P. 119 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11801-11900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fsum0diaglem 11801*	Lemma for fisum0diag 11802. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 𝑘))))
Theorem	fisum0diag 11802*	Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 𝑀 ≤ 𝑗, 𝑀 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁". (Contributed by NM, 31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...(𝑁 − 𝑘))𝐴)
Theorem	mptfzshft 11803*	1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗 − 𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁))
Theorem	fsumrev 11804*	Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑗 = (𝐾 − 𝑘) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))𝐵)
Theorem	fsumshft 11805*	Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) (Proof shortened by AV, 8-Sep-2019.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑗 = (𝑘 − 𝐾) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)
Theorem	fsumshftm 11806*	Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾))𝐵)
Theorem	fisumrev2 11807*	Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
Theorem	fisum0diag2 11808*	Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 0 ≤ 𝑗, 0 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁". (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)    &   ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶)
Theorem	fsummulc2 11809*	A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
Theorem	fsummulc1 11810*	A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶))
Theorem	fsumdivapc 11811*	A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 # 0)    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 / 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝐶))
Theorem	fsumneg 11812*	Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 -𝐵 = -Σ𝑘 ∈ 𝐴 𝐵)
Theorem	fsumsub 11813*	Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 − 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 − Σ𝑘 ∈ 𝐴 𝐶))
Theorem	fsum2mul 11814*	Separate the nested sum of the product 𝐶(𝑗) · 𝐷(𝑘). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 (𝐶 · 𝐷) = (Σ𝑗 ∈ 𝐴 𝐶 · Σ𝑘 ∈ 𝐵 𝐷))
Theorem	fsumconst 11815*	The sum of constant terms (𝑘 is not free in 𝐵). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))
Theorem	fsumdifsnconst 11816*	The sum of constant terms (𝑘 is not free in 𝐶) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑘 ∈ (𝐴 ∖ {𝐵})𝐶 = (((♯‘𝐴) − 1) · 𝐶))
Theorem	modfsummodlem1 11817*	Lemma for modfsummod 11819. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)
Theorem	modfsummodlemstep 11818*	Induction step for modfsummod 11819. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ)    &   ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)    &   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 mod 𝑁) = (Σ𝑘 ∈ 𝐴 (𝐵 mod 𝑁) mod 𝑁))    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 mod 𝑁) = (Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) mod 𝑁))
Theorem	modfsummod 11819*	A finite sum modulo a positive integer equals the finite sum of their summands modulo the positive integer, modulo the positive integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 mod 𝑁) = (Σ𝑘 ∈ 𝐴 (𝐵 mod 𝑁) mod 𝑁))
Theorem	fsumge0 11820*	If all of the terms of a finite sum are nonnegative, so is the sum. (Contributed by NM, 26-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 𝐵)
Theorem	fsumlessfi 11821*	A shorter sum of nonnegative terms is no greater than a longer one. (Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon, 12-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵)
Theorem	fsumge1 11822*	A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ (𝑘 = 𝑀 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ≤ Σ𝑘 ∈ 𝐴 𝐵)
Theorem	fsum00 11823*	A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 = 0 ↔ ∀𝑘 ∈ 𝐴 𝐵 = 0))
Theorem	fsumle 11824*	If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶)
Theorem	fsumlt 11825*	If every term in one finite sum is less than the corresponding term in another, then the first sum is less than the second. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 < Σ𝑘 ∈ 𝐴 𝐶)
Theorem	fsumabs 11826*	Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘Σ𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (abs‘𝐵))
Theorem	telfsumo 11827*	Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)    &   ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)    &   ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 − 𝐶) = (𝐷 − 𝐸))
Theorem	telfsumo2 11828*	Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)    &   ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)    &   ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷))
Theorem	telfsum 11829*	Sum of a telescoping series. (Contributed by Scott Fenton, 24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)    &   ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐵 − 𝐶) = (𝐷 − 𝐸))
Theorem	telfsum2 11830*	Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵)    &   ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷)    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷))
Theorem	fsumparts 11831*	Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
⊢ (𝑘 = 𝑗 → (𝐴 = 𝐵 ∧ 𝑉 = 𝑊))    &   ⊢ (𝑘 = (𝑗 + 1) → (𝐴 = 𝐶 ∧ 𝑉 = 𝑋))    &   ⊢ (𝑘 = 𝑀 → (𝐴 = 𝐷 ∧ 𝑉 = 𝑌))    &   ⊢ (𝑘 = 𝑁 → (𝐴 = 𝐸 ∧ 𝑉 = 𝑍))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑉 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 · (𝑋 − 𝑊)) = (((𝐸 · 𝑍) − (𝐷 · 𝑌)) − Σ𝑗 ∈ (𝑀..^𝑁)((𝐶 − 𝐵) · 𝑋)))
Theorem	fsumrelem 11832*	Lemma for fsumre 11833, fsumim 11834, and fsumcj 11835. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝐹:ℂ⟶ℂ    &   ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))
Theorem	fsumre 11833*	The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℜ‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (ℜ‘𝐵))
Theorem	fsumim 11834*	The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℑ‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (ℑ‘𝐵))
Theorem	fsumcj 11835*	The complex conjugate of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (∗‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (∗‘𝐵))
Theorem	iserabs 11836*	Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵)
Theorem	cvgcmpub 11837*	An upper bound for the limit of a real infinite series. This theorem can also be used to compare two infinite series. (Contributed by Mario Carneiro, 24-Mar-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 𝐵 ≤ 𝐴)
Theorem	fsumiun 11838*	Sum over a disjoint indexed union. (Contributed by Mario Carneiro, 1-Jul-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶)
Theorem	hashiun 11839*	The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵))
