑀))
 
Theoremsitgfval 32317* 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))       (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))))
 
Theoremsitgclg 32318* Closure of the Bochner integral on simple functions, generic version. See sitgclbn 32319 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   𝐺 = (Scalar‘𝑊)    &   𝐷 = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ CMnd)    &   (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &   ((𝜑𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥𝐵) → (𝑚 · 𝑥) ∈ 𝐵)       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitgclbn 32319 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ Ban)    &   (𝜑 → (Scalar‘𝑊) ∈ ℝExt )       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitgclcn 32320 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ Ban)    &   (𝜑 → (Scalar‘𝑊) = ℂfld)       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitgclre 32321 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝑊 ∈ Ban)    &   (𝜑 → (Scalar‘𝑊) = ℝfld)       (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
 
Theoremsitg0 32322 The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ Mnd)       (𝜑 → ((𝑊sitg𝑀)‘( dom 𝑀 × { 0 })) = 0 )
 
Theoremsitgf 32323* 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   ((𝜑𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)       (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
 
Theoremsitgaddlemb 32324 Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑆 = (sigaGen‘𝐽)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐻 = (ℝHom‘(Scalar‘𝑊))    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑 → (𝑊v (𝐻 “ (0[,)+∞))) ∈ SLMod)    &   (𝜑𝐽 ∈ Fre)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))    &   (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &    + = (+g𝑊)       ((𝜑𝑝 ∈ ((ran 𝐹 × ran 𝐺) ∖ {⟨ 0 , 0 ⟩})) → ((𝐻‘(𝑀‘((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))) · (2nd𝑝)) ∈ 𝐵)
 
Theoremsitmval 32325* 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)    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)       (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝑓f 𝐷𝑔))))
 
Theoremsitmfval 32326 Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐷 = (dist‘𝑊)    &   (𝜑𝑊𝑉)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))       (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠s (0[,]+∞))sitg𝑀)‘(𝐹f 𝐷𝐺)))
 
Theoremsitmcl 32327 Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
(𝜑𝑊 ∈ Mnd)    &   (𝜑𝑊 ∈ ∞MetSp)    &   (𝜑𝑀 ran measures)    &   (𝜑𝐹 ∈ dom (𝑊sitg𝑀))    &   (𝜑𝐺 ∈ dom (𝑊sitg𝑀))       (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) ∈ (0[,]+∞))
 
Theoremsitmf 32328 The integral metric as a function. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝜑𝑊 ∈ Mnd)    &   (𝜑𝑊 ∈ ∞MetSp)    &   (𝜑𝑀 ran measures)       (𝜑 → (𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))
 
Definitiondf-itgm 32329* 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 (𝑤sitm𝑚).

He then defines the Bochner integral in chapter 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 24796.

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

itgm = (𝑤 ∈ V, 𝑚 ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
 
20.3.19  Euler's partition theorem
 
Theoremoddpwdc 32330* Lemma for eulerpart 32358. 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 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))       𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
 
Theoremoddpwdcv 32331* Lemma for eulerpart 32358: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.)
𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))       (𝑊 ∈ (𝐽 × ℕ0) → (𝐹𝑊) = ((2↑(2nd𝑊)) · (1st𝑊)))
 
Theoremeulerpartlemsv1 32332* Lemma for eulerpart 32358. Value of the sum of a partition 𝐴. (Contributed by Thierry Arnoux, 26-Aug-2018.)
𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ ℕ ((𝐴𝑘) · 𝑘))
 
Theoremeulerpartlemelr 32333* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 8-Aug-2018.)
𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
 
Theoremeulerpartlemsv2 32334* Lemma for eulerpart 32358. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.)
𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘))
 
Theoremeulerpartlemsf 32335* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 8-Aug-2018.)
𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       𝑆:((ℕ0m ℕ) ∩ 𝑅)⟶ℕ0
 
Theoremeulerpartlems 32336* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 6-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))) → (𝐴𝑡) = 0)
 
