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Theorem fiblem 28534
Description: Lemma for fib0 28535, fib1 28536 and fibp1 28537. (Contributed by Thierry Arnoux, 25-Apr-2019.)
Assertion
Ref Expression
fiblem  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= ` 
2 ) ) ) 
|->  ( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) ) ) : (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) ) --> NN0

Proof of Theorem fiblem
StepHypRef Expression
1 s2len 12864 . . . . . . 7  |-  ( # `  <" 0 1 "> )  =  2
21eqcomi 2470 . . . . . 6  |-  2  =  ( # `  <" 0 1 "> )
32fveq2i 5875 . . . . 5  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( # `  <" 0 1 "> ) )
43imaeq2i 5345 . . . 4  |-  ( `' # " ( ZZ>= `  2
) )  =  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) )
54ineq2i 3693 . . 3  |-  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  2 ) ) )  =  (Word  NN0  i^i  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) )
6 eqid 2457 . . 3  |-  ( ( w `  ( (
# `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )  =  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) )
75, 6mpteq12i 4541 . 2  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= ` 
2 ) ) ) 
|->  ( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) ) )  =  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  |->  ( ( w `  (
( # `  w )  -  2 ) )  +  ( w `  ( ( # `  w
)  -  1 ) ) ) )
8 elin 3683 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  <->  ( w  e. Word  NN0  /\  w  e.  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) ) ) )
98simplbi 460 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w  e. Word  NN0 )
10 wrdf 12558 . . . . 5  |-  ( w  e. Word  NN0  ->  w :
( 0..^ ( # `  w ) ) --> NN0 )
119, 10syl 16 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w : ( 0..^ (
# `  w )
) --> NN0 )
128simprbi 464 . . . . . . . . 9  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  ->  w  e.  ( `' #
" ( ZZ>= `  ( # `
 <" 0 1 "> ) ) ) )
13 hashf 12415 . . . . . . . . . 10  |-  # : _V
--> ( NN0  u.  { +oo } )
14 ffn 5737 . . . . . . . . . 10  |-  ( # : _V --> ( NN0  u.  { +oo } )  ->  #  Fn  _V )
15 elpreima 6008 . . . . . . . . . 10  |-  ( #  Fn  _V  ->  ( w  e.  ( `' # " ( ZZ>=
`  ( # `  <" 0 1 "> ) ) )  <->  ( w  e.  _V  /\  ( # `  w )  e.  (
ZZ>= `  ( # `  <" 0 1 "> ) ) ) ) )
1613, 14, 15mp2b 10 . . . . . . . . 9  |-  ( w  e.  ( `' # " ( ZZ>= `  ( # `  <" 0 1 "> ) ) )  <->  ( w  e.  _V  /\  ( # `  w )  e.  (
ZZ>= `  ( # `  <" 0 1 "> ) ) ) )
1712, 16sylib 196 . . . . . . . 8  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w  e.  _V  /\  ( # `  w
)  e.  ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )
1817simprd 463 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  ( # `
 <" 0 1 "> ) ) )
1918, 3syl6eleqr 2556 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  2
) )
20 uznn0sub 11137 . . . . . 6  |-  ( (
# `  w )  e.  ( ZZ>= `  2 )  ->  ( ( # `  w
)  -  2 )  e.  NN0 )
2119, 20syl 16 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  e.  NN0 )
22 1zzd 10916 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
1  e.  ZZ )
23 1p1e2 10670 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
2423fveq2i 5875 . . . . . . . 8  |-  ( ZZ>= `  ( 1  +  1 ) )  =  (
ZZ>= `  2 )
2519, 24syl6eleqr 2556 . . . . . . 7  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  (
1  +  1 ) ) )
26 peano2uzr 11161 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  ( # `  w )  e.  ( ZZ>= `  (
1  +  1 ) ) )  ->  ( # `
 w )  e.  ( ZZ>= `  1 )
)
2722, 25, 26syl2anc 661 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  ( ZZ>= `  1
) )
28 nnuz 11141 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
2927, 28syl6eleqr 2556 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  NN )
3029nnred 10571 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( # `  w )  e.  RR )
31 2rp 11250 . . . . . . 7  |-  2  e.  RR+
3231a1i 11 . . . . . 6  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
2  e.  RR+ )
3330, 32ltsubrpd 11309 . . . . 5  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  <  ( # `  w
) )
34 elfzo0 11862 . . . . 5  |-  ( ( ( # `  w
)  -  2 )  e.  ( 0..^ (
# `  w )
)  <->  ( ( (
# `  w )  -  2 )  e. 
NN0  /\  ( # `  w
)  e.  NN  /\  ( ( # `  w
)  -  2 )  <  ( # `  w
) ) )
3521, 29, 33, 34syl3anbrc 1180 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  2 )  e.  ( 0..^ (
# `  w )
) )
3611, 35ffvelrnd 6033 . . 3  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w `  (
( # `  w )  -  2 ) )  e.  NN0 )
37 fzo0end 11907 . . . . 5  |-  ( (
# `  w )  e.  NN  ->  ( ( # `
 w )  - 
1 )  e.  ( 0..^ ( # `  w
) ) )
3829, 37syl 16 . . . 4  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( # `  w
)  -  1 )  e.  ( 0..^ (
# `  w )
) )
3911, 38ffvelrnd 6033 . . 3  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( w `  (
( # `  w )  -  1 ) )  e.  NN0 )
4036, 39nn0addcld 10877 . 2  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) )  -> 
( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) )  e. 
NN0 )
417, 40fmpti 6055 1  |-  ( w  e.  (Word  NN0  i^i  ( `' # " ( ZZ>= ` 
2 ) ) ) 
|->  ( ( w `  ( ( # `  w
)  -  2 ) )  +  ( w `
 ( ( # `  w )  -  1 ) ) ) ) : (Word  NN0  i^i  ( `' # " ( ZZ>= `  ( # `  <" 0
1 "> )
) ) ) --> NN0
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1819   _Vcvv 3109    u. cun 3469    i^i cin 3470   {csn 4032   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   +oocpnf 9642    < clt 9645    - cmin 9824   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245  ..^cfzo 11821   #chash 12408  Word cword 12538   <"cs2 12818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825
This theorem is referenced by:  fib0  28535  fib1  28536  fibp1  28537
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