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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmodirr 13401 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 13402* 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 13403 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 13404 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 13405 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 15770. (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
 
Theoremom2uz0i 13406* 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 13407* The value of 𝐺 (see om2uz0i 13406) 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 13408* The value 𝐺 (see om2uz0i 13406) 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 13409* Less-than relation for 𝐺 (see om2uz0i 13406). (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 13410* The mapping 𝐺 (see om2uz0i 13406) 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 13411* Range of 𝐺 (see om2uz0i 13406). (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 13412* 𝐺 (see om2uz0i 13406) 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 13413* 𝐺 (see om2uz0i 13406) 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 13414* An alternative definition of 𝐺 in terms of df-oi 9047. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝐶 ∈ ℤ    &   𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)       𝐺 = OrdIso( < , (ℤ𝐶))
 
Theoremom2uzrdg 13415* 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 13406. (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 13416* 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 13417* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13415. (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 13418* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 13415. (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 13419* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13415. (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 13420 < 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 13421 Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
< We ℕ
 
Theoremltwefz 13422 Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
< We (𝑀...𝑁)
 
Theoremuzenom 13423 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → 𝑍 ≈ ω)
 
Theoremuzinf 13424 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 13425 The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
¬ ℕ ∈ Fin
 
Theoremuzrdgxfr 13426* 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 13427 The cardinality of a finite set of sequential integers. (See om2uz0i 13406 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 13428 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 13429 The cardinality of a finite set of sequential integers. (See om2uz0i 13406 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 13430 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 13431 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 13432 Commonly used special case of fzfi 13431. (Contributed by Mario Carneiro, 25-May-2014.)
(𝜑 → (𝑀...𝑁) ∈ Fin)
 
Theoremfzofi 13433 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝑀..^𝑁) ∈ Fin
 
Theoremfsequb 13434* 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 13435* 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 13436 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 13437 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 13438 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
0 ≈ ℕ
 
Theoremnnenom 13439 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 13440 is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
ℕ ≼ ω
 
Theoremuzindi 13441* 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 13442* Lemma for axdc4uz 13443. (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 13443* A version of axdc4 9956 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 13444* 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 13445* 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 13446* 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 13447* 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 13448* 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 13449* Lemma for fsuppmapnn0fiub 13450 and fsuppmapnn0fiubex 13451. (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 13450* 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 13451* 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 13452* 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 13453* 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 13454* 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 13455* 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 13456* 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 13457* 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 13458* 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 13459 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 13460 Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹)
 
Definitiondf-seq 13461* 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 13473 and seqp1 13475. 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 14999), by climdm 15001 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 13406 through om2uzf1oi 13412, originally proved by Raph Levien for use with df-exp 13522 and later generalized for arbitrary recursive sequences. Definition df-sum 15136 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 13462 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
seq𝑀( + , 𝐹) ∈ V
 
Theoremseqeq1 13463 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
 
Theoremseqeq2 13464 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
 
Theoremseqeq3 13465 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
 
Theoremseqeq1d 13466 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
 
Theoremseqeq2d 13467 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
 
Theoremseqeq3d 13468 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵))
 
Theoremseqeq123d 13469 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 13470 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 13471* 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 13472 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 13473 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 13474 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 13475 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 13476 Weak version of seqex 13462 that holds without ax-rep 5154. 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 13477 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 13478 Obsolete version of seqp1d 13477 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 13479 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 13480* 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 13481* 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 13482* 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 13483* Range of the recursive sequence builder (special case of seqf2 13481). (Contributed by Mario Carneiro, 24-Jun-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹):𝑍𝑆)
 
Theoremseqfveq2 13484* 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 13485* 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 13486* 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 13487* 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 13488* 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 13489 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 13490* 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 13491* 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 13492* 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 13493* 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 13494* 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 13495* 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 13496* 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 13497* Lemma for seqcaopr2 13498. (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 13498* 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)))
 
Theoremseqcaopr 13499* 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 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
 
Theoremseqf1olem2a 13500* Lemma for seqf1o 13503. (Contributed by Mario Carneiro, 24-Apr-2016.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐶𝑆)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐾𝐴)    &   (𝜑 → (𝑀...𝑁) ⊆ 𝐴)       (𝜑 → ((𝐺𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺𝐾)))
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