P. 154 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15301-15400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	climub 15301*	The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝐴)
Theorem	climserle 15302*	The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴)
Theorem	isershft 15303	Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.)
⊢ 𝐹 ∈ V    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴))
Theorem	isercolllem1 15304*	Lemma for isercoll 15307. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    ⇒   ⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → (𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆)))
Theorem	isercolllem2 15305*	Lemma for isercoll 15307. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)))
Theorem	isercolllem3 15306*	Lemma for isercoll 15307. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))
Theorem	isercoll 15307*	Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴))
Theorem	isercoll2 15308*	Generalize isercoll 15307 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑊) → (𝐹‘𝑛) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))
Theorem	climsup 15309*	A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ sup(ran 𝐹, ℝ, < ))
Theorem	climcau 15310*	A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
Theorem	climbdd 15311*	A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)
Theorem	caucvgrlem 15312*	Lemma for caurcvgr 15313. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by AV, 12-Sep-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅))))
Theorem	caurcvgr 15313*	A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.) (Revised by AV, 12-Sep-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ⇝𝑟 (lim sup‘𝐹))
Theorem	caucvgrlem2 15314*	Lemma for caucvgr 15315. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    &   ⊢ 𝐻:ℂ⟶ℝ    &   ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))    ⇒   ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))) ⇝𝑟 ( ⇝𝑟 ‘(𝐻 ∘ 𝐹)))
Theorem	caucvgr 15315*	A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑟 )
Theorem	caurcvg 15316*	A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by AV, 12-Sep-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹))
Theorem	caurcvg2 15317*	A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
Theorem	caucvg 15318*	A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
Theorem	caucvgb 15319*	A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
Theorem	serf0 15320*	If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 0)
Theorem	iseraltlem1 15321*	Lemma for iseralt 15324. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → 0 ≤ (𝐺‘𝑁))
Theorem	iseraltlem2 15322*	Lemma for iseralt 15324. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example 𝑆(1) ≤ 𝑆(3) ≤ 𝑆(5) ≤ ... and ... ≤ 𝑆(4) ≤ 𝑆(2) ≤ 𝑆(0) (assuming 𝑀 = 0 so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ0) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))
Theorem	iseraltlem3 15323*	Lemma for iseralt 15324. From iseraltlem2 15322, we have (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) and (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1), and we also have (-1↑𝑛) · 𝑆(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) for each 𝑛 by the definition of the partial sum 𝑆, so combining the inequalities we get (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − 𝐺(𝑛 + 2𝑘 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) ≤ (-1↑𝑛) · 𝑆(𝑛) + 𝐺(𝑛 + 1), so ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘 + 1) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1) and ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1). Thus, both even and odd partial sums are Cauchy if 𝐺 converges to 0. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ0) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1))))
Theorem	iseralt 15324*	The alternating series test. If 𝐺(𝑘) is a decreasing sequence that converges to 0, then Σ𝑘 ∈ 𝑍(-1↑𝑘) · 𝐺(𝑘) is a convergent series. (Note that the first term is positive if 𝑀 is even, and negative if 𝑀 is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by -1 using isermulc2 15297.) (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 9-Jul-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
5.10.3  Finite and infinite sums
Syntax	csu 15325	Extend class notation to include finite and infinite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
class Σ𝑘 ∈ 𝐴 𝐵
Definition	df-sum 15326*	Define the sum of a series with an index set of integers 𝐴. The variable 𝑘 is normally a free variable in 𝐵, i.e., 𝐵 can be thought of as 𝐵(𝑘). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers) by summo 15357. Examples: Σ𝑘 ∈ {1, 2, 4}𝑘 means 1 + 2 + 4 = 7, and Σ𝑘 ∈ ℕ(1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 15522). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
Theorem	sumex 15327	A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ Σ𝑘 ∈ 𝐴 𝐵 ∈ V
Theorem	sumeq1 15328	Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
Theorem	nfsum1 15329	Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ Ⅎ𝑘𝐴    ⇒   ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵
Theorem	nfsum 15330*	Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. Version of nfsum 15330 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 11-Dec-2005.) (Revised by Gino Giotto, 24-Feb-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵
Theorem	nfsumOLD 15331	Obsolete version of nfsum 15330 as of 24-Feb-2024. Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵
Theorem	sumeq2w 15332	Equality theorem for sum, when the class expressions 𝐵 and 𝐶 are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ (∀𝑘 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)
Theorem	sumeq2ii 15333*	Equality theorem for sum, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Mario Carneiro, 13-Jun-2019.)
⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)
Theorem	sumeq2 15334*	Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)
Theorem	cbvsum 15335*	Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)    &   ⊢ Ⅎ𝑘𝐴    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ Ⅎ𝑘𝐵    &   ⊢ Ⅎ𝑗𝐶    ⇒   ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
Theorem	cbvsumv 15336*	Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)    ⇒   ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
Theorem	cbvsumi 15337*	Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
⊢ Ⅎ𝑘𝐵    &   ⊢ Ⅎ𝑗𝐶    &   ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)    ⇒   ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
Theorem	sumeq1i 15338*	Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶
Theorem	sumeq2i 15339*	Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
⊢ (𝑘 ∈ 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
Theorem	sumeq12i 15340*	Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑘 ∈ 𝐴 → 𝐶 = 𝐷)    ⇒   ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷
Theorem	sumeq1d 15341*	Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
Theorem	sumeq2d 15342*	Equality deduction for sum. Note that unlike sumeq2dv 15343, 𝑘 may occur in 𝜑. (Contributed by NM, 1-Nov-2005.)
