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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmodsubdir 13301 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) ↔ ((𝐴𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶))))

Theoremmodeqmodmin 13302 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) = ((𝐴𝑀) mod 𝑀))

Theoremmodirr 13303 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)

Theoremmodfzo0difsn 13304* 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))

Theoremmodsumfzodifsn 13305 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}))

Theoremmodlteq 13306 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))

Theoremaddmodlteq 13307 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 15664. (Contributed by AV, 20-Mar-2021.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))

Theoremom2uz0i 13308* 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 0 or 1 for the upper integers ), 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 0 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

Theoremom2uzsuci 13309* The value of 𝐺 (see om2uz0i 13308) 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) = ((𝐺𝐴) + 1))

Theoremom2uzuzi 13310* The value 𝐺 (see om2uz0i 13308) 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 → (𝐺𝐴) ∈ (ℤ𝐶))

Theoremom2uzlti 13311* Less-than relation for 𝐺 (see om2uz0i 13308). (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) → (𝐴𝐵 → (𝐺𝐴) < (𝐺𝐵)))

Theoremom2uzlt2i 13312* The mapping 𝐺 (see om2uz0i 13308) 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) → (𝐴𝐵 ↔ (𝐺𝐴) < (𝐺𝐵)))

Theoremom2uzrani 13313* Range of 𝐺 (see om2uz0i 13308). (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 = (ℤ𝐶)

Theoremom2uzf1oi 13314* 𝐺 (see om2uz0i 13308) 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→(ℤ𝐶)

Theoremom2uzisoi 13315* 𝐺 (see om2uz0i 13308) 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, (ℤ𝐶))

Theoremom2uzoi 13316* An alternative definition of 𝐺 in terms of df-oi 8958. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝐶 ∈ ℤ    &   𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)       𝐺 = OrdIso( < , (ℤ𝐶))

Theoremom2uzrdg 13317* A helper lemma for the value of a recursive definition generator on upper integers (typically either or 0) 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 13308. (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 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)

Theoremuzrdglem 13318* 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 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)

Theoremuzrdgfni 13319* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13317. (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 (ℤ𝐶)

Theoremuzrdg0i 13320* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 13317. (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       (𝑆𝐶) = 𝐴

Theoremuzrdgsuci 13321* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13317. (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 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆𝐵)))

Theoremltweuz 13322 < 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 (ℤ𝐴)

Theoremltwenn 13323 Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
< We ℕ

Theoremltwefz 13324 Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
< We (𝑀...𝑁)

Theoremuzenom 13325 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → 𝑍 ≈ ω)

Theoremuzinf 13326 An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → ¬ 𝑍 ∈ Fin)

Theoremnnnfi 13327 The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
¬ ℕ ∈ Fin

Theoremuzrdgxfr 13328* 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 → (𝐺𝑁) = ((𝐻𝑁) + (𝐴𝐵)))

Theoremfzennn 13329 The cardinality of a finite set of sequential integers. (See om2uz0i 13308 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) ≈ (𝐺𝑁))

Theoremfzen2 13330 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 ∈ (ℤ𝑀) → (𝑀...𝑁) ≈ (𝐺‘((𝑁 + 1) − 𝑀)))

Theoremcardfz 13331 The cardinality of a finite set of sequential integers. (See om2uz0i 13308 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)) = (𝐺𝑁))

Theoremhashgf1o 13332 𝐺 maps ω one-to-one onto 0. (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

Theoremfzfi 13333 A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
(𝑀...𝑁) ∈ Fin

Theoremfzfid 13334 Commonly used special case of fzfi 13333. (Contributed by Mario Carneiro, 25-May-2014.)
(𝜑 → (𝑀...𝑁) ∈ Fin)

Theoremfzofi 13335 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝑀..^𝑁) ∈ Fin

Theoremfsequb 13336* 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)(𝐹𝑘) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) < 𝑥)

Theoremfsequb2 13337* 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 𝑦𝑥)

Theoremfseqsupcl 13338 The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)

Theoremfseqsupubi 13339 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) → (𝐹𝐾) ≤ sup(ran 𝐹, ℝ, < ))

Theoremnn0ennn 13340 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
0 ≈ ℕ

Theoremnnenom 13341 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

Theoremnnct 13342 is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
ℕ ≼ ω

Theoremuzindi 13343* Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝑇 ∈ (ℤ𝐿))    &   ((𝜑𝑅 ∈ (𝐿...𝑇) ∧ ∀𝑦(𝑆 ∈ (𝐿..^𝑅) → 𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝑥 = 𝑦𝑅 = 𝑆)    &   (𝑥 = 𝐴𝑅 = 𝑇)       (𝜑𝜃)

