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Theorem fibp1 29060
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.
StepHypRef Expression
1 df-fib 29056 . . . 4  |- Fibci  =  (
<" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) )
21fveq1i 5882 . . 3  |-  (Fibci `  ( N  +  1
) )  =  ( ( <" 0
1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) ) `
 ( N  + 
1 ) )
32a1i 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 10875 . . . 4  |-  NN0  e.  _V
54a1i 11 . . 3  |-  ( N  e.  NN  ->  NN0  e.  _V )
6 0nn0 10884 . . . . 5  |-  0  e.  NN0
76a1i 11 . . . 4  |-  ( N  e.  NN  ->  0  e.  NN0 )
8 1nn0 10885 . . . . 5  |-  1  e.  NN0
98a1i 11 . . . 4  |-  ( N  e.  NN  ->  1  e.  NN0 )
107, 9s2cld 12950 . . 3  |-  ( N  e.  NN  ->  <" 0
1 ">  e. Word  NN0 )
11 eqid 2429 . . 3  |-  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) )  =  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )
12 fiblem 29057 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= ` 
2 ) ) ) 
|->  ( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) ) ) : (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) ) --> NN0
1312a1i 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 11184 . . . . 5  |-  ( N  e.  ( ZZ>= `  1
)  ->  ( N  +  1 )  e.  ( ZZ>= `  ( 1  +  1 ) ) )
15 nnuz 11194 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
1614, 15eleq2s 2537 . . . 4  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  (
1  +  1 ) ) )
17 s2len 12968 . . . . . 6  |-  ( # `  <" 0 1 "> )  =  2
18 1p1e2 10723 . . . . . 6  |-  ( 1  +  1 )  =  2
1917, 18eqtr4i 2461 . . . . 5  |-  ( # `  <" 0 1 "> )  =  ( 1  +  1 )
2019fveq2i 5884 . . . 4  |-  ( ZZ>= `  ( # `  <" 0
1 "> )
)  =  ( ZZ>= `  ( 1  +  1 ) )
2116, 20syl6eleqr 2528 . . 3  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  ( # `
 <" 0 1 "> ) ) )
225, 10, 11, 13, 21sseqp1 29054 . 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 23 . . . . . . 7  |-  ( w  =  t  ->  w  =  t )
24 fveq2 5881 . . . . . . . 8  |-  ( w  =  t  ->  ( # `
 w )  =  ( # `  t
) )
2524oveq1d 6320 . . . . . . 7  |-  ( w  =  t  ->  (
( # `  w )  -  2 )  =  ( ( # `  t
)  -  2 ) )
2623, 25fveq12d 5887 . . . . . 6  |-  ( w  =  t  ->  (
w `  ( ( # `
 w )  - 
2 ) )  =  ( t `  (
( # `  t )  -  2 ) ) )
2724oveq1d 6320 . . . . . . 7  |-  ( w  =  t  ->  (
( # `  w )  -  1 )  =  ( ( # `  t
)  -  1 ) )
2823, 27fveq12d 5887 . . . . . 6  |-  ( w  =  t  ->  (
w `  ( ( # `
 w )  - 
1 ) )  =  ( t `  (
( # `  t )  -  1 ) ) )
2926, 28oveq12d 6323 . . . . 5  |-  ( w  =  t  ->  (
( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )  =  ( ( t `  (
( # `  t )  -  2 ) )  +  ( t `  ( ( # `  t
)  -  1 ) ) ) )
3029cbvmptv 4518 . . . 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 ) ) ) )
3130a1i 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 462 . . . . 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 ) ) ) )
331a1i 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 ) ) ) ) ) )
3433reseq1d 5124 . . . . 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 ) ) ) )
3532, 34eqtr4d 2473 . . . 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 462 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  t  =  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) )
3736fveq2d 5885 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  t
)  =  ( # `  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) ) )
385, 10, 11, 13sseqf 29051 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( <" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) ) : NN0 --> NN0 )
391a1i 11 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  -> Fibci  =  (
<" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) ) )
4039feq1d 5732 . . . . . . . . . . . . 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 )
)
4138, 40mpbird 235 . . . . . . . . . . . 12  |-  ( N  e.  NN  -> Fibci : NN0 --> NN0 )
42 nnnn0 10876 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  NN0 )
4342, 9nn0addcld 10929 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN0 )
445, 41, 43subiwrdlen 29045 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( # `
 (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  =  ( N  +  1 ) )
