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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxccarsg 31201 Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
 
Definitiondf-carsg 31202* Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 31203 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.)
toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 dom 𝑚((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒)})
 
Theoremcarsgval 31203* Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
 
Theoremcarsgcl 31204 Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
 
Theoremelcarsg 31205* Property of being a Catatheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
(𝜑𝑂𝑉)    &   (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))       (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
 
Theorembaselcarsg 31206 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 31207 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 31208 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 31209 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 31210 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 31211* 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 31212* 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 31213* 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 31214* 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 31215* 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 31216* 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 31217* Lemma for carsgclctun 31221. (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 31218* 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 31219* Lemma for carsgclctun 31221. (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 31220* Lemma for carsgclctun 31221. (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 31221* 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 31222* 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 31223 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))
 
Theorempmeasmono 31224* 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 31225* 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 31226* 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 31227 Extend class notation with the (measure) Bochner integral.
class itgm
 
Syntaxcsitm 31228 Extend class notation with the integral metric for simple functions.
class sitm
 
Syntaxcsitg 31229 Extend class notation with the integral of simple functions.
class sitg
 
Definitiondf-sitg 31230* 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 31231* 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 31232* 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 31233* 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 31234 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 31235 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 31236 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 31237 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 31238 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 31239 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 31240 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 31241* 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 31242* Closure of the Bochner integral on simple functions, generic version. See sitgclbn 31243 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 31243 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 31244 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 31245 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 31246 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 31247* 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 31248 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 31249* 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 31250 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 31251 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 31252 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 31253* 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 23927.

(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 31254* Lemma for eulerpart 31282. 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 31255* Lemma for eulerpart 31282: 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 31256* Lemma for eulerpart 31282. 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 31257* Lemma for eulerpart 31282. (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 31258* Lemma for eulerpart 31282. 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 31259* Lemma for eulerpart 31282. (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 31260* Lemma for eulerpart 31282. (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 31261* Lemma for eulerpart 31282. 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 31262* Lemma for eulerpart 31282. (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 31263* Lemma for eulerpart 31282. (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 31264* Lemma for eulerpart 31282. (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 31265* Lemma for eulerpart 31282: 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 31266* Lemma for eulerpart 31282: 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 31267* Lemma for eulerpart 31282. (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 31268* Lemma for eulerpart 31282. 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 31269* Lemma for eulerpart 31282. (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 31270* Lemma for eulerpart 31282: 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 31271* Lemma for eulerpart 31282. (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 31272* Lemma for eulerpart 31282: 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 31273* Lemma for eulerpart 31282: 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 31274* Lemma for eulerpart 31282. (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 31275* Lemma for eulerpart 31282. (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 31276* Lemma for eulerpart 31282: 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 31277* Lemma for eulerpart 31282: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 8695. (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 31278* Lemma for eulerpart 31282: 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 31279* Lemma for eulerpart 31282: 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 31280* Lemma for eulerpart 31282: 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 31281* Lemma for eulerpart 31282. (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 31282* 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 31283 Sequences defined by strong recursion.
class seqstr
 
Definitiondf-sseq 31284* 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 31285 Lemma for sseqp1 31296. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word 𝑆)
 
Theoremsubiwrdlen 31286 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 31287 Lemma for sseqp1 31296. (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)”⟩))
 
TheoremiwrdsplitOLD 31288 Obsolete version of iwrdsplit 31287 as of 12-Oct-2022. (Contributed by Thierry Arnoux, 25-Apr-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝑆 ∈ V)    &   (𝜑𝐹:ℕ0𝑆)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐹 ↾ (0..^(𝑁 + 1))) = ((𝐹 ↾ (0..^𝑁)) ++ ⟨“(𝐹𝑁)”⟩))
 
Theoremsseqval 31289* 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 31290 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 31291 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 31292 Lemma for sseqf 31293 amd sseqp1 31296. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝜑𝑆 ∈ V)    &   (𝜑𝑀 ∈ Word 𝑆)    &   𝑊 = (Word 𝑆 ∩ (♯ “ (ℤ‘(♯‘𝑀))))    &   (𝜑𝐹:𝑊𝑆)       (𝜑𝑀𝑊)
 
Theoremsseqf 31293 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 31294 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 31295* 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 31296 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 31297 The Fibonacci sequence.
class Fibci
 
Definitiondf-fib 31298 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 31299 Lemma for fib0 31300, fib1 31301 and fibp1 31302. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(𝑤 ∈ (Word ℕ0 ∩ (♯ “ (ℤ‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))):(Word ℕ0 ∩ (♯ “ (ℤ‘(♯‘⟨“01”⟩))))⟶ℕ0
 
Theoremfib0 31300 Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.)
(Fibci‘0) = 0
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44271
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