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Theorem fibp1 29234
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 29230 . . . 4  |- Fibci  =  (
<" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) )
21fveq1i 5866 . . 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 12965 . . 3  |-  ( N  e.  NN  ->  <" 0
1 ">  e. Word  NN0 )
11 eqid 2451 . . 3  |-  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) )  =  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )
12 fiblem 29231 . . . 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 2547 . . . 4  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  (
1  +  1 ) ) )
17 s2len 12983 . . . . . 6  |-  ( # `  <" 0 1 "> )  =  2
18 1p1e2 10723 . . . . . 6  |-  ( 1  +  1 )  =  2
1917, 18eqtr4i 2476 . . . . 5  |-  ( # `  <" 0 1 "> )  =  ( 1  +  1 )
2019fveq2i 5868 . . . 4  |-  ( ZZ>= `  ( # `  <" 0
1 "> )
)  =  ( ZZ>= `  ( 1  +  1 ) )
2116, 20syl6eleqr 2540 . . 3  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  ( # `
 <" 0 1 "> ) ) )
225, 10, 11, 13, 21sseqp1 29228 . 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 5865 . . . . . . . 8  |-  ( w  =  t  ->  ( # `
 w )  =  ( # `  t
) )
2524oveq1d 6305 . . . . . . 7  |-  ( w  =  t  ->  (
( # `  w )  -  2 )  =  ( ( # `  t
)  -  2 ) )
2623, 25fveq12d 5871 . . . . . 6  |-  ( w  =  t  ->  (
w `  ( ( # `
 w )  - 
2 ) )  =  ( t `  (
( # `  t )  -  2 ) ) )
2724oveq1d 6305 . . . . . . 7  |-  ( w  =  t  ->  (
( # `  w )  -  1 )  =  ( ( # `  t
)  -  1 ) )
2823, 27fveq12d 5871 . . . . . 6  |-  ( w  =  t  ->  (
w `  ( ( # `
 w )  - 
1 ) )  =  ( t `  (
( # `  t )  -  1 ) ) )
2926, 28oveq12d 6308 . . . . 5  |-  ( w  =  t  ->  (
( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )  =  ( ( t `  (
( # `  t )  -  2 ) )  +  ( t `  ( ( # `  t
)  -  1 ) ) ) )
3029cbvmptv 4495 . . . 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 463 . . . . 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 5104 . . . . 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 2488 . . . 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 463 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  t  =  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) )
3736fveq2d 5869 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  t
)  =  ( # `  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) ) )
385, 10, 11, 13sseqf 29225 . . . . . . . . . . . . 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 5714 . . . . . . . . . . . . 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 236 . . . . . . . . . . . 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 29219 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( # `
 (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  =  ( N  +  1 ) )
4544adantr 467 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) )  =  ( N  +  1 ) )
4637, 45eqtrd 2485 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( # `  t
)  =  ( N  +  1 ) )
4746oveq1d 6305 . . . . . . . 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 9659 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
50 2cnd 10682 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  2  e.  CC )
5148, 49, 50addsubassd 10006 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  2 )  =  ( N  +  ( 1  -  2 ) ) )
5248, 50, 49subsub2d 10015 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  ( 2  -  1 ) )  =  ( N  +  ( 1  -  2 ) ) )
53 2m1e1 10724 . . . . . . . . . . . 12  |-  ( 2  -  1 )  =  1
5453oveq2i 6301 . . . . . . . . . . 11  |-  ( N  -  ( 2  -  1 ) )  =  ( N  -  1 )
5554a1i 11 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  ( 2  -  1 ) )  =  ( N  - 
1 ) )
5651, 52, 553eqtr2d 2491 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  2 )  =  ( N  - 
1 ) )
5756adantr 467 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( N  +  1 )  -  2 )  =  ( N  -  1 ) )
5847, 57eqtrd 2485 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
2 )  =  ( N  -  1 ) )
5958fveq2d 5869 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  2 ) )  =  ( t `
 ( N  - 
1 ) ) )
6036fveq1d 5867 . . . . . 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 9670 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  RR )
67 1red 9658 . . . . . . . . . . 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 10225 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  1 )  <  ( ( N  +  2 )  - 
1 ) )
7248, 50, 49addsubassd 10006 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
7353oveq2i 6301 . . . . . . . . . . 11  |-  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 )
7472, 73syl6eq 2501 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  1 )  =  ( N  + 
1 ) )
7571, 74breqtrd 4427 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  <  ( N  + 
1 ) )
76 elfzo0 11956 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) )  <->  ( ( N  -  1 )  e. 
