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

Theoremindf 24401 An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
𝟭

Theoremindfval 24402 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
𝟭

Theorempr01ssre 24403 The range of the indicator function is a subset of . (Contributed by Thierry Arnoux, 14-Aug-2017.)

Theoremind1 24404 Value of the indicator function where it is . (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremind0 24405 Value of the indicator function where it is . (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremind1a 24406 Value of the indicator function where it is . (Contributed by Thierry Arnoux, 22-Aug-2017.)
𝟭

Theoremindpi1 24407 Preimage of the singleton by the indicator function. See i1f1lem 19569. (Contributed by Thierry Arnoux, 21-Aug-2017.)
𝟭

Theoremindsum 24408* Finite sum of a product with the indicator function / cross-product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremindf1o 24409 The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
𝟭

Theoremindpreima 24410 A function with range as an indicator of the preimage of (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝟭

Theoremindf1ofs 24411* The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
𝟭

19.3.11.3  Extended sum

Syntaxcesum 24412 Extend class notation to include infinite summations.
Σ*

Definitiondf-esum 24413 Define a short-hand for the possibly infinite sum over the extended non-negative reals. Σ* is relying on the properties of the tsums, developped by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Σ* s tsums

Theoremesumex 24414 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Σ*

Theoremesumcl 24415* Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumeq12dvaf 24416 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
Σ* Σ*

Theoremesumeq12dva 24417* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)
Σ* Σ*

Theoremesumeq12d 24418* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
Σ* Σ*

Theoremesumeq1 24419* Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
Σ* Σ*

Theoremesumeq1d 24420 Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ* Σ*

Theoremesumeq2 24421* Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
Σ* Σ*

Theoremesumeq2d 24422 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Σ* Σ*

Theoremesumeq2dv 24423* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ* Σ*

Theoremesumeq2sdv 24424* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Σ* Σ*

Theoremnfesum1 24425 Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ*

Theoremcbvesum 24426* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ* Σ*

Theoremcbvesumv 24427* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ* Σ*

Theoremesumid 24428 Identify the extended sum as any limit points of the infinite sum. (Contributed by Thierry Arnoux, 9-May-2017.)
s tsums        Σ*

Theoremesumval 24429* Develop the value of the extended sum. (Contributed by Thierry Arnoux, 4-Jan-2017.)
s g        Σ*

Theoremesumel 24430* The extended sum is a limit point of the corresponding infinite group sum. (Contributed by Thierry Arnoux, 24-Mar-2017.)
Σ* s tsums

Theoremesumnul 24431 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
Σ*

Theoremesum0 24432* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
Σ*

Theoremesumf1o 24433* Re-index an extended sum using a bijection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Σ* Σ*

Theoremesumc 24434* Convert from the collection form to the class-variable form of a sum. (Contributed by Thierry Arnoux, 10-May-2017.)
Σ* Σ*

Theoremesumsplit 24435 Split an extended sum into two parts. (Contributed by Thierry Arnoux, 9-May-2017.)
Σ* Σ* Σ*

Theoremesumadd 24436* Addition of infinite sums. (Contributed by Thierry Arnoux, 24-Mar-2017.)
Σ* Σ* Σ*

Theoremesumle 24437* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Σ* Σ*

Theoremesummono 24438* Extended sum is monotonic. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ* Σ*

Theoremgsumesum 24439* Relate a group sum on s to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
s g Σ*

Theoremesumlub 24440* The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Σ*        Σ*

Theoremesumaddf 24441* Addition of infinite sums. (Contributed by Thierry Arnoux, 22-Jun-2017.)
Σ* Σ* Σ*

Theoremesumlef 24442* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
Σ* Σ*

Theoremesumcst 24443* The extended sum of a constant. (Contributed by Thierry Arnoux, 3-Mar-2017.) (Revised by Thierry Arnoux, 5-Jul-2017.)
Σ*

Theoremesumsn 24444* The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumpr 24445* Extended sum over a pair. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumpr2 24446* Extended sum over a pair, with a relaxed condition compared to esumpr 24445. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Σ*

Theoremesumfzf 24447* Formulating a partial extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
Σ*

Theoremesumfsup 24448 Formulating an extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
Σ*

Theoremesumfsupre 24449 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 24450 Change the index set to a subset by adding zeroes. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Σ* Σ*

Theoremesumpinfval 24451* 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 24452 Lemma for esumpfinval 24453 (Contributed by Thierry Arnoux, 28-Jun-2017.)
fld g s g

