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	fprodconst 15901*	The product of constant terms (𝑘 is not free in 𝐵). (Contributed by Scott Fenton, 12-Jan-2018.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴)))
Theorem	fprodn0 15902*	A finite product of nonzero terms is nonzero. (Contributed by Scott Fenton, 15-Jan-2018.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≠ 0)
Theorem	fprod2dlem 15903*	Lemma for fprod2d 15904- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)
⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥)    &   ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)    &   ⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
Theorem	fprod2d 15904*	Write a double product as a product over a two-dimensional region. Compare fsum2d 15694. (Contributed by Scott Fenton, 30-Jan-2018.)
⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷)
Theorem	fprodxp 15905*	Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)
⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ (𝐴 × 𝐵)𝐷)
Theorem	fprodcnv 15906*	Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
⊢ (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)    &   ⊢ (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → Rel 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑥 ∈ 𝐴 𝐵 = ∏𝑦 ∈ ◡ 𝐴𝐶)
Theorem	fprodcom2 15907*	Interchange order of multiplication. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸)
Theorem	fprodcom 15908*	Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ 𝐵 ∏𝑗 ∈ 𝐴 𝐶)
Theorem	fprod0diag 15909*	Two ways to express "the product of 𝐴(𝑗, 𝑘) over the triangular region 𝑀 ≤ 𝑗, 𝑀 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁. Compare fsum0diag 15700. (Contributed by Scott Fenton, 2-Feb-2018.)
⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑁)∏𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = ∏𝑘 ∈ (0...𝑁)∏𝑗 ∈ (0...(𝑁 − 𝑘))𝐴)
Theorem	fproddivf 15910*	The quotient of two finite products. A version of fproddiv 15884 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 / 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 / ∏𝑘 ∈ 𝐴 𝐶))
Theorem	fprodsplitf 15911*	Split a finite product into two parts. A version of fprodsplit 15889 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑈 𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ 𝐵 𝐶))
Theorem	fprodsplitsn 15912*	Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐷    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
Theorem	fprodsplit1f 15913*	Separate out a term in a finite product. A version of fprodsplit1 45835 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → Ⅎ𝑘𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
Theorem	fprodn0f 15914*	A finite product of nonzero terms is nonzero. A version of fprodn0 15902 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≠ 0)
Theorem	fprodclf 15915*	Closure of a finite product of complex numbers. A version of fprodcl 15875 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
Theorem	fprodge0 15916*	If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵)
Theorem	fprodeq0g 15917*	Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0)
Theorem	fprodge1 15918*	If all of the terms of a finite product are greater than or equal to 1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵)
Theorem	fprodle 15919*	If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ≤ ∏𝑘 ∈ 𝐴 𝐶)
Theorem	fprodmodd 15920*	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 15921*	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 15922*	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 15923*	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 15924*	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 15925*	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 15926*	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 15927	Declare the syntax for the falling factorial.
class FallFac
Syntax	crisefac 15928	Declare the syntax for the rising factorial.
class RiseFac
Definition	df-risefac 15929*	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 15930*	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 15931*	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 15932*	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 15933*	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 15934*	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 15935*	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 15936*	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 15937*	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 15938	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℂ)
Theorem	fallfaccl 15939	Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℂ)
Theorem	rerisefaccl 15940	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℝ)
Theorem	refallfaccl 15941	Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℝ)
Theorem	nnrisefaccl 15942	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℕ)
Theorem	zrisefaccl 15943	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℤ)
Theorem	zfallfaccl 15944	Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) ∈ ℤ)
Theorem	nn0risefaccl 15945	Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℕ0)
Theorem	rprisefaccl 15946	Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) ∈ ℝ+)
Theorem	risefallfac 15947	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 15948	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 15949	Lemma for risefac0 15950 and fallfac0 15951. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (0...(0 − 1)) = ∅
Theorem	risefac0 15950	The value of the rising factorial when 𝑁 = 0. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 RiseFac 0) = 1)
Theorem	fallfac0 15951	The value of the falling factorial when 𝑁 = 0. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
Theorem	risefacp1 15952	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 15953	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 15954	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 15955	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 15956	The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 RiseFac 1) = 𝐴)
Theorem	fallfac1 15957	The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 1) = 𝐴)
Theorem	risefacfac 15958	Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ (𝑁 ∈ ℕ0 → (1 RiseFac 𝑁) = (!‘𝑁))
Theorem	fallfacfwd 15959	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 15960	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 15961	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 15962	Lemma for binomfallfac 15964. 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 15963*	Lemma for binomfallfac 15964. 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 15964*	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 15965*	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 15966	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 15967	Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾)))
Theorem	fallfacfac 15968	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 15969	Declare the constant for the Bernoulli polynomial operator.
class BernPoly
Definition	df-bpoly 15970*	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 15971*	Lemma for bpolyval 15972. (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 15972*	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 15973	The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
Theorem	bpoly1 15974	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 15975	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 15976*	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 15977*	Lemma for bpolydif 15978. (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 15978	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 15979*	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 15980	The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))
Theorem	bpoly3 15981	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 15982	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 15983*	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 15984	Extend class notation to include the exponential function.
class exp
Syntax	ceu 15985	Extend class notation to include Euler's constant e = 2.71828....
class e
Syntax	csin 15986	Extend class notation to include the sine function.
class sin
Syntax	ccos 15987	Extend class notation to include the cosine function.
class cos
Syntax	ctan 15988	Extend class notation to include the tangent function.
class tan
Syntax	cpi 15989	Extend class notation to include the constant pi, π = 3.14159....
class π
Definition	df-ef 15990*	Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 16002. (Contributed by NM, 14-Mar-2005.)
⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))
Definition	df-e 15991	Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
⊢ e = (exp‘1)
Definition	df-sin 15992	Define the sine function. (Contributed by NM, 14-Mar-2005.)
⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
Definition	df-cos 15993	Define the cosine function. (Contributed by NM, 14-Mar-2005.)
⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
Definition	df-tan 15994	Define the tangent function. We define it this way for cmpt 5179, 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 15995	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 15996	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 15997	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 15998	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 15999*	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 16000*	Lemma for efcl 16005. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 15806 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 ⇝ )