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Theorem List for Metamath Proof Explorer - 14701-14800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfprodsplitsn 14701* Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   𝑘𝐷    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝑉)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝑘 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘𝐴 𝐶 · 𝐷))

Theoremfprodsplit1f 14702* Separate out a term in a finite product. A version of fprodsplit1 39625 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝑘𝐷)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑘 = 𝐶) → 𝐵 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))

Theoremfprodn0f 14703* A finite product of nonzero terms is nonzero. A version of fprodn0 14690 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ≠ 0)       (𝜑 → ∏𝑘𝐴 𝐵 ≠ 0)

Theoremfprodclf 14704* Closure of a finite product of complex numbers. A version of fprodcl 14663 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℂ)

Theoremfprodge0 14705* If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)       (𝜑 → 0 ≤ ∏𝑘𝐴 𝐵)

Theoremfprodeq0g 14706* Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑘 = 𝐶) → 𝐵 = 0)       (𝜑 → ∏𝑘𝐴 𝐵 = 0)

Theoremfprodge1 14707* If all of the terms of a finite product are larger or equal to 1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 1 ≤ 𝐵)       (𝜑 → 1 ≤ ∏𝑘𝐴 𝐵)

Theoremfprodle 14708* If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 ≤ ∏𝑘𝐴 𝐶)

Theoremfprodmodd 14709* If all factors of two finite products are equal modulo 𝑀, the products are equal modulo 𝑀. (Contributed by AV, 7-Jul-2021.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑘𝐴) → (𝐵 mod 𝑀) = (𝐶 mod 𝑀))       (𝜑 → (∏𝑘𝐴 𝐵 mod 𝑀) = (∏𝑘𝐴 𝐶 mod 𝑀))

5.10.12.5  Infinite products

Theoremiprodclim 14710* An infinite product equals the value its sequence converges to. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝐵)       (𝜑 → ∏𝑘𝑍 𝐴 = 𝐵)

Theoremiprodclim2 14711* A converging product converges to its infinite product. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)       (𝜑 → seq𝑀( · , 𝐹) ⇝ ∏𝑘𝑍 𝐴)

Theoremiprodclim3 14712* The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘𝑍𝐴)) ⇝ 𝑦))    &   (𝜑𝐹 ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴)       (𝜑𝐹 ⇝ ∏𝑘𝑍 𝐴)

Theoremiprodcl 14713* The product of a non-trivially converging infinite sequence is a complex number. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘𝑍 𝐴 ∈ ℂ)

Theoremiprodrecl 14714* The product of a non-trivially converging infinite real sequence is a real number. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)       (𝜑 → ∏𝑘𝑍 𝐴 ∈ ℝ)

Theoremiprodmul 14715* Multiplication of infinite sums. (Contributed by Scott Fenton, 18-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → ∃𝑚𝑍𝑧(𝑧 ≠ 0 ∧ seq𝑚( · , 𝐺) ⇝ 𝑧))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝑍 (𝐴 · 𝐵) = (∏𝑘𝑍 𝐴 · ∏𝑘𝑍 𝐵))

5.10.13  Falling and Rising Factorial

Syntaxcfallfac 14716 Declare the syntax for the falling factorial.
class FallFac

Syntaxcrisefac 14717 Declare the syntax for the rising factorial.
class RiseFac

Definitiondf-risefac 14718* Define the rising factorial function. This is the function (𝐴 · (𝐴 + 1) · ...(𝐴 + 𝑁)) for complex 𝐴 and nonnegative integers 𝑁. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))

Definitiondf-fallfac 14719* Define the falling factorial function. This is the function (𝐴 · (𝐴 − 1) · ...(𝐴𝑁)) for complex 𝐴 and nonnegative integers 𝑁. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))

Theoremrisefacval 14720* The value of the rising factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))

Theoremfallfacval 14721* The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘))

Theoremrisefacval2 14722* One-based value of rising factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) = ∏𝑘 ∈ (1...𝑁)(𝐴 + (𝑘 − 1)))

Theoremfallfacval2 14723* One-based value of falling factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (1...𝑁)(𝐴 − (𝑘 − 1)))

Theoremfallfacval3 14724* A product representation of falling factorial when 𝐴 is a nonnegative integer. (Contributed by Scott Fenton, 20-Mar-2018.)
(𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘)

Theoremrisefaccllem 14725* Lemma for rising factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
𝑆 ⊆ ℂ    &   1 ∈ 𝑆    &   ((𝑥𝑆𝑦𝑆) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝐴𝑆𝑘 ∈ ℕ0) → (𝐴 + 𝑘) ∈ 𝑆)       ((𝐴𝑆𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ 𝑆)

Theoremfallfaccllem 14726* Lemma for falling factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
𝑆 ⊆ ℂ    &   1 ∈ 𝑆    &   ((𝑥𝑆𝑦𝑆) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝐴𝑆𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ 𝑆)       ((𝐴𝑆𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ 𝑆)

