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Theorem dfseq3-2 9916
 Description: Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as or ) 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 9941, seq3-1 9938 and seq3p1 9945. 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 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence with values 1, 3/2, 7/4, 15/8,.., so that , 3/2, etc. In other words, transforms a sequence into an infinite series. Internally, the frec function generates as its values a set of ordered pairs starting at , 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. Eventually, this will be the definition of , replacing df-iseq 9914 and df-seq3 9915. (Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)
Assertion
Ref Expression
dfseq3-2 frec
Distinct variable groups:   , ,   ,,   ,,

Proof of Theorem dfseq3-2
StepHypRef Expression
1 df-seq3 9915 . 2
2 df-iseq 9914 . 2 frec
31, 2eqtri 2109 1 frec
 Colors of variables: wff set class Syntax hints:   wceq 1290  cvv 2620  cop 3453   crn 4453  cfv 5028  (class class class)co 5666   cmpt2 5668  freccfrec 6169  c1 7412   caddc 7414  cuz 9080   cseq4 9912   cseq 9913 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-iseq 9914  df-seq3 9915 This theorem is referenced by:  seq3val  9935
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