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Definition df-seqfrec 10060
 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 10075, seq3-1 10074 and seq3p1 10076. 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 10901), by climdm 10903 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 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
Distinct variable groups:   𝑥, + ,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦

Detailed syntax breakdown of Definition df-seqfrec
StepHypRef Expression
1 c.pl . . 3 class +
2 cF . . 3 class 𝐹
3 cM . . 3 class 𝑀
41, 2, 3cseq 10059 . 2 class seq𝑀( + , 𝐹)
5 vx . . . . 5 setvar 𝑥
6 vy . . . . 5 setvar 𝑦
7 cuz 9176 . . . . . 6 class
83, 7cfv 5059 . . . . 5 class (ℤ𝑀)
9 cvv 2641 . . . . 5 class V
105cv 1298 . . . . . . 7 class 𝑥
11 c1 7501 . . . . . . 7 class 1
12 caddc 7503 . . . . . . 7 class +
1310, 11, 12co 5706 . . . . . 6 class (𝑥 + 1)
146cv 1298 . . . . . . 7 class 𝑦
1513, 2cfv 5059 . . . . . . 7 class (𝐹‘(𝑥 + 1))
1614, 15, 1co 5706 . . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
1713, 16cop 3477 . . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
185, 6, 8, 9, 17cmpo 5708 . . . 4 class (𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
193, 2cfv 5059 . . . . 5 class (𝐹𝑀)
203, 19cop 3477 . . . 4 class 𝑀, (𝐹𝑀)⟩
2118, 20cfrec 6217 . . 3 class frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2221crn 4478 . 2 class ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
234, 22wceq 1299 1 wff seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
 Colors of variables: wff set class This definition is referenced by:  seqex  10061  seqeq1  10062  seqeq2  10063  seqeq3  10064  nfseq  10069  seq3val  10072  seqvalcd  10073
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