P. 129 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ltwenn 12801	Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
⊢ < We ℕ
Theorem	ltwefz 12802	Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
⊢ < We (𝑀...𝑁)
Theorem	uzenom 12803	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
Theorem	uzinf 12804	An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → ¬ 𝑍 ∈ Fin)
Theorem	nnnfi 12805	The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ¬ ℕ ∈ Fin
Theorem	uzrdgxfr 12806*	Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.)
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐴) ↾ ω)    &   ⊢ 𝐻 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐵) ↾ ω)    &   ⊢ 𝐴 ∈ ℤ    &   ⊢ 𝐵 ∈ ℤ    ⇒   ⊢ (𝑁 ∈ ω → (𝐺‘𝑁) = ((𝐻‘𝑁) + (𝐴 − 𝐵)))
Theorem	fzennn 12807	The cardinality of a finite set of sequential integers. (See om2uz0i 12786 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))
Theorem	fzen2 12808	The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.)
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀)))
Theorem	cardfz 12809	The cardinality of a finite set of sequential integers. (See om2uz0i 12786 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡𝐺‘𝑁))
Theorem	hashgf1o 12810	𝐺 maps ω one-to-one onto ℕ0. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)    ⇒   ⊢ 𝐺:ω–1-1-onto→ℕ0
Theorem	fzfi 12811	A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
⊢ (𝑀...𝑁) ∈ Fin
Theorem	fzfid 12812	Commonly used special case of fzfi 12811. (Contributed by Mario Carneiro, 25-May-2014.)
⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
Theorem	fzofi 12813	Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝑀..^𝑁) ∈ Fin
Theorem	fsequb 12814*	The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥)
Theorem	fsequb2 12815*	The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
Theorem	fseqsupcl 12816	The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
Theorem	fseqsupubi 12817	The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.)
⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → (𝐹‘𝐾) ≤ sup(ran 𝐹, ℝ, < ))
Theorem	nn0ennn 12818	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
⊢ ℕ0 ≈ ℕ
Theorem	nnenom 12819	The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ ℕ ≈ ω
Theorem	nnct 12820	ℕ is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ ℕ ≼ ω
Theorem	uzindi 12821*	Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ (ℤ≥‘𝐿))    &   ⊢ ((𝜑 ∧ 𝑅 ∈ (𝐿...𝑇) ∧ ∀𝑦(𝑆 ∈ (𝐿..^𝑅) → 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)    &   ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	axdc4uzlem 12822*	Lemma for axdc4uz 12823. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐺 = (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω)    &   ⊢ 𝐻 = (𝑛 ∈ ω, 𝑥 ∈ 𝐴 ↦ ((𝐺‘𝑛)𝐹𝑥))    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
Theorem	axdc4uz 12823*	A version of axdc4 9316 that works on an upper set of integers instead of ω. (Contributed by Mario Carneiro, 8-Jan-2014.)
⊢ 𝑀 ∈ ℤ    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
Theorem	ssnn0fi 12824*	A subset of the nonnegative integers is finite if and only if there is a nonnegative integer so that all integers greater than this integer are not contained in the subset. (Contributed by AV, 3-Oct-2019.)
⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆)))
Theorem	rabssnn0fi 12825*	A subset of the nonnegative integers defined by a restricted class abstraction is finite if there is a nonnegative integer so that for all integers greater than this integer the condition of the class abstraction is not fulfilled. (Contributed by AV, 3-Oct-2019.)
⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑))
5.6.4  Strong induction over upper sets of integers
Theorem	uzsinds 12826*	Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒)
Theorem	nnsinds 12827*	Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜓 → 𝜑))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝜒)
Theorem	nn0sinds 12828*	Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ℕ0 → (∀𝑦 ∈ (0...(𝑥 − 1))𝜓 → 𝜑))    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝜒)
5.6.5  Finitely supported functions over the nonnegative integers
Theorem	fsuppmapnn0fiublem 12829*	Lemma for fsuppmapnn0fiub 12830 and fsuppmapnn0fiubex 12832. (Contributed by AV, 2-Oct-2019.)
⊢ 𝑈 = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)    &   ⊢ 𝑆 = sup(𝑈, ℝ, < )    ⇒   ⊢ ((𝑀 ⊆ (𝑅 ↑𝑚 ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ 𝑈 ≠ ∅) → 𝑆 ∈ ℕ0))
Theorem	fsuppmapnn0fiub 12830*	If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0 and ending with the supremum of the union of the support of these functions. (Contributed by AV, 2-Oct-2019.) (Proof shortened by JJ, 2-Aug-2021.)