Theoremesumpfinval 24453* The value of the extended sum of a finite set of non-negative finite terms (Contributed by Thierry Arnoux, 28-Jun-2017.)
Σ*

Theoremesumpfinvalf 24454 Same as esumpfinval 24453, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Σ*

Theoremesumpinfsum 24455* The value of the extended sum of infinitely many terms greater than one. (Contributed by Thierry Arnoux, 29-Jun-2017.)
Σ*

Theoremesumpcvgval 24456* The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)
Σ*

Theoremesumpmono 24457* The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Σ* Σ*

Theoremesumcocn 24458* Lemma for esummulc2 24460 and co. Composing with a continuous function preserves extended sums (Contributed by Thierry Arnoux, 29-Jun-2017.)
ordTop t                                           Σ* Σ*

Theoremesummulc1 24459* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* Σ*

Theoremesummulc2 24460* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* Σ*

Theoremesumdivc 24461* An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Σ* /𝑒 Σ* /𝑒

Theoremhashf2 24462 Lemma for hasheuni 24463 (Contributed by Thierry Arnoux, 19-Nov-2016.)

Theoremhasheuni 24463* The cardinality of a disjoint union, not necessarily finite. cf. hashuni 12592. (Contributed by Thierry Arnoux, 19-Nov-2016.) (Revised by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 20-Jun-2017.)
Disj Σ*

Theoremesumcvg 24464* The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 12509. (Contributed by Thierry Arnoux, 5-Sep-2017.)
s        Σ*                      Σ*

Theoremesumcvg2 24465* Simpler version of esumcvg 24464. (Contributed by Thierry Arnoux, 5-Sep-2017.)
s                             Σ* Σ*

19.3.12  Mixed Function/Constant operation

Syntaxcofc 24466 Extend class notation to include mapping of an operation to an operation for a function and a constant.
𝑓/𝑐

Definitiondf-ofc 24467* 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 24468 Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
𝑓/𝑐 𝑓/𝑐

Theoremofcfval 24469* Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
𝑓/𝑐

Theoremofcval 24470 Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcfn 24471 The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcfeqd2 24472* Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐 𝑓/𝑐

Theoremofcfval3 24473* General value of 𝑓/𝑐 with no assumptions on functionality of . (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcf 24474* The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
𝑓/𝑐

Theoremofcfval2 24475* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcfval4 24476* The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)
𝑓/𝑐

Theoremofcc 24477 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 24478 Extend class notation to include the function giving the sigma-algebras on a given base set.
sigAlgebra

Definitiondf-siga 24479* 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 24480* Lemma for issiga 24482 and isrnsiga 24484 The set of sigma algebra with base set is a set. Note: a more generic version with could be useful for sigaval 24481. (Contributed by Thierry Arnoux, 24-Oct-2016.)

Theoremsigaval 24481* The set of sigma-algebra with a given base set. (Contributed by Thierry Arnoux, 23-Sep-2016.)
sigAlgebra

Theoremissiga 24482* An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)
sigAlgebra

TheoremisrnsigaOLD 24483* 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 24484* 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 24485 A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
sigAlgebra

Theorembaselsiga 24486 A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
sigAlgebra

Theoremsigasspw 24487 A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
sigAlgebra

Theoremsigaclcu 24488 A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
sigAlgebra

Theoremsigaclcuni 24489* A sigma-algebra is closed under countable union: indexed union version (Contributed by Thierry Arnoux, 8-Jun-2017.)
sigAlgebra

Theoremsigaclfu 24490 A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
sigAlgebra

Theoremsigaclcu2 24491* A sigma-algebra is closed under countable union - indexing on (Contributed by Thierry Arnoux, 29-Dec-2016.)
sigAlgebra

Theoremsigaclfu2 24492* A sigma-algebra is closed under finite union - indexing on ..^ (Contributed by Thierry Arnoux, 28-Dec-2016.)
sigAlgebra ..^ ..^

Theoremsigaclcu3 24493* A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
sigAlgebra       ..^

Theoremissgon 24494 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 24495 A sigma alebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra sigAlgebra

Theoremelsigass 24496 An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra

Theoremelrnsiga 24497 Dropping the base information off a sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)
sigAlgebra sigAlgebra

Theoremisrnsigau 24498* The property of being a sigma algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
sigAlgebra

Theoremunielsiga 24499 A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
sigAlgebra

Theoremdmvlsiga 24500 Lebesgue-measurable subsets of form a sigma-algebra (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
sigAlgebra

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