P. 342 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34101-34200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	esumpinfsum 34101*	The value of the extended sum of infinitely many terms greater than one. (Contributed by Thierry Arnoux, 29-Jun-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐴    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ ℝ*)    &   ⊢ (𝜑 → 0 < 𝑀)    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = +∞)
Theorem	esumpcvgval 34102*	The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))    &   ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
Theorem	esumpmono 34103*	The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑁)𝐴)
Theorem	esumcocn 34104*	Lemma for esummulc2 34106 and co. Composing with a continuous function preserves extended sums. (Contributed by Thierry Arnoux, 29-Jun-2017.)
⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐽))    &   ⊢ (𝜑 → (𝐶‘0) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶‘𝐵))
Theorem	esummulc1 34105*	An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 ·e 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 ·e 𝐶))
Theorem	esummulc2 34106*	An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (𝐶 ·e Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶 ·e 𝐵))
Theorem	esumdivc 34107*	An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐵 /𝑒 𝐶) = Σ*𝑘 ∈ 𝐴(𝐵 /𝑒 𝐶))
Theorem	hashf2 34108	Lemma for hasheuni 34109. (Contributed by Thierry Arnoux, 19-Nov-2016.)
⊢ ♯:V⟶(0[,]+∞)
Theorem	hasheuni 34109*	The cardinality of a disjoint union, not necessarily finite. cf. hashuni 15743. (Contributed by Thierry Arnoux, 19-Nov-2016.) (Revised by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 20-Jun-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) → (♯‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(♯‘𝑥))
Theorem	esumcvg 34110*	The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 15644. (Contributed by Thierry Arnoux, 5-Sep-2017.)
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞)))    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)Σ*𝑘 ∈ ℕ𝐴)
Theorem	esumcvg2 34111*	Simpler version of esumcvg 34110. (Contributed by Thierry Arnoux, 5-Sep-2017.)
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞)))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)    &   ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)(⇝𝑡‘𝐽)Σ*𝑘 ∈ ℕ𝐴)
Theorem	esumcvgsum 34112*	The value of the extended sum when the corresponding sum is convergent. (Contributed by Thierry Arnoux, 29-Oct-2019.)
⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴)    &   ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝐿)    &   ⊢ (𝜑 → 𝐿 ∈ ℝ)    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
Theorem	esumsup 34113*	Express an extended sum as a supremum of extended sums. (Contributed by Thierry Arnoux, 24-May-2020.)
⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < ))
Theorem	esumgect 34114*	"Send 𝑛 to +∞ " in an inequality with an extended sum. (Contributed by Thierry Arnoux, 24-May-2020.)
⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 ≤ 𝐵)
Theorem	esumcvgre 34115*	All terms of a converging extended sum shall be finite. (Contributed by Thierry Arnoux, 23-Sep-2019.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ)    ⇒   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
Theorem	esum2dlem 34116*	Lemma for esum2d 34117 (finite case). (Contributed by Thierry Arnoux, 17-May-2020.) (Proof shortened by AV, 17-Sep-2021.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
Theorem	esum2d 34117*	Write a double extended sum as a sum over a two-dimensional region. Note that 𝐵(𝑗) is a function of 𝑗. This can be seen as "slicing" the relation 𝐴. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
Theorem	esumiun 34118*	Sum over a nonnecessarily disjoint indexed union. The inequality is strict in the case where the sets B(x) overlap. (Contributed by Thierry Arnoux, 21-Sep-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)
21.3.17  Mixed Function/Constant operation
Syntax	cofc 34119	Extend class notation to include mapping of an operation to an operation for a function and a constant.
class ∘f/c 𝑅
Definition	df-ofc 34120*	Define the function/constant operation map. The definition is designed so that if 𝑅 is a binary operation, then ∘f/c 𝑅 is the analogous operation on functions and constants. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
Theorem	ofceq 34121	Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)
Theorem	ofcfval 34122*	Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
Theorem	ofcval 34123	Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))
Theorem	ofcfn 34124	The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) Fn 𝐴)
Theorem	ofcfeqd2 34125*	Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f/c 𝑃𝐶))
Theorem	ofcfval3 34126*	General value of (𝐹 ∘f/c 𝑅𝐶) with no assumptions on functionality of 𝐹. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
Theorem	ofcf 34127*	The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶):𝐴⟶𝑈)
Theorem	ofcfval2 34128*	The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    ⇒   ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
Theorem	ofcfval4 34129*	The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))
Theorem	ofcc 34130	Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))
Theorem	ofcof 34131	Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶})))
21.3.18  Abstract measure
21.3.18.1  Sigma-Algebra
Syntax	csiga 34132	Extend class notation to include the function giving the sigma-algebras on a given base set.
class sigAlgebra
Definition	df-siga 34133*	Define a sigma-algebra, i.e. a set closed under complement and countable union. Literature usually uses capital greek sigma and omega letters for the algebra set, and the base set respectively. We are using 𝑆 and 𝑂 as a parallel. (Contributed by Thierry Arnoux, 3-Sep-2016.)
⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
Theorem	sigaex 34134*	Lemma for issiga 34136 and isrnsiga 34137. The class of sigma-algebras with base set 𝑜 is a set. Note: a more generic version with (𝑂 ∈ V → ...) could be useful for sigaval 34135. (Contributed by Thierry Arnoux, 24-Oct-2016.)
⊢ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))} ∈ V
Theorem	sigaval 34135*	The set of sigma-algebra with a given base set. (Contributed by Thierry Arnoux, 23-Sep-2016.)
⊢ (𝑂 ∈ V → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
Theorem	issiga 34136*	An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)
⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
Theorem	isrnsiga 34137*	The property of being a sigma-algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (Proof shortened by Thierry Arnoux, 23-Oct-2016.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
Theorem	0elsiga 34138	A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆)
Theorem	baselsiga 34139	A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴 ∈ 𝑆)
Theorem	sigasspw 34140	A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴)
Theorem	sigaclcu 34141	A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝐴 ∈ 𝑆)
Theorem	sigaclcuni 34142*	A sigma-algebra is closed under countable union: indexed union version. (Contributed by Thierry Arnoux, 8-Jun-2017.)
⊢ Ⅎ𝑘𝐴    ⇒   ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
Theorem	sigaclfu 34143	A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ∈ Fin) → ∪ 𝐴 ∈ 𝑆)
Theorem	sigaclcu2 34144*	A sigma-algebra is closed under countable union - indexing on ℕ (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ ℕ 𝐴 ∈ 𝑆)
Theorem	sigaclfu2 34145*	A sigma-algebra is closed under finite union - indexing on (1..^𝑁). (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑁)𝐴 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑁)𝐴 ∈ 𝑆)
Theorem	sigaclcu3 34146*	A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑁 𝐴 ∈ 𝑆)
Theorem	issgon 34147	Property of being a sigma-algebra with a given base set, noting that the base set of a sigma-algebra is actually its union set. (Contributed by Thierry Arnoux, 24-Sep-2016.) (Revised by Thierry Arnoux, 23-Oct-2016.)
⊢ (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑂 = ∪ 𝑆))
Theorem	sgon 34148	A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
Theorem	elsigass 34149	An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
Theorem	elrnsiga 34150	Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
⊢ (𝑆 ∈ (sigAlgebra‘𝑂) → 𝑆 ∈ ∪ ran sigAlgebra)
Theorem	isrnsigau 34151*	The property of being a sigma-algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
Theorem	unielsiga 34152	A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
Theorem	dmvlsiga 34153	Lebesgue-measurable subsets of ℝ form a sigma-algebra. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
⊢ dom vol ∈ (sigAlgebra‘ℝ)
Theorem	pwsiga 34154	Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
Theorem	prsiga 34155	The smallest possible sigma-algebra containing 𝑂. (Contributed by Thierry Arnoux, 13-Sep-2016.)
⊢ (𝑂 ∈ 𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))
Theorem	sigaclci 34156	A sigma-algebra is closed under countable intersections. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)
⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ∩ 𝐴 ∈ 𝑆)
Theorem	difelsiga 34157	A sigma-algebra is closed under class differences. (Contributed by Thierry Arnoux, 13-Sep-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
Theorem	unelsiga 34158	A sigma-algebra is closed under pairwise unions. (Contributed by Thierry Arnoux, 13-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)
Theorem	inelsiga 34159	A sigma-algebra is closed under pairwise intersections. (Contributed by Thierry Arnoux, 13-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
Theorem	sigainb 34160	Building a sigma-algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (𝑆 ∩ 𝒫 𝐴) ∈ (sigAlgebra‘𝐴))
Theorem	insiga 34161	The intersection of a collection of sigma-algebras of same base is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))
21.3.18.2  Generated sigma-Algebra
Syntax	csigagen 34162	Extend class notation to include the sigma-algebra generator.
class sigaGen
Definition	df-sigagen 34163*	Define the sigma-algebra generated by a given collection of sets as the intersection of all sigma-algebra containing that set. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
Theorem	sigagenval 34164*	Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
Theorem	sigagensiga 34165	A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))
Theorem	sgsiga 34166	A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)
Theorem	unisg 34167	The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on ∪ 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴)
Theorem	dmsigagen 34168	A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ dom sigaGen = V
Theorem	sssigagen 34169	A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
Theorem	sssigagen2 34170	A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))
Theorem	elsigagen 34171	Any element of a set is also an element of the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (sigaGen‘𝐴))
Theorem	elsigagen2 34172	Any countable union of elements of a set is also in the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))
Theorem	sigagenss 34173	The generated sigma-algebra is a subset of all sigma-algebras containing the generating set, i.e. the generated sigma-algebra is the smallest sigma-algebra containing the generating set, here 𝐴. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ ((𝑆 ∈ (sigAlgebra‘∪ 𝐴) ∧ 𝐴 ⊆ 𝑆) → (sigaGen‘𝐴) ⊆ 𝑆)
Theorem	sigagenss2 34174	Sufficient condition for inclusion of sigma-algebras. This is used to prove equality of sigma-algebras. (Contributed by Thierry Arnoux, 10-Oct-2017.)
