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Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhash2iun1dif1 11301* The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
Disj        Disj

Theoremhashrabrex 11302* The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
Disj

Theoremhashuni 11303* The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
Disj

4.8.3  The binomial theorem

Theorembinomlem 11304* Lemma for binom 11305 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)

Theorembinom 11305* The binomial theorem: is the sum from to of . Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 11304. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)

Theorembinom1p 11306* Special case of the binomial theorem for . (Contributed by Paul Chapman, 10-May-2007.)

Theorembinom11 11307* Special case of the binomial theorem for . (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembinom1dif 11308* A summation for the difference between and . (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)

Theorembcxmaslem1 11309 Lemma for bcxmas 11310. (Contributed by Paul Chapman, 18-May-2007.)

Theorembcxmas 11310* Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)

4.8.4  Infinite sums (cont.)

Theoremisumshft 11311* Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)

Theoremisumsplit 11312* Split off the first terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)

Theoremisum1p 11313* The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)

Theoremisumnn0nn 11314* Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)

Theoremisumrpcl 11315* The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)

Theoremisumle 11316* Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)

Theoremisumlessdc 11317* A finite sum of nonnegative numbers is less than or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)
DECID

4.8.5  Miscellaneous converging and diverging sequences

Theoremdivcnv 11318* The sequence of reciprocals of positive integers, multiplied by the factor , converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)

4.8.6  Arithmetic series

Theoremarisum 11319* Arithmetic series sum of the first positive integers. This is Metamath 100 proof #68. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.)

Theoremarisum2 11320* Arithmetic series sum of the first nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.)

Theoremtrireciplem 11321 Lemma for trirecip 11322. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)

Theoremtrirecip 11322 The sum of the reciprocals of the triangle numbers converge to two. This is Metamath 100 proof #42. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)

4.8.7  Geometric series

Theoremexpcnvap0 11323* A sequence of powers of a complex number with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 23-Oct-2022.)
#

Theoremexpcnvre 11324* A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)

Theoremexpcnv 11325* A sequence of powers of a complex number with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)

Theoremexplecnv 11326* A sequence of terms converges to zero when it is less than powers of a number whose absolute value is smaller than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeosergap 11327* The value of the finite geometric series ... . (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)
#                      ..^

Theoremgeoserap 11328* The value of the finite geometric series ... . This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Revised by Jim Kingdon, 24-Oct-2022.)
#

Theorempwm1geoserap1 11329* The n-th power of a number decreased by 1 expressed by the finite geometric series ... . (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.)
#

Theoremabsltap 11330 Less-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
#

Theoremabsgtap 11331 Greater-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
#

Theoremgeolim 11332* The partial sums in the infinite series ... converge to . (Contributed by NM, 15-May-2006.)

Theoremgeolim2 11333* The partial sums in the geometric series ... converge to . (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoreclim 11334* The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeo2sum 11335* The value of the finite geometric series ... , multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeo2sum2 11336* The value of the finite geometric series ... . (Contributed by Mario Carneiro, 7-Sep-2016.)
..^

Theoremgeo2lim 11337* The value of the infinite geometric series ... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremgeoisum 11338* The infinite sum of ... is . (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoisumr 11339* The infinite sum of reciprocals ... is . (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoisum1 11340* The infinite sum of ... is . (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremgeoisum1c 11341* The infinite sum of ... is . (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theorem0.999... 11342 The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. , is exactly equal to 1. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.)
;

Theoremgeoihalfsum 11343 Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 11340. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 11342 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by AV, 9-Jul-2022.)

4.8.8  Ratio test for infinite series convergence

Theoremcvgratnnlembern 11344 Lemma for cvgratnn 11352. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.)

Theoremcvgratnnlemnexp 11345* Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 15-Nov-2022.)

Theoremcvgratnnlemmn 11346* Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 15-Nov-2022.)

Theoremcvgratnnlemseq 11347* Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 21-Nov-2022.)

Theoremcvgratnnlemabsle 11348* Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 21-Nov-2022.)

