P. 160 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fprodmodd 15901*	If all factors of two finite products are equal modulo 𝑀, the products are equal modulo 𝑀. (Contributed by AV, 7-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 mod 𝑀) = (𝐶 mod 𝑀))    ⇒   ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐵 mod 𝑀) = (∏𝑘 ∈ 𝐴 𝐶 mod 𝑀))
5.10.12.5  Infinite products
Theorem	iprodclim 15902*	An infinite product equals the value its sequence converges to. (Contributed by Scott Fenton, 18-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = 𝐵)
Theorem	iprodclim2 15903*	A converging product converges to its infinite product. (Contributed by Scott Fenton, 18-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ∏𝑘 ∈ 𝑍 𝐴)
Theorem	iprodclim3 15904*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑦))    &   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴)
Theorem	iprodcl 15905*	The product of a non-trivially converging infinite sequence is a complex number. (Contributed by Scott Fenton, 18-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
Theorem	iprodrecl 15906*	The product of a non-trivially converging infinite real sequence is a real number. (Contributed by Scott Fenton, 18-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 ∈ ℝ)
Theorem	iprodmul 15907*	Multiplication of infinite sums. (Contributed by Scott Fenton, 18-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∃𝑧(𝑧 ≠ 0 ∧ seq𝑚( · , 𝐺) ⇝ 𝑧))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (𝐴 · 𝐵) = (∏𝑘 ∈ 𝑍 𝐴 · ∏𝑘 ∈ 𝑍 𝐵))
5.10.13  Falling and Rising Factorial
Syntax	cfallfac 15908	Declare the syntax for the falling factorial.
class FallFac
Syntax	crisefac 15909	Declare the syntax for the rising factorial.
class RiseFac
Definition	df-risefac 15910*	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))
Definition	df-fallfac 15911*	Define the falling 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.)
⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘))
Theorem	risefacval 15912*	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))
Theorem	fallfacval 15913*	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))(𝐴 − 𝑘))
Theorem	risefacval2 15914*	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)))
Theorem	fallfacval2 15915*	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)))
Theorem	fallfacval3 15916*	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)
Theorem	risefaccllem 15917*	Lemma for rising factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ 𝑆 ⊆ ℂ    &   ⊢ 1 ∈ 𝑆    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐴 + 𝑘) ∈ 𝑆)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ 𝑆)
Theorem	fallfaccllem 15918*	Lemma for falling factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ 𝑆 ⊆ ℂ    &   ⊢ 1 ∈ 𝑆    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐴 − 𝑘) ∈ 𝑆)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ 𝑆)
Theorem	risefaccl 15919	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℂ)
Theorem	fallfaccl 15920	Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℂ)
Theorem	rerisefaccl 15921	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℝ)
Theorem	refallfaccl 15922	Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℝ)
Theorem	nnrisefaccl 15923	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℕ)
Theorem	zrisefaccl 15924	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℤ)
Theorem	zfallfaccl 15925	Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℤ)
Theorem	nn0risefaccl 15926	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℕ0)
Theorem	rprisefaccl 15927	Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℝ+)
Theorem	risefallfac 15928	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)))
Theorem	fallrisefac 15929	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)))
Theorem	risefall0lem 15930	Lemma for risefac0 15931 and fallfac0 15932. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (0...(0 − 1)) = ∅
Theorem	risefac0 15931	The value of the rising factorial when 𝑁 = 0. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 RiseFac 0) = 1)
Theorem	fallfac0 15932	The value of the falling factorial when 𝑁 = 0. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
Theorem	risefacp1 15933	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)))
Theorem	fallfacp1 15934	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) · (𝐴 − 𝑁)))
Theorem	risefacp1d 15935	The value of the rising factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴 RiseFac (𝑁 + 1)) = ((𝐴 RiseFac 𝑁) · (𝐴 + 𝑁)))
Theorem	fallfacp1d 15936	The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴 FallFac (𝑁 + 1)) = ((𝐴 FallFac 𝑁) · (𝐴 − 𝑁)))
Theorem	risefac1 15937	The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 RiseFac 1) = 𝐴)
Theorem	fallfac1 15938	The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)
Theorem	risefacfac 15939	Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝑁 ∈ ℕ0 → (1 RiseFac 𝑁) = (!‘𝑁))
Theorem	fallfacfwd 15940	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))))
Theorem	0fallfac 15941	The value of the zero falling factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0)
Theorem	0risefac 15942	The value of the zero rising factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.)
⊢ (𝑁 ∈ ℕ → (0 RiseFac 𝑁) = 0)
Theorem	binomfallfaclem1 15943	Lemma for binomfallfac 15945. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → ((𝑁C𝐾) · ((𝐴 FallFac (𝑁 − 𝐾)) · (𝐵 FallFac (𝐾 + 1)))) ∈ ℂ)
Theorem	binomfallfaclem2 15944*	Lemma for binomfallfac 15945. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜓 → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵) FallFac (𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴 FallFac ((𝑁 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))
Theorem	binomfallfac 15945*	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 (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
Theorem	binomrisefac 15946*	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 (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))))
Theorem	fallfacval4 15947	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) / (!‘(𝐴 − 𝑁))))
Theorem	bcfallfac 15948	Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))
Theorem	fallfacfac 15949	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
Syntax	cbp 15950	Declare the constant for the Bernoulli polynomial operator.
class BernPoly
Definition	df-bpoly 15951*	Define the Bernoulli polynomials. Here we use well-founded recursion to define the Bernoulli polynomials. This agrees with most textbook definitions, although explicit formulas do exist. (Contributed by Scott Fenton, 22-May-2014.)
