Type  Label  Description 
Statement 

Theorem  hash2iun1dif1 11301* 
The cardinality of a nested disjoint indexed union. (Contributed by AV,
9Jan2022.)

Disj
Disj
♯ ♯
♯ ♯ 

Theorem  hashrabrex 11302* 
The number of elements in a class abstraction with a restricted
existential quantification. (Contributed by Alexander van der Vekens,
29Jul2018.)

Disj ♯ ♯ 

Theorem  hashuni 11303* 
The cardinality of a disjoint union. (Contributed by Mario Carneiro,
24Jan2015.)

Disj ♯
♯ 

4.8.3 The binomial theorem


Theorem  binomlem 11304* 
Lemma for binom 11305 (binomial theorem). Inductive step.
(Contributed by
NM, 6Dec2005.) (Revised by Mario Carneiro, 24Apr2014.)



Theorem  binom 11305* 
The binomial theorem: is the sum from to
of . Theorem
152.8 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, 7Dec2005.) (Proof shortened by Mario Carneiro,
24Apr2014.)



Theorem  binom1p 11306* 
Special case of the binomial theorem for .
(Contributed by Paul Chapman, 10May2007.)



Theorem  binom11 11307* 
Special case of the binomial theorem for .
(Contributed by
Mario Carneiro, 13Mar2014.)



Theorem  binom1dif 11308* 
A summation for the difference between and
.
(Contributed by Scott Fenton, 9Apr2014.) (Revised by
Mario Carneiro, 22May2014.)



Theorem  bcxmaslem1 11309 
Lemma for bcxmas 11310. (Contributed by Paul Chapman,
18May2007.)



Theorem  bcxmas 11310* 
Parallel summation (Christmas Stocking) theorem for Pascal's Triangle.
(Contributed by Paul Chapman, 18May2007.) (Revised by Mario Carneiro,
24Apr2014.)



4.8.4 Infinite sums (cont.)


Theorem  isumshft 11311* 
Index shift of an infinite sum. (Contributed by Paul Chapman,
31Oct2007.) (Revised by Mario Carneiro, 24Apr2014.)



Theorem  isumsplit 11312* 
Split off the first
terms of an infinite sum. (Contributed by
Paul Chapman, 9Feb2008.) (Revised by Jim Kingdon, 21Oct2022.)



Theorem  isum1p 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,
2Jan2006.) (Revised by Mario Carneiro, 24Apr2014.)



Theorem  isumnn0nn 11314* 
Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed
by NM, 2Jan2006.) (Revised by Mario Carneiro, 24Apr2014.)



Theorem  isumrpcl 11315* 
The infinite sum of positive reals is positive. (Contributed by Paul
Chapman, 9Feb2008.) (Revised by Mario Carneiro, 24Apr2014.)



Theorem  isumle 11316* 
Comparison of two infinite sums. (Contributed by Paul Chapman,
13Nov2007.) (Revised by Mario Carneiro, 24Apr2014.)



Theorem  isumlessdc 11317* 
A finite sum of nonnegative numbers is less than or equal to its limit.
(Contributed by Mario Carneiro, 24Apr2014.)

DECID


4.8.5 Miscellaneous converging and diverging
sequences


Theorem  divcnv 11318* 
The sequence of reciprocals of positive integers, multiplied by the
factor ,
converges to zero. (Contributed by NM, 6Feb2008.)
(Revised by Jim Kingdon, 22Oct2022.)



4.8.6 Arithmetic series


Theorem  arisum 11319* 
Arithmetic series sum of the first positive integers. This is
Metamath 100 proof #68. (Contributed by FL, 16Nov2006.) (Proof
shortened by Mario Carneiro, 22May2014.)



Theorem  arisum2 11320* 
Arithmetic series sum of the first nonnegative integers.
(Contributed by Mario Carneiro, 17Apr2015.) (Proof shortened by AV,
2Aug2021.)



Theorem  trireciplem 11321 
Lemma for trirecip 11322. Show that the sum converges. (Contributed
by
Scott Fenton, 22Apr2014.) (Revised by Mario Carneiro,
22May2014.)