Theoremcvgratnnlemsumlt 11349* Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 23-Nov-2022.)

Theoremcvgratnnlemfm 11350* Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 23-Nov-2022.)

Theoremcvgratnnlemrate 11351* Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 21-Nov-2022.)

Theoremcvgratnn 11352* Ratio test for convergence of a complex infinite series. If the ratio of the absolute values of successive terms in an infinite sequence is less than 1 for all terms, then the infinite sum of the terms of converges to a complex number. Although this theorem is similar to cvgratz 11353 and cvgratgt0 11354, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11171 is sensitive to how a sequence is indexed. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 12-Nov-2022.)

Theoremcvgratz 11353* Ratio test for convergence of a complex infinite series. If the ratio of the absolute values of successive terms in an infinite sequence is less than 1 for all terms, then the infinite sum of the terms of converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)

Theoremcvgratgt0 11354* Ratio test for convergence of a complex infinite series. If the ratio of the absolute values of successive terms in an infinite sequence is less than 1 for all terms beyond some index , then the infinite sum of the terms of converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)

4.8.9  Mertens' theorem

Theoremmertenslemub 11355* Lemma for mertensabs 11358. An upper bound for . (Contributed by Jim Kingdon, 3-Dec-2022.)

Theoremmertenslemi1 11356* Lemma for mertensabs 11358. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)

Theoremmertenslem2 11357* Lemma for mertensabs 11358. (Contributed by Mario Carneiro, 28-Apr-2014.)

Theoremmertensabs 11358* Mertens' theorem. If is an absolutely convergent series and is convergent, then (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 8-Dec-2022.)

4.8.10  Finite and infinite products

4.8.10.1  Product sequences

Theoremprodf 11359* An infinite product of complex terms is a function from an upper set of integers to . (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremclim2prod 11360* The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremclim2divap 11361* The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)
#

Theoremprod3fmul 11362* The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.)

Theoremprodf1 11363 The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodf1f 11364 A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodfclim1 11365 The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)

Theoremprodfap0 11366* The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)
#        #

Theoremprodfrecap 11367* The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
#

Theoremprodfdivap 11368* The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
#

4.8.10.2  Non-trivial convergence

Theoremntrivcvgap 11369* A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)
#

Theoremntrivcvgap0 11370* A product that converges to a value apart from zero converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
#        #

4.8.10.3  Complex products

Syntaxcprod 11371 Extend class notation to include complex products.

Definitiondf-proddc 11372* Define the product of a series with an index set of integers . This definition takes most of the aspects of df-sumdc 11175 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 21-Mar-2024.)
DECID #

Theoremprodeq1f 11373 Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremprodeq1 11374* Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)

Theoremnfcprod1 11375* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremnfcprod 11376* 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 11377* Equality theorem for product, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2 11378* Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprod 11379* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprodv 11380* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremcbvprodi 11381* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1i 11382* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2i 11383* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12i 11384* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq1d 11385* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2d 11386* Equality deduction for product. Note that unlike prodeq2dv 11387, may occur in . (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2dv 11387* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq2sdv 11388* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theorem2cprodeq2dv 11389* Equality deduction for double product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12dv 11390* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodeq12rdv 11391* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Theoremprodrbdclem 11392* Lemma for prodrbdc 11395. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.)
DECID

Theoremfproddccvg 11393* The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)
DECID

Theoremprodrbdclem2 11394* Lemma for prodrbdc 11395. (Contributed by Scott Fenton, 4-Dec-2017.)
DECID        DECID

Theoremprodrbdc 11395* Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)
DECID        DECID

Theoremprodmodclem3 11396* Lemma for prodmodc 11399. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)

Theoremprodmodclem2a 11397* Lemma for prodmodc 11399. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
DECID

Theoremprodmodclem2 11398* Lemma for prodmodc 11399. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
DECID #

Theoremprodmodc 11399* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.) (Modified by Jim Kingdon, 14-Apr-2024.)
DECID #

Theoremzproddc 11400* Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)
#               DECID

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