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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembinom 14401* The binomial theorem: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)). 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 14400. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))

Theorembinom1p 14402* Special case of the binomial theorem for (1 + 𝐴)↑𝑁. (Contributed by Paul Chapman, 10-May-2007.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((1 + 𝐴)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (𝐴𝑘)))

Theorembinom11 14403* Special case of the binomial theorem for 2↑𝑁. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ0 → (2↑𝑁) = Σ𝑘 ∈ (0...𝑁)(𝑁C𝑘))

Theorembinom1dif 14404* A summation for the difference between ((𝐴 + 1)↑𝑁) and (𝐴𝑁). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝐴 + 1)↑𝑁) − (𝐴𝑁)) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴𝑘)))

Theorembcxmaslem1 14405 Lemma for bcxmas 14406. (Contributed by Paul Chapman, 18-May-2007.)
(𝐴 = 𝐵 → ((𝑁 + 𝐴)C𝐴) = ((𝑁 + 𝐵)C𝐵))

Theorembcxmas 14406* Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))

5.10.5  The inclusion/exclusion principle

Theoremincexclem 14407* Lemma for incexc 14408. (Contributed by Mario Carneiro, 7-Aug-2017.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐵) − (#‘(𝐵 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(𝐵 𝑠))))

Theoremincexc 14408* The inclusion/exclusion principle for counting the elements of a finite union of finite sets. This is Metamath 100 proof #96. (Contributed by Mario Carneiro, 7-Aug-2017.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘ 𝑠)))

Theoremincexc2 14409* The inclusion/exclusion principle for counting the elements of a finite union of finite sets. (Contributed by Mario Carneiro, 7-Aug-2017.)
((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘ 𝐴) = Σ𝑛 ∈ (1...(#‘𝐴))((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} (#‘ 𝑠)))

5.10.6  Infinite sums (cont.)

Theoremisumshft 14410* Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ‘(𝑀 + 𝐾))    &   (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑗𝑊) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗𝑊 𝐴 = Σ𝑘𝑍 𝐵)

Theoremisumsplit 14411* Split off the first 𝑁 terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘𝑊 𝐴))

Theoremisum1p 14412* 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.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 = ((𝐹𝑀) + Σ𝑘 ∈ (ℤ‘(𝑀 + 1))𝐴))

Theoremisumnn0nn 14413* 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.)
(𝑘 = 0 → 𝐴 = 𝐵)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴))

Theoremisumrpcl 14414* The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ+)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑊 𝐴 ∈ ℝ+)

Theoremisumle 14415* Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 𝐴𝐵)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ≤ Σ𝑘𝑍 𝐵)

Theoremisumless 14416* A finite sum of nonnegative numbers is less or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐵)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝐴 𝐵 ≤ Σ𝑘𝑍 𝐵)

Theoremisumsup2 14417* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑍 = (ℤ𝑀)    &   𝐺 = seq𝑀( + , 𝐹)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐺𝑗) ≤ 𝑥)       (𝜑𝐺 ⇝ sup(ran 𝐺, ℝ, < ))

Theoremisumsup 14418* An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.)
𝑍 = (ℤ𝑀)    &   𝐺 = seq𝑀( + , 𝐹)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐴)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐺𝑗) ≤ 𝑥)       (𝜑 → Σ𝑘𝑍 𝐴 = sup(ran 𝐺, ℝ, < ))

Theoremisumltss 14419* A partial sum of a series with positive terms is less than the infinite sum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ+)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝐴 𝐵 < Σ𝑘𝑍 𝐵)

Theoremclimcndslem1 14420* Lemma for climcnds 14422: bound the original series by the condensed series. (Contributed by Mario Carneiro, 18-Jul-2014.)
((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))       ((𝜑𝑁 ∈ ℕ0) → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))

Theoremclimcndslem2 14421* Lemma for climcnds 14422: bound the condensed series by the original series. (Contributed by Mario Carneiro, 18-Jul-2014.)
((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))       ((𝜑𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))

