Theorem fibp1 30463

Description: Value of the Fibonacci sequence at higher indices. (Contributed by Thierry Arnoux, 25-Apr-2019.)

Assertion

Ref	Expression
fibp1	|- ( N e. NN -> (Fibci ` ( N + 1 ) ) = ( (Fibci ` ( N - 1 ) ) + (Fibci ` N ) ) )

Proof of Theorem fibp1

Dummy variables w t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fib 30459	. . . 4 |- Fibci = ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) )
2	1	fveq1i 6192	. . 3 |- (Fibci ` ( N + 1 ) ) = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) ` ( N + 1 ) )
3	2	a1i 11	. 2 |- ( N e. NN -> (Fibci ` ( N + 1 ) ) = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) ` ( N + 1 ) ) )
4		nn0ex 11298	. . . 4 |- NN0 e. _V
5	4	a1i 11	. . 3 |- ( N e. NN -> NN0 e. _V )
6		0nn0 11307	. . . . 5 |- 0 e. NN0
7	6	a1i 11	. . . 4 |- ( N e. NN -> 0 e. NN0 )
8		1nn0 11308	. . . . 5 |- 1 e. NN0
9	8	a1i 11	. . . 4 |- ( N e. NN -> 1 e. NN0 )
10	7, 9	s2cld 13616	. . 3 |- ( N e. NN -> <" 0 1 "> e. Word NN0 )
11		eqid 2622	. . 3 |- (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) = (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) )
12		fiblem 30460	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) : (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) --> NN0
13	12	a1i 11	. . 3 |- ( N e. NN -> ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) : (Word NN0 i^i ( `' # " ( ZZ>= ` ( # ` <" 0 1 "> ) ) ) ) --> NN0 )
14		eluzp1p1 11713	. . . . 5 |- ( N e. ( ZZ>= ` 1 ) -> ( N + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) )
15		nnuz 11723	. . . . 5 |- NN = ( ZZ>= ` 1 )
16	14, 15	eleq2s 2719	. . . 4 |- ( N e. NN -> ( N + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) )
17		s2len 13634	. . . . . 6 |- ( # ` <" 0 1 "> ) = 2
18		1p1e2 11134	. . . . . 6 |- ( 1 + 1 ) = 2
19	17, 18	eqtr4i 2647	. . . . 5 |- ( # ` <" 0 1 "> ) = ( 1 + 1 )
20	19	fveq2i 6194	. . . 4 |- ( ZZ>= ` ( # ` <" 0 1 "> ) ) = ( ZZ>= ` ( 1 + 1 ) )
21	16, 20	syl6eleqr 2712	. . 3 |- ( N e. NN -> ( N + 1 ) e. ( ZZ>= ` ( # ` <" 0 1 "> ) ) )
22	5, 10, 11, 13, 21	sseqp1 30457	. 2 |- ( N e. NN -> ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) ` ( N + 1 ) ) = ( ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ` ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) ) )
23		id 22	. . . . . . 7 |- ( w = t -> w = t )
24		fveq2 6191	. . . . . . . 8 |- ( w = t -> ( # ` w ) = ( # ` t ) )
25	24	oveq1d 6665	. . . . . . 7 |- ( w = t -> ( ( # ` w ) - 2 ) = ( ( # ` t ) - 2 ) )
26	23, 25	fveq12d 6197	. . . . . 6 |- ( w = t -> ( w ` ( ( # ` w ) - 2 ) ) = ( t ` ( ( # ` t ) - 2 ) ) )
27	24	oveq1d 6665	. . . . . . 7 |- ( w = t -> ( ( # ` w ) - 1 ) = ( ( # ` t ) - 1 ) )
28	23, 27	fveq12d 6197	. . . . . 6 |- ( w = t -> ( w ` ( ( # ` w ) - 1 ) ) = ( t ` ( ( # ` t ) - 1 ) ) )
29	26, 28	oveq12d 6668	. . . . 5 |- ( w = t -> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) = ( ( t ` ( ( # ` t ) - 2 ) ) + ( t ` ( ( # ` t ) - 1 ) ) ) )
30	29	cbvmptv 4750	. . . 4 |- ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) = ( t e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( t ` ( ( # ` t ) - 2 ) ) + ( t ` ( ( # ` t ) - 1 ) ) ) )
31	30	a1i 11	. . 3 |- ( N e. NN -> ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) = ( t e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( t ` ( ( # ` t ) - 2 ) ) + ( t ` ( ( # ` t ) - 1 ) ) ) ) )