4544adantr 466 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) )  =  ( N  +  1 ) )
4637, 45eqtrd 2470 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  t
)  =  ( N  +  1 ) )
4746oveq1d 6320 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
2 )  =  ( ( N  +  1 )  -  2 ) )
48 nncn 10617 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
49 1cnd 9658 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
50 2cnd 10682 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  2  e.  CC )
5148, 49, 50addsubassd 10005 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  2 )  =  ( N  +  ( 1  -  2 ) ) )
5248, 50, 49subsub2d 10014 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  ( 2  -  1 ) )  =  ( N  +  ( 1  -  2 ) ) )
53 2m1e1 10724 . . . . . . . . . . . 12  |-  ( 2  -  1 )  =  1
5453oveq2i 6316 . . . . . . . . . . 11  |-  ( N  -  ( 2  -  1 ) )  =  ( N  -  1 )
5554a1i 11 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  ( 2  -  1 ) )  =  ( N  - 
1 ) )
5651, 52, 553eqtr2d 2476 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  2 )  =  ( N  - 
1 ) )
5756adantr 466 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( N  +  1 )  -  2 )  =  ( N  -  1 ) )
5847, 57eqtrd 2470 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
2 )  =  ( N  -  1 ) )
5958fveq2d 5885 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  2 ) )  =  ( t `
 ( N  - 
1 ) ) )
6036fveq1d 5883 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( N  -  1 ) )  =  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  ( N  -  1 ) ) )
61 nnm1nn0 10911 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
62 peano2nn 10621 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
63 nnre 10616 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  RR )
64 2re 10679 . . . . . . . . . . . . 13  |-  2  e.  RR
6564a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  2  e.  RR )
6663, 65readdcld 9669 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  RR )
67 1red 9657 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  RR )
68 2rp 11307 . . . . . . . . . . . . 13  |-  2  e.  RR+
6968a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  2  e.  RR+ )
7063, 69ltaddrpd 11371 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  <  ( N  +  2 ) )
7163, 66, 67, 70ltsub1dd 10224 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  1 )  <  ( ( N  +  2 )  - 
1 ) )
7248, 50, 49addsubassd 10005 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
7353oveq2i 6316 . . . . . . . . . . 11  |-  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 )
7472, 73syl6eq 2486 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  1 )  =  ( N  + 
1 ) )
7571, 74breqtrd 4450 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  <  ( N  + 
1 ) )
76 elfzo0 11954 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) )  <->  ( ( N  -  1 )  e. 
NN0  /\  ( N  +  1 )  e.  NN  /\  ( N  -  1 )  < 
( N  +  1 ) ) )
7761, 62, 75, 76syl3anbrc 1189 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) ) )
7877adantr 466 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) ) )
79 fvres 5895 . . . . . . 7  |-  ( ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  ( N  -  1 ) )  =  (Fibci `  ( N  -  1 ) ) )
8078, 79syl 17 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  ( N  -  1 ) )  =  (Fibci `  ( N  -  1 ) ) )
8159, 60, 803eqtrd 2474 . . . . 5  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  2 ) )  =  (Fibci `  ( N  -  1
) ) )
8246oveq1d 6320 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
1 )  =  ( ( N  +  1 )  -  1 ) )
83 simpl 458 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  N  e.  NN )
8483nncnd 10625 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  N  e.  CC )
85 1cnd 9658 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  1  e.  CC )
8684, 85pncand 9986 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
8782, 86eqtrd 2470 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
1 )  =  N )
8887fveq2d 5885 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  1 ) )  =  ( t `
 N ) )
8936fveq1d 5883 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  N )  =  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  N
) )