⊢ ((∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵 ∈ 𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
Theorem	sigagenid 34175	The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ (𝑆 ∈ ∪ ran sigAlgebra → (sigaGen‘𝑆) = 𝑆)
21.3.18.3  lambda and pi-Systems, Rings of Sets Because they are not widely used outside of measure theory, we do not introduce specific definitions for lambda- and pi-systems. Instead, we define 𝑃 and 𝐿 respectively as the classes of pi- and lambda-systems in 𝑂 throughout this section.
Theorem	ispisys 34176*	The property of being a pi-system. (Contributed by Thierry Arnoux, 10-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
Theorem	ispisys2 34177*	The property of being a pi-system, expanded version. Pi-systems are closed under finite intersections. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
Theorem	inelpisys 34178*	Pi-systems are closed under pairwise intersections. (Contributed by Thierry Arnoux, 6-Jul-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ ((𝑆 ∈ 𝑃 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
Theorem	sigapisys 34179*	All sigma-algebras are pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    ⇒   ⊢ (sigAlgebra‘𝑂) ⊆ 𝑃
Theorem	isldsys 34180*	The property of being a lambda-system or Dynkin system. Lambda-systems contain the empty set, are closed under complement, and closed under countable disjoint union. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆))))
Theorem	pwldsys 34181*	The power set of the universe set 𝑂 is always a lambda-system. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)
Theorem	unelldsys 34182*	Lambda-systems are closed under disjoint set unions. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑆 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑆)
Theorem	sigaldsys 34183*	All sigma-algebras are lambda-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿
Theorem	ldsysgenld 34184*	The intersection of all lambda-systems containing a given collection of sets 𝐴, which is called the lambda-system generated by 𝐴, is itself also a lambda-system. (Contributed by Thierry Arnoux, 16-Jun-2020.)
⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝐿)
Theorem	sigapildsyslem 34185*	Lemma for sigapildsys 34186. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝜑 → 𝑡 ∈ (𝑃 ∩ 𝐿))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑡)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐵 ∈ 𝑡)    ⇒   ⊢ (𝜑 → (𝐴 ∖ ∪ 𝑛 ∈ 𝑁 𝐵) ∈ 𝑡)
Theorem	sigapildsys 34186*	Sigma-algebra are exactly classes which are both lambda and pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    ⇒   ⊢ (sigAlgebra‘𝑂) = (𝑃 ∩ 𝐿)
Theorem	ldgenpisyslem1 34187*	Lemma for ldgenpisys 34190. (Contributed by Thierry Arnoux, 29-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐸)    ⇒   ⊢ (𝜑 → {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸} ∈ 𝐿)
Theorem	ldgenpisyslem2 34188*	Lemma for ldgenpisys 34190. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑇 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})    ⇒   ⊢ (𝜑 → 𝐸 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})
Theorem	ldgenpisyslem3 34189*	Lemma for ldgenpisys 34190. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    ⇒   ⊢ (𝜑 → 𝐸 ⊆ {𝑏 ∈ 𝒫 𝑂 ∣ (𝐴 ∩ 𝑏) ∈ 𝐸})
Theorem	ldgenpisys 34190*	The lambda system 𝐸 generated by a pi-system 𝑇 is also a pi-system. (Contributed by Thierry Arnoux, 18-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ 𝐸 = ∩ {𝑡 ∈ 𝐿 ∣ 𝑇 ⊆ 𝑡}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐸 ∈ 𝑃)
Theorem	dynkin 34191*	Dynkin's lambda-pi theorem: if a lambda-system contains a pi-system, it also contains the sigma-algebra generated by that pi-system. (Contributed by Thierry Arnoux, 16-Jun-2020.)
⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}    &   ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → ∩ {𝑢 ∈ (sigAlgebra‘𝑂) ∣ 𝑇 ⊆ 𝑢} ⊆ 𝑆)
Theorem	isros 34192*	The property of being a rings of sets, i.e. containing the empty set, and closed under finite union and set complement. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))
Theorem	rossspw 34193*	A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)
Theorem	0elros 34194*	A ring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
Theorem	unelros 34195*	A ring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)
Theorem	difelros 34196*	A ring of sets is closed under set complement. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
Theorem	inelros 34197*	A ring of sets is closed under intersection. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    ⇒   ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
Theorem	fiunelros 34198*	A ring of sets is closed under finite union. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑆 ∈ 𝑄)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^𝑁)) → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑁)𝐵 ∈ 𝑆)
Theorem	issros 34199*	The property of being a semirings of sets, i.e., collections of sets containing the empty set, closed under finite intersection, and where complements can be written as finite disjoint unions. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ (𝑆 ∈ 𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))))
Theorem	srossspw 34200*	A semiring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)
⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}    ⇒   ⊢ (𝑆 ∈ 𝑁 → 𝑆 ⊆ 𝒫 𝑂)