Theoremaxdc4uzlem 13344* Lemma for axdc4uz 13345. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)    &   𝐴 ∈ V    &   𝐺 = (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω)    &   𝐻 = (𝑛 ∈ ω, 𝑥𝐴 ↦ ((𝐺𝑛)𝐹𝑥))       ((𝐶𝐴𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍𝐴 ∧ (𝑔𝑀) = 𝐶 ∧ ∀𝑘𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremaxdc4uz 13345* A version of axdc4 9863 that works on an upper set of integers instead of ω. (Contributed by Mario Carneiro, 8-Jan-2014.)
𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)       ((𝐴𝑉𝐶𝐴𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍𝐴 ∧ (𝑔𝑀) = 𝐶 ∧ ∀𝑘𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremssnn0fi 13346* 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 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))

Theoremrabssnn0fi 13347* 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 ∈ ℕ0𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑))

5.6.4  Strong induction over upper sets of integers

Theoremuzsinds 13348* Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑁 → (𝜑𝜒))    &   (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))       (𝑁 ∈ (ℤ𝑀) → 𝜒)

Theoremnnsinds 13349* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑁 → (𝜑𝜒))    &   (𝑥 ∈ ℕ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜓𝜑))       (𝑁 ∈ ℕ → 𝜒)

Theoremnn0sinds 13350* Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑁 → (𝜑𝜒))    &   (𝑥 ∈ ℕ0 → (∀𝑦 ∈ (0...(𝑥 − 1))𝜓𝜑))       (𝑁 ∈ ℕ0𝜒)

5.6.5  Finitely supported functions over the nonnegative integers

Theoremfsuppmapnn0fiublem 13351* Lemma for fsuppmapnn0fiub 13352 and fsuppmapnn0fiubex 13353. (Contributed by AV, 2-Oct-2019.)
𝑈 = 𝑓𝑀 (𝑓 supp 𝑍)    &   𝑆 = sup(𝑈, ℝ, < )       ((𝑀 ⊆ (𝑅m0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → ((∀𝑓𝑀 𝑓 finSupp 𝑍𝑈 ≠ ∅) → 𝑆 ∈ ℕ0))

Theoremfsuppmapnn0fiub 13352* 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 = 𝑓𝑀 (𝑓 supp 𝑍)    &   𝑆 = sup(𝑈, ℝ, < )       ((𝑀 ⊆ (𝑅m0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → ((∀𝑓𝑀 𝑓 finSupp 𝑍𝑈 ≠ ∅) → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑆)))

Theoremfsuppmapnn0fiubex 13353* 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 ⊆ (𝑅m0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚)))

Theoremfsuppmapnn0fiub0 13354* 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 ⊆ (𝑅m0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝑓𝑥) = 𝑍)))

Theoremsuppssfz 13355* 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.)
(𝜑𝑍𝑉)    &   (𝜑𝐹 ∈ (𝐵m0))    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍))       (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))

Theoremfsuppmapnn0ub 13356* 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 ∈ (𝑅m0) ∧ 𝑍𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹𝑥) = 𝑍)))

Theoremfsuppmapnn0fz 13357* 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 ∈ (𝑅m0) ∧ 𝑍𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 (𝐹 supp 𝑍) ⊆ (0...𝑚)))

Theoremmptnn0fsupp 13358* A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.)
(𝜑0𝑉)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐶𝐵)    &   (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥 / 𝑘𝐶 = 0 ))       (𝜑 → (𝑘 ∈ ℕ0𝐶) finSupp 0 )

Theoremmptnn0fsuppd 13359* A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 2-Dec-2019.) (Revised by AV, 23-Dec-2019.)
(𝜑0𝑉)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐶𝐵)    &   (𝑘 = 𝑥𝐶 = 𝐷)    &   (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝐷 = 0 ))       (𝜑 → (𝑘 ∈ ℕ0𝐶) finSupp 0 )

Theoremmptnn0fsuppr 13360* A finitely supported mapping from the nonnegative integers fulfills certain conditions. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 23-Dec-2019.)
(𝜑0𝑉)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐶𝐵)    &   (𝜑 → (𝑘 ∈ ℕ0𝐶) finSupp 0 )       (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥 / 𝑘𝐶 = 0 ))

