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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Σ* Σ* Σ*

Theoremesumle 24402* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Σ* Σ*

Theoremesummono 24403* Extended sum is monotonic. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ* Σ*

Theoremgsumesum 24404* Relate a group sum on s to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
s g Σ*

Theoremesumlub 24405* The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ*        Σ*

Σ* Σ* Σ*

Theoremesumlef 24407* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Σ* Σ*

Theoremesumcst 24408* The extended sum of a constant. (Contributed by Thierry Arnoux, 3-Mar-2017.) (Revised by Thierry Arnoux, 5-Jul-2017.)
Σ*

Theoremesumsn 24409* The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumpr 24410* Extended sum over a pair. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumpr2 24411* Extended sum over a pair, with a relaxed condition compared to esumpr 24410. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumfzf 24412* Formulating a partial extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
Σ*

Theoremesumfsup 24413 Formulating an extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
Σ*

Theoremesumfsupre 24414 Formulating an extended sum over integers using the recursive sequence builder. This version is limited to real valued functions. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ*

Theoremesumss 24415 Change the index set to a subset by adding zeroes. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ* Σ*

Theoremesumpinfval 24416* The value of the extended sum of non-negative terms, with at least one infinite term. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ*

Theoremesumpfinvallem 24417 Lemma for esumpfinval 24418 (Contributed by Thierry Arnoux, 28-Jun-2017.)
fld g s g

Theoremesumpfinval 24418* The value of the extended sum of a finite set of non-negative finite terms (Contributed by Thierry Arnoux, 28-Jun-2017.)
Σ*

Theoremesumpfinvalf 24419 Same as esumpfinval 24418, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Σ*

Theoremesumpinfsum 24420* The value of the extended sum of infinitely many terms greater than one. (Contributed by Thierry Arnoux, 29-Jun-2017.)
Σ*

Theoremesumpcvgval 24421* The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)
Σ*

Theoremesumpmono 24422* The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Σ* Σ*

Theoremesumcocn 24423* Lemma for esummulc2 24425 and co. Composing with a continuous function preserves extended sums (Contributed by Thierry Arnoux, 29-Jun-2017.)
ordTop t                                           Σ* Σ*

Theoremesummulc1 24424* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* Σ*

Theoremesummulc2 24425* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* Σ*

Theoremesumdivc 24426* An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* /𝑒 Σ* /𝑒

Theoremhashf2 24427 Lemma for hasheuni 24428 (Contributed by Thierry Arnoux, 19-Nov-2016.)

Theoremhasheuni 24428* The cardinality of a disjoint union, not necessarily finite. cf. hashuni 12559. (Contributed by Thierry Arnoux, 19-Nov-2016.) (Revised by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 20-Jun-2017.)
Disj Σ*

Theoremesumcvg 24429* The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 12476. (Contributed by Thierry Arnoux, 5-Sep-2017.)
s        Σ*                      Σ*

Theoremesumcvg2 24430* Simpler version of esumcvg 24429. (Contributed by Thierry Arnoux, 5-Sep-2017.)
s                             Σ* Σ*

19.3.12  Mixed Function/Constant operation

Syntaxcofc 24431 Extend class notation to include mapping of an operation to an operation for a function and a constant.
𝑓/𝑐

Definitiondf-ofc 24432* Define the function/constant operation map. The definition is designed so that if is a binary operation, then ∘𝑓/𝑐 is the analogous operation on functions and constants. (Contributed by Thierry Arnoux, 21-Jan-2017.)
𝑓/𝑐

Theoremofceq 24433 Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
𝑓/𝑐 𝑓/𝑐

Theoremofcfval 24434* Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
𝑓/𝑐

Theoremofcval 24435 Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcfn 24436 The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcfeqd2 24437* Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐 𝑓/𝑐

Theoremofcfval3 24438* General value of 𝑓/𝑐 with no assumptions on functionality of . (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcf 24439* The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
𝑓/𝑐

Theoremofcfval2 24440* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcfval4 24441* The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcc 24442 Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

19.3.13  Abstract measure

19.3.13.1  Sigma-Algebra

Syntaxcsiga 24443 Extend class notation to include the function giving the sigma-algebras on a given base set.
sigAlgebra

Definitiondf-siga 24444* Define a sigma-algebra, i.e. a set closed under complement and countable union. Litterature usually uses capital greek sigma and omega letters for the algebra set, and the base set respectively. We are using and as a parallel. (Contributed by Thierry Arnoux, 3-Sep-2016.)
sigAlgebra

Theoremsigaex 24445* Lemma for issiga 24447 and isrnsiga 24449 The set of sigma algebra with base set is a set. Note: a more generic version with could be useful for sigaval 24446. (Contributed by Thierry Arnoux, 24-Oct-2016.)

Theoremsigaval 24446* The set of sigma-algebra with a given base set. (Contributed by Thierry Arnoux, 23-Sep-2016.)
sigAlgebra

Theoremissiga 24447* An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)
sigAlgebra

TheoremisrnsigaOLD 24448* The property of being a sigma algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
sigAlgebra

Theoremisrnsiga 24449* 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.)
sigAlgebra

Theorem0elsiga 24450 A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
sigAlgebra

Theorembaselsiga 24451 A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
sigAlgebra

Theoremsigasspw 24452 A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
sigAlgebra

Theoremsigaclcu 24453 A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
sigAlgebra

Theoremsigaclcuni 24454* A sigma-algebra is closed under countable union: indexed union version (Contributed by Thierry Arnoux, 8-Jun-2017.)
sigAlgebra

Theoremsigaclfu 24455 A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
sigAlgebra