Theoremclimcnds 14422* The Cauchy condensation test. If 𝑎(𝑘) is a decreasing sequence of nonnegative terms, then Σ𝑘 ∈ ℕ𝑎(𝑘) converges iff Σ𝑛 ∈ ℕ02↑𝑛 · 𝑎(2↑𝑛) converges. (Contributed by Mario Carneiro, 18-Jul-2014.)
((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))       (𝜑 → (seq1( + , 𝐹) ∈ dom ⇝ ↔ seq0( + , 𝐺) ∈ dom ⇝ ))

5.10.7  Miscellaneous converging and diverging sequences

Theoremdivrcnv 14423* The sequence of reciprocals of real numbers, multiplied by the factor 𝐴, converges to zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℂ → (𝑛 ∈ ℝ+ ↦ (𝐴 / 𝑛)) ⇝𝑟 0)

Theoremdivcnv 14424* The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0)

Theoremflo1 14425 The floor function satisfies ⌊(𝑥) = 𝑥 + 𝑂(1). (Contributed by Mario Carneiro, 21-May-2016.)
(𝑥 ∈ ℝ ↦ (𝑥 − (⌊‘𝑥))) ∈ 𝑂(1)

Theoremdivcnvshft 14426* Limit of a ratio function. (Contributed by Scott Fenton, 16-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐴 / (𝑘 + 𝐵)))       (𝜑𝐹 ⇝ 0)

Theoremsupcvg 14427* Extract a sequence 𝑓 in 𝑋 such that the image of the points in the bounded set 𝐴 converges to the supremum 𝑆 of the set. Similar to Equation 4 of [Kreyszig] p. 144. The proof uses countable choice ax-cc 9140. (Contributed by Mario Carneiro, 15-Feb-2013.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
𝑋 ∈ V    &   𝑆 = sup(𝐴, ℝ, < )    &   𝑅 = (𝑛 ∈ ℕ ↦ (𝑆 − (1 / 𝑛)))    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐹:𝑋onto𝐴)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)       (𝜑 → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ (𝐹𝑓) ⇝ 𝑆))

Theoreminfcvgaux1i 14428* Auxiliary theorem for applications of supcvg 14427. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-2008.)
𝑅 = {𝑥 ∣ ∃𝑦𝑋 𝑥 = -𝐴}    &   (𝑦𝑋𝐴 ∈ ℝ)    &   𝑍𝑋    &   𝑧 ∈ ℝ ∀𝑤𝑅 𝑤𝑧       (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤𝑅 𝑤𝑧)

Theoreminfcvgaux2i 14429* Auxiliary theorem for applications of supcvg 14427. (Contributed by NM, 4-Mar-2008.)
𝑅 = {𝑥 ∣ ∃𝑦𝑋 𝑥 = -𝐴}    &   (𝑦𝑋𝐴 ∈ ℝ)    &   𝑍𝑋    &   𝑧 ∈ ℝ ∀𝑤𝑅 𝑤𝑧    &   𝑆 = -sup(𝑅, ℝ, < )    &   (𝑦 = 𝐶𝐴 = 𝐵)       (𝐶𝑋𝑆𝐵)

Theoremharmonic 14430 The harmonic series 𝐻 diverges. This fact follows from the stronger emcl 24529, which establishes that the harmonic series grows as log𝑛 + γ + o(1), but this uses a more elementary method, attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof #34. (Contributed by Mario Carneiro, 11-Jul-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (1 / 𝑛))    &   𝐻 = seq1( + , 𝐹)        ¬ 𝐻 ∈ dom ⇝

5.10.8  Arithmetic series

Theoremarisum 14431* Arithmetic series sum of the first 𝑁 positive integers. This is Metamath 100 proof #68. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 = (((𝑁↑2) + 𝑁) / 2))

Theoremarisum2 14432* Arithmetic series sum of the first 𝑁 nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))

Theoremtrireciplem 14433 Lemma for trirecip 14434. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))       seq1( + , 𝐹) ⇝ 1

Theoremtrirecip 14434 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.)
Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2