⊢ 𝑈 = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)    &   ⊢ 𝑆 = sup(𝑈, ℝ, < )    ⇒   ⊢ ((𝑀 ⊆ (𝑅 ↑𝑚 ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ 𝑈 ≠ ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑆)))
Theorem	fsuppmapnn0fiubOLD 12831*	Obsolete proof of fsuppmapnn0fiub 12830 as of 2-Aug-2021. (Contributed by AV, 2-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑈 = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)    &   ⊢ 𝑆 = sup(𝑈, ℝ, < )    ⇒   ⊢ ((𝑀 ⊆ (𝑅 ↑𝑚 ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ 𝑈 ≠ ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑆)))
Theorem	fsuppmapnn0fiubex 12832*	If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0. (Contributed by AV, 2-Oct-2019.)
⊢ ((𝑀 ⊆ (𝑅 ↑𝑚 ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚)))
Theorem	fsuppmapnn0fiub0 12833*	If all functions of a finite set of functions over the nonnegative integers are finitely supported, then all these functions are zero for all integers greater than a fixed integer. (Contributed by AV, 3-Oct-2019.)
⊢ ((𝑀 ⊆ (𝑅 ↑𝑚 ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝑓‘𝑥) = 𝑍)))
Theorem	suppssfz 12834*	Condition for a function over the nonnegative integers to have a support contained in a finite set of sequential integers. (Contributed by AV, 9-Oct-2019.)
⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑𝑚 ℕ0))    &   ⊢ (𝜑 → 𝑆 ∈ ℕ0)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍))    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))
Theorem	fsuppmapnn0ub 12835*	If a function over the nonnegative integers is finitely supported, then there is an upper bound for the arguments resulting in nonzero values. (Contributed by AV, 6-Oct-2019.)
⊢ ((𝐹 ∈ (𝑅 ↑𝑚 ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍)))
Theorem	fsuppmapnn0fz 12836*	If a function over the nonnegative integers is finitely supported, then there is an upper bound for a finite set of sequential integers containing the support of the function. (Contributed by AV, 30-Sep-2019.) (Proof shortened by AV, 6-Oct-2019.)
⊢ ((𝐹 ∈ (𝑅 ↑𝑚 ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 (𝐹 supp 𝑍) ⊆ (0...𝑚)))
Theorem	mptnn0fsupp 12837*	A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.)
⊢ (𝜑 → 0 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))    ⇒   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 )
Theorem	mptnn0fsuppd 12838*	A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 2-Dec-2019.) (Revised by AV, 23-Dec-2019.)
⊢ (𝜑 → 0 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵)    &   ⊢ (𝑘 = 𝑥 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝐷 = 0 ))    ⇒   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 )
Theorem	mptnn0fsuppr 12839*	A finitely supported mapping from the nonnegative integers fulfills certain conditions. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 23-Dec-2019.)
⊢ (𝜑 → 0 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 )    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))
Theorem	f13idfv 12840	A one-to-one function with the domain { 0, 1 ,2 } in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
⊢ 𝐴 = (0...2)    ⇒   ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ((𝐹‘0) ≠ (𝐹‘1) ∧ (𝐹‘0) ≠ (𝐹‘2) ∧ (𝐹‘1) ≠ (𝐹‘2))))
5.6.6  The infinite sequence builder "seq" - extension
Syntax	cseq 12841	Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹)
Definition	df-seq 12842*	Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers ℕ or some other upper integer set) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seq1 12854 and seqp1 12856. Typically, those are the main theorems that would be used in practice. The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹)‘1) = 1, (seq1( + , 𝐹)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹) transforms a sequence 𝐹 into an infinite series. seq𝑀( + , 𝐹) ⇝ 2 means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 14327), by climdm 14329 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ⇝ ‘seq1( + , 𝐹)) (provided the sequence converges) and evaluates to 2 in this example. Internally, the rec function generates as its values a set of ordered pairs starting at ⟨𝑀, (𝐹‘𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of rec is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain. This definition has its roots in a series of theorems from om2uz0i 12786 through om2uzf1oi 12792, originally proved by Raph Levien for use with df-exp 12901 and later generalized for arbitrary recursive sequences. Definition df-sum 14461 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)
⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
Theorem	seqex 12843	Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ seq𝑀( + , 𝐹) ∈ V
Theorem	seqeq1 12844	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Theorem	seqeq2 12845	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Theorem	seqeq3 12846	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Theorem	seqeq1d 12847	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
Theorem	seqeq2d 12848	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
Theorem	seqeq3d 12849	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵))
Theorem	seqeq123d 12850	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝑀 = 𝑁)    &   ⊢ (𝜑 → + = 𝑄)    &   ⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))
Theorem	nfseq 12851	Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝑀    &   ⊢ Ⅎ𝑥 +    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)
Theorem	seqval 12852*	Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)    ⇒   ⊢ seq𝑀( + , 𝐹) = ran 𝑅
Theorem	seqfn 12853	The sequence builder function is a function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
Theorem	seq1 12854	Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
Theorem	seq1i 12855	Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 30-Apr-2014.)