Theoremsigaclcu2 24456* A sigma-algebra is closed under countable union - indexing on (Contributed by Thierry Arnoux, 29-Dec-2016.)
sigAlgebra

Theoremsigaclfu2 24457* A sigma-algebra is closed under finite union - indexing on ..^ (Contributed by Thierry Arnoux, 28-Dec-2016.)
sigAlgebra ..^ ..^

Theoremsigaclcu3 24458* A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
sigAlgebra       ..^

Theoremissgon 24459 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.)
sigAlgebra sigAlgebra

Theoremsgon 24460 A sigma alebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra sigAlgebra

Theoremelsigass 24461 An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra

Theoremelrnsiga 24462 Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
sigAlgebra sigAlgebra

Theoremisrnsigau 24463* The property of being a sigma algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
sigAlgebra

Theoremunielsiga 24464 A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra

Theoremdmvlsiga 24465 Lebesgue-measurable subsets of form a sigma-algebra (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
sigAlgebra

Theorempwsiga 24466 Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
sigAlgebra

Theoremprsiga 24467 The smallest possible sigma-algebra containing (Contributed by Thierry Arnoux, 13-Sep-2016.)
sigAlgebra

Theoremsigaclci 24468 A sigma-algebra is closed under countable intersection. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)
sigAlgebra

Theoremdifelsiga 24469 A sigma algebra is closed under set difference. (Contributed by Thierry Arnoux, 13-Sep-2016.)
sigAlgebra

Theoremunelsiga 24470 A sigma algebra is closed under set union. (Contributed by Thierry Arnoux, 13-Dec-2016.)
sigAlgebra

Theoreminelsiga 24471 A sigma algebra is closed under set intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.)
sigAlgebra

Theoremsigainb 24472 Building a sigma algebra from intersections with a given set. (Contributed by Thierry Arnoux, 26-Dec-2016.)
sigAlgebra sigAlgebra

Theoreminsiga 24473 The intersection of a collection of sigma-algebras of same base is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigAlgebra sigAlgebra

19.3.13.2  Generated Sigma-Algebra

Syntaxcsigagen 24474 Extend class notation to include the sigma-algebra generator.
sigaGen

Definitiondf-sigagen 24475* 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 sigAlgebra

Theoremsigagenval 24476* Value of the generated sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigaGen sigAlgebra

Theoremsigagensiga 24477 A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigaGen sigAlgebra

Theoremsgsiga 24478 A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
sigaGen sigAlgebra

Theoremunisg 24479 The sigma algebra generated by a collection is a sigma algebra on . (Contributed by Thierry Arnoux, 27-Dec-2016.)
sigaGen

Theoremdmsigagen 24480 A sigma algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.)
sigaGen

Theoremsssigagen 24481 A set is a subset of the sigma algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
sigaGen

Theoremsssigagen2 24482 A subset of the generating set is also a subset of the generated sigma algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
sigaGen

Theoremelsigagen 24483 Any element of set is also an element of the sigma algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
sigaGen

Theoremelsigagen2 24484 Any countable union of elements of a set is also in the sigma algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
sigaGen

Theoremsigagenss 24485 The generated sigma-algebra is a subset of all sigma algebra containing the generating set, i.e. the generated sigma-algebra is the smallest sigma algebra containing the generating set, here . (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra sigaGen

Theoremsigagenss2 24486 Sufficient condition for inclusion of sigma algebra. This is used to prove equality of sigma algebra. (Contributed by Thierry Arnoux, 10-Oct-2017.)
sigaGen sigaGen sigaGen

Theoremsigagenid 24487 The sigma-algebra generated by a sigma-algebra is itself. (Contributed by Thierry Arnoux, 4-Jun-2017.)
sigAlgebra sigaGen

19.3.13.3  The Borel algebra on the real numbers

Syntaxcbrsiga 24488 The Borel Algebra on real numbers, usually a gothic B
𝔅

Definitiondf-brsiga 24489 A Borel Algebra is defined as a sigma algebra generated by a topology. 'The' Borel sigma algebra here refers to the sigma algebra generated by the topology of open intervals on real numbers. The Borel algebra of a given topology is the sigma-algebra generated by , sigaGen, so there is no need to introduce a special constant function for Borel sigma Algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 sigaGen

Theorembrsiga 24490 The Borel Algebra on real numbers is a Borel sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 sigaGen

Theorembrsigarn 24491 The Borel Algebra is a sigma algebra on the real numbers. (Contributed by Thierry Arnoux, 27-Dec-2016.)
𝔅 sigAlgebra

Theorembrsigasspwrn 24492 The Borel Algebra is a set of subsets of the real numbers. (Contributed by Thierry Arnoux, 19-Jan-2017.)
𝔅

Theoremunibrsiga 24493 The union of the Borel Algebra is the set of real numbers. (Contributed by Thierry Arnoux, 21-Jan-2017.)
𝔅

Theoremcldssbrsiga 24494 A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.)
sigaGen

19.3.13.4  Product Sigma-Algebra

Syntaxcsx 24495 Extend class notation with the product sigma-algebra operation.
×s

Definitiondf-sx 24496* Define the product sigma-algebra operation, analogue to df-tx 17547. (Contributed by Thierry Arnoux, 1-Jun-2017.)
×s sigaGen

Theoremsxval 24497* Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
×s sigaGen

Theoremsxsiga 24498 A product sigma-algebra is a sigma-algebra (Contributed by Thierry Arnoux, 1-Jun-2017.)
sigAlgebra sigAlgebra ×s sigAlgebra

Theoremsxsigon 24499 A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
sigAlgebra sigAlgebra ×s sigAlgebra

Theoremsxuni 24500 The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
sigAlgebra sigAlgebra ×s

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