P. 140 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	modmulmod 13901	The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) · 𝐵) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))
Theorem	modmulmodr 13902	The product of an integer and a real number modulo a positive real number equals the product of the integer and the real number modulo the positive real number. (Contributed by Alexander van der Vekens, 9-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))
Theorem	modaddmulmod 13903	The sum of a real number and the product of a second real number modulo a positive real number and an integer equals the sum of the real number and the product of the other real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℤ) ∧ 𝑀 ∈ ℝ+) → ((𝐴 + ((𝐵 mod 𝑀) · 𝐶)) mod 𝑀) = ((𝐴 + (𝐵 · 𝐶)) mod 𝑀))
Theorem	moddi 13904	Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶)))
Theorem	modsubdir 13905	Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → ((𝐵 mod 𝐶) ≤ (𝐴 mod 𝐶) ↔ ((𝐴 − 𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶))))
Theorem	modeqmodmin 13906	A real number equals the difference of the real number and a positive real number modulo the positive real number. (Contributed by AV, 3-Nov-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (𝐴 mod 𝑀) = ((𝐴 − 𝑀) mod 𝑀))
Theorem	modirr 13907	A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐴 / 𝐵) ∈ (ℝ ∖ ℚ)) → (𝐴 mod 𝐵) ≠ 0)
Theorem	modfzo0difsn 13908*	For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑖 ∈ (1..^𝑁)𝐾 = ((𝑖 + 𝐽) mod 𝑁))
Theorem	modsumfzodifsn 13909	The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ (1..^𝑁)) → ((𝐾 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))
Theorem	modlteq 13910	Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐼 mod 𝑁) = (𝐽 mod 𝑁) ↔ 𝐼 = 𝐽))
Theorem	addmodlteq 13911	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. A much shorter proof exists if the "divides" relation ∥ can be used, see addmodlteqALT 16295. (Contributed by AV, 20-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))
5.6.3  Miscellaneous theorems about integers
Theorem	om2uz0i 13912*	The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ0 or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (This series of theorems generalizes an earlier series for ℕ0 contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ (𝐺‘∅) = 𝐶
Theorem	om2uzsuci 13913*	The value of 𝐺 (see om2uz0i 13912) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))
Theorem	om2uzuzi 13914*	The value 𝐺 (see om2uz0i 13912) at an ordinal natural number is in the upper integers. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶))
Theorem	om2uzlti 13915*	Less-than relation for 𝐺 (see om2uz0i 13912). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) < (𝐺‘𝐵)))
Theorem	om2uzlt2i 13916*	The mapping 𝐺 (see om2uz0i 13912) preserves order. (Contributed by NM, 4-May-2005.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵)))
Theorem	om2uzrani 13917*	Range of 𝐺 (see om2uz0i 13912). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ ran 𝐺 = (ℤ≥‘𝐶)
Theorem	om2uzf1oi 13918*	𝐺 (see om2uz0i 13912) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
Theorem	om2uzisoi 13919*	𝐺 (see om2uz0i 13912) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶))
Theorem	om2uzoi 13920*	An alternative definition of 𝐺 in terms of df-oi 9463. (Contributed by Mario Carneiro, 2-Jun-2015.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    ⇒   ⊢ 𝐺 = OrdIso( < , (ℤ≥‘𝐶))
Theorem	om2uzrdg 13921*	A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. Normally 𝐹 is a function on the partition, and 𝐴 is a member of the partition. See also comment in om2uz0i 13912. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)    ⇒   ⊢ (𝐵 ∈ ω → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
Theorem	uzrdglem 13922*	A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)    ⇒   ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
Theorem	uzrdgfni 13923*	The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13921. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)    &   ⊢ 𝑆 = ran 𝑅    ⇒   ⊢ 𝑆 Fn (ℤ≥‘𝐶)
Theorem	uzrdg0i 13924*	Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 13921. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)    &   ⊢ 𝑆 = ran 𝑅    ⇒   ⊢ (𝑆‘𝐶) = 𝐴
Theorem	uzrdgsuci 13925*	Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13921. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ 𝐶 ∈ ℤ    &   ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)    &   ⊢ 𝑆 = ran 𝑅    ⇒   ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆‘𝐵)))
Theorem	ltweuz 13926	< is a well-founded relation on any sequence of upper integers. (Contributed by Andrew Salmon, 13-Nov-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ < We (ℤ≥‘𝐴)
Theorem	ltwenn 13927	Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
⊢ < We ℕ
Theorem	ltwefz 13928	Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
⊢ < We (𝑀...𝑁)
Theorem	uzenom 13929	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
Theorem	uzinf 13930	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 13931	The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ¬ ℕ ∈ Fin
Theorem	uzrdgxfr 13932*	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 13933	The cardinality of a finite set of sequential integers. (See om2uz0i 13912 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 13934	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 13935	The cardinality of a finite set of sequential integers. (See om2uz0i 13912 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 13936	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 13937	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 13938	Commonly used special case of fzfi 13937. (Contributed by Mario Carneiro, 25-May-2014.)
⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
Theorem	fzofi 13939	Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (𝑀..^𝑁) ∈ Fin
Theorem	fsequb 13940*	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 13941*	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 13942	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 13943	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 13944	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
⊢ ℕ0 ≈ ℕ
Theorem	nnenom 13945	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 13946	ℕ is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ ℕ ≼ ω
Theorem	uzindi 13947*	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 13948*	Lemma for axdc4uz 13949. (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 13949*	A version of axdc4 10409 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 13950*	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 13951*	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 13952*	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 13953*	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 13954*	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 13955*	Lemma for fsuppmapnn0fiub 13956 and fsuppmapnn0fiubex 13957. (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 13956*	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	fsuppmapnn0fiubex 13957*	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 13958*	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 13959*	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 13960*	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 13961*	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 13962*	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 13963*	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 13964*	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 13965	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 13966	Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹)
Definition	df-seq 13967*	Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as ℕ or ℕ0) 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 13979 and seqp1 13981. 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 15518), by climdm 15520 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 13912 through om2uzf1oi 13918, originally proved by Raph Levien for use with df-exp 14027 and later generalized for arbitrary recursive sequences. Definition df-sum 15653 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 13968	Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ seq𝑀( + , 𝐹) ∈ V
Theorem	seqeq1 13969	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Theorem	seqeq2 13970	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Theorem	seqeq3 13971	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Theorem	seqeq1d 13972	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
Theorem	seqeq2d 13973	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 13974	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵))
Theorem	seqeq123d 13975	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 13976	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 13977*	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 13978	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 13979	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 13980	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 13981	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	seqexw 13982	Weak version of seqex 13968 that holds without ax-rep 5234. A sequence builder exists when its binary operation input exists and its starting index is an integer. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ + ∈ V    &   ⊢ 𝑀 ∈ ℤ    ⇒   ⊢ seq𝑀( + , 𝐹) ∈ V
Theorem	seqp1d 13983	Value of the sequence builder function at a successor, deduction form. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by AV, 3-May-2024.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ 𝐾 = (𝑁 + 1)    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝐾) = 𝐵)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐴 + 𝐵))
Theorem	seqm1 13984	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 13985*	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 13986*	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 13987*	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 13988*	Range of the recursive sequence builder (special case of seqf2 13986). (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 13989*	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 13990*	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 13991*	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 13992*	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 13993*	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 13994	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 13995*	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 13996*	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 13997*	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 13998*	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 13999*	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 14000*	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)))