Theoremeulerpartlemsv3 32337* Lemma for eulerpart 32358. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.)
𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
 
Theoremeulerpartlemgc 32338* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 9-Aug-2018.)
𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆𝐴))
 
Theoremeulerpartleme 32339* Lemma for eulerpart 32358. (Contributed by Mario Carneiro, 26-Jan-2015.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}       (𝐴𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴𝑘) · 𝑘) = 𝑁))
 
Theoremeulerpartlemv 32340* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 19-Aug-2018.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}       (𝐴𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ (𝐴 “ ℕ)((𝐴𝑘) · 𝑘) = 𝑁))
 
Theoremeulerpartlemo 32341* Lemma for eulerpart 32358: 𝑂 is the set of odd partitions of 𝑁. (Contributed by Thierry Arnoux, 10-Aug-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}       (𝐴𝑂 ↔ (𝐴𝑃 ∧ ∀𝑛 ∈ (𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
 
Theoremeulerpartlemd 32342* Lemma for eulerpart 32358: 𝐷 is the set of distinct part. of 𝑁. (Contributed by Thierry Arnoux, 11-Aug-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}       (𝐴𝐷 ↔ (𝐴𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1}))
 
Theoremeulerpartlem1 32343* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})       𝑀:𝐻1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
 
Theoremeulerpartlemb 32344* Lemma for eulerpart 32358. The set of all partitions of 𝑁 is finite. (Contributed by Mario Carneiro, 26-Jan-2015.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})       𝑃 ∈ Fin
 
Theoremeulerpartlemt0 32345* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}       (𝐴 ∈ (𝑇𝑅) ↔ (𝐴 ∈ (ℕ0m ℕ) ∧ (𝐴 “ ℕ) ∈ Fin ∧ (𝐴 “ ℕ) ⊆ 𝐽))
 
Theoremeulerpartlemf 32346* Lemma for eulerpart 32358: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}       ((𝐴 ∈ (𝑇𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴𝑡) = 0)
 
Theoremeulerpartlemt 32347* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}       ((ℕ0m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇𝑅) ↦ (𝑚𝐽))
 
Theoremeulerpartgbij 32348* Lemma for eulerpart 32358: The 𝐺 function is a bijection. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))       𝐺:(𝑇𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅)
 
Theoremeulerpartlemgv 32349* Lemma for eulerpart 32358: value of the function 𝐺. (Contributed by Thierry Arnoux, 13-Nov-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))       (𝐴 ∈ (𝑇𝑅) → (𝐺𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴𝐽))))))
 
Theoremeulerpartlemr 32350* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 13-Nov-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))       𝑂 = ((𝑇𝑅) ∩ 𝑃)
 
Theoremeulerpartlemmf 32351* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 30-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))       (𝐴 ∈ (𝑇𝑅) → (bits ∘ (𝐴𝐽)) ∈ 𝐻)
 
Theoremeulerpartlemgvv 32352* Lemma for eulerpart 32358: value of the function 𝐺 evaluated. (Contributed by Thierry Arnoux, 10-Aug-2018.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))       ((𝐴 ∈ (𝑇𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺𝐴)‘𝐵) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝐵, 1, 0))
 
Theoremeulerpartlemgu 32353* Lemma for eulerpart 32358: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9204. (Contributed by Thierry Arnoux, 10-Aug-2018.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))    &   𝑈 = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))       (𝐴 ∈ (𝑇𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})
 
Theoremeulerpartlemgh 32354* Lemma for eulerpart 32358: The 𝐹 function is a bijection on the 𝑈 subsets. (Contributed by Thierry Arnoux, 15-Aug-2018.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))    &   𝑈 = 𝑡 ∈ ((𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴𝑡)))       (𝐴 ∈ (𝑇𝑅) → (𝐹𝑈):𝑈1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴𝑡))((2↑𝑛) · 𝑡) = 𝑚})
 
