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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrecuzrdgrclt 10401* The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. Similar to frecuzrdgrcl 10396 except that 𝑆 and 𝑇 need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
 
Theoremfrecuzrdgg 10402* Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating 𝑅 at a natural number gives an ordered pair whose first element is the mapping of that natural number via 𝐺. (Contributed by Jim Kingdon, 23-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑁 ∈ ω)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → (1st ‘(𝑅𝑁)) = (𝐺𝑁))
 
Theoremfrecuzrdgdomlem 10403* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → dom ran 𝑅 = (ℤ𝐶))
 
Theoremfrecuzrdgdom 10404* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑 → dom ran 𝑅 = (ℤ𝐶))
 
Theoremfrecuzrdgfunlem 10405* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → Fun ran 𝑅)
 
Theoremfrecuzrdgfun 10406* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑 → Fun ran 𝑅)
 
Theoremfrecuzrdgtclt 10407* The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑃 = ran 𝑅)       (𝜑𝑃:(ℤ𝐶)⟶𝑆)
 
Theoremfrecuzrdg0t 10408* Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑃 = ran 𝑅)       (𝜑 → (𝑃𝐶) = 𝐴)
 
Theoremfrecuzrdgsuctlem 10409* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10385 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 29-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑃 = ran 𝑅)       ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃𝐵)))
 
Theoremfrecuzrdgsuct 10410* Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑃 = ran 𝑅)       ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃𝐵)))
 
Theoremuzenom 10411 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → 𝑍 ≈ ω)
 
Theoremfrecfzennn 10412 The cardinality of a finite set of sequential integers. (See frec2uz0d 10385 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
 
Theoremfrecfzen2 10413 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) ≈ (𝐺‘((𝑁 + 1) − 𝑀)))
 
Theoremfrechashgf1o 10414 𝐺 maps ω one-to-one onto 0. (Contributed by Jim Kingdon, 19-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       𝐺:ω–1-1-onto→ℕ0
 
Theoremfrec2uzled 10415* The mapping 𝐺 (see frec2uz0d 10385) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝐴𝐵 ↔ (𝐺𝐴) ≤ (𝐺𝐵)))
 
Theoremfzfig 10416 A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin)
 
Theoremfzfigd 10417 Deduction form of fzfig 10416. (Contributed by Jim Kingdon, 21-May-2020.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝑀...𝑁) ∈ Fin)
 
Theoremfzofig 10418 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin)
 
Theoremnn0ennn 10419 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
0 ≈ ℕ
 
Theoremnnenom 10420 The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of natural numbers as ordinals). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
ℕ ≈ ω
 
Theoremnnct 10421 is dominated by ω. (Contributed by Thierry Arnoux, 29-Dec-2016.)
ℕ ≼ ω
 
Theoremuzennn 10422 An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝑀 ∈ ℤ → (ℤ𝑀) ≈ ℕ)
 
Theoremfnn0nninf 10423* A function from 0 into . (Contributed by Jim Kingdon, 16-Jul-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))       (𝐹𝐺):ℕ0⟶ℕ
 
Theoremfxnn0nninf 10424* A function from 0* into . (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7116 instead of infnninfOLD 7117. More generally, this theorem and most theorems in this section could use an extended 𝐺 defined by 𝐺 = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ ⟨ω, +∞⟩) and 𝐹 = (𝑛 ∈ suc ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) as in nnnninf2 7119.
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))    &   𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})       𝐼:ℕ0*⟶ℕ
 
Theorem0tonninf 10425* The mapping of zero into is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))    &   𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})       (𝐼‘0) = (𝑥 ∈ ω ↦ ∅)
 
Theorem1tonninf 10426* The mapping of one into is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))    &   𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})       (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
 
Theoreminftonninf 10427* The mapping of +∞ into is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))    &   𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})       (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o)
 
4.6.4  Strong induction over upper sets of integers
 
Theoremuzsinds 10428* 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 10429* 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 10430* 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)
 
4.6.5  The infinite sequence builder "seq"
 
Syntaxcseq 10431 Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹)
 
Definitiondf-seqfrec 10432* 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 seqf 10447, seq3-1 10446 and seq3p1 10448. 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 11285), by climdm 11287 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 frec 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 frec is exactly the sequence we want, and we just extract the range and throw away the domain.

(Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)

seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
 
Theoremseqex 10433 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
seq𝑀( + , 𝐹) ∈ V
 
Theoremseqeq1 10434 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
 
Theoremseqeq2 10435 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
 
Theoremseqeq3 10436 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
 
Theoremseqeq1d 10437 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
 
Theoremseqeq2d 10438 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
 
Theoremseqeq3d 10439 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵))
 
Theoremseqeq123d 10440 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 10441 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)
 
Theoremiseqovex 10442* Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)
 
Theoremiseqvalcbv 10443* Changing the bound variables in an expression which appears in some seq related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.)
frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹𝑀)⟩)
 
Theoremseq3val 10444* Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10447, seq3-1 10446 and seq3p1 10448, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
 
Theoremseqvalcd 10445* Value of the sequence builder function. Similar to seq3val 10444 but the classes 𝐷 (type of each term) and 𝐶 (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)    &   (𝜑 → (𝐹𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)       (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)
 
Theoremseq3-1 10446* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
 
Theoremseqf 10447* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹):𝑍𝑆)
 
Theoremseq3p1 10448* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
 
Theoremseqovcd 10449* A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10450 and seq1cd 10451 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)       ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝐶)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝐶)
 
Theoremseqf2 10450* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
(𝜑 → (𝐹𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)       (𝜑 → seq𝑀( + , 𝐹):𝑍𝐶)
 
Theoremseq1cd 10451* Initial value of the recursive sequence builder. A version of seq3-1 10446 which provides two classes 𝐷 and 𝐶 for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
(𝜑 → (𝐹𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
 
Theoremseqp1cd 10452* Value of the sequence builder function at a successor. A version of seq3p1 10448 which provides two classes 𝐷 and 𝐶 for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → (𝐹𝑀) ∈ 𝐶)    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝐹𝑥) ∈ 𝐷)       (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
 
Theoremseq3clss 10453* Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑇)    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥𝑇𝑦𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆)
 
Theoremseq3m1 10454* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹𝑁)))
 
Theoremseq3fveq2 10455* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))
 
Theoremseq3feq2 10456* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ‘(𝐾 + 1))) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ𝐾)) = seq𝐾( + , 𝐺))
 
Theoremseq3fveq 10457* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺𝑘))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 
Theoremseq3feq 10458* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = (𝐺𝑘))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
 
Theoremseq3shft2 10459* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ ℤ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
 
Theoremserf 10460* 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 10461* 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 10462* 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 10463* 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))
 
Theoremser3mono 10464* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ ℝ)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹𝑥))       (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ≤ (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremseq3split 10465* Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑀 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
 
Theoremseq3-1p 10466* Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((𝐹𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
 
Theoremseq3caopr3 10467* Lemma for seq3caopr2 10468. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (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)))
 
Theoremseq3caopr2 10468* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
 
Theoremseq3caopr 10469* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
 
Theoremiseqf1olemkle 10470* Lemma for seq3f1o 10490. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)       (𝜑𝐾 ≤ (𝐽𝐾))
 
Theoremiseqf1olemklt 10471* Lemma for seq3f1o 10490. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))       (𝜑𝐾 < (𝐽𝐾))
 
Theoremiseqf1olemqcl 10472 Lemma for seq3f1o 10490. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))       (𝜑 → if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)) ∈ (𝑀...𝑁))
 
Theoremiseqf1olemqval 10473* Lemma for seq3f1o 10490. Value of the function 𝑄. (Contributed by Jim Kingdon, 28-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑 → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))
 
Theoremiseqf1olemnab 10474* Lemma for seq3f1o 10490. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))
 
Theoremiseqf1olemab 10475* Lemma for seq3f1o 10490. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑𝐴 ∈ (𝐾...(𝐽𝐾)))    &   (𝜑𝐵 ∈ (𝐾...(𝐽𝐾)))       (𝜑𝐴 = 𝐵)
 
Theoremiseqf1olemnanb 10476* Lemma for seq3f1o 10490. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑 → ¬ 𝐴 ∈ (𝐾...(𝐽𝐾)))    &   (𝜑 → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))       (𝜑𝐴 = 𝐵)
 
Theoremiseqf1olemqf 10477* Lemma for seq3f1o 10490. Domain and codomain of 𝑄. (Contributed by Jim Kingdon, 26-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))
 