⊢ BernPoly = (𝑚 ∈ ℕ0, 𝑥 ∈ ℂ ↦ (wrecs( < , ℕ0, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚))
Theorem	bpolylem 15952*	Lemma for bpolyval 15953. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐺 = (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))    &   ⊢ 𝐹 = wrecs( < , ℕ0, 𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
Theorem	bpolyval 15953*	The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
Theorem	bpoly0 15954	The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
Theorem	bpoly1 15955	The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
Theorem	bpolycl 15956	Closure law for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)
Theorem	bpolysum 15957*	A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (𝑋↑𝑁))
Theorem	bpolydiflem 15958*	Lemma for bpolydif 15959. (Contributed by Scott Fenton, 12-Jun-2014.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (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))))
Theorem	bpolydif 15959	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))))
Theorem	fsumkthpow 15960*	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 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → Σ𝑛 ∈ (0...𝑀)(𝑛↑𝐾) = ((((𝐾 + 1) BernPoly (𝑀 + 1)) − ((𝐾 + 1) BernPoly 0)) / (𝐾 + 1)))
Theorem	bpoly2 15961	The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))
Theorem	bpoly3 15962	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)))
Theorem	bpoly4 15963	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)))
Theorem	fsumcube 15964*	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
Syntax	ce 15965	Extend class notation to include the exponential function.
class exp
Syntax	ceu 15966	Extend class notation to include Euler's constant e = 2.71828....
class e
Syntax	csin 15967	Extend class notation to include the sine function.
class sin
Syntax	ccos 15968	Extend class notation to include the cosine function.
class cos
Syntax	ctan 15969	Extend class notation to include the tangent function.
class tan
Syntax	cpi 15970	Extend class notation to include the constant pi, π = 3.14159....
class π
Definition	df-ef 15971*	Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 15983. (Contributed by NM, 14-Mar-2005.)
⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))
Definition	df-e 15972	Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
⊢ e = (exp‘1)
Definition	df-sin 15973	Define the sine function. (Contributed by NM, 14-Mar-2005.)
⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
Definition	df-cos 15974	Define the cosine function. (Contributed by NM, 14-Mar-2005.)
⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
Definition	df-tan 15975	Define the tangent function. We define it this way for cmpt 5172, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
Definition	df-pi 15976	Define the constant pi, π = 3.14159..., which is the smallest positive number whose sine is zero. Definition of π in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
⊢ π = inf((ℝ+ ∩ (◡sin “ {0})), ℝ, < )
Theorem	eftcl 15977	Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → ((𝐴↑𝐾) / (!‘𝐾)) ∈ ℂ)
Theorem	reeftcl 15978	The terms of the series expansion of the exponential function at a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐾 ∈ ℕ0) → ((𝐴↑𝐾) / (!‘𝐾)) ∈ ℝ)
Theorem	eftabs 15979	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‘((𝐴↑𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾)))
Theorem	eftval 15980*	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 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))
Theorem	efcllem 15981*	Lemma for efcl 15986. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 15787 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Proof shortened by AV, 9-Jul-2022.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
Theorem	ef0lem 15982*	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 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 = 0 → seq0( + , 𝐹) ⇝ 1)
Theorem	efval 15983*	Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘)))
Theorem	esum 15984	Value of Euler's constant e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
⊢ e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))
Theorem	eff 15985	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
Theorem	efcl 15986	Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
Theorem	efcld 15987	Closure law for the exponential function, deduction version. (Contributed by Thierry Arnoux, 1-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (exp‘𝐴) ∈ ℂ)
Theorem	efval2 15988*	Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘))
Theorem	efcvg 15989*	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 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ⇝ (exp‘𝐴))
Theorem	efcvgfsum 15990*	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)((𝐴↑𝑘) / (!‘𝑘)))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ (exp‘𝐴))
Theorem	reefcl 15991	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)
Theorem	reefcld 15992	The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (exp‘𝐴) ∈ ℝ)
Theorem	ere 15993	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
Theorem	ege2le3 15994	Lemma for egt2lt3 16112. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))    ⇒   ⊢ (2 ≤ e ∧ e ≤ 3)
Theorem	ef0 15995	Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ (exp‘0) = 1
Theorem	efcj 15996	The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ (𝐴 ∈ ℂ → (exp‘(∗‘𝐴)) = (∗‘(exp‘𝐴)))
Theorem	efaddlem 15997*	Lemma for efadd 15998 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ ((𝐵↑𝑛) / (!‘𝑛)))    &   ⊢ 𝐻 = (𝑛 ∈ ℕ0 ↦ (((𝐴 + 𝐵)↑𝑛) / (!‘𝑛)))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))
Theorem	efadd 15998	Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))
Theorem	fprodefsum 15999*	Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))
Theorem	efcan 16000	Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) · (exp‘-𝐴)) = 1)