90 nn0fz0 11888 . . . . . . . . . 10  |-  ( N  e.  NN0  <->  N  e.  (
0 ... N ) )
9142, 90sylib 199 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ( 0 ... N
) )
92 nnz 10959 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
93 fzval3 11980 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 ... N )  =  ( 0..^ ( N  +  1 ) ) )
9492, 93syl 17 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
0 ... N )  =  ( 0..^ ( N  +  1 ) ) )
9591, 94eleqtrd 2519 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ( 0..^ ( N  +  1 ) ) )
9695adantr 466 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  N  e.  ( 0..^ ( N  + 
1 ) ) )
97 fvres 5895 . . . . . . 7  |-  ( N  e.  ( 0..^ ( N  +  1 ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  N )  =  (Fibci `  N
) )
9896, 97syl 17 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  N )  =  (Fibci `  N
) )
9988, 89, 983eqtrd 2474 . . . . 5  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  1 ) )  =  (Fibci `  N ) )
10081, 99oveq12d 6323 . . . 4  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( (
t `  ( ( # `
 t )  - 
2 ) )  +  ( t `  (
( # `  t )  -  1 ) ) )  =  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N ) ) )
10135, 100syldan 472 . . 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
) ) )
10239reseq1d 5124 . . . 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 ) ) ) )
1035, 41, 43subiwrd 29044 . . . . 5  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e. Word  NN0 )
104 ovex 6333 . . . . . . . . 9  |-  ( <" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) )  e.  _V
1051, 104eqeltri 2513 . . . . . . . 8  |- Fibci  e.  _V
106105resex 5168 . . . . . . 7  |-  (Fibci  |`  (
0..^ ( N  + 
1 ) ) )  e.  _V
107106a1i 11 . . . . . 6  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V )
10818fveq2i 5884 . . . . . . . 8  |-  ( ZZ>= `  ( 1  +  1 ) )  =  (
ZZ>= `  2 )
10916, 108syl6eleq 2527 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  2
) )
11044, 109eqeltrd 2517 . . . . . 6  |-  ( N  e.  NN  ->  ( # `
 (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  e.  ( ZZ>= ` 
2 ) )
111 hashf 12519 . . . . . . 7  |-  # : _V
--> ( NN0  u.  { +oo } )
112 ffn 5746 . . . . . . 7  |-  ( # : _V --> ( NN0  u.  { +oo } )  ->  #  Fn  _V )
113 elpreima 6017 . . . . . . 7  |-  ( #  Fn  _V  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  ( `' # " ( ZZ>= `  2
) )  <->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V  /\  ( # `  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  e.  ( ZZ>= ` 
2 ) ) ) )
114111, 112, 113mp2b 10 . . . . . 6  |-  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  ( `' # " ( ZZ>= `  2
) )  <->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V  /\  ( # `  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  e.  ( ZZ>= ` 
2 ) ) )
115107, 110, 114sylanbrc 668 . . . . 5  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  ( `' # " ( ZZ>= `  2
) ) )
116103, 115elind 3656 . . . 4  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  2 ) ) ) )
117102, 116eqeltrrd 2518 . . 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 6333 . . . 4  |-  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N ) )  e. 
_V
119118a1i 11 . . 3  |-  ( N  e.  NN  ->  (
(Fibci `  ( N  -  1 ) )  +  (Fibci `  N
) )  e.  _V )
12031, 101, 117, 119fvmptd 5970 . 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
) ) )
1213, 22, 1203eqtrd 2474 1  |-  ( N  e.  NN  ->  (Fibci `  ( N  +  1 ) )  =  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    u. cun 3440    i^i cin 3441   {csn 4002   class class class wbr 4426    |-> cmpt 4484   `'ccnv 4853    |` cres 4856   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541   +oocpnf 9671    < clt 9674    - cmin 9859   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   ...cfz 11782  ..^cfzo 11913   #chash 12512  Word cword 12643   <"cs2 12922  seqstrcsseq 29042  Fibcicfib 29055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-word 12651  df-lsw 12652  df-concat 12653  df-s1 12654  df-substr 12655  df-s2 12929  df-sseq 29043  df-fib 29056
This theorem is referenced by:  fib2  29061  fib3  29062  fib4  29063  fib5  29064  fib6  29065
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