Theorem  trirecip 11322 
The sum of the reciprocals of the triangle numbers converge to two.
This is Metamath 100 proof #42. (Contributed by Scott Fenton,
23Apr2014.) (Revised by Mario Carneiro, 22May2014.)



4.8.7 Geometric series


Theorem  expcnvap0 11323* 
A sequence of powers of a complex number with absolute value
smaller than 1 converges to zero. (Contributed by NM, 8May2006.)
(Revised by Jim Kingdon, 23Oct2022.)

#


Theorem  expcnvre 11324* 
A sequence of powers of a nonnegative real number less than one
converges to zero. (Contributed by Jim Kingdon, 28Oct2022.)



Theorem  expcnv 11325* 
A sequence of powers of a complex number with absolute value
smaller than 1 converges to zero. (Contributed by NM, 8May2006.)
(Revised by Jim Kingdon, 28Oct2022.)



Theorem  explecnv 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, 19Jul2008.) (Revised by Mario Carneiro, 26Apr2014.)



Theorem  geosergap 11327* 
The value of the finite geometric series ...
. (Contributed by Mario Carneiro, 2May2016.)
(Revised by Jim Kingdon, 24Oct2022.)

# ..^ 

Theorem  geoserap 11328* 
The value of the finite geometric series
...
. This is Metamath 100 proof #66. (Contributed by
NM, 12May2006.) (Revised by Jim Kingdon, 24Oct2022.)

# 

Theorem  pwm1geoserap1 11329* 
The nth power of a number decreased by 1 expressed by the finite
geometric series
... .
(Contributed by AV, 14Aug2021.) (Revised by Jim Kingdon,
24Oct2022.)

#


Theorem  absltap 11330 
Lessthan of absolute value implies apartness. (Contributed by Jim
Kingdon, 29Oct2022.)

# 

Theorem  absgtap 11331 
Greaterthan of absolute value implies apartness. (Contributed by Jim
Kingdon, 29Oct2022.)

# 

Theorem  geolim 11332* 
The partial sums in the infinite series
...
converge to . (Contributed by NM,
15May2006.)



Theorem  geolim2 11333* 
The partial sums in the geometric series ...
converge to .
(Contributed by NM,
6Jun2006.) (Revised by Mario Carneiro, 26Apr2014.)



Theorem  georeclim 11334* 
The limit of a geometric series of reciprocals. (Contributed by Paul
Chapman, 28Dec2007.) (Revised by Mario Carneiro, 26Apr2014.)



Theorem  geo2sum 11335* 
The value of the finite geometric series ...
,
multiplied by a constant. (Contributed by Mario
Carneiro, 17Mar2014.) (Revised by Mario Carneiro, 26Apr2014.)



Theorem  geo2sum2 11336* 
The value of the finite geometric series
...
. (Contributed by Mario Carneiro, 7Sep2016.)

..^


Theorem  geo2lim 11337* 
The value of the infinite geometric series
... , multiplied by a constant. (Contributed
by Mario Carneiro, 15Jun2014.)



Theorem  geoisum 11338* 
The infinite sum of ... is
.
(Contributed by NM, 15May2006.) (Revised by Mario Carneiro,
26Apr2014.)



Theorem  geoisumr 11339* 
The infinite sum of reciprocals
... is .
(Contributed by rpenner, 3Nov2007.) (Revised by Mario Carneiro,
26Apr2014.)



Theorem  geoisum1 11340* 
The infinite sum of ... is .
(Contributed by NM, 1Nov2007.) (Revised by Mario Carneiro,
26Apr2014.)



Theorem  geoisum1c 11341* 
The infinite sum of
... is
. (Contributed by NM, 2Nov2007.) (Revised
by Mario Carneiro, 26Apr2014.)



Theorem  0.999... 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, 2Nov2007.)
(Revised by AV, 8Sep2021.)

; 

Theorem  geoihalfsum 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, 4Jan2017.)
(Proof shortened by AV, 9Jul2022.)



