Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fibp1 Structured version   Unicode version

Theorem fibp1 26782
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 26778 . . . 4  |- Fibci  =  (
<" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) )
21fveq1i 5690 . . 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 10583 . . . 4  |-  NN0  e.  _V
54a1i 11 . . 3  |-  ( N  e.  NN  ->  NN0  e.  _V )
6 0nn0 10592 . . . . 5  |-  0  e.  NN0
76a1i 11 . . . 4  |-  ( N  e.  NN  ->  0  e.  NN0 )
8 1nn0 10593 . . . . 5  |-  1  e.  NN0
98a1i 11 . . . 4  |-  ( N  e.  NN  ->  1  e.  NN0 )
107, 9s2cld 12494 . . 3  |-  ( N  e.  NN  ->  <" 0
1 ">  e. Word  NN0 )
11 eqid 2441 . . 3  |-  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) )  =  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )
12 fiblem 26779 . . . 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 10884 . . . . 5  |-  ( N  e.  ( ZZ>= `  1
)  ->  ( N  +  1 )  e.  ( ZZ>= `  ( 1  +  1 ) ) )
15 nnuz 10894 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
1614, 15eleq2s 2533 . . . 4  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  (
1  +  1 ) ) )
17 s2len 12512 . . . . . 6  |-  ( # `  <" 0 1 "> )  =  2
18 1p1e2 10433 . . . . . 6  |-  ( 1  +  1 )  =  2
1917, 18eqtr4i 2464 . . . . 5  |-  ( # `  <" 0 1 "> )  =  ( 1  +  1 )
2019fveq2i 5692 . . . 4  |-  ( ZZ>= `  ( # `  <" 0
1 "> )
)  =  ( ZZ>= `  ( 1  +  1 ) )
2116, 20syl6eleqr 2532 . . 3  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  ( # `
 <" 0 1 "> ) ) )
225, 10, 11, 13, 21sseqp1 26776 . 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 5689 . . . . . . . 8  |-  ( w  =  t  ->  ( # `
 w )  =  ( # `  t
) )
2524oveq1d 6104 . . . . . . 7  |-  ( w  =  t  ->  (
( # `  w )  -  2 )  =  ( ( # `  t
)  -  2 ) )
2623, 25fveq12d 5695 . . . . . 6  |-  ( w  =  t  ->  (
w `  ( ( # `
 w )  - 
2 ) )  =  ( t `  (
( # `  t )  -  2 ) ) )
2724oveq1d 6104 . . . . . . 7  |-  ( w  =  t  ->  (
( # `  w )  -  1 )  =  ( ( # `  t
)  -  1 ) )
2823, 27fveq12d 5695 . . . . . 6  |-  ( w  =  t  ->  (
w `  ( ( # `
 w )  - 
1 ) )  =  ( t `  (
( # `  t )  -  1 ) ) )
2926, 28oveq12d 6107 . . . . 5  |-  ( w  =  t  ->  (
( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )  =  ( ( t `  (
( # `  t )  -  2 ) )  +  ( t `  ( ( # `  t
)  -  1 ) ) ) )
3029cbvmptv 4381 . . . 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 simpl 457 . . . 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 ) ) ) )  ->  N  e.  NN )
33 simpr 461 . . . . 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 ) ) ) )
341a1i 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 ) ) ) ) ) )
3534reseq1d 5107 . . . . 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 ) ) ) )
3633, 35eqtr4d 2476 . . . 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 ) ) ) )
37 simpr 461 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  t  =  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) )
3837fveq2d 5693 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  t
)  =  ( # `  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) ) )
395, 10, 11, 13sseqf 26773 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( <" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) ) : NN0 --> NN0 )
401a1i 11 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  -> Fibci  =  (
<" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) ) )
4140feq1d 5544 . . . . . . . . . . . . 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 )
)
4239, 41mpbird 232 . . . . . . . . . . . 12  |-  ( N  e.  NN  -> Fibci : NN0 --> NN0 )
43 nnnn0 10584 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  NN0 )
4443, 9nn0addcld 10638 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN0 )
455, 42, 44subiwrdlen 26767 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( # `
 (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  =  ( N  +  1 ) )
4645adantr 465 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) )  =  ( N  +  1 ) )
4738, 46eqtrd 2473 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  t
)  =  ( N  +  1 ) )
4847oveq1d 6104 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
2 )  =  ( ( N  +  1 )  -  2 ) )
49 nncn 10328 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
50 1cnd 9400 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