Theoremeulerpartlemgf 32355* Lemma for eulerpart 32358: Images under 𝐺 have finite support. (Contributed by Thierry Arnoux, 29-Aug-2018.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))       (𝐴 ∈ (𝑇𝑅) → ((𝐺𝐴) “ ℕ) ∈ Fin)
 
Theoremeulerpartlemgs2 32356* Lemma for eulerpart 32358: The 𝐺 function also preserves partition sums. (Contributed by Thierry Arnoux, 10-Sep-2017.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       (𝐴 ∈ (𝑇𝑅) → (𝑆‘(𝐺𝐴)) = (𝑆𝐴))
 
Theoremeulerpartlemn 32357* Lemma for eulerpart 32358. (Contributed by Thierry Arnoux, 30-Aug-2018.)
𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}    &   𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})    &   𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑇 = {𝑓 ∈ (ℕ0m ℕ) ∣ (𝑓 “ ℕ) ⊆ 𝐽}    &   𝐺 = (𝑜 ∈ (𝑇𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜𝐽))))))    &   𝑆 = (𝑓 ∈ ((ℕ0m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))       (𝐺𝑂):𝑂1-1-onto𝐷
 
Theoremeulerpart 32358* 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 to 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 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}    &   𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}       (♯‘𝑂) = (♯‘𝐷)
 
20.3.20  Sequences defined by strong recursion
 
Syntaxcsseq 32359 Sequences defined by strong recursion.
class seqstr
 
Definitiondf-sseq 32360* 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 = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓𝑚)”⟩)})))))
 
Theoremsubiwrd 32361 Lemma for sseqp1 32371. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word 𝑆)
 
Theoremsubiwrdlen 32362 Length of a subword of an infinite word. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
 
Theoremiwrdsplit 32363 Lemma for sseqp1 32371. (Contributed by Thierry Arnoux, 25-Apr-2019.) (Proof shortened by AV, 14-Oct-2022.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐹 ↾ (0..^(𝑁 + 1))) = ((𝐹 ↾ (0..^𝑁)) ++ ⟨“(𝐹𝑁)”⟩))
 
Theoremsseqval 32364* Value of the strong sequence builder function. The set 𝑊 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.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)})))))
 
Theoremsseqfv1 32365 Value of the strong sequence builder function at one of its initial values. (Contributed by Thierry Arnoux, 21-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)    &   (𝜑𝑁 ∈ (0..^(♯‘𝑀)))       (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀𝑁))
 
Theoremsseqfn 32366 A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑 → (𝑀seqstr𝐹) Fn ℕ0)
 
Theoremsseqmw 32367 Lemma for sseqf 32368 amd sseqp1 32371. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑𝑀𝑊)
 
Theoremsseqf 32368 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.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑 → (𝑀seqstr𝐹):ℕ0𝑆)
 
Theoremsseqfres 32369 The first elements in the strong recursive sequence are the sequence initializer. (Contributed by Thierry Arnoux, 23-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑 → ((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀)
 
Theoremsseqfv2 32370* Value of the strong sequence builder function. (Contributed by Thierry Arnoux, 21-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)    &   (𝜑𝑁 ∈ (ℤ‘(♯‘𝑀)))       (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹𝑀)”⟩)}))‘𝑁)))
 
Theoremsseqp1 32371 Value of the strong sequence builder function at a successor. (Contributed by Thierry Arnoux, 24-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)    &   (𝜑𝑁 ∈ (ℤ‘(♯‘𝑀)))       (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝐹‘((𝑀seqstr𝐹) ↾ (0..^𝑁))))
 
20.3.21  Fibonacci Numbers
 
Syntaxcfib 32372 The Fibonacci sequence.
class Fibci
 
Definitiondf-fib 32373 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 = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (♯ “ (ℤ‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))
 