4.8.8 Ratio test for infinite series
convergence


Theorem  cvgratnnlembern 11344 
Lemma for cvgratnn 11352. Upper bound for a geometric progression of
positive ratio less than one. (Contributed by Jim Kingdon,
24Nov2022.)



Theorem  cvgratnnlemnexp 11345* 
Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 15Nov2022.)



Theorem  cvgratnnlemmn 11346* 
Lemma for cvgratnn 11352. (Contributed by Jim Kingdon,
15Nov2022.)



Theorem  cvgratnnlemseq 11347* 
Lemma for cvgratnn 11352. (Contributed by Jim Kingdon,
21Nov2022.)



Theorem  cvgratnnlemabsle 11348* 
Lemma for cvgratnn 11352. (Contributed by Jim Kingdon,
21Nov2022.)



Theorem  cvgratnnlemsumlt 11349* 
Lemma for cvgratnn 11352. (Contributed by Jim Kingdon,
23Nov2022.)



Theorem  cvgratnnlemfm 11350* 
Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 23Nov2022.)



Theorem  cvgratnnlemrate 11351* 
Lemma for cvgratnn 11352. (Contributed by Jim Kingdon, 21Nov2022.)



Theorem  cvgratnn 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, 26Apr2005.) (Revised by Jim Kingdon,
12Nov2022.)



Theorem  cvgratz 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,
26Apr2005.) (Revised by Jim Kingdon, 11Nov2022.)



Theorem  cvgratgt0 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, 26Apr2005.) (Revised by Jim Kingdon,
11Nov2022.)



4.8.9 Mertens' theorem


Theorem  mertenslemub 11355* 
Lemma for mertensabs 11358. An upper bound for . (Contributed by
Jim Kingdon, 3Dec2022.)



Theorem  mertenslemi1 11356* 
Lemma for mertensabs 11358. (Contributed by Mario Carneiro,
29Apr2014.) (Revised by Jim Kingdon, 2Dec2022.)



Theorem  mertenslem2 11357* 
Lemma for mertensabs 11358. (Contributed by Mario Carneiro,
28Apr2014.)



Theorem  mertensabs 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, 29Apr2014.) (Revised by Jim Kingdon,
8Dec2022.)



4.8.10 Finite and infinite
products


4.8.10.1 Product sequences


Theorem  prodf 11359* 
An infinite product of complex terms is a function from an upper set of
integers to .
(Contributed by Scott Fenton, 4Dec2017.)



Theorem  clim2prod 11360* 
The limit of an infinite product with an initial segment added.
(Contributed by Scott Fenton, 18Dec2017.)



Theorem  clim2divap 11361* 
The limit of an infinite product with an initial segment removed.
(Contributed by Scott Fenton, 20Dec2017.)

#


Theorem  prod3fmul 11362* 
The product of two infinite products. (Contributed by Scott Fenton,
18Dec2017.) (Revised by Jim Kingdon, 22Mar2024.)



Theorem  prodf1 11363 
The value of the partial products in a onevalued infinite product.
(Contributed by Scott Fenton, 5Dec2017.)



Theorem  prodf1f 11364 
A onevalued infinite product is equal to the constant one function.
(Contributed by Scott Fenton, 5Dec2017.)



Theorem  prodfclim1 11365 
The constant one product converges to one. (Contributed by Scott
Fenton, 5Dec2017.)



Theorem  prodfap0 11366* 
The product of finitely many terms apart from zero is apart from zero.
(Contributed by Scott Fenton, 14Jan2018.) (Revised by Jim Kingdon,
23Mar2024.)

# # 

Theorem  prodfrecap 11367* 
The reciprocal of a finite product. (Contributed by Scott Fenton,
15Jan2018.) (Revised by Jim Kingdon, 24Mar2024.)

#


Theorem  prodfdivap 11368* 
The quotient of two products. (Contributed by Scott Fenton,
15Jan2018.) (Revised by Jim Kingdon, 24Mar2024.)

