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

Theoremsupfz 25201 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)

Theoreminffz 25202 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)

Theorembcnm1 25203 The binomial coefficent of is . (Contributed by Scott Fenton, 16-May-2014.)

Theoremfz0n 25204 The sequence is empty iff is zero. (Contributed by Scott Fenton, 16-May-2014.)

Theorem4bc3eq4 25205 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)

Theorem4bc2eq6 25206 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)

Theoremhalfthird 25207 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)

Theorem5recm6rec 25208 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
;

Theorempnpncand 25209 Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremshftvalg 25210 Value of a sequence shifted by . (Contributed by Scott Fenton, 16-Dec-2017.)

Theorempossumd 25211 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfzp1nel 25212 One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremdivcnvshft 25213* Limit of a ratio function. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremdivcnvlin 25214* Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)

Theoremmuls1d 25215 Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.)

Theoremclimlec3 25216* Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)

19.7.6  Product sequences

Theoremprodf 25217* An infinite product of complex terms is a function from an upper set of integers to . (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremclim2prod 25218* The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremclim2div 25219* The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)

Theoremprodfmul 25220* The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremprodf1 25221 The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodf1f 25222 A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodfclim1 25223 The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodfn0 25224* No term of a non-zero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.)

Theoremprodfrec 25225* The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.)

Theoremprodfdiv 25226* The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.)

19.7.7  Non-trivial convergence

Theoremntrivcvg 25227* A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremntrivcvgn0 25228* A product that converges to a non-zero value converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremntrivcvgfvn0 25229* Any value of a product sequence that converges to a non-zero value is itself non-zero. (Contributed by Scott Fenton, 20-Dec-2017.)

Theoremntrivcvgtail 25230* A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.)

Theoremntrivcvgmullem 25231* Lemma for ntrivcvgmul 25232. (Contributed by Scott Fenton, 19-Dec-2017.)

Theoremntrivcvgmul 25232* The product of two non-trivially converging products converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)

19.7.8  Complex products

Syntaxcprod 25233 Extend class notation to include complex products.

Definitiondf-prod 25234* Define the product of a series with an index set of integers . This definition takes most of the aspects of df-sum 12482 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a non-zero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodex 25235 A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1f 25236 Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremprodeq1 25237* Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremnfcprod1 25238* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremnfcprod 25239* Bound-variable hypothesis builder for product: if is (effectively) not free in and , it is not free in . (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremprodeq2w 25240* Equality theorem for product, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2ii 25241* Equality theorem for product, with the class expressions and guarded by to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2 25242* Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprod 25243* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprodv 25244* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprodi 25245* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1i 25246* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2i 25247* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12i 25248* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1d 25249* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2d 25250* Equality deduction for sum. Note that unlike prodeq2dv 25251, may occur in . (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2dv 25251* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2sdv 25252* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theorem2cprodeq2dv 25253* Equality deduction for double sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12dv 25254* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12rdv 25255* Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprod2id 25256* 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 25257* Lemma for prodrb 25260. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremfprodcvg 25258* The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrblem2 25259* Lemma for prodrb 25260. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrb 25260* Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem3 25261* Lemma for prodmo 25264. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem2a 25262* Lemma for prodmo 25264. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmolem2 25263* Lemma for prodmo 25264. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodmo 25264* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremzprod 25265* Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremiprod 25266* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremzprodn0 25267* Non-zero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)

Theoremiprodn0 25268* 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 25269* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.)

Theoremfprodntriv 25270* A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremprod0 25271 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprod1 25272* Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremprodfc 25273* 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 25274* Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)

Theoremprodss 25275* Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.)

Theoremfprodss 25276* Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfprodser 25277* 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 25278* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcllem 25279* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodcl 25280* Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodrecl 25281* Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodzcl 25282* Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodnncl 25283* Closure of a finite product of natural numbers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodrpcl 25284* Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodnn0cl 25285* Closure of a finite product of non-negative integers. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprodmul 25286* The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfproddiv 25287* The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)

Theoremprodsn 25288* A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremfprod1 25289* A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.)

Theoremclimprod1 25290 The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfprodsplit 25291* Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfprodm1 25292* Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremfprod1p 25293* Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)

Theoremfprodp1 25294* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)

Theoremfprodm1s 25295* Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)

Theoremfprodp1s 25296* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)

Theoremprodsns 25297* A product of the singleton is the term. (Contributed by Scott Fenton, 25-Dec-2017.)

Theoremfprodfac 25298* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfprodabs 25299* The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)

Theoremfprodefsum 25300* Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)

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