32		simpr 477	. . . . 5 |- ( ( N e. NN /\ t = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) ) -> t = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) )
33	1	a1i 11	. . . . . 6 |- ( ( N e. NN /\ t = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) ) -> Fibci = ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) )
34	33	reseq1d 5395	. . . . 5 |- ( ( N e. NN /\ t = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) ) -> (Fibci |` ( 0..^ ( N + 1 ) ) ) = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) )
35	32, 34	eqtr4d 2659	. . . 4 |- ( ( N e. NN /\ t = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) ) -> t = (Fibci |` ( 0..^ ( N + 1 ) ) ) )
36		simpr 477	. . . . . . . . . . 11 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> t = (Fibci |` ( 0..^ ( N + 1 ) ) ) )
37	36	fveq2d 6195	. . . . . . . . . 10 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( # ` t ) = ( # ` (Fibci |` ( 0..^ ( N + 1 ) ) ) ) )
38	5, 10, 11, 13	sseqf 30454	. . . . . . . . . . . . 13 |- ( N e. NN -> ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) : NN0 --> NN0 )
39	1	a1i 11	. . . . . . . . . . . . . 14 |- ( N e. NN -> Fibci = ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) )
40	39	feq1d 6030	. . . . . . . . . . . . 13 |- ( N e. NN -> (Fibci : NN0 --> NN0 <-> ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) : NN0 --> NN0 ) )
41	38, 40	mpbird 247	. . . . . . . . . . . 12 |- ( N e. NN -> Fibci : NN0 --> NN0 )
42		nnnn0 11299	. . . . . . . . . . . . 13 |- ( N e. NN -> N e. NN0 )
43	42, 9	nn0addcld 11355	. . . . . . . . . . . 12 |- ( N e. NN -> ( N + 1 ) e. NN0 )
44	5, 41, 43	subiwrdlen 30448	. . . . . . . . . . 11 |- ( N e. NN -> ( # ` (Fibci |` ( 0..^ ( N + 1 ) ) ) ) = ( N + 1 ) )
45	44	adantr 481	. . . . . . . . . 10 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( # ` (Fibci |` ( 0..^ ( N + 1 ) ) ) ) = ( N + 1 ) )
46	37, 45	eqtrd 2656	. . . . . . . . 9 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( # ` t ) = ( N + 1 ) )
47	46	oveq1d 6665	. . . . . . . 8 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( # ` t ) - 2 ) = ( ( N + 1 ) - 2 ) )
48		nncn 11028	. . . . . . . . . . 11 |- ( N e. NN -> N e. CC )
49		1cnd 10056	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. CC )
50		2cnd 11093	. . . . . . . . . . 11 |- ( N e. NN -> 2 e. CC )
51	48, 49, 50	addsubassd 10412	. . . . . . . . . 10 |- ( N e. NN -> ( ( N + 1 ) - 2 ) = ( N + ( 1 - 2 ) ) )
52	48, 50, 49	subsub2d 10421	. . . . . . . . . 10 |- ( N e. NN -> ( N - ( 2 - 1 ) ) = ( N + ( 1 - 2 ) ) )
53		2m1e1 11135	. . . . . . . . . . . 12 |- ( 2 - 1 ) = 1
54	53	oveq2i 6661	. . . . . . . . . . 11 |- ( N - ( 2 - 1 ) ) = ( N - 1 )
55	54	a1i 11	. . . . . . . . . 10 |- ( N e. NN -> ( N - ( 2 - 1 ) ) = ( N - 1 ) )
56	51, 52, 55	3eqtr2d 2662	. . . . . . . . 9 |- ( N e. NN -> ( ( N + 1 ) - 2 ) = ( N - 1 ) )
57	56	adantr 481	. . . . . . . 8 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( N + 1 ) - 2 ) = ( N - 1 ) )
58	47, 57	eqtrd 2656	. . . . . . 7 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( # ` t ) - 2 ) = ( N - 1 ) )
59	58	fveq2d 6195	. . . . . 6 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( t ` ( ( # ` t ) - 2 ) ) = ( t ` ( N - 1 ) ) )
60	36	fveq1d 6193	. . . . . 6 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( t ` ( N - 1 ) ) = ( (Fibci |` ( 0..^ ( N + 1 ) ) ) ` ( N - 1 ) ) )
61		nnm1nn0 11334	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. NN0 )
