Definition df-fib 30459

Description: Define the Fibonacci sequence, where that each element is the sum of the two preceding ones, starting from 0 and 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)

Assertion

Ref	Expression
df-fib	⊢ Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))))

Detailed syntax breakdown of Definition df-fib

Step	Hyp	Ref	Expression
1		cfib 30458	. 2 class Fibci
2		cc0 9936	. . . 4 class 0
3		c1 9937	. . . 4 class 1
4	2, 3	cs2 13586	. . 3 class ⟨“01”⟩
5		vw	. . . 4 setvar 𝑤
6		cn0 11292	. . . . . 6 class ℕ0
7	6	cword 13291	. . . . 5 class Word ℕ0
8		chash 13117	. . . . . . 7 class #
9	8	ccnv 5113	. . . . . 6 class ◡#
10		c2 11070	. . . . . . 7 class 2
11		cuz 11687	. . . . . . 7 class ℤ≥
12	10, 11	cfv 5888	. . . . . 6 class (ℤ≥‘2)
13	9, 12	cima 5117	. . . . 5 class (◡# “ (ℤ≥‘2))
14	7, 13	cin 3573	. . . 4 class (Word ℕ0 ∩ (◡# “ (ℤ≥‘2)))
15	5	cv 1482	. . . . . . . 8 class 𝑤
16	15, 8	cfv 5888	. . . . . . 7 class (#‘𝑤)
17		cmin 10266	. . . . . . 7 class −
18	16, 10, 17	co 6650	. . . . . 6 class ((#‘𝑤) − 2)
19	18, 15	cfv 5888	. . . . 5 class (𝑤‘((#‘𝑤) − 2))
20	16, 3, 17	co 6650	. . . . . 6 class ((#‘𝑤) − 1)
21	20, 15	cfv 5888	. . . . 5 class (𝑤‘((#‘𝑤) − 1))
22		caddc 9939	. . . . 5 class +
23	19, 21, 22	co 6650	. . . 4 class ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))
24	5, 14, 23	cmpt 4729	. . 3 class (𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1))))
25		csseq 30445	. . 3 class seqstr
26	4, 24, 25	co 6650	. 2 class (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))))
27	1, 26	wceq 1483	1 wff Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))))

This definition is referenced by:  fib0  30461  fib1  30462  fibp1  30463