Theoremrisefaccl 14727 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℂ)

Theoremfallfaccl 14728 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℂ)

Theoremrerisefaccl 14729 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℝ)

Theoremrefallfaccl 14730 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℝ)

Theoremnnrisefaccl 14731 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℕ)

Theoremzrisefaccl 14732 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℤ)

Theoremzfallfaccl 14733 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℤ)

Theoremnn0risefaccl 14734 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℕ0)

Theoremrprisefaccl 14735 Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
((𝐴 ∈ ℝ+𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℝ+)

Theoremrisefallfac 14736 A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.)
((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑋 RiseFac 𝑁) = ((-1↑𝑁) · (-𝑋 FallFac 𝑁)))

Theoremfallrisefac 14737 A relationship between falling and rising factorials. (Contributed by Scott Fenton, 17-Jan-2018.)
((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑋 FallFac 𝑁) = ((-1↑𝑁) · (-𝑋 RiseFac 𝑁)))

Theoremrisefall0lem 14738 Lemma for risefac0 14739 and fallfac0 14740. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)
(0...(0 − 1)) = ∅

Theoremrisefac0 14739 The value of the rising factorial when 𝑁 = 0. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝐴 ∈ ℂ → (𝐴 RiseFac 0) = 1)

Theoremfallfac0 14740 The value of the falling factorial when 𝑁 = 0. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)

Theoremrisefacp1 14741 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac (𝑁 + 1)) = ((𝐴 RiseFac 𝑁) · (𝐴 + 𝑁)))

Theoremfallfacp1 14742 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac (𝑁 + 1)) = ((𝐴 FallFac 𝑁) · (𝐴𝑁)))

Theoremrisefacp1d 14743 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴 RiseFac (𝑁 + 1)) = ((𝐴 RiseFac 𝑁) · (𝐴 + 𝑁)))

Theoremfallfacp1d 14744 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴 FallFac (𝑁 + 1)) = ((𝐴 FallFac 𝑁) · (𝐴𝑁)))

Theoremrisefac1 14745 The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝐴 ∈ ℂ → (𝐴 RiseFac 1) = 𝐴)

Theoremfallfac1 14746 The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)

Theoremrisefacfac 14747 Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝑁 ∈ ℕ0 → (1 RiseFac 𝑁) = (!‘𝑁))

Theoremfallfacfwd 14748 The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) FallFac 𝑁) − (𝐴 FallFac 𝑁)) = (𝑁 · (𝐴 FallFac (𝑁 − 1))))

Theorem0fallfac 14749 The value of the zero falling factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
(𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0)

Theorem0risefac 14750 The value of the zero rising factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
(𝑁 ∈ ℕ → (0 RiseFac 𝑁) = 0)

Theorembinomfallfaclem1 14751 Lemma for binomfallfac 14753. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝐾 ∈ (0...𝑁)) → ((𝑁C𝐾) · ((𝐴 FallFac (𝑁𝐾)) · (𝐵 FallFac (𝐾 + 1)))) ∈ ℂ)

Theorembinomfallfaclem2 14752* Lemma for binomfallfac 14753. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁𝑘)) · (𝐵 FallFac 𝑘))))       ((𝜑𝜓) → ((𝐴 + 𝐵) FallFac (𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴 FallFac ((𝑁 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))

Theorembinomfallfac 14753* A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁𝑘)) · (𝐵 FallFac 𝑘))))

Theorembinomrisefac 14754* A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵) RiseFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 RiseFac (𝑁𝑘)) · (𝐵 RiseFac 𝑘))))

Theoremfallfacval4 14755 Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)
(𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ((!‘𝐴) / (!‘(𝐴𝑁))))

Theorembcfallfac 14756 Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
(𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))

Theoremfallfacfac 14757 Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝑁 ∈ ℕ0 → (𝑁 FallFac 𝑁) = (!‘𝑁))

5.10.14  Bernoulli polynomials and sums of k-th powers

Syntaxcbp 14758 Declare the constant for the Bernoulli polynomial operator.
class BernPoly

Definitiondf-bpoly 14759* Define the Bernoulli polynomials. Here we use well-founded recursion to define the Bernoulli polynomials. This agrees with most textbook definitions, although explicit formulae do exist. (Contributed by Scott Fenton, 22-May-2014.)
BernPoly = (𝑚 ∈ ℕ0, 𝑥 ∈ ℂ ↦ (wrecs( < , ℕ0, (𝑔 ∈ V ↦ (#‘dom 𝑔) / 𝑛((𝑥𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔𝑘) / ((𝑛𝑘) + 1))))))‘𝑚))

