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Theorem List for Metamath Proof Explorer - 34301-34400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcarsgclctunlem3 34301* Lemma for carsgclctun 34302. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))    &   (𝜑𝐴 ≼ ω)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))    &   (𝜑𝐸 ∈ 𝒫 𝑂)       (𝜑 → ((𝑀‘(𝐸 𝐴)) +𝑒 (𝑀‘(𝐸 𝐴))) ≤ (𝑀𝐸))
 
Theoremcarsgclctun 34302* The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))    &   (𝜑𝐴 ≼ ω)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))       (𝜑 𝐴 ∈ (toCaraSiga‘𝑀))
 
Theoremcarsgsiga 34303* The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))       (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))
 
Theoremomsmeas 34304 The restriction of a constructed outer measure to Caratheodory measurable sets is a measure. This theorem allows to construct measures from pre-measures with the required characteristics, as for the Lebesgue measure. (Contributed by Thierry Arnoux, 17-May-2020.)
𝑀 = (toOMeas‘𝑅)    &   𝑆 = (toCaraSiga‘𝑀)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑 → ∅ ∈ dom 𝑅)    &   (𝜑 → (𝑅‘∅) = 0)       (𝜑 → (𝑀𝑆) ∈ (measures‘𝑆))
 
Theorempmeasmono 34305* This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(𝜑𝑃:𝑅⟶(0[,]+∞))    &   (𝜑 → (𝑃‘∅) = 0)    &   ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥𝑅Disj 𝑦𝑥 𝑦)) → (𝑃 𝑥) = Σ*𝑦𝑥(𝑃𝑦))    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑅)    &   (𝜑 → (𝐵𝐴) ∈ 𝑅)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑃𝐴) ≤ (𝑃𝐵))
 
Theorempmeasadd 34306* 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))
 
21.3.19  Integration
 
21.3.19.1  Lebesgue integral - misc additions
 
Theoremitgeq12dv 34307* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → 𝐶 = 𝐷)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
 
21.3.19.2  Bochner integral
 
Syntaxcitgm 34308 Extend class notation with the (measure) Bochner integral.
class itgm
 
Syntaxcsitm 34309 Extend class notation with the integral metric for simple functions.
class sitm
 
Syntaxcsitg 34310 Extend class notation with the integral of simple functions.
class sitg
 
Definitiondf-sitg 34311* 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 34312* 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 34313* 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 34314* 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 34315 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 34316 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 34317 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 34318 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 34319 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 34320 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 34321 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 34322* 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 34323* Closure of the Bochner integral on simple functions, generic version. See sitgclbn 34324 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 34324 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 34325 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 34326 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 34327 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 34328* 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 34329 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 34330* 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 34331 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 34332 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 34333 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 34334* 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 25671.

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

itgm = (𝑤 ∈ V, 𝑚 ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
 
21.3.20  Euler's partition theorem
 
Theoremoddpwdc 34335* Lemma for eulerpart 34363. 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 34336* Lemma for eulerpart 34363: 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 34337* Lemma for eulerpart 34363. 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 34338* Lemma for eulerpart 34363. (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 34339* Lemma for eulerpart 34363. 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 34340* Lemma for eulerpart 34363. (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 34341* Lemma for eulerpart 34363. (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 34342* Lemma for eulerpart 34363. 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 34343* Lemma for eulerpart 34363. (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 34344* Lemma for eulerpart 34363. (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 34345* Lemma for eulerpart 34363. (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 34346* Lemma for eulerpart 34363: 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 34347* Lemma for eulerpart 34363: 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 34348* Lemma for eulerpart 34363. (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 34349* Lemma for eulerpart 34363. 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 34350* Lemma for eulerpart 34363. (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 34351* Lemma for eulerpart 34363: 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 34352* Lemma for eulerpart 34363. (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 34353* Lemma for eulerpart 34363: 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 34354* Lemma for eulerpart 34363: 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 34355* Lemma for eulerpart 34363. (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 34356* Lemma for eulerpart 34363. (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 34357* Lemma for eulerpart 34363: 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 34358* Lemma for eulerpart 34363: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9473. (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 34359* Lemma for eulerpart 34363: 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 34360* Lemma for eulerpart 34363: 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 34361* Lemma for eulerpart 34363: 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 34362* Lemma for eulerpart 34363. (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 34363* 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)
 
21.3.21  Sequences defined by strong recursion
 
Syntaxcsseq 34364 Sequences defined by strong recursion.
class seqstr
 
Definitiondf-sseq 34365* 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 34366 Lemma for sseqp1 34376. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word 𝑆)
 
Theoremsubiwrdlen 34367 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 34368 Lemma for sseqp1 34376. (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 34369* 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 34370 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 34371 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 34372 Lemma for sseqf 34373 amd sseqp1 34376. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑𝑀𝑊)
 
Theoremsseqf 34373 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 34374 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 34375* 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 34376 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))))
 
21.3.22  Fibonacci Numbers
 
Syntaxcfib 34377 The Fibonacci sequence.
class Fibci
 
Definitiondf-fib 34378 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 34379 Lemma for fib0 34380, fib1 34381 and fibp1 34382. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝑤 ∈ (Word ℕ0 ∩ (♯ “ (ℤ‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))):(Word ℕ0 ∩ (♯ “ (ℤ‘(♯‘⟨“01”⟩))))⟶ℕ0
 
Theoremfib0 34380 Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘0) = 0
 
Theoremfib1 34381 Value of the Fibonacci sequence at index 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘1) = 1
 
Theoremfibp1 34382 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 34383 Value of the Fibonacci sequence at index 2. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘2) = 1
 
Theoremfib3 34384 Value of the Fibonacci sequence at index 3. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘3) = 2
 
Theoremfib4 34385 Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘4) = 3
 
Theoremfib5 34386 Value of the Fibonacci sequence at index 5. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘5) = 5
 
Theoremfib6 34387 Value of the Fibonacci sequence at index 6. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘6) = 8
 
21.3.23  Probability
 
21.3.23.1  Probability Theory
 
Syntaxcprb 34388 Extend class notation to include the class of probability measures.
class Prob
 
Definitiondf-prob 34389 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 34390 The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
 
Theoremdomprobmeas 34391 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
(𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
 
Theoremdomprobsiga 34392 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
(𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
 
Theoremprobtot 34393 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 34394 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 34395 The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑃 ∈ Prob → (𝑃‘∅) = 0)
 
Theoremunveldomd 34396 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 34397 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 34398 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝑃 ∈ Prob → ∅ ∈ dom 𝑃)
 
Theoremprobcun 34399* 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 34400 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))))
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