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Definition df-seq 11279
 Description: Define a general-purpose operation that builds a recursive sequence (i.e. a function on the natural numbers 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 seq1 11291 and seqp1 11293. 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. means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 12301), by climdm 12303 the "sum of F(n) from n = 1 to infinity" can be expressed as (provided the sequence converges) and evaluates to 2 in this example. Internally, the 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 is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain. This definition has its roots in a series of theorems from om2uz0i 11242 through om2uzf1oi 11248, originally proved by Raph Levien for use with df-exp 11338 and later generalized for arbitrary recursive sequences. Definition df-sum 12435 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
df-seq
Distinct variable groups:   , ,   ,,   ,,

Detailed syntax breakdown of Definition df-seq
StepHypRef Expression
1 c.pl . . 3
2 cF . . 3
3 cM . . 3
41, 2, 3cseq 11278 . 2
5 vx . . . . 5
6 vy . . . . 5
7 cvv 2916 . . . . 5
85cv 1648 . . . . . . 7
9 c1 8947 . . . . . . 7
10 caddc 8949 . . . . . . 7
118, 9, 10co 6040 . . . . . 6
126cv 1648 . . . . . . 7
1311, 2cfv 5413 . . . . . . 7
1412, 13, 1co 6040 . . . . . 6
1511, 14cop 3777 . . . . 5
165, 6, 7, 7, 15cmpt2 6042 . . . 4
173, 2cfv 5413 . . . . 5
183, 17cop 3777 . . . 4
1916, 18crdg 6626 . . 3
20 com 4804 . . 3
2119, 20cima 4840 . 2
224, 21wceq 1649 1
 Colors of variables: wff set class This definition is referenced by:  seqex  11280  seqeq1  11281  seqeq2  11282  seqeq3  11283  nfseq  11288  seqval  11289
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