5.10.9  Geometric series

Theoremexpcnv 14435* A sequence of powers of a complex number 𝐴 with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (abs‘𝐴) < 1)       (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴𝑛)) ⇝ 0)

Theoremexplecnv 14436* A sequence of terms converges to zero when it is less than powers of a number 𝐴 whose absolute value is smaller than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) < 1)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (abs‘(𝐹𝑘)) ≤ (𝐴𝑘))       (𝜑𝐹 ⇝ 0)

Theoremgeoserg 14437* The value of the finite geometric series 𝐴𝑀 + 𝐴↑(𝑀 + 1) +... + 𝐴↑(𝑁 − 1). (Contributed by Mario Carneiro, 2-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 1)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴𝑘) = (((𝐴𝑀) − (𝐴𝑁)) / (1 − 𝐴)))

Theoremgeoser 14438* The value of the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Proof shortened by Mario Carneiro, 15-Jun-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 1)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘) = ((1 − (𝐴𝑁)) / (1 − 𝐴)))

Theorempwm1geoser 14439* The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). (Contributed by AV, 14-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘)))

Theoremgeolim 14440* The partial sums in the infinite series 1 + 𝐴↑1 + 𝐴↑2... converge to (1 / (1 − 𝐴)). (Contributed by NM, 15-May-2006.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (abs‘𝐴) < 1)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐹𝑘) = (𝐴𝑘))       (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − 𝐴)))

Theoremgeolim2 14441* The partial sums in the geometric series 𝐴𝑀 + 𝐴↑(𝑀 + 1)... converge to ((𝐴𝑀) / (1 − 𝐴)). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (abs‘𝐴) < 1)    &   (𝜑𝑀 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = (𝐴𝑘))       (𝜑 → seq𝑀( + , 𝐹) ⇝ ((𝐴𝑀) / (1 − 𝐴)))

Theoremgeoreclim 14442* The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → 1 < (abs‘𝐴))    &   ((𝜑𝑘 ∈ ℕ0) → (𝐹𝑘) = ((1 / 𝐴)↑𝑘))       (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1)))

Theoremgeo2sum 14443* The value of the finite geometric series 2↑-1 + 2↑-2 +... + 2↑-𝑁, multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)(𝐴 / (2↑𝑘)) = (𝐴 − (𝐴 / (2↑𝑁))))

Theoremgeo2sum2 14444* The value of the finite geometric series 1 + 2 + 4 + 8 +... + 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 7-Sep-2016.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1))

Theoremgeo2lim 14445* The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)
𝐹 = (𝑘 ∈ ℕ ↦ (𝐴 / (2↑𝑘)))       (𝐴 ∈ ℂ → seq1( + , 𝐹) ⇝ 𝐴)

Theoremgeomulcvg 14446* The geometric series converges even if it is multiplied by 𝑘 to result in the larger series 𝑘 · 𝐴𝑘. (Contributed by Mario Carneiro, 27-Mar-2015.)
𝐹 = (𝑘 ∈ ℕ0 ↦ (𝑘 · (𝐴𝑘)))       ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , 𝐹) ∈ dom ⇝ )

Theoremgeoisum 14447* The infinite sum of 1 + 𝐴↑1 + 𝐴↑2... is (1 / (1 − 𝐴)). (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ0 (𝐴𝑘) = (1 / (1 − 𝐴)))

Theoremgeoisumr 14448* The infinite sum of reciprocals 1 + (1 / 𝐴)↑1 + (1 / 𝐴)↑2... is 𝐴 / (𝐴 − 1). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → Σ𝑘 ∈ ℕ0 ((1 / 𝐴)↑𝑘) = (𝐴 / (𝐴 − 1)))

Theoremgeoisum1 14449* The infinite sum of 𝐴↑1 + 𝐴↑2... is (𝐴 / (1 − 𝐴)). (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ (𝐴𝑘) = (𝐴 / (1 − 𝐴)))