Theorem	hash2iun 11840*	The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶)    ⇒   ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 (♯‘𝐶))
Theorem	hash2iun1dif1 11841*	The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ 𝐵 = (𝐴 ∖ {𝑥})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (♯‘𝐶) = 1)    ⇒   ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = ((♯‘𝐴) · ((♯‘𝐴) − 1)))
Theorem	hashrabrex 11842*	The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → {𝑥 ∈ 𝑋 ∣ 𝜓} ∈ Fin)    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓})    ⇒   ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ 𝜓}))
Theorem	hashuni 11843*	The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ Fin)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥)    ⇒   ⊢ (𝜑 → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
4.9.3  The binomial theorem
Theorem	binomlem 11844*	Lemma for binom 11845 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))
Theorem	binom 11845*	The binomial theorem: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). 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 11844. 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↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
Theorem	binom1p 11846*	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) · (𝐴↑𝑘)))
Theorem	binom11 11847*	Special case of the binomial theorem for 2↑𝑁. (Contributed by Mario Carneiro, 13-Mar-2014.)
⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) = Σ𝑘 ∈ (0...𝑁)(𝑁C𝑘))
Theorem	binom1dif 11848*	A summation for the difference between ((𝐴 + 1)↑𝑁) and (𝐴↑𝑁). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝐴 + 1)↑𝑁) − (𝐴↑𝑁)) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘)))
Theorem	bcxmaslem1 11849	Lemma for bcxmas 11850. (Contributed by Paul Chapman, 18-May-2007.)
⊢ (𝐴 = 𝐵 → ((𝑁 + 𝐴)C𝐴) = ((𝑁 + 𝐵)C𝐵))
Theorem	bcxmas 11850*	Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
4.9.4  Infinite sums (cont.)
Theorem	isumshft 11851*	Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝐾))    &   ⊢ (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑊) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ 𝑊 𝐴 = Σ𝑘 ∈ 𝑍 𝐵)
Theorem	isumsplit 11852*	Split off the first 𝑁 terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ 𝑊 𝐴))
Theorem	isum1p 11853*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴))
Theorem	isumnn0nn 11854*	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)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴))
Theorem	isumrpcl 11855*	The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ+)
Theorem	isumle 11856*	Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ≤ Σ𝑘 ∈ 𝑍 𝐵)
Theorem	isumlessdc 11857*	A finite sum of nonnegative numbers is less than or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 DECID 𝑘 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵)
4.9.5  Miscellaneous converging and diverging sequences
Theorem	divcnv 11858*	The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)
⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0)
4.9.6  Arithmetic series
Theorem	arisum 11859*	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))
Theorem	arisum2 11860*	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))
Theorem	trireciplem 11861	Lemma for trirecip 11862. 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
Theorem	trirecip 11862	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
4.9.7  Geometric series
Theorem	expcnvap0 11863*	A sequence of powers of a complex number 𝐴 with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 23-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐴) < 1)    &   ⊢ (𝜑 → 𝐴 # 0)    ⇒   ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0)
Theorem	expcnvre 11864*	A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0)
Theorem	expcnv 11865*	A sequence of powers of a complex number 𝐴 with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐴) < 1)    ⇒   ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0)
Theorem	explecnv 11866*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐴) < 1)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ≤ (𝐴↑𝑘))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 0)
Theorem	geosergap 11867*	The value of the finite geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1) +... + 𝐴↑(𝑁 − 1). (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 # 1)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴)))
Theorem	geoserap 11868*	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.) (Revised by Jim Kingdon, 24-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 # 1)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴)))
Theorem	pwm1geoserap1 11869*	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.) (Revised by Jim Kingdon, 24-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 # 1)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘)))
Theorem	absltap 11870	Less-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐴) < 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 # 𝐵)
Theorem	absgtap 11871	Greater-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 < (abs‘𝐴))    ⇒   ⊢ (𝜑 → 𝐴 # 𝐵)
Theorem	geolim 11872*	The partial sums in the infinite series 1 + 𝐴↑1 + 𝐴↑2... converge to (1 / (1 − 𝐴)). (Contributed by NM, 15-May-2006.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐴) < 1)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐴↑𝑘))    ⇒   ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − 𝐴)))
Theorem	geolim2 11873*	The partial sums in the geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1)... converge to ((𝐴↑𝑀) / (1 − 𝐴)). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐴) < 1)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐴↑𝑘))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ((𝐴↑𝑀) / (1 − 𝐴)))
Theorem	georeclim 11874*	The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 1 < (abs‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘))    ⇒   ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1)))
Theorem	geo2sum 11875*	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))))
Theorem	geo2sum2 11876*	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))
Theorem	geo2lim 11877*	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)
Theorem	geoisum 11878*	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 (𝐴↑𝑘) = (1 / (1 − 𝐴)))
Theorem	geoisumr 11879*	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)))
Theorem	geoisum1 11880*	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 (𝐴↑𝑘) = (𝐴 / (1 − 𝐴)))
Theorem	geoisum1c 11881*	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 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅)))
Theorem	0.999... 11882	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. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.)
⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1
Theorem	geoihalfsum 11883	Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 11880. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 11882 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by AV, 9-Jul-2022.)
⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1
4.9.8  Ratio test for infinite series convergence
Theorem	cvgratnnlembern 11884	Lemma for cvgratnn 11892. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑀) < ((1 / ((1 / 𝐴) − 1)) / 𝑀))
Theorem	cvgratnnlemnexp 11885*	Lemma for cvgratnn 11892. (Contributed by Jim Kingdon, 15-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))
Theorem	cvgratnnlemmn 11886*	Lemma for cvgratnn 11892. (Contributed by Jim Kingdon, 15-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))
Theorem	cvgratnnlemseq 11887*	Lemma for cvgratnn 11892. (Contributed by Jim Kingdon, 21-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → ((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀)) = Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖))
Theorem	cvgratnnlemabsle 11888*	Lemma for cvgratnn 11892. (Contributed by Jim Kingdon, 21-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → (abs‘Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑖)) ≤ ((abs‘(𝐹‘𝑀)) · Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀))))
Theorem	cvgratnnlemsumlt 11889*	Lemma for cvgratnn 11892. (Contributed by Jim Kingdon, 23-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → Σ𝑖 ∈ ((𝑀 + 1)...𝑁)(𝐴↑(𝑖 − 𝑀)) < (𝐴 / (1 − 𝐴)))
Theorem	cvgratnnlemfm 11890*	Lemma for cvgratnn 11892. (Contributed by Jim Kingdon, 23-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    ⇒   ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) < ((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) / 𝑀))
Theorem	cvgratnnlemrate 11891*	Lemma for cvgratnn 11892. (Contributed by Jim Kingdon, 21-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → (abs‘((seq1( + , 𝐹)‘𝑁) − (seq1( + , 𝐹)‘𝑀))) < (((((1 / ((1 / 𝐴) − 1)) / 𝐴) · ((abs‘(𝐹‘1)) + 1)) · (𝐴 / (1 − 𝐴))) / 𝑀))
Theorem	cvgratnn 11892*	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, then the infinite sum of the terms of 𝐹 converges to a complex number. Although this theorem is similar to cvgratz 11893 and cvgratgt0 11894, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11711 is sensitive to how a sequence is indexed. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 12-Nov-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    ⇒   ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
Theorem	cvgratz 11893*	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, then the infinite sum of the terms of 𝐹 converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Theorem	cvgratgt0 11894*	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. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 1)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
4.9.9  Mertens' theorem
Theorem	mertenslemub 11895*	Lemma for mertensabs 11898. An upper bound for 𝑇. (Contributed by Jim Kingdon, 3-Dec-2022.)
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑇)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
Theorem	mertenslemi1 11896*	Lemma for mertensabs 11898. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))    &   ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}    &   ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))))    &   ⊢ (𝜑 → 𝑃 ∈ ℝ)    &   ⊢ (𝜑 → (𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (𝑃 + 1)))))    &   ⊢ (𝜑 → 0 ≤ 𝑃)    &   ⊢ (𝜑 → ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑃)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
Theorem	mertenslem2 11897*	Lemma for mertensabs 11898. (Contributed by Mario Carneiro, 28-Apr-2014.)
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))    &   ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}    &   ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
Theorem	mertensabs 11898*	Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐵(𝑘) is convergent, then (Σ𝑗 ∈ ℕ0𝐴(𝑗) · Σ𝑘 ∈ ℕ0𝐵(𝑘)) = Σ𝑘 ∈ ℕ0Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘 − 𝑗)) (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.) (Revised by Jim Kingdon, 8-Dec-2022.)
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))    &   ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )    &   ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )    &   ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
4.9.10  Finite and infinite products
4.9.10.1  Product sequences
Theorem	prodf 11899*	An infinite product of complex terms is a function from an upper set of integers to ℂ. (Contributed by Scott Fenton, 4-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)
Theorem	clim2prod 11900*	The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴)    ⇒   ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴))