Definition df-iseq 10002

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 iseq1 10021 and iseqp1 10028. 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, ℚ) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹, ℚ)‘1) = 1, (seq1( + , 𝐹, ℚ)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹, ℚ) transforms a sequence 𝐹 into an infinite series.

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 Jim Kingdon, 29-May-2020.) Use df-seq3 10003 or dfseq3-2 10004 instead. (New usage is discouraged.)

Assertion

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

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

Detailed syntax breakdown of Definition df-iseq

Step	Hyp	Ref	Expression
1		c.pl	. . 3 class +
2		cS	. . 3 class 𝑆
3		cF	. . 3 class 𝐹
4		cM	. . 3 class 𝑀
5	1, 2, 3, 4	cseq4 10000	. 2 class seq𝑀( + , 𝐹, 𝑆)
6		vx	. . . . 5 setvar 𝑥
7		vy	. . . . 5 setvar 𝑦
8		cuz 9118	. . . . . 6 class ℤ≥
9	4, 8	cfv 5049	. . . . 5 class (ℤ≥‘𝑀)
10	6	cv 1295	. . . . . . 7 class 𝑥
11		c1 7448	. . . . . . 7 class 1
12		caddc 7450	. . . . . . 7 class +
13	10, 11, 12	co 5690	. . . . . 6 class (𝑥 + 1)
14	7	cv 1295	. . . . . . 7 class 𝑦
15	13, 3	cfv 5049	. . . . . . 7 class (𝐹‘(𝑥 + 1))
16	14, 15, 1	co 5690	. . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
17	13, 16	cop 3469	. . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
18	6, 7, 9, 2, 17	cmpt2 5692	. . . 4 class (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
19	4, 3	cfv 5049	. . . . 5 class (𝐹‘𝑀)
20	4, 19	cop 3469	. . . 4 class ⟨𝑀, (𝐹‘𝑀)⟩
21	18, 20	cfrec 6193	. . 3 class frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
22	21	crn 4468	. 2 class ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
23	5, 22	wceq 1296	1 wff seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)

This definition is referenced by:  dfseq3-2  10004  iseqeq3  10006  nfiseq  10014  iseqval  10017  iseqvalt  10019