⊢ 𝑀 ∈ ℤ    &   ⊢ (𝜑 → (𝐹‘𝑀) = 𝐴)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝐴)
Theorem	seqp1 12856	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
Theorem	seqp1i 12857	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 30-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑁 ∈ 𝑍    &   ⊢ 𝐾 = (𝑁 + 1)    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝐾) = 𝐵)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐴 + 𝐵))
Theorem	seqm1 12858	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁)))
Theorem	seqcl2 12859*	Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)
Theorem	seqf2 12860*	Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝐶)
Theorem	seqcl 12861*	Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆)
Theorem	seqf 12862*	Range of the recursive sequence builder (special case of seqf2 12860). (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝑆)
Theorem	seqfveq2 12863*	Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))
Theorem	seqfeq2 12864*	Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺))
Theorem	seqfveq 12865*	Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Theorem	seqfeq 12866*	Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Theorem	seqshft2 12867*	Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
Theorem	seqres 12868	Restricting its characteristic function to (ℤ≥‘𝑀) does not affect the seq function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝐹 ↾ (ℤ≥‘𝑀))) = seq𝑀( + , 𝐹))
Theorem	serf 12869*	An infinite series of complex terms is a function from ℕ to ℂ. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
Theorem	serfre 12870*	An infinite series of real numbers is a function from ℕ to ℝ. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
Theorem	monoord 12871*	Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))
Theorem	monoord2 12872*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))
Theorem	sermono 12873*	The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ≤ (seq𝑀( + , 𝐹)‘𝑁))
Theorem	seqsplit 12874*	Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
Theorem	seq1p 12875*	Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((𝐹‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
Theorem	seqcaopr3 12876*	Lemma for seqcaopr2 12877. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqcaopr2 12877*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqcaopr 12878*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqf1olem2a 12879*	Lemma for seqf1o 12882. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))
Theorem	seqf1olem1 12880*	Lemma for seqf1o 12882. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))    &   ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶)    &   ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))    &   ⊢ 𝐾 = (◡𝐹‘(𝑁 + 1))    ⇒   ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
Theorem	seqf1olem2 12881*	Lemma for seqf1o 12882. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))    &   ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶)    &   ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))    &   ⊢ 𝐾 = (◡𝐹‘(𝑁 + 1))    &   ⊢ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = (seq𝑀( + , 𝐺)‘(𝑁 + 1)))
Theorem	seqf1o 12882*	Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Theorem	seradd 12883*	The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	sersub 12884*	The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) − (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqid3 12885*	A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a + -idempotent sums (or "+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.)
⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
Theorem	seqid 12886*	Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for +) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (𝐹‘𝑁) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹))
Theorem	seqid2 12887*	The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for +) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥)    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
Theorem	seqhomo 12888*	Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
Theorem	seqz 12889*	If the operation + has an absorbing element 𝑍 (a.k.a. zero element), then any sequence containing a 𝑍 evaluates to 𝑍. (Contributed by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐹‘𝐾) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
Theorem	seqfeq4 12890*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁))
Theorem	seqfeq3 12891*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Theorem	seqdistr 12892*	The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
Theorem	ser0 12893	The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0)
Theorem	ser0f 12894	A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0}))
Theorem	serge0 12895*	A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
Theorem	serle 12896*	Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁))
Theorem	ser1const 12897	Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))
Theorem	seqof 12898*	Distribute function operation through a sequence. Note that 𝐺(𝑧) is an implicit function on 𝑧. (Contributed by Mario Carneiro, 3-Mar-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))    ⇒   ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqof2 12899*	Distribute function operation through a sequence. Maps-to notation version of seqof 12898. (Contributed by Mario Carneiro, 7-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) → 𝑋 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)))
5.6.7  Integer powers
Syntax	cexp 12900	Extend class notation to include exponentiation of a complex number to an integer power.
class ↑