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 NN or NN0) 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 F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq 1 ( + , F ) with values 1, 3/2, 7/4, 15/8,.., so that ( seq 1 ( + , F ) ` 1 ) = 1, ( seq 1 ( + , F ) ` 2 ) = 3/2, etc. In other words, seq M ( + , F ) 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.

Eventually, this will be the definition of seq, 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	|- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )

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

Proof of Theorem dfseq3-2

Step	Hyp	Ref	Expression
1		df-seq3 9915	. 2 |- seq M ( .+ , F ) = seq M ( .+ , F , _V )
2		df-iseq 9914	. 2 |- seq M ( .+ , F , _V ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
3	1, 2	eqtri 2109	1 |- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )

Syntax hints:   = wceq 1290   _Vcvv 2620   <.cop 3453   ran crn 4453   ` cfv 5028  (class class class)co 5666   |-> cmpt2 5668  freccfrec 6169   1c1 7412   + caddc 7414   ZZ>=cuz 9080   seqcseq4 9912   seqcseq 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