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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprodeq2d 25201* Equality deduction for sum. Note that unlike prodeq2dv 25202, may occur in . (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2dv 25202* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2sdv 25203* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theorem2cprodeq2dv 25204* Equality deduction for double sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12dv 25205* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12rdv 25206* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprod2id 25207* The second class argument to a sum can be chosen so that it is always a set. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrblem 25208* Lemma for prodrb 25211. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremfprodcvg 25209* The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrblem2 25210* Lemma for prodrb 25211. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrb 25211* Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem3 25212* Lemma for prodmo 25215. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem2a 25213* Lemma for prodmo 25215. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem2 25214* Lemma for prodmo 25215. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmo 25215* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremzprod 25216* Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremiprod 25217* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremzprodn0 25218* Non-zero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)

Theoremiprodn0 25219* Non-zero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)

19.7.9  Finite products

Theoremfprod 25220* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.)

Theoremfprodntriv 25221* A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremprod0 25222 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprod1 25223* Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremprodfc 25224* A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremfprodf1o 25225* Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremprodss 25226* Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.)

Theoremfprodss 25227* Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfprodser 25228* A finite product expressed in terms of a partial product of an infinite sequence. The recursive definition of a finite product follows from here. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcl2lem 25229* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcllem 25230* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcl 25231* Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodrecl 25232* Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodzcl 25233* Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodnncl 25234* Closure of a finite product of natural numbers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodrpcl 25235* Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodnn0cl 25236* Closure of a finite product of non-negative integers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodmul 25237* The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfproddiv 25238* The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)

Theoremprodsn 25239* A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprod1 25240* A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremclimprod1 25241 The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfprodsplit 25242* Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfprodm1 25243* Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfprod1p 25244* Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)

Theoremfprodp1 25245* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)

Theoremfprodm1s 25246* Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)

Theoremfprodp1s 25247* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)

Theoremprodsns 25248* A product of the singleton is the term. (Contributed by Scott Fenton, 25-Dec-2017.)

Theoremfprodfac 25249* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfprodabs 25250* The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)

Theoremfprodefsum 25251* Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)

Theoremfprodeq0 25252* Anything finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)

Theoremfprodshft 25253* Shift the index of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)

Theoremfprodrev 25254* Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)

Theoremfprodconst 25255* The product of constant terms ( is not free in .) (Contributed by Scott Fenton, 12-Jan-2018.)

Theoremfprodn0 25256* A finite product of non-zero terms is non-zero. (Contributed by Scott Fenton, 15-Jan-2018.)

Theoremfprod2dlem 25257* Lemma for fprod2d 25258- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)

Theoremfprod2d 25258* Write a double product as a product over a two dimensional region. Compare fsum2d 12510. (Contributed by Scott Fenton, 30-Jan-2018.)

Theoremfprodxp 25259* Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)

Theoremfprodcnv 25260* Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)

Theoremfprodcom2 25261* Interchange order of multiplication. Note that and are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.)

Theoremfprodcom 25262* Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)

Theoremfprod0diag 25263* Two ways to express "the product of over the the triangular region , , . Compare fsum0diag 12516. (Contributed by Scott Fenton, 2-Feb-2018.)

19.7.10  Infinite products

Theoremiprodclim 25264* An infinite product equals the value its sequence converges to. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodclim2 25265* A converging product converges to its infinite product. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodclim3 25266* The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that must not occur in . (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodcl 25267* The product of a non-trivially converging infinite sequence is a complex number. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodrecl 25268* The product of a non-trivially converging infinite real sequence is a real number. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodmul 25269* Multiplication of infinite sums. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodefisumlem 25270 Lemma for iprodefisum 25271. (Contributed by Scott Fenton, 11-Feb-2018.)

Theoremiprodefisum 25271* Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)

Theoremiprodgam 25272* An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)

19.7.11  Falling and Rising Factorial

Syntaxcfallfac 25273 Declare the syntax for the falling factorial.
FallFac

Syntaxcrisefac 25274 Declare the syntax for the rising factorial.
RiseFac

Definitiondf-risefac 25275* Define the rising factorial function. This is the function for complex and non-negative integers . (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Definitiondf-fallfac 25276* Define the falling factorial function. This is the function for complex and non-negative integers . (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefacval 25277* The value of the rising factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfacval 25278* The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefacval2 25279* One-based value of rising factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
RiseFac

Theoremfallfacval2 25280* One-based value of falling factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
FallFac

Theoremrisefaccllem 25281* Lemma for rising factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfaccllem 25282* Lemma for falling factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefaccl 25283 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfaccl 25284 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrerisefaccl 25285 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremrefallfaccl 25286 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremnnrisefaccl 25287 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremzrisefaccl 25288 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremzfallfaccl 25289 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremnn0risefaccl 25290 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremrprisefaccl 25291 Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
RiseFac

Theoremrisefallfac 25292 A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.)
RiseFac FallFac

Theoremfallrisefac 25293 A relationship between falling and rising factorials. (Contributed by Scott Fenton, 17-Jan-2018.)
FallFac RiseFac

Theoremrisefall0lem 25294 Lemma for risefac0 25295 and fallfac0 25296. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)

Theoremrisefac0 25295 The value of the rising factorial when . (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfac0 25296 The value of the falling factorial when . (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefacp1 25297 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac RiseFac

Theoremfallfacp1 25298 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac FallFac

Theoremrisefac1 25299 The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfac1 25300 The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

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