Definition df-seqfrec 10817

Description: 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 seqf 10833, seq3-1 10831 and seq3p1 10834. 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 11986), by climdm 11988 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, (𝐹‘𝑀)⟩, 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.)

Assertion

Ref	Expression
df-seqfrec	⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)

Distinct variable groups:   𝑥, + ,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦

Detailed syntax breakdown of Definition df-seqfrec

Step	Hyp	Ref	Expression
1		c.pl	. . 3 class +
2		cF	. . 3 class 𝐹
3		cM	. . 3 class 𝑀
4	1, 2, 3	cseq 10816	. 2 class seq𝑀( + , 𝐹)
5		vx	. . . . 5 setvar 𝑥
6		vy	. . . . 5 setvar 𝑦
7		cuz 9859	. . . . . 6 class ℤ≥
8	3, 7	cfv 5354	. . . . 5 class (ℤ≥‘𝑀)
9		cvv 2815	. . . . 5 class V
10	5	cv 1397	. . . . . . 7 class 𝑥
11		c1 8133	. . . . . . 7 class 1
12		caddc 8135	. . . . . . 7 class +
13	10, 11, 12	co 6052	. . . . . 6 class (𝑥 + 1)
14	6	cv 1397	. . . . . . 7 class 𝑦
15	13, 2	cfv 5354	. . . . . . 7 class (𝐹‘(𝑥 + 1))
16	14, 15, 1	co 6052	. . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
17	13, 16	cop 3694	. . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
18	5, 6, 8, 9, 17	cmpo 6054	. . . 4 class (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
19	3, 2	cfv 5354	. . . . 5 class (𝐹‘𝑀)
20	3, 19	cop 3694	. . . 4 class ⟨𝑀, (𝐹‘𝑀)⟩
21	18, 20	cfrec 6623	. . 3 class frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
22	21	crn 4752	. 2 class ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
23	4, 22	wceq 1398	1 wff seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)

This definition is referenced by:  seqex  10818  seqeq1  10819  seqeq2  10820  seqeq3  10821  nfseq  10826  seq3val  10829  seqvalcd  10830