62		peano2nn 11032	. . . . . . . . 9 |- ( N e. NN -> ( N + 1 ) e. NN )
63		nnre 11027	. . . . . . . . . . 11 |- ( N e. NN -> N e. RR )
64		2re 11090	. . . . . . . . . . . . 13 |- 2 e. RR
65	64	a1i 11	. . . . . . . . . . . 12 |- ( N e. NN -> 2 e. RR )
66	63, 65	readdcld 10069	. . . . . . . . . . 11 |- ( N e. NN -> ( N + 2 ) e. RR )
67		1red 10055	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. RR )
68		2rp 11837	. . . . . . . . . . . . 13 |- 2 e. RR+
69	68	a1i 11	. . . . . . . . . . . 12 |- ( N e. NN -> 2 e. RR+ )
70	63, 69	ltaddrpd 11905	. . . . . . . . . . 11 |- ( N e. NN -> N < ( N + 2 ) )
71	63, 66, 67, 70	ltsub1dd 10639	. . . . . . . . . 10 |- ( N e. NN -> ( N - 1 ) < ( ( N + 2 ) - 1 ) )
72	48, 50, 49	addsubassd 10412	. . . . . . . . . . 11 |- ( N e. NN -> ( ( N + 2 ) - 1 ) = ( N + ( 2 - 1 ) ) )
73	53	oveq2i 6661	. . . . . . . . . . 11 |- ( N + ( 2 - 1 ) ) = ( N + 1 )
74	72, 73	syl6eq 2672	. . . . . . . . . 10 |- ( N e. NN -> ( ( N + 2 ) - 1 ) = ( N + 1 ) )
75	71, 74	breqtrd 4679	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) < ( N + 1 ) )
76		elfzo0 12508	. . . . . . . . 9 |- ( ( N - 1 ) e. ( 0..^ ( N + 1 ) ) <-> ( ( N - 1 ) e. NN0 /\ ( N + 1 ) e. NN /\ ( N - 1 ) < ( N + 1 ) ) )
77	61, 62, 75, 76	syl3anbrc 1246	. . . . . . . 8 |- ( N e. NN -> ( N - 1 ) e. ( 0..^ ( N + 1 ) ) )
78	77	adantr 481	. . . . . . 7 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( N - 1 ) e. ( 0..^ ( N + 1 ) ) )
79		fvres 6207	. . . . . . 7 |- ( ( N - 1 ) e. ( 0..^ ( N + 1 ) ) -> ( (Fibci |` ( 0..^ ( N + 1 ) ) ) ` ( N - 1 ) ) = (Fibci ` ( N - 1 ) ) )
80	78, 79	syl 17	. . . . . 6 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( (Fibci |` ( 0..^ ( N + 1 ) ) ) ` ( N - 1 ) ) = (Fibci ` ( N - 1 ) ) )
81	59, 60, 80	3eqtrd 2660	. . . . 5 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( t ` ( ( # ` t ) - 2 ) ) = (Fibci ` ( N - 1 ) ) )
82	46	oveq1d 6665	. . . . . . . 8 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( # ` t ) - 1 ) = ( ( N + 1 ) - 1 ) )
83		simpl 473	. . . . . . . . . 10 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> N e. NN )
84	83	nncnd 11036	. . . . . . . . 9 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> N e. CC )
85		1cnd 10056	. . . . . . . . 9 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> 1 e. CC )
86	84, 85	pncand 10393	. . . . . . . 8 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( N + 1 ) - 1 ) = N )
87	82, 86	eqtrd 2656	. . . . . . 7 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( # ` t ) - 1 ) = N )
88	87	fveq2d 6195	. . . . . 6 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( t ` ( ( # ` t ) - 1 ) ) = ( t ` N ) )
89	36	fveq1d 6193	. . . . . 6 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( t ` N ) = ( (Fibci |` ( 0..^ ( N + 1 ) ) ) ` N ) )
90		nn0fz0 12437	. . . . . . . . . 10 |- ( N e. NN0 <-> N e. ( 0 ... N ) )
91	42, 90	sylib 208	. . . . . . . . 9 |- ( N e. NN -> N e. ( 0 ... N ) )
92		nnz 11399	. . . . . . . . . 10 |- ( N e. NN -> N e. ZZ )
93		fzval3 12536	. . . . . . . . . 10 |- ( N e. ZZ -> ( 0 ... N ) = ( 0..^ ( N + 1 ) ) )
94	92, 93	syl 17	. . . . . . . . 9 |- ( N e. NN -> ( 0 ... N ) = ( 0..^ ( N + 1 ) ) )
95	91, 94	eleqtrd 2703	. . . . . . . 8 |- ( N e. NN -> N e. ( 0..^ ( N + 1 ) ) )
96	95	adantr 481	. . . . . . 7 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> N e. ( 0..^ ( N + 1 ) ) )
97		fvres 6207	. . . . . . 7 |- ( N e. ( 0..^ ( N + 1 ) ) -> ( (Fibci |` ( 0..^ ( N + 1 ) ) ) ` N ) = (Fibci ` N ) )