Theoremf13idfv 13361 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:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ((𝐹‘0) ≠ (𝐹‘1) ∧ (𝐹‘0) ≠ (𝐹‘2) ∧ (𝐹‘1) ≠ (𝐹‘2))))

5.6.6  The infinite sequence builder "seq" - extension

Syntaxcseq 13362 Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹)

Definitiondf-seq 13363* Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as or 0) 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 13375 and seqp1 13377. 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 14898), by climdm 14900 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, (𝐹𝑀)⟩, 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 13308 through om2uzf1oi 13314, originally proved by Raph Levien for use with df-exp 13424 and later generalized for arbitrary recursive sequences. Definition df-sum 15032 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, (𝐹𝑀)⟩) “ ω)

Theoremseqex 13364 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
seq𝑀( + , 𝐹) ∈ V

Theoremseqeq1 13365 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

Theoremseqeq2 13366 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Theoremseqeq3 13367 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Theoremseqeq1d 13368 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Theoremseqeq2d 13369 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))

Theoremseqeq3d 13370 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵))

Theoremseqeq123d 13371 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝑀 = 𝑁)    &   (𝜑+ = 𝑄)    &   (𝜑𝐹 = 𝐺)       (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

Theoremnfseq 13372 Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝑀    &   𝑥 +    &   𝑥𝐹       𝑥seq𝑀( + , 𝐹)

Theoremseqval 13373* 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, (𝐹𝑀)⟩) ↾ ω)       seq𝑀( + , 𝐹) = ran 𝑅

Theoremseqfn 13374 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 (ℤ𝑀))

Theoremseq1 13375 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) = (𝐹𝑀))

Theoremseq1i 13376 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 30-Apr-2014.)
𝑀 ∈ ℤ    &   (𝜑 → (𝐹𝑀) = 𝐴)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝐴)

Theoremseqp1 13377 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
(𝑁 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))

Theoremseqexw 13378 Weak version of seqex 13364 that holds without ax-rep 5171. 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

Theoremseqp1d 13379 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 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   𝐾 = (𝑁 + 1)    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝐴)    &   (𝜑 → (𝐹𝐾) = 𝐵)       (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐴 + 𝐵))

Theoremseqp1iOLD 13380 Obsolete version of seqp1d 13379 as of 3-May-2024. Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑍 = (ℤ𝑀)    &   𝑁𝑍    &   𝐾 = (𝑁 + 1)    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝐴)    &   (𝜑 → (𝐹𝐾) = 𝐵)       (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐴 + 𝐵))

Theoremseqm1 13381 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)) + (𝐹𝑁)))

Theoremseqcl2 13382* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑 → (𝐹𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹𝑥) ∈ 𝐷)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)

Theoremseqf2 13383* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑 → (𝐹𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)       (𝜑 → seq𝑀( + , 𝐹):𝑍𝐶)

Theoremseqcl 13384* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆)

Theoremseqf 13385* Range of the recursive sequence builder (special case of seqf2 13383). (Contributed by Mario Carneiro, 24-Jun-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹):𝑍𝑆)

Theoremseqfveq2 13386* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))

Theoremseqfeq2 13387* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))    &   ((𝜑𝑘 ∈ (ℤ‘(𝐾 + 1))) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ𝐾)) = seq𝐾( + , 𝐺))

Theoremseqfveq 13388* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Theoremseqfeq 13389* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Theoremseqshft2 13390* Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ ℤ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))

Theoremseqres 13391 Restricting its characteristic function to (ℤ𝑀) does not affect the seq function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
(𝑀 ∈ ℤ → seq𝑀( + , (𝐹 ↾ (ℤ𝑀))) = seq𝑀( + , 𝐹))

Theoremserf 13392* An infinite series of complex terms is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)

Theoremserfre 13393* An infinite series of real numbers is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)       (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)

Theoremmonoord 13394* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))       (𝜑 → (𝐹𝑀) ≤ (𝐹𝑁))

Theoremmonoord2 13395* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))       (𝜑 → (𝐹𝑁) ≤ (𝐹𝑀))

Theoremsermono 13396* 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.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ ℝ)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹𝑥))       (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ≤ (seq𝑀( + , 𝐹)‘𝑁))

Theoremseqsplit 13397* Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑀 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (𝐾...𝑁)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))

Theoremseq1p 13398* Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((𝐹𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))

Theoremseqcaopr3 13399* Lemma for seqcaopr2 13400. (Contributed by Mario Carneiro, 25-Apr-2016.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))    &   ((𝜑𝑛 ∈ (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)))

Theoremseqcaopr2 13400* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

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