Theoremfiblem 32374 Lemma for fib0 32375, fib1 32376 and fibp1 32377. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝑤 ∈ (Word ℕ0 ∩ (♯ “ (ℤ‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))):(Word ℕ0 ∩ (♯ “ (ℤ‘(♯‘⟨“01”⟩))))⟶ℕ0
 
Theoremfib0 32375 Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘0) = 0
 
Theoremfib1 32376 Value of the Fibonacci sequence at index 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘1) = 1
 
Theoremfibp1 32377 Value of the Fibonacci sequence at higher indices. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝑁 ∈ ℕ → (Fibci‘(𝑁 + 1)) = ((Fibci‘(𝑁 − 1)) + (Fibci‘𝑁)))
 
Theoremfib2 32378 Value of the Fibonacci sequence at index 2. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘2) = 1
 
Theoremfib3 32379 Value of the Fibonacci sequence at index 3. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘3) = 2
 
Theoremfib4 32380 Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘4) = 3
 
Theoremfib5 32381 Value of the Fibonacci sequence at index 5. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘5) = 5
 
Theoremfib6 32382 Value of the Fibonacci sequence at index 6. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘6) = 8
 
20.3.22  Probability
 
20.3.22.1  Probability Theory
 
Syntaxcprb 32383 Extend class notation to include the class of probability measures.
class Prob
 
Definitiondf-prob 32384 Define the class of probability measures as the set of measures with total measure 1. (Contributed by Thierry Arnoux, 14-Sep-2016.)
Prob = {𝑝 ran measures ∣ (𝑝 dom 𝑝) = 1}
 
Theoremelprob 32385 The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
 
Theoremdomprobmeas 32386 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
(𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
 
Theoremdomprobsiga 32387 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
(𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
 
Theoremprobtot 32388 The probability of the universe set is 1. Second axiom of Kolmogorov. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝑃 ∈ Prob → (𝑃 dom 𝑃) = 1)
 
Theoremprob01 32389 A probability is an element of [ 0 , 1 ]. First axiom of Kolmogorov. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) → (𝑃𝐴) ∈ (0[,]1))
 
Theoremprobnul 32390 The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑃 ∈ Prob → (𝑃‘∅) = 0)
 
Theoremunveldomd 32391 The universe is an element of the domain of the probability, the universe (entire probability space) being dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝜑𝑃 ∈ Prob)       (𝜑 dom 𝑃 ∈ dom 𝑃)
 
Theoremunveldom 32392 The universe is an element of the domain of the probability, the universe (entire probability space) being dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝑃 ∈ Prob → dom 𝑃 ∈ dom 𝑃)
 
Theoremnuleldmp 32393 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝑃 ∈ Prob → ∅ ∈ dom 𝑃)
 
Theoremprobcun 32394* The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the Σ construct cannot be used as it can handle infinite indexing set only if they are subsets of , which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
 
Theoremprobun 32395 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 𝑃𝐵 ∈ dom 𝑃) → ((𝐴𝐵) = ∅ → (𝑃‘(𝐴𝐵)) = ((𝑃𝐴) + (𝑃𝐵))))
 
Theoremprobdif 32396 The probability of the difference of two event sets. (Contributed by Thierry Arnoux, 12-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (𝑃‘(𝐴𝐵)) = ((𝑃𝐴) − (𝑃‘(𝐴𝐵))))
 
Theoremprobinc 32397 A probability law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
(((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ 𝐴𝐵) → (𝑃𝐴) ≤ (𝑃𝐵))
 
Theoremprobdsb 32398 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) → (𝑃‘( dom 𝑃𝐴)) = (1 − (𝑃𝐴)))
 
Theoremprobmeasd 32399 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(𝜑𝑃 ∈ Prob)       (𝜑𝑃 ran measures)
 
Theoremprobvalrnd 32400 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝐴 ∈ dom 𝑃)       (𝜑 → (𝑃𝐴) ∈ ℝ)
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