51 2cnd 10392 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  2  e.  CC )
5249, 50, 51addsubassd 9737 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  2 )  =  ( N  +  ( 1  -  2 ) ) )
5349, 51, 50subsub2d 9746 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  ( 2  -  1 ) )  =  ( N  +  ( 1  -  2 ) ) )
54 2m1e1 10434 . . . . . . . . . . . 12  |-  ( 2  -  1 )  =  1
5554oveq2i 6100 . . . . . . . . . . 11  |-  ( N  -  ( 2  -  1 ) )  =  ( N  -  1 )
5655a1i 11 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  ( 2  -  1 ) )  =  ( N  - 
1 ) )
5752, 53, 563eqtr2d 2479 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  2 )  =  ( N  - 
1 ) )
5857adantr 465 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( N  +  1 )  -  2 )  =  ( N  -  1 ) )
5948, 58eqtrd 2473 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
2 )  =  ( N  -  1 ) )
6059fveq2d 5693 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  2 ) )  =  ( t `
 ( N  - 
1 ) ) )
6137fveq1d 5691 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( N  -  1 ) )  =  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  ( N  -  1 ) ) )
62 nnm1nn0 10619 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
63 peano2nn 10332 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
64 nnre 10327 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  RR )
65 2re 10389 . . . . . . . . . . . . . 14  |-  2  e.  RR
6665a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  2  e.  RR )
6764, 66readdcld 9411 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  RR )
68 1red 9399 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  1  e.  RR )
69 2rp 10994 . . . . . . . . . . . . . 14  |-  2  e.  RR+
7069a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  2  e.  RR+ )
7164, 70ltaddrpd 11054 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  <  ( N  +  2 ) )
7264, 67, 68, 71ltsub1dd 9949 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  -  1 )  <  ( ( N  +  2 )  - 
1 ) )
7349, 51, 50addsubassd 9737 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
7454oveq2i 6100 . . . . . . . . . . . 12  |-  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 )
7573, 74syl6eq 2489 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  1 )  =  ( N  + 
1 ) )
7672, 75breqtrd 4314 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  1 )  <  ( N  + 
1 ) )
7762, 63, 763jca 1168 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  -  1 )  e.  NN0  /\  ( N  +  1
)  e.  NN  /\  ( N  -  1
)  <  ( N  +  1 ) ) )
78 elfzo0 11585 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) )  <->  ( ( N  -  1 )  e. 
NN0  /\  ( N  +  1 )  e.  NN  /\  ( N  -  1 )  < 
( N  +  1 ) ) )
7977, 78sylibr 212 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) ) )
8079adantr 465 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) ) )
81 fvres 5702 . . . . . . 7  |-  ( ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  ( N  -  1 ) )  =  (Fibci `  ( N  -  1 ) ) )
8280, 81syl 16 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  ( N  -  1 ) )  =  (Fibci `  ( N  -  1 ) ) )
8360, 61, 823eqtrd 2477 . . . . 5  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  2 ) )  =  (Fibci `  ( N  -  1
) ) )
8447oveq1d 6104 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
1 )  =  ( ( N  +  1 )  -  1 ) )
85 simpl 457 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  N  e.  NN )
8685nncnd 10336 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  N  e.  CC )
87 1cnd 9400 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  1  e.  CC )
8886, 87pncand 9718 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
8984, 88eqtrd 2473 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
1 )  =  N )
9089fveq2d 5693 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  1 ) )  =  ( t `
 N ) )
9137fveq1d 5691 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  N )  =  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  N
) )
92 nn0fz0 11523 . . . . . . . . . 10  |-  ( N  e.  NN0  <->  N  e.  (
0 ... N ) )
9343, 92sylib 196 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ( 0 ... N
) )
94 nnz 10666 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
95 fzval3 11603 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 ... N )  =  ( 0..^ ( N  +  1 ) ) )
