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Definition df-iseq 9212
 Description: Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers ℕ or some other upper integer set) 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 9222 and iseqp1 9225. 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.)
Assertion
Ref Expression
df-iseq seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
Distinct variable groups:   𝑥, + ,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦

Detailed syntax breakdown of Definition df-iseq
StepHypRef Expression
1 c.pl . . 3 class +
2 cS . . 3 class 𝑆
3 cF . . 3 class 𝐹
4 cM . . 3 class 𝑀
51, 2, 3, 4cseq 9211 . 2 class seq𝑀( + , 𝐹, 𝑆)
6 vx . . . . 5 setvar 𝑥
7 vy . . . . 5 setvar 𝑦
8 cuz 8473 . . . . . 6 class
94, 8cfv 4902 . . . . 5 class (ℤ𝑀)
106cv 1242 . . . . . . 7 class 𝑥
11 c1 6890 . . . . . . 7 class 1
12 caddc 6892 . . . . . . 7 class +
1310, 11, 12co 5512 . . . . . 6 class (𝑥 + 1)
147cv 1242 . . . . . . 7 class 𝑦
1513, 3cfv 4902 . . . . . . 7 class (𝐹‘(𝑥 + 1))
1614, 15, 1co 5512 . . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
1713, 16cop 3378 . . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
186, 7, 9, 2, 17cmpt2 5514 . . . 4 class (𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
194, 3cfv 4902 . . . . 5 class (𝐹𝑀)
204, 19cop 3378 . . . 4 class 𝑀, (𝐹𝑀)⟩
2118, 20cfrec 5977 . . 3 class frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2221crn 4346 . 2 class ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
235, 22wceq 1243 1 wff seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
 Colors of variables: wff set class This definition is referenced by:  iseqex  9213  iseqeq1  9214  iseqeq2  9215  iseqeq3  9216  iseqeq4  9217  nfiseq  9218  iseqval  9220
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