NN0  /\  ( N  +  1 )  e.  NN  /\  ( N  -  1 )  < 
( N  +  1 ) ) )
7761, 62, 75, 76syl3anbrc 1192 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) ) )
7877adantr 467 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( N  -  1 )  e.  ( 0..^ ( N  +  1 ) ) )
79 fvres 5879 . . . . . . 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 2489 . . . . 5  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  2 ) )  =  (Fibci `  ( N  -  1
) ) )
8246oveq1d 6305 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
1 )  =  ( ( N  +  1 )  -  1 ) )
83 simpl 459 . . . . . . . . . 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 9659 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  1  e.  CC )
8684, 85pncand 9987 . . . . . . . 8  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
8782, 86eqtrd 2485 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( ( # `
 t )  - 
1 )  =  N )
8887fveq2d 5869 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  1 ) )  =  ( t `
 N ) )
8936fveq1d 5867 . . . . . 6  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  N )  =  ( (Fibci  |`  ( 0..^ ( N  +  1 ) ) ) `  N
) )
90 nn0fz0 11890 . . . . . . . . . 10  |-  ( N  e.  NN0  <->  N  e.  (
0 ... N ) )
9142, 90sylib 200 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ( 0 ... N
) )
92 nnz 10959 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  ZZ )
93 fzval3 11983 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 ... N )  =  ( 0..^ ( N  +  1 ) ) )
9492, 93syl 17 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
0 ... N )  =  ( 0..^ ( N  +  1 ) ) )
9591, 94eleqtrd 2531 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ( 0..^ ( N  +  1 ) ) )
9695adantr 467 . . . . . . 7  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  N  e.  ( 0..^ ( N  + 
1 ) ) )
97 fvres 5879 . . . . . . 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 2489 . . . . 5  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( t `  ( ( # `  t
)  -  1 ) )  =  (Fibci `  N ) )
10081, 99oveq12d 6308 . . . 4  |-  ( ( N  e.  NN  /\  t  =  (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  ->  ( (
t `  ( ( # `
 t )  - 
2 ) )  +  ( t `  (
( # `  t )  -  1 ) ) )  =  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N ) ) )
10135, 100syldan 473 . . 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 5104 . . . 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 29218 . . . . 5  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e. Word  NN0 )
104 ovex 6318 . . . . . . . . 9  |-  ( <" 0 1 ">seqstr ( w  e.  (Word 
NN0  i^i  ( `' #
" ( ZZ>= `  2
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) ) )  e.  _V
1051, 104eqeltri 2525 . . . . . . . 8  |- Fibci  e.  _V
106105resex 5148 . . . . . . 7  |-  (Fibci  |`  (
0..^ ( N  + 
1 ) ) )  e.  _V
107106a1i 11 . . . . . 6  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  _V )
10818fveq2i 5868 . . . . . . . 8  |-  ( ZZ>= `  ( 1  +  1 ) )  =  (
ZZ>= `  2 )
10916, 108syl6eleq 2539 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  ( ZZ>= `  2
) )
11044, 109eqeltrd 2529 . . . . . 6  |-  ( N  e.  NN  ->  ( # `
 (Fibci  |`  (
0..^ ( N  + 
1 ) ) ) )  e.  ( ZZ>= ` 
2 ) )
111 hashf 12522 . . . . . . 7  |-  # : _V
--> ( NN0  u.  { +oo } )
112 ffn 5728 . . . . . . 7  |-  ( # : _V --> ( NN0  u.  { +oo } )  ->  #  Fn  _V )
113 elpreima 6002 . . . . . . 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 670 . . . . 5  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  ( `' # " ( ZZ>= `  2
) ) )
116103, 115elind 3618 . . . 4  |-  ( N  e.  NN  ->  (Fibci  |`  ( 0..^ ( N  +  1 ) ) )  e.  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  2 ) ) ) )
117102, 116eqeltrrd 2530 . . 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 6318 . . . 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 5954 . 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 2489 1  |-  ( N  e.  NN  ->  (Fibci `  ( N  +  1 ) )  =  ( (Fibci `  ( N  -  1 ) )  +  (Fibci `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    u. cun 3402    i^i cin 3403   {csn 3968   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833    |` cres 4836   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542   +oocpnf 9672    < clt 9675    - cmin 9860   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   ...cfz 11784  ..^cfzo 11915   #chash 12515  Word cword 12656   <"cs2 12937  seqstrcsseq 29216  Fibcicfib 29229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-s2 12944  df-sseq 29217  df-fib 29230
This theorem is referenced by:  fib2  29235  fib3  29236  fib4  29237  fib5  29238  fib6  29239
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