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)
Theorem	sumeq2dv 15343*	Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)
Theorem	sumeq2sdv 15344*	Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Proof shortened by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)
Theorem	2sumeq2dv 15345*	Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐷)
Theorem	sumeq12dv 15346*	Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)
Theorem	sumeq12rdv 15347*	Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)
Theorem	sum2id 15348*	The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 ( I ‘𝐵)
Theorem	sumfc 15349*	A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
⊢ Σ𝑗 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = Σ𝑘 ∈ 𝐴 𝐵
Theorem	fz1f1o 15350*	A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)
⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
Theorem	sumrblem 15351*	Lemma for sumrb 15353. (Contributed by Mario Carneiro, 12-Aug-2013.)
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹))
Theorem	fsumcvg 15352*	The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))
Theorem	sumrb 15353*	Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹) ⇝ 𝐶))
Theorem	summolem3 15354*	Lemma for summo 15357. (Contributed by Mario Carneiro, 29-Mar-2014.)
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)    &   ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Theorem	summolem2a 15355*	Lemma for summo 15357. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.)
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))
Theorem	summolem2 15356*	Lemma for summo 15357. (Contributed by Mario Carneiro, 3-Apr-2014.)
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)    ⇒   ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
Theorem	summo 15357*	A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)    ⇒   ⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))))
Theorem	zsum 15358*	Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Theorem	isum 15359*	Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Theorem	fsum 15360*	The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))
Theorem	sum0 15361	Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
⊢ Σ𝑘 ∈ ∅ 𝐴 = 0
Theorem	sumz 15362*	Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0)
Theorem	fsumf1o 15363*	Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷)
Theorem	sumss 15364*	Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)    &   ⊢ (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
Theorem	fsumss 15365*	Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
Theorem	sumss2 15366*	Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ) ∧ (𝐵 ⊆ (ℤ≥‘𝑀) ∨ 𝐵 ∈ Fin)) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 𝐶, 0))
Theorem	fsumcvg2 15367*	The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))
Theorem	fsumsers 15368*	Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 21-Apr-2014.)
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁))    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁))
Theorem	fsumcvg3 15369*	A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Theorem	fsumser 15370*	A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 15387 and fsump1i 15409, which should make our notation clear and from which, along with closure fsumcl 15373, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁))
Theorem	fsumcl2lem 15371*	- Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
Theorem	fsumcllem 15372*	- Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 0 ∈ 𝑆)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
Theorem	fsumcl 15373*	Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
Theorem	fsumrecl 15374*	Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ)
Theorem	fsumzcl 15375*	Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ)
Theorem	fsumnn0cl 15376*	Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ0)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ0)
Theorem	fsumrpcl 15377*	Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ+)
Theorem	fsumclf 15378*	Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsumcl 15373 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 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
Theorem	fsumzcl2 15379*	A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ)
Theorem	fsumadd 15380*	The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-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	fsumsplit 15381*	Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)
⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶))
Theorem	fsumsplitf 15382*	Split a sum into two parts. A version of fsumsplit 15381 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 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶))
Theorem	sumsnf 15383*	A sum of a singleton is the term. A version of sumsn 15386 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝐵    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)
Theorem	fsumsplitsn 15384*	Separate out a term in a finite sum. (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 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷))
Theorem	fsumsplit1 15385*	Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐷    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝑘 = 𝐶 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
Theorem	sumsn 15386*	A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)
Theorem	fsum1 15387*	The finite sum of 𝐴(𝑘) from 𝑘 = 𝑀 to 𝑀 (i.e. a sum with only one term) is 𝐵 i.e. 𝐴(𝑀). (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)𝐴 = 𝐵)
Theorem	sumpr 15388*	A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)    &   ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
Theorem	sumtp 15389*	A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.)
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸)    &   ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹)    &   ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺)    &   ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺))
Theorem	sumsns 15390*	A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
⊢ ((𝑀 ∈ 𝑉 ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
Theorem	fsumm1 15391*	Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))
Theorem	fzosump1 15392*	Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵))
Theorem	fsum1p 15393*	Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
Theorem	fsummsnunz 15394*	A finite sum all of whose summands are integers is itself an integer (case where the summation set is the union of a finite set and a singleton). (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ)
Theorem	fsumsplitsnun 15395*	Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
⊢ ((𝐴 ∈ Fin ∧ (𝑍 ∈ 𝑉 ∧ 𝑍 ∉ 𝐴) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 = (Σ𝑘 ∈ 𝐴 𝐵 + ⦋𝑍 / 𝑘⦌𝐵))
Theorem	fsump1 15396*	The addition of the next term in a finite sum of 𝐴(𝑘) is the current term plus 𝐵 i.e. 𝐴(𝑁 + 1). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝑁)𝐴 + 𝐵))
Theorem	isumclim 15397*	An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = 𝐵)
Theorem	isumclim2 15398*	A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴)
Theorem	isumclim3 15399*	The sequence of partial finite sums of a converging infinite series converges to the infinite sum of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ Σ𝑘 ∈ 𝑍 𝐴)
Theorem	sumnul 15400*	The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (Revised by Mario Carneiro, 23-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 = ∅)