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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembaselcarsg 29501 The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑𝑂 ∈ (toCaraSiga‘𝑀))
 
Theorem0elcarsg 29502 The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑 → ∅ ∈ (toCaraSiga‘𝑀))
 
Theoremcarsguni 29503 The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)       (𝜑 (toCaraSiga‘𝑀) = 𝑂)
 
Theoremelcarsgss 29504 Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))       (𝜑𝐴𝑂)
 
Theoremdifelcarsg 29505 The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝑂𝐴) ∈ (toCaraSiga‘𝑀))
 
Theoreminelcarsg 29506* The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))    &   ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))    &   (𝜑𝐵 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
 
Theoremunelcarsg 29507* The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))    &   ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))    &   (𝜑𝐵 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
 
Theoremdifelcarsg2 29508* The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴 ∈ (toCaraSiga‘𝑀))    &   ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))    &   (𝜑𝐵 ∈ (toCaraSiga‘𝑀))       (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
 
Theoremcarsgmon 29509* Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ 𝒫 𝑂)    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
 
Theoremcarsgsigalem 29510* Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))       ((𝜑𝑒 ∈ 𝒫 𝑂𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒𝑓)) ≤ ((𝑀𝑒) +𝑒 (𝑀𝑓)))
 
Theoremfiunelcarsg 29511* The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))       (𝜑 𝐴 ∈ (toCaraSiga‘𝑀))
 
Theoremcarsgclctunlem1 29512* Lemma for carsgclctun 29516. (Contributed by Thierry Arnoux, 23-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))    &   (𝜑Disj 𝑦𝐴 𝑦)    &   (𝜑𝐸 ∈ 𝒫 𝑂)       (𝜑 → (𝑀‘(𝐸 𝐴)) = Σ*𝑦𝐴(𝑀‘(𝐸𝑦)))
 
Theoremcarsggect 29513* The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))    &   (𝜑 → (𝑀‘∅) = 0)    &   ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))    &   (𝜑 → ¬ ∅ ∈ 𝐴)    &   (𝜑𝐴 ≼ ω)    &   (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))    &   (𝜑Disj 𝑦𝐴 𝑦)    &   ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))       (𝜑 → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
 
Theoremcarsgclctunlem2 29514* Lemma for carsgclctun 29516. (Contributed by Thierry Arnoux, 25-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))    &   (𝜑Disj 𝑘 ∈ ℕ 𝐴)    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))    &   (𝜑𝐸 ∈ 𝒫 𝑂)    &   (𝜑 → (𝑀𝐸) ≠ +∞)       (𝜑 → ((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀𝐸))
 
Theoremcarsgclctunlem3 29515* Lemma for carsgclctun 29516. (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 29516* 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 29517* 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 29518 The restriction of a constructed outer measure to Catatheodory 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))
 
TheoremomsmeasOLD 29519 The restriction of a constructed outer measure to Catatheodory 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.) Obsolete version of omsmeas 29518 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑀 = (toOMeas‘𝑅)    &   𝑆 = (toCaraSiga‘𝑀)    &   (𝜑𝑄𝑉)    &   (𝜑𝑅:𝑄⟶(0[,]+∞))    &   (𝜑 → ∅ ∈ dom 𝑅)    &   (𝜑 → (𝑅‘∅) = 0)       (𝜑 → (𝑀𝑆) ∈ (measures‘𝑆))
 
Theorempmeasmono 29520* 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 29521* 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))
 
20.3.16  Integration
 
20.3.16.1  Lebesgue integral - misc additions
 
Theoremitgeq12dv 29522* Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → 𝐶 = 𝐷)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
 
20.3.16.2  Bochner integral
 
Syntaxcitgm 29523 Extend class notation with the (measure) Bochner integral.
class itgm
 
Syntaxcsitm 29524 Extend class notation with the integral metric for simple functions.
class sitm
 
Syntaxcsitg 29525 Extend class notation with the integral of simple functions.
class sitg
 
Definitiondf-sitg 29526* 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 29527* 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 29528* 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 29529* 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 29530 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 29531 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 29532 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 29533 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 29534 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 29535 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 29536 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 29537* 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 29538* Closure of the Bochner integral on simple functions, generic version. See sitgclbn 29539 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 29539 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 29540 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 29541 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 29542 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 29543* 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 29544 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 29545* 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 29546 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 29547 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 29548 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 29549* 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 23073.

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

itgm = (𝑤 ∈ V, 𝑚 ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
 
20.3.17  Euler's partition theorem
 
Theoremoddpwdc 29550* Lemma for eulerpart 29578. 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 29551* Lemma for eulerpart 29578: 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 29552* Lemma for eulerpart 29578. 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 29553* Lemma for eulerpart 29578. (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 29554* Lemma for eulerpart 29578. 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 29555* Lemma for eulerpart 29578. (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 29556* Lemma for eulerpart 29578. (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 29557* Lemma for eulerpart 29578. 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 29558* Lemma for eulerpart 29578. (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 29559* Lemma for eulerpart 29578. (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 29560* Lemma for eulerpart 29578. (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 29561* Lemma for eulerpart 29578: 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 29562* Lemma for eulerpart 29578: 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 29563* Lemma for eulerpart 29578. (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 29564* Lemma for eulerpart 29578. 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 29565* Lemma for eulerpart 29578. (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 29566* Lemma for eulerpart 29578: 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 29567* Lemma for eulerpart 29578. (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 29568* Lemma for eulerpart 29578: 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 29569* Lemma for eulerpart 29578: 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 29570* Lemma for eulerpart 29578. (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 29571* Lemma for eulerpart 29578. (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 29572* Lemma for eulerpart 29578: 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 29573* Lemma for eulerpart 29578: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 8101. (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 29574* Lemma for eulerpart 29578: 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 29575* Lemma for eulerpart 29578: 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 29576* Lemma for eulerpart 29578: 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 29577* Lemma for eulerpart 29578. (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 29578* 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)
 
20.3.18  Sequences defined by strong recursion
 
Syntaxcsseq 29579 Sequences defined by strong recursion.
class seqstr
 
Definitiondf-sseq 29580* 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 29581 Lemma for sseqp1 29591. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word 𝑆)
 
Theoremsubiwrdlen 29582 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 29583 Lemma for sseqp1 29591. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐹 ↾ (0..^(𝑁 + 1))) = ((𝐹 ↾ (0..^𝑁)) ++ ⟨“(𝐹𝑁)”⟩))
 
Theoremsseqval 29584* 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 29585 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 29586 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 29587 Lemma for sseqf 29588 amd sseqp1 29591. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (# “ (ℤ‘(#‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑𝑀𝑊)
 
Theoremsseqf 29588 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):ℕ0𝑆)
 
Theoremsseqfres 29589 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 29590* 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 29591 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))))
 
20.3.19  Fibonacci Numbers
 
Syntaxcfib 29592 The Fibonacci sequence.
class Fibci
 
Definitiondf-fib 29593 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 29594 Lemma for fib0 29595, fib1 29596 and fibp1 29597. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝑤 ∈ (Word ℕ0 ∩ (# “ (ℤ‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))):(Word ℕ0 ∩ (# “ (ℤ‘(#‘⟨“01”⟩))))⟶ℕ0
 
Theoremfib0 29595 Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘0) = 0
 
Theoremfib1 29596 Value of the Fibonacci sequence at index 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘1) = 1
 
Theoremfibp1 29597 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 29598 Value of the Fibonacci sequence at index 2. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘2) = 1
 
Theoremfib3 29599 Value of the Fibonacci sequence at index 3. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘3) = 2
 
Theoremfib4 29600 Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘4) = 3
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