Theoremgeoisum1c 14450* The infinite sum of 𝐴 · (𝑅↑1) + 𝐴 · (𝑅↑2)... is (𝐴 · 𝑅) / (1 − 𝑅). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅)))

Theorem0.999... 14451 The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 10↑1 + 9 / 10↑2 + 9 / 10↑3 + ..., is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.)
Σ𝑘 ∈ ℕ (9 / (10↑𝑘)) = 1

Theorem0.999...OLD 14452 Obsolete version of 0.999... 14451 as of 8-Sep-2021. (Contributed by NM, 2-Nov-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
Σ𝑘 ∈ ℕ (9 / (10↑𝑘)) = 1

Theoremgeoihalfsum 14453 Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 14449. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 14451 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.)
Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1

5.10.10  Ratio test for infinite series convergence

Theoremcvgrat 14454* Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms beyond some index 𝐵, then the infinite sum of the terms of 𝐹 converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 27-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 1)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹𝑘))))       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

5.10.11  Mertens' theorem

Theoremmertenslem1 14455* Lemma for mertens 14457. (Contributed by Mario Carneiro, 29-Apr-2014.)
((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)    &   ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))    &   ((𝜑𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))    &   (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   (𝜑𝐸 ∈ ℝ+)    &   𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))}    &   (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ𝑠)(abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))))    &   (𝜑 → (𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ𝑡)(𝐾𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))))    &   (𝜑 → (0 ≤ sup(𝑇, ℝ, < ) ∧ (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤𝑇 𝑤𝑧)))       (𝜑 → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝐸)

Theoremmertenslem2 14456* Lemma for mertens 14457. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)    &   ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))    &   ((𝜑𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))    &   (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   (𝜑𝐸 ∈ ℝ+)    &   𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))}    &   (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ𝑠)(abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))))       (𝜑 → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝐸)

Theoremmertens 14457* Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐵(𝑘) is convergent, then 𝑗 ∈ ℕ0𝐴(𝑗) · Σ𝑘 ∈ ℕ0𝐵(𝑘)) = Σ𝑘 ∈ ℕ0Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘𝑗)) (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.)
((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)    &   ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))    &   ((𝜑𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))    &   (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )       (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))

5.10.12  Finite and infinite products

5.10.12.1  Product sequences

Theoremprodf 14458* An infinite product of complex terms is a function from an upper set of integers to . (Contributed by Scott Fenton, 4-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)

Theoremclim2prod 14459* The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴)       (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴))

Theoremclim2div 14460* The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝐴)    &   (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)       (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ (𝐴 / (seq𝑀( · , 𝐹)‘𝑁)))

Theoremprodfmul 14461* The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘) · (𝐺𝑘)))       (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , 𝐺)‘𝑁)))

Theoremprodf1 14462 The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (seq𝑀( · , (𝑍 × {1}))‘𝑁) = 1)

Theoremprodf1f 14463 A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) = (𝑍 × {1}))

Theoremprodfclim1 14464 The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) ⇝ 1)

Theoremprodfn0 14465* No term of a nonzero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ≠ 0)       (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)

Theoremprodfrec 14466* The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ≠ 0)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) = (1 / (𝐹𝑘)))       (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))

Theoremprodfdiv 14467* The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ≠ 0)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘) / (𝐺𝑘)))       (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁)))

5.10.12.2  Non-trivial convergence

Theoremntrivcvg 14468* A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → seq𝑀( · , 𝐹) ∈ dom ⇝ )

Theoremntrivcvgn0 14469* A product that converges to a nonzero value converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)    &   (𝜑𝑋 ≠ 0)       (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))

Theoremntrivcvgfvn0 14470* Any value of a product sequence that converges to a nonzero value is itself nonzero. (Contributed by Scott Fenton, 20-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)    &   (𝜑𝑋 ≠ 0)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)

Theoremntrivcvgtail 14471* A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)    &   (𝜑𝑋 ≠ 0)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))))

