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Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminfcvgaux2i 14301* Auxiliary theorem for applications of supcvg 14299. (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 14302 The harmonic series 𝐻 diverges. This fact follows from the stronger emcl 24410, 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 14303* 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 14304* 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 14305 Lemma for trirecip 14306. 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 14306 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 14307* 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 14308* 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 14309* 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 14310* 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 14311* 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 14312* 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 14313* 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 14314* 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 14315* 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 14316* 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 14317* 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 14318* 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 14319* 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 14320* 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 14321* 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 14322* 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... 14323 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 14324 Obsolete version of 0.999... 14323 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 14325 Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 14321. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 14323 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 14326* 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 14327* Lemma for mertens 14329. (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 14328* Lemma for mertens 14329. (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 14329* 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 14330* 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 14331* 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 14332* 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 14333* 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 14334 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 14335 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 14336 The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) ⇝ 1)

Theoremprodfn0 14337* 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 14338* 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 14339* 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 14340* 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 14341* 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 14342* 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 14343* 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 14344* Lemma for ntrivcvgmul 14345. (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 14345* 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 14346 Extend class notation to include complex products.
class 𝑘𝐴 𝐵

Definitiondf-prod 14347* Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sum 14137 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 14348 A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)
𝑘𝐴 𝐵 ∈ V

Theoremprodeq1f 14349 Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
𝑘𝐴    &   𝑘𝐵       (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremprodeq1 14350* Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
(𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremnfcprod1 14351* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝑘𝐴       𝑘𝑘𝐴 𝐵

Theoremnfcprod 14352* 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 14353* 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 14354* 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 14355* Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(∀𝑘𝐴 𝐵 = 𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremcbvprod 14356* 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 14357* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)       𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶

Theoremcbvprodi 14358* Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝑘𝐵    &   𝑗𝐶    &   (𝑗 = 𝑘𝐵 = 𝐶)       𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶

Theoremprodeq1i 14359* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐴 = 𝐵       𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶

Theoremprodeq2i 14360* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝑘𝐴𝐵 = 𝐶)       𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶

Theoremprodeq12i 14361* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐴 = 𝐵    &   (𝑘𝐴𝐶 = 𝐷)       𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷

Theoremprodeq1d 14362* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremprodeq2d 14363* Equality deduction for product. Note that unlike prodeq2dv 14364, 𝑘 may occur in 𝜑. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremprodeq2dv 14364* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theoremprodeq2sdv 14365* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)

Theorem2cprodeq2dv 14366* Equality deduction for double product. (Contributed by Scott Fenton, 4-Dec-2017.)
((𝜑𝑗𝐴𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐷)

Theoremprodeq12dv 14367* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷)

Theoremprodeq12rdv 14368* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷)

Theoremprod2id 14369* 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 14370* Lemma for prodrb 14373. (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 14371* 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 14372* Lemma for prodrb 14373. (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))

Theoremprodrb 14373* Rebase the starting point of a product. (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))       (𝜑 → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶))

Theoremprodmolem3 14374* Lemma for prodmo 14377. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)    &   𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)    &   (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)       (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

Theoremprodmolem2a 14375* Lemma for prodmo 14377. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)    &   𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)    &   (𝜑𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴))       (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))

Theoremprodmolem2 14376* Lemma for prodmo 14377. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)       ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))

Theoremprodmo 14377* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)       (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))))

Theoremzprod 14378* Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))

Theoremiprod 14379* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝑍 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))

Theoremzprodn0 14380* Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑋 ≠ 0)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 = 𝑋)

Theoremiprodn0 14381* Nonzero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑋 ≠ 0)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝑍 𝐵 = 𝑋)

5.10.12.4  Finite products

Theoremfprod 14382* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = (seq1( · , 𝐺)‘𝑀))

Theoremfprodntriv 14383* A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘𝑍 ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))

Theoremprod0 14384 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑘 ∈ ∅ 𝐴 = 1

Theoremprod1 14385* Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)
((𝐴 ⊆ (ℤ𝑀) ∨ 𝐴 ∈ Fin) → ∏𝑘𝐴 1 = 1)

Theoremprodfc 14386* A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)
𝑗𝐴 ((𝑘𝐴𝐵)‘𝑗) = ∏𝑘𝐴 𝐵

Theoremfprodf1o 14387* Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)
(𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)

Theoremprodss 14388* Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → ∃𝑛 ∈ (ℤ𝑀)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ (ℤ𝑀) ↦ if(𝑘𝐵, 𝐶, 1))) ⇝ 𝑦))    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)    &   (𝜑𝐵 ⊆ (ℤ𝑀))       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremfprodss 14389* Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)

Theoremfprodser 14390* 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.)
((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( · , 𝐹)‘𝑁))

Theoremfprodcl2lem 14391* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ∏𝑘𝐴 𝐵𝑆)

Theoremfprodcllem 14392* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑 → 1 ∈ 𝑆)       (𝜑 → ∏𝑘𝐴 𝐵𝑆)

Theoremfprodcl 14393* Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℂ)

Theoremfprodrecl 14394* Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℝ)

Theoremfprodzcl 14395* Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℤ)

Theoremfprodnncl 14396* Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℕ)

Theoremfprodrpcl 14397* Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ+)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℝ+)

Theoremfprodnn0cl 14398* Closure of a finite product of nonnegative integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ0)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℕ0)

Theoremfprodcllemf 14399* Finite product closure lemma. A version of fprodcllem 14392 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑 → 1 ∈ 𝑆)       (𝜑 → ∏𝑘𝐴 𝐵𝑆)

Theoremfprodreclf 14400* Closure of a finite product of real numbers. A version of fprodrecl 14394 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℝ)

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