#


4.8.10.2 Nontrivial convergence


Theorem  ntrivcvgap 11369* 
A nontrivially converging infinite product converges. (Contributed by
Scott Fenton, 18Dec2017.)

#


Theorem  ntrivcvgap0 11370* 
A product that converges to a value apart from zero converges
nontrivially. (Contributed by Scott Fenton, 18Dec2017.)

#
#


4.8.10.3 Complex products


Syntax  cprod 11371 
Extend class notation to include complex products.



Definition  dfproddc 11372* 
Define the product of a series with an index set of integers .
This definition takes most of the aspects of dfsumdc 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, 4Dec2017.) (Revised by Jim Kingdon,
21Mar2024.)

DECID
#


Theorem  prodeq1f 11373 
Equality theorem for a product. (Contributed by Scott Fenton,
1Dec2017.)



Theorem  prodeq1 11374* 
Equality theorem for a product. (Contributed by Scott Fenton,
1Dec2017.)



Theorem  nfcprod1 11375* 
Boundvariable hypothesis builder for product. (Contributed by Scott
Fenton, 4Dec2017.)



Theorem  nfcprod 11376* 
Boundvariable hypothesis builder for product: if is (effectively)
not free in
and , it is not free
in .
(Contributed by Scott Fenton, 1Dec2017.)



Theorem  prodeq2w 11377* 
Equality theorem for product, when the class expressions and
are equal everywhere. Proved using only Extensionality. (Contributed
by Scott Fenton, 4Dec2017.)



Theorem  prodeq2 11378* 
Equality theorem for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  cbvprod 11379* 
Change bound variable in a product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  cbvprodv 11380* 
Change bound variable in a product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  cbvprodi 11381* 
Change bound variable in a product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq1i 11382* 
Equality inference for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq2i 11383* 
Equality inference for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq12i 11384* 
Equality inference for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq1d 11385* 
Equality deduction for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq2d 11386* 
Equality deduction for product. Note that unlike prodeq2dv 11387,
may occur in . (Contributed by Scott Fenton, 4Dec2017.)



Theorem  prodeq2dv 11387* 
Equality deduction for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq2sdv 11388* 
Equality deduction for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  2cprodeq2dv 11389* 
Equality deduction for double product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq12dv 11390* 
Equality deduction for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodeq12rdv 11391* 
Equality deduction for product. (Contributed by Scott Fenton,
4Dec2017.)



Theorem  prodrbdclem 11392* 
Lemma for prodrbdc 11395. (Contributed by Scott Fenton, 4Dec2017.)
(Revised by Jim Kingdon, 4Apr2024.)

DECID


Theorem  fproddccvg 11393* 
The sequence of partial products of a finite product converges to
the whole product. (Contributed by Scott Fenton, 4Dec2017.)

DECID 

Theorem  prodrbdclem2 11394* 
Lemma for prodrbdc 11395. (Contributed by Scott Fenton,
4Dec2017.)

DECID
DECID


Theorem  prodrbdc 11395* 
Rebase the starting point of a product. (Contributed by Scott Fenton,
4Dec2017.)

DECID
DECID


Theorem  prodmodclem3 11396* 
Lemma for prodmodc 11399. (Contributed by Scott Fenton, 4Dec2017.)
(Revised by Jim Kingdon, 11Apr2024.)

♯
♯


Theorem  prodmodclem2a 11397* 
Lemma for prodmodc 11399. (Contributed by Scott Fenton, 4Dec2017.)
(Revised by Jim Kingdon, 11Apr2024.)

♯
♯
DECID ♯


Theorem  prodmodclem2 11398* 
Lemma for prodmodc 11399. (Contributed by Scott Fenton, 4Dec2017.)
(Revised by Jim Kingdon, 13Apr2024.)

♯
DECID #


Theorem  prodmodc 11399* 
A product has at most one limit. (Contributed by Scott Fenton,
4Dec2017.) (Modified by Jim Kingdon, 14Apr2024.)

♯
DECID
#


Theorem  zproddc 11400* 
Series product with index set a subset of the upper integers.
(Contributed by Scott Fenton, 5Dec2017.)

#
DECID
