P. 313 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	ccarsg 31201	Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
Definition	df-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‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
Theorem	carsgval 31203*	Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})
Theorem	carsgcl 31204	Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
Theorem	elcarsg 31205*	Property of being a Catatheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒))))
Theorem	baselcarsg 31206	The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
Theorem	0elcarsg 31207	The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀))
Theorem	carsguni 31208	The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂)
Theorem	elcarsgss 31209	Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)
Theorem	difelcarsg 31210	The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝑂 ∖ 𝐴) ∈ (toCaraSiga‘𝑀))
Theorem	inelcarsg 31211*	The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	unelcarsg 31212*	The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	difelcarsg2 31213*	The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	carsgmon 31214*	Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	carsgsigalem 31215*	Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓)))
Theorem	fiunelcarsg 31216*	The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))
Theorem	carsgclctunlem1 31217*	Lemma for carsgclctun 31221. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
Theorem	carsggect 31218*	The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → ¬ ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))
Theorem	carsgclctunlem2 31219*	Lemma for carsgclctun 31221. (Contributed by Thierry Arnoux, 25-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    &   ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞)    ⇒   ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))
Theorem	carsgclctunlem3 31220*	Lemma for carsgclctun 31221. (Contributed by Thierry Arnoux, 24-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝐴))) ≤ (𝑀‘𝐸))
Theorem	carsgclctun 31221*	The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))
Theorem	carsgsiga 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.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))
Theorem	omsmeas 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)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → ∅ ∈ dom 𝑅)    &   ⊢ (𝜑 → (𝑅‘∅) = 0)    ⇒   ⊢ (𝜑 → (𝑀 ↾ 𝑆) ∈ (measures‘𝑆))
Theorem	pmeasmono 31224*	This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑃‘∅) = 0)    &   ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
Theorem	pmeasadd 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.)
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑃‘∅) = 0)    &   ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦))    &   ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑅 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵))
20.3.18  Integration
20.3.18.1  Lebesgue integral - misc additions
Theorem	itgeq12dv 31226*	Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
20.3.18.2  Bochner integral
Syntax	citgm 31227	Extend class notation with the (measure) Bochner integral.
class itgm
Syntax	csitm 31228	Extend class notation with the integral metric for simple functions.
class sitm
Syntax	csitg 31229	Extend class notation with the integral of simple functions.
class sitg
Definition	df-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 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠 ‘𝑤)𝑥)))))
Definition	df-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 ℝ*, the range structure for this integral is (ℝ*𝑠 ↾s (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*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘𝑓 (dist‘𝑤)𝑔))))
Theorem	sitgval 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ 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)))))
Theorem	issibf 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
Theorem	sibf0 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ Mnd)    ⇒   ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))
Theorem	sibfmbl 31235	A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
Theorem	sibff 31236	A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
Theorem	sibfrn 31237	A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ Fin)
Theorem	sibfima 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))
Theorem	sibfinima 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    ⇒   ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞))
Theorem	sibfof 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ 𝐶 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐾 ∈ TopSp)    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    &   ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀))
Theorem	sitgfval 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))))
Theorem	sitgclg 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ 𝐺 = (Scalar‘𝑊)    &   ⊢ 𝐷 = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ CMnd)    &   ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &   ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitgclbn 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ Ban)    &   ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitgclcn 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ Ban)    &   ⊢ (𝜑 → (Scalar‘𝑊) = ℂfld)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitgclre 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ Ban)    &   ⊢ (𝜑 → (Scalar‘𝑊) = ℝfld)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitg0 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ Mnd)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘(∪ dom 𝑀 × { 0 })) = 0 )
Theorem	sitgf 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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
Theorem	sitgaddlemb 31248	Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → (𝑊 ↾v (𝐻 “ (0[,)+∞))) ∈ SLMod)    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑝 ∈ ((ran 𝐹 × ran 𝐺) ∖ {⟨ 0 , 0 ⟩})) → ((𝐻‘(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))) · (2nd ‘𝑝)) ∈ 𝐵)
Theorem	sitmval 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)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘𝑓 𝐷𝑔))))
Theorem	sitmfval 31250	Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐷 = (dist‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘𝑓 𝐷𝐺)))
Theorem	sitmcl 31251	Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
⊢ (𝜑 → 𝑊 ∈ Mnd)    &   ⊢ (𝜑 → 𝑊 ∈ ∞MetSp)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) ∈ (0[,]+∞))
Theorem	sitmf 31252	The integral metric as a function. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝜑 → 𝑊 ∈ Mnd)    &   ⊢ (𝜑 → 𝑊 ∈ ∞MetSp)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))
Definition	df-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
Theorem	oddpwdc 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 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    ⇒   ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ
Theorem	oddpwdcv 31255*	Lemma for eulerpart 31282: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    ⇒   ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊)))
Theorem	eulerpartlemsv1 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 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘))
Theorem	eulerpartlemelr 31257*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 8-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
Theorem	eulerpartlemsv2 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 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘))
Theorem	eulerpartlemsf 31259*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 8-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ 𝑆:((ℕ0 ↑𝑚 ℕ) ∩ 𝑅)⟶ℕ0
Theorem	eulerpartlems 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 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)
Theorem	eulerpartlemsv3 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 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
Theorem	eulerpartlemgc 31262*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 9-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴))
Theorem	eulerpartleme 31263*	Lemma for eulerpart 31282. (Contributed by Mario Carneiro, 26-Jan-2015.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    ⇒   ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = 𝑁))
Theorem	eulerpartlemv 31264*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 19-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    ⇒   ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘) = 𝑁))
Theorem	eulerpartlemo 31265*	Lemma for eulerpart 31282: 𝑂 is the set of odd partitions of 𝑁. (Contributed by Thierry Arnoux, 10-Aug-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    ⇒   ⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛))
Theorem	eulerpartlemd 31266*	Lemma for eulerpart 31282: 𝐷 is the set of distinct part. of 𝑁. (Contributed by Thierry Arnoux, 11-Aug-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    ⇒   ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1}))
Theorem	eulerpartlem1 31267*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    ⇒   ⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)
Theorem	eulerpartlemb 31268*	Lemma for eulerpart 31282. The set of all partitions of 𝑁 is finite. (Contributed by Mario Carneiro, 26-Jan-2015.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    ⇒   ⊢ 𝑃 ∈ Fin
Theorem	eulerpartlemt0 31269*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 19-Sep-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽))
Theorem	eulerpartlemf 31270*	Lemma for eulerpart 31282: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    ⇒   ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0)
Theorem	eulerpartlemt 31271*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 19-Sep-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    ⇒   ⊢ ((ℕ0 ↑𝑚 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽))
Theorem	eulerpartgbij 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 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    ⇒   ⊢ 𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑𝑚 ℕ) ∩ 𝑅)
Theorem	eulerpartlemgv 31273*	Lemma for eulerpart 31282: value of the function 𝐺. (Contributed by Thierry Arnoux, 13-Nov-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽))))))
Theorem	eulerpartlemr 31274*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 13-Nov-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    ⇒   ⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃)
Theorem	eulerpartlemmf 31275*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 30-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (bits ∘ (𝐴 ↾ 𝐽)) ∈ 𝐻)
Theorem	eulerpartlemgvv 31276*	Lemma for eulerpart 31282: value of the function 𝐺 evaluated. (Contributed by Thierry Arnoux, 10-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    ⇒   ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺‘𝐴)‘𝐵) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝐵, 1, 0))
Theorem	eulerpartlemgu 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 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})
Theorem	eulerpartlemgh 31278*	Lemma for eulerpart 31282: The 𝐹 function is a bijection on the 𝑈 subsets. (Contributed by Thierry Arnoux, 15-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
Theorem	eulerpartlemgf 31279*	Lemma for eulerpart 31282: Images under 𝐺 have finite support. (Contributed by Thierry Arnoux, 29-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ∈ Fin)
Theorem	eulerpartlemgs2 31280*	Lemma for eulerpart 31282: The 𝐺 function also preserves partition sums. (Contributed by Thierry Arnoux, 10-Sep-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = (𝑆‘𝐴))
Theorem	eulerpartlemn 31281*	Lemma for eulerpart 31282. (Contributed by Thierry Arnoux, 30-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷
Theorem	eulerpart 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 ℕ to ℕ0 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 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    ⇒   ⊢ (♯‘𝑂) = (♯‘𝐷)
20.3.20  Sequences defined by strong recursion
Syntax	csseq 31283	Sequences defined by strong recursion.
class seqstr
Definition	df-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 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
Theorem	subiwrd 31285	Lemma for sseqp1 31296. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word 𝑆)
Theorem	subiwrdlen 31286	Length of a subword of an infinite word. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
Theorem	iwrdsplit 31287	Lemma for sseqp1 31296. (Contributed by Thierry Arnoux, 25-Apr-2019.) (Proof shortened by AV, 14-Oct-2022.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐹 ↾ (0..^(𝑁 + 1))) = ((𝐹 ↾ (0..^𝑁)) ++ ⟨“(𝐹‘𝑁)”⟩))
Theorem	iwrdsplitOLD 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.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐹 ↾ (0..^(𝑁 + 1))) = ((𝐹 ↾ (0..^𝑁)) ++ ⟨“(𝐹‘𝑁)”⟩))
Theorem	sseqval 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.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))
Theorem	sseqfv1 31290	Value of the strong sequence builder function at one of its initial values. (Contributed by Thierry Arnoux, 21-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑀)))    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁))
Theorem	sseqfn 31291	A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0)
Theorem	sseqmw 31292	Lemma for sseqf 31293 amd sseqp1 31296. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → 𝑀 ∈ 𝑊)
Theorem	sseqf 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.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → (𝑀seqstr𝐹):ℕ0⟶𝑆)
Theorem	sseqfres 31294	The first elements in the strong recursive sequence are the sequence initializer. (Contributed by Thierry Arnoux, 23-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀)
Theorem	sseqfv2 31295*	Value of the strong sequence builder function. (Contributed by Thierry Arnoux, 21-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(♯‘𝑀)))    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))‘𝑁)))
Theorem	sseqp1 31296	Value of the strong sequence builder function at a successor. (Contributed by Thierry Arnoux, 24-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(♯‘𝑀)))    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝐹‘((𝑀seqstr𝐹) ↾ (0..^𝑁))))
20.3.21  Fibonacci Numbers
Syntax	cfib 31297	The Fibonacci sequence.
class Fibci
Definition	df-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)))))
Theorem	fiblem 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
Theorem	fib0 31300	Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘0) = 0