Theoremntrivcvgmullem 14472* Lemma for ntrivcvgmul 14473. (Contributed by Scott Fenton, 19-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑𝑃𝑍)    &   (𝜑𝑋 ≠ 0)    &   (𝜑𝑌 ≠ 0)    &   (𝜑 → seq𝑁( · , 𝐹) ⇝ 𝑋)    &   (𝜑 → seq𝑃( · , 𝐺) ⇝ 𝑌)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   (𝜑𝑁𝑃)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) · (𝐺𝑘)))       (𝜑 → ∃𝑞𝑍𝑤(𝑤 ≠ 0 ∧ seq𝑞( · , 𝐻) ⇝ 𝑤))

Theoremntrivcvgmul 14473* The product of two non-trivially converging products converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   (𝜑 → ∃𝑚𝑍𝑧(𝑧 ≠ 0 ∧ seq𝑚( · , 𝐺) ⇝ 𝑧))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐻𝑘) = ((𝐹𝑘) · (𝐺𝑘)))       (𝜑 → ∃𝑝𝑍𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , 𝐻) ⇝ 𝑤))

5.10.12.3  Complex products

Syntaxcprod 14474 Extend class notation to include complex products.
class 𝑘𝐴 𝐵

Definitiondf-prod 14475* Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sum 14265 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.)
𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))

Theoremprodex 14476 A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)
𝑘𝐴 𝐵 ∈ V

Theoremprodeq1f 14477 Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
𝑘𝐴    &   𝑘𝐵       (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremprodeq1 14478* Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
(𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremnfcprod1 14479* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝑘𝐴       𝑘𝑘𝐴 𝐵

Theoremnfcprod 14480* Bound-variable hypothesis builder for product: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in 𝑘𝐴𝐵. (Contributed by Scott Fenton, 1-Dec-2017.)
𝑥𝐴    &   𝑥𝐵       𝑥𝑘𝐴 𝐵

Theoremprodeq2w 14481* Equality theorem for product, when the class expressions 𝐵 and 𝐶 are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.)
(∀𝑘 𝐵 = 𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremprodeq2ii 14482* Equality theorem for product, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.)
(∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremprodeq2 14483* Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(∀𝑘𝐴 𝐵 = 𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremcbvprod 14484* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)    &   𝑘𝐴    &   𝑗𝐴    &   𝑘𝐵    &   𝑗𝐶       𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶

Theoremcbvprodv 14485* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)       𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶

Theoremcbvprodi 14486* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝑘𝐵    &   𝑗𝐶    &   (𝑗 = 𝑘𝐵 = 𝐶)       𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶

Theoremprodeq1i 14487* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐴 = 𝐵       𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶

Theoremprodeq2i 14488* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝑘𝐴𝐵 = 𝐶)       𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶

Theoremprodeq12i 14489* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐴 = 𝐵    &   (𝑘𝐴𝐶 = 𝐷)       𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷

Theoremprodeq1d 14490* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremprodeq2d 14491* Equality deduction for product. Note that unlike prodeq2dv 14492, 𝑘 may occur in 𝜑. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremprodeq2dv 14492* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremprodeq2sdv 14493* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theorem2cprodeq2dv 14494* Equality deduction for double product. (Contributed by Scott Fenton, 4-Dec-2017.)
((𝜑𝑗𝐴𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐷)

Theoremprodeq12dv 14495* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷)

Theoremprodeq12rdv 14496* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷)

Theoremprod2id 14497* The second class argument to a product can be chosen so that it is always a set. (Contributed by Scott Fenton, 4-Dec-2017.)
𝑘𝐴 𝐵 = ∏𝑘𝐴 ( I ‘𝐵)

Theoremprodrblem 14498* Lemma for prodrb 14501. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ (ℤ𝑀))       ((𝜑𝐴 ⊆ (ℤ𝑁)) → (seq𝑀( · , 𝐹) ↾ (ℤ𝑁)) = seq𝑁( · , 𝐹))

Theoremfprodcvg 14499* The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘𝑁))

Theoremprodrblem2 14500* Lemma for prodrb 14501. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (ℤ𝑁))       ((𝜑𝑁 ∈ (ℤ𝑀)) → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶))

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