98	96, 97	syl 17	. . . . . 6 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( (Fibci |` ( 0..^ ( N + 1 ) ) ) ` N ) = (Fibci ` N ) )
99	88, 89, 98	3eqtrd 2660	. . . . 5 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( t ` ( ( # ` t ) - 1 ) ) = (Fibci ` N ) )
100	81, 99	oveq12d 6668	. . . 4 |- ( ( N e. NN /\ t = (Fibci |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( t ` ( ( # ` t ) - 2 ) ) + ( t ` ( ( # ` t ) - 1 ) ) ) = ( (Fibci ` ( N - 1 ) ) + (Fibci ` N ) ) )
101	35, 100	syldan 487	. . 3 |- ( ( N e. NN /\ t = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) ) -> ( ( t ` ( ( # ` t ) - 2 ) ) + ( t ` ( ( # ` t ) - 1 ) ) ) = ( (Fibci ` ( N - 1 ) ) + (Fibci ` N ) ) )
102	39	reseq1d 5395	. . . 4 |- ( N e. NN -> (Fibci |` ( 0..^ ( N + 1 ) ) ) = ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) )
103	5, 41, 43	subiwrd 30447	. . . . 5 |- ( N e. NN -> (Fibci |` ( 0..^ ( N + 1 ) ) ) e. Word NN0 )
104		ovex 6678	. . . . . . . . 9 |- ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) e. _V
105	1, 104	eqeltri 2697	. . . . . . . 8 |- Fibci e. _V
106	105	resex 5443	. . . . . . 7 |- (Fibci |` ( 0..^ ( N + 1 ) ) ) e. _V
107	106	a1i 11	. . . . . 6 |- ( N e. NN -> (Fibci |` ( 0..^ ( N + 1 ) ) ) e. _V )
108	18	fveq2i 6194	. . . . . . . 8 |- ( ZZ>= ` ( 1 + 1 ) ) = ( ZZ>= ` 2 )
109	16, 108	syl6eleq 2711	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. ( ZZ>= ` 2 ) )
110	44, 109	eqeltrd 2701	. . . . . 6 |- ( N e. NN -> ( # ` (Fibci |` ( 0..^ ( N + 1 ) ) ) ) e. ( ZZ>= ` 2 ) )
111		hashf 13125	. . . . . . 7 |- # : _V --> ( NN0 u. { +oo } )
112		ffn 6045	. . . . . . 7 |- ( # : _V --> ( NN0 u. { +oo } ) -> # Fn _V )
113		elpreima 6337	. . . . . . 7 |- ( # Fn _V -> ( (Fibci |` ( 0..^ ( N + 1 ) ) ) e. ( `' # " ( ZZ>= ` 2 ) ) <-> ( (Fibci |` ( 0..^ ( N + 1 ) ) ) e. _V /\ ( # ` (Fibci |` ( 0..^ ( N + 1 ) ) ) ) e. ( ZZ>= ` 2 ) ) ) )
114	111, 112, 113	mp2b 10	. . . . . 6 |- ( (Fibci |` ( 0..^ ( N + 1 ) ) ) e. ( `' # " ( ZZ>= ` 2 ) ) <-> ( (Fibci |` ( 0..^ ( N + 1 ) ) ) e. _V /\ ( # ` (Fibci |` ( 0..^ ( N + 1 ) ) ) ) e. ( ZZ>= ` 2 ) ) )
115	107, 110, 114	sylanbrc 698	. . . . 5 |- ( N e. NN -> (Fibci |` ( 0..^ ( N + 1 ) ) ) e. ( `' # " ( ZZ>= ` 2 ) ) )
116	103, 115	elind 3798	. . . 4 |- ( N e. NN -> (Fibci |` ( 0..^ ( N + 1 ) ) ) e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) )
117	102, 116	eqeltrrd 2702	. . 3 |- ( N e. NN -> ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) )
118		ovex 6678	. . . 4 |- ( (Fibci ` ( N - 1 ) ) + (Fibci ` N ) ) e. _V
119	118	a1i 11	. . 3 |- ( N e. NN -> ( (Fibci ` ( N - 1 ) ) + (Fibci ` N ) ) e. _V )
120	31, 101, 117, 119	fvmptd 6288	. 2 |- ( N e. NN -> ( ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ` ( ( <" 0 1 ">seqstr ( w e. (Word NN0 i^i ( `' # " ( ZZ>= ` 2 ) ) ) |-> ( ( w ` ( ( # ` w ) - 2 ) ) + ( w ` ( ( # ` w ) - 1 ) ) ) ) ) |` ( 0..^ ( N + 1 ) ) ) ) = ( (Fibci ` ( N - 1 ) ) + (Fibci ` N ) ) )
121	3, 22, 120	3eqtrd 2660	1 |- ( N e. NN -> (Fibci ` ( N + 1 ) ) = ( (Fibci ` ( N - 1 ) ) + (Fibci ` N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   < clt 10074   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs2 13586  seq_strcsseq 30445  Fibcicfib 30458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-s2 13593  df-sseq 30446  df-fib 30459

This theorem is referenced by:  fib2  30464  fib3  30465  fib4  30466  fib5  30467  fib6  30468