Theorembpolylem 14760* Lemma for bpolyval 14761. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐺 = (𝑔 ∈ V ↦ (#‘dom 𝑔) / 𝑛((𝑋𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔𝑘) / ((𝑛𝑘) + 1)))))    &   𝐹 = wrecs( < , ℕ0, 𝐺)       ((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁𝑘) + 1)))))

Theorembpolyval 14761* The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁𝑘) + 1)))))

Theorembpoly0 14762 The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
(𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)

Theorembpoly1 14763 The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
(𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))

Theorembpolycl 14764 Closure law for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

Theorembpolysum 14765* A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁𝑘) + 1))) = (𝑋𝑁))

Theorembpolydiflem 14766* Lemma for bpolydif 14767. (Contributed by Scott Fenton, 12-Jun-2014.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1))))       (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))

Theorembpolydif 14767 Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))

Theoremfsumkthpow 14768* A closed-form expression for the sum of 𝐾-th powers. (Contributed by Scott Fenton, 16-May-2014.) This is Metamath 100 proof #77. (Revised by Mario Carneiro, 16-Jun-2014.)
((𝐾 ∈ ℕ0𝑀 ∈ ℕ0) → Σ𝑛 ∈ (0...𝑀)(𝑛𝐾) = ((((𝐾 + 1) BernPoly (𝑀 + 1)) − ((𝐾 + 1) BernPoly 0)) / (𝐾 + 1)))

Theorembpoly2 14769 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
(𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Theorembpoly3 14770 The Bernoulli polynomials at three. (Contributed by Scott Fenton, 8-Jul-2015.)
(𝑋 ∈ ℂ → (3 BernPoly 𝑋) = (((𝑋↑3) − ((3 / 2) · (𝑋↑2))) + ((1 / 2) · 𝑋)))

Theorembpoly4 14771 The Bernoulli polynomials at four. (Contributed by Scott Fenton, 8-Jul-2015.)
(𝑋 ∈ ℂ → (4 BernPoly 𝑋) = ((((𝑋↑4) − (2 · (𝑋↑3))) + (𝑋↑2)) − (1 / 30)))

Theoremfsumcube 14772* Express the sum of cubes in closed terms. (Contributed by Scott Fenton, 16-Jun-2015.)
(𝑇 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑇)(𝑘↑3) = (((𝑇↑2) · ((𝑇 + 1)↑2)) / 4))

5.11  Elementary trigonometry

5.11.1  The exponential, sine, and cosine functions

Syntaxce 14773 Extend class notation to include the exponential function.
class exp

Syntaxceu 14774 Extend class notation to include Euler's constant = 2.7182818....
class e

Syntaxcsin 14775 Extend class notation to include the sine function.
class sin

Syntaxccos 14776 Extend class notation to include the cosine function.
class cos

Syntaxctan 14777 Extend class notation to include the tangent function.
class tan

Syntaxcpi 14778 Extend class notation to include pi = 3.14159....
class π

Definitiondf-ef 14779* Define the exponential function. (Contributed by NM, 14-Mar-2005.)
exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))

Definitiondf-e 14780 Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)
e = (exp‘1)

Definitiondf-sin 14781 Define the sine function. (Contributed by NM, 14-Mar-2005.)
sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))

Definitiondf-cos 14782 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))

Definitiondf-tan 14783 Define the tangent function. We define it this way for cmpt 4720, which requires the form (𝑥𝐴𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)
tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))

Definitiondf-pi 14784 Define pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
π = inf((ℝ+ ∩ (sin “ {0})), ℝ, < )

Theoremeftcl 14785 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → ((𝐴𝐾) / (!‘𝐾)) ∈ ℂ)

Theoremreeftcl 14786 The terms of the series expansion of the exponential function of a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
((𝐴 ∈ ℝ ∧ 𝐾 ∈ ℕ0) → ((𝐴𝐾) / (!‘𝐾)) ∈ ℝ)

Theoremeftabs 14787 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾)))

Theoremeftval 14788* The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))

Theoremefcllem 14789* Lemma for efcl 14794. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 14596 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )

Theoremef0lem 14790* The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 = 0 → seq0( + , 𝐹) ⇝ 1)

Theoremefval 14791* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴𝑘) / (!‘𝑘)))

Theoremesum 14792 Value of Euler's constant e = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)
e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))

Theoremeff 14793 Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
exp:ℂ⟶ℂ

Theoremefcl 14794 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)

Theoremefval2 14795* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹𝑘))

Theoremefcvg 14796* The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → seq0( + , 𝐹) ⇝ (exp‘𝐴))

Theoremefcvgfsum 14797* Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) / (!‘𝑘)))       (𝐴 ∈ ℂ → 𝐹 ⇝ (exp‘𝐴))

Theoremreefcl 14798 The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
(𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ)

Theoremreefcld 14799 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (exp‘𝐴) ∈ ℝ)

Theoremere 14800 Euler's constant e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)
e ∈ ℝ

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