Theoremiseqf1olemmo 10478* Lemma for seq3f1o 10490. Showing that 𝑄 is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremiseqf1olemqf1o 10479* Lemma for seq3f1o 10490. 𝑄 is a permutation of (𝑀...𝑁). 𝑄 is formed from the constant portion of 𝐽, followed by the single element 𝐾 (at position 𝐾), followed by the rest of J (with the 𝐾 deleted and the elements before 𝐾 moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
 
Theoremiseqf1olemqk 10480* Lemma for seq3f1o 10490. 𝑄 is constant for one more position than 𝐽 is. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)       (𝜑 → ∀𝑥 ∈ (𝑀...𝐾)(𝑄𝑥) = 𝑥)
 
Theoremiseqf1olemjpcl 10481* Lemma for seq3f1o 10490. A closure lemma involving 𝐽 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐽 / 𝑓𝑃𝑥) ∈ 𝑆)
 
Theoremiseqf1olemqpcl 10482* Lemma for seq3f1o 10490. A closure lemma involving 𝑄 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝑄 / 𝑓𝑃𝑥) ∈ 𝑆)
 
Theoremiseqf1olemfvp 10483* Lemma for seq3f1o 10490. (Contributed by Jim Kingdon, 30-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝑇:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (𝑇 / 𝑓𝑃𝐴) = (𝐺‘(𝑇𝐴)))
 
Theoremseq3f1olemqsumkj 10484* Lemma for seq3f1o 10490. 𝑄 gives the same sum as 𝐽 in the range (𝐾...(𝐽𝐾)). (Contributed by Jim Kingdon, 29-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝐾( + , 𝐽 / 𝑓𝑃)‘(𝐽𝐾)) = (seq𝐾( + , 𝑄 / 𝑓𝑃)‘(𝐽𝐾)))
 
Theoremseq3f1olemqsumk 10485* Lemma for seq3f1o 10490. 𝑄 gives the same sum as 𝐽 in the range (𝐾...𝑁). (Contributed by Jim Kingdon, 22-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝐾( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝐾( + , 𝑄 / 𝑓𝑃)‘𝑁))
 
Theoremseq3f1olemqsum 10486* Lemma for seq3f1o 10490. 𝑄 gives the same sum as 𝐽. (Contributed by Jim Kingdon, 21-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝑄 / 𝑓𝑃)‘𝑁))
 
Theoremseq3f1olemstep 10487* Lemma for seq3f1o 10490. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
 
Theoremseq3f1olemp 10488* Lemma for seq3f1o 10490. Existence of a constant permutation of (𝑀...𝑁) which leads to the same sum as the permutation 𝐹 itself. (Contributed by Jim Kingdon, 18-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
 
Theoremseq3f1oleml 10489* Lemma for seq3f1o 10490. This is more or less the result, but stated in terms of 𝐹 and 𝐺 without 𝐻. 𝐿 and 𝐻 may differ in terms of what happens to terms after 𝑁. The terms after 𝑁 don't matter for the value at 𝑁 but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 
Theoremseq3f1o 10490* Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = (𝐺‘(𝐹𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 
Theoremser3add 10491* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 4-Oct-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
 
Theoremser3sub 10492* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) − (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) − (seq𝑀( + , 𝐺)‘𝑁)))
 
Theoremseq3id3 10493* 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.) (Revised by Jim Kingdon, 8-Apr-2023.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)    &   (𝜑𝑍𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
 
Theoremseq3id 10494* 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 Jim Kingdon, 8-Apr-2023.)
((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑥)    &   (𝜑𝑍𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → (𝐹𝑁) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑥) = 𝑍)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ𝑁)) = seq𝑁( + , 𝐹))
 
Theoremseq3id2 10495* 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 Jim Kingdon, 12-Nov-2022.)
((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)    &   (𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremseq3homo 10496* Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)       (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
 
Theoremseq3z 10497* 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.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)    &   ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → (𝐹𝐾) = 𝑍)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
 
Theoremseqfeq3 10498* 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.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))       (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
 
Theoremseq3distr 10499* The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) = (𝐶𝑇(𝐺𝑥)))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
 
Theoremser0 10500 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 = (ℤ𝑀)       (𝑁𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0)
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