9694, 95syl 16 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
0 ... N )  =  ( 0..^ ( N  +  1 ) ) )
9793, 96eleqtrd 2517 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ( 0..^ ( N  +  1 ) ) )
9897adantr 465 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  N  e.  ( 0..^ ( N  + 
1 ) ) )
99 fvres 5702 . . . . . . 7  |-  ( N  e.  ( 0..^ ( N  +  1 ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  N )  =  (Fibci `  N
) )
10098, 99syl 16 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  N )  =  (Fibci `  N
) )
10190, 91, 1003eqtrd 2477 . . . . 5  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  1 ) )  =  (Fibci `  N ) )
10283, 101oveq12d 6107 . . . 4  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( (
t `  ( ( # `
 t )  - 
2 ) )  +  ( t `  (
( # `  t )  -  1 ) ) )  =  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N ) ) )
10332, 36, 102syl2anc 661 . . 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
) ) )
10440reseq1d 5107 . . . 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 ) ) ) )
1055, 42, 44subiwrd 26766 . . . . 5  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e. Word  NN0 )
106 ovex 6114 . . . . . . . . . 10  |-  ( <" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) )  e.  _V
1071, 106eqeltri 2511 . . . . . . . . 9  |- Fibci  e.  _V
108107resex 5148 . . . . . . . 8  |-  (Fibci  |`  (
0..^ ( N  + 
1 ) ) )  e.  _V
109108a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V )
11018fveq2i 5692 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  +  1 ) )  =  (
ZZ>= `  2 )
11116, 110syl6eleq 2531 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  2
) )
11245, 111eqeltrd 2515 . . . . . . 7  |-  ( N  e.  NN  ->  ( # `
 (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  e.  ( ZZ>= ` 
2 ) )
113109, 112jca 532 . . . . . 6  |-  ( N  e.  NN  ->  (
(Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V  /\  ( # `  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) )  e.  (
ZZ>= `  2 ) ) )
114 hashf 12108 . . . . . . 7  |-  # : _V
--> ( NN0  u.  { +oo } )
115 ffn 5557 . . . . . . 7  |-  ( # : _V --> ( NN0  u.  { +oo } )  ->  #  Fn  _V )
116 elpreima 5821 . . . . . . 7  |-  ( #  Fn  _V  ->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  ( `' # " ( ZZ>= `  2
) )  <->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V  /\  ( # `  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  e.  ( ZZ>= ` 
2 ) ) ) )
117114, 115, 116mp2b 10 . . . . . 6  |-  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  ( `' # " ( ZZ>= `  2
) )  <->  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V  /\  ( # `  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  e.  ( ZZ>= ` 
2 ) ) )
118113, 117sylibr 212 . . . . 5  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  ( `' # " ( ZZ>= `  2
) ) )
119105, 118elind 3538 . . . 4  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  2 ) ) ) )
120104, 119eqeltrrd 2516 . . 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 ) ) ) )
121 ovex 6114 . . . 4  |-  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N ) )  e. 
_V
122121a1i 11 . . 3  |-  ( N  e.  NN  ->  (
(Fibci `  ( N  -  1 ) )  +  (Fibci `  N
) )  e.  _V )
12331, 103, 120, 122fvmptd 5777 . 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
) ) )
1243, 22, 1233eqtrd 2477 1  |-  ( N  e.  NN  ->  (Fibci `  ( N  +  1 ) )  =  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2970    u. cun 3324    i^i cin 3325   {csn 3875   class class class wbr 4290    e. cmpt 4348   `'ccnv 4837    |` cres 4840   "cima 4841    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089   RRcr 9279   0cc0 9280   1c1 9281    + caddc 9283   +oocpnf 9413    < clt 9416    - cmin 9593   NNcn 10320   2c2 10369   NN0cn0 10577   ZZcz 10644   ZZ>=cuz 10859   RR+crp 10989   ...cfz 11435  ..^cfzo 11546   #chash 12101  Word cword 12219   <"cs2 12466  seqstrcsseq 26764  Fibcicfib 26777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-seq 11805  df-hash 12102  df-word 12227  df-lsw 12228  df-concat 12229  df-s1 12230  df-substr 12231  df-s2 12473  df-sseq 26765  df-fib 26778
This theorem is referenced by:  fib2  26783  fib3  26784  fib4  26785  fib5  26786  fib6  